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Dan Hagon committed cb2c564

Finally fixed the Unicode problem.

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  • Parent commits 17eb781

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File PvsNP_timeline.html

      <script src="timeline.js" type="text/javascript"></script>
    </head>
    <body onload="onLoad();" onresize="onResize();">
-     <div id="my-timeline" style="height: 240px; border: 1px solid #aaa"></div>
+     <div id="my-timeline" style="height: 500px; border: 1px solid #aaa"></div>
      <noscript>
      This page uses Javascript to show you a Timeline. Please enable Javascript in your browser to see the full page. Thank you.
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File PvsNP_timeline.xml

-<?xml version="1.0" encoding="UTF-8"?>
-    <data>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/08/07/p-n-np/" start="Sun Aug 08 2010 03:21:15 GMT+0000" title="P &amp;#8800; NP">
-    An email I was recently forwarded (a couple of steps removed) from Vinay Deolalikar from HP Labs: Dear Fellow Researchers, I am pleased to announce a proof that P is not equal to NP, which is attached in 10pt and 12pt fonts. The proof required the piecing together of principles from multiple areas within mathematics. [...]
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/06/24/everything-i-know-about-databases-is-wrong-also-right/" start="Thu Jun 24 2010 19:48:44 GMT+0000" title="Everything I know about databases is wrong.  Also, right.">
-    I have been teaching CMPT 470 for six years now, with my 13th offering going on right now. Anybody doing that is going to pick up a thing or two about web systems. I was there for the rise of the MVC frameworks and greeted them with open arms. I watched Web 2.0 proclaim &amp;#8220;screw [...]
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/06/10/computer-woes/" start="Thu Jun 10 2010 17:32:46 GMT+0000" title="Computer Woes">
-    My computer at home has been locking up occasionally for the last few weeks. This has been happening since my upgrade to Ubuntu 10.04/Lucid, but I suspect this is a coincidence. (1) The lockups are hard: even the SysRq magic doesn&amp;#8217;t do anything, so I deduce that the problem is in the kernel or below. [...]
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/06/03/future-epic-culinary-journeys/" start="Fri Jun 04 2010 06:46:31 GMT+0000" title="Future epic culinary journeys?">
-    Now that me fried rice journey has ended, I find myself contemplating the next epic food journey. Having gone through one, I think I can handle another. The key is to only ask individual people, namely either my Aunt Daisy or my mom, very specific questions about whatever dish I am trying to make. I [...]
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/06/01/how-fried-rice-drove-me-insane-part-3/" start="Wed Jun 02 2010 04:55:58 GMT+0000" title="How fried rice drove me insane (Part 3)">
-    Following Saturday&amp;#8217;s soul-crushing family conversation about my Ama&amp;#8217;s fried rice, during which I was informed of additonal, never-before-mentioned fried rice ingredients and cooking techniques, I have to say that I was at my wit&amp;#8217;s end. There may have been a few instances that I sat in a ball and ricked back and forth. However, Sunday [...]
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/05/30/how-fried-rice-drove-me-insane-part-2/" start="Sun May 30 2010 18:20:30 GMT+0000" title="How fried rice drove me insane (Part 2)">
-    My mom is in town, so Greg and I went to have dinner at my Ama&amp;#8217;s house. Inevitably the conversation turned to food, which led me to ask about my Ama&amp;#8217;s fried rice for the 100th time. This time there were more people around, and everyone had an opinion about what went into Ama&amp;#8217;s fried [...]
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/05/29/how-fried-rice-drove-me-insane-part-1/" start="Sun May 30 2010 05:32:09 GMT+0000" title="How fried rice drove me insane (Part 1)">
-    For the last ~5+ years I have been attempting to make fried rice. Not just any fried rice though, my Ama&amp;#8217;s fried rice. I have wonderful memories of visiting Vancouver in the summer and for Christmas and having my Ama&amp;#8217;s fried rice for dinner, usually with an accompanying soup of some sort. The combination of [...]
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/05/28/how-to-not-attend-a-lecture/" start="Fri May 28 2010 07:06:10 GMT+0000" title="How to not attend a lecture">
-    I teach at a university. That comes with certain parameters: most of my students are in their late teens or early twenties, the average student is reasonably bright but occasionally unmotivated, and I don&amp;#8217;t really have any way to compel students to come to lectures. I do my best to give interesting, informative, and entertaining [...]
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/04/19/cmpt-383-for-real-this-time/" start="Tue Apr 20 2010 01:09:07 GMT+0000" title="CMPT 383: for real this time">
-    I have mentioned here before that I was planning to teach CMPT 383. It ended up being a no-go this semester because of a very productive capstone project team (more on that later). But, I&amp;#8217;m on-deck to teach it in the summer. The class is full; the waiting list is full; must be time to [...]
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/03/19/the-history-of-html/" start="Fri Mar 19 2010 15:46:21 GMT+0000" title="The History of HTML">
-    After a simple query from a colleague about the differences between HTML versions, I wrote this. I thought I might as well post it. Everything was from-memory, so there may be some minor errors. HTML 1 never existed (it was the informal &amp;#8220;standard&amp;#8221; that the first documentation implied). HTML 2 was a really minimal initial [...]
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/" start="Sun Aug 15 2010 18:30:20 GMT+0000" title="The P&amp;#8800;NP &amp;#8220;Proof&amp;#8221; Is One Week Old">
-    Another update on progress understanding the proof of Deolalikar Terence Tao is leading, as often is the case, with some very insightful comments on the status of Vinay Deolalikar&amp;#8217;s proof. Today Ken Regan and I want to try to summarize the main issues with Deolalikar&amp;#8217;s proof of PNP raised by Tao and many many others. [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5273&amp;subd=rjlipton&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/08/12/fatal-flaws-in-deolalikars-proof/" start="Fri Aug 13 2010 04:15:21 GMT+0000" title="Fatal Flaws in Deolalikar&amp;#8217;s Proof?">
-    Possible fatal flaws in the finite model part of Deolalikar&amp;#8217;s proof Neil Immerman is one of the world&amp;#8217;s experts on Finite Model Theory. He used insights from this area to co-discover the great result that is closed under complement. Today I had planned not to discuss the proof, but I just received a note from [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5260&amp;subd=rjlipton&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/08/11/deolalikar-responds-to-issues-about-his-p%e2%89%a0np-proof/" start="Thu Aug 12 2010 02:14:13 GMT+0000" title="Deolalikar Responds To Issues About His P&amp;#8800;NP Proof">
-    A principle for checking proofs with an application to the proof Vinay Deolalikar is standing by his claim and proof. He and I have been exchanging e-mails, and as noted in the several comments, he has updated his paper several times. The updates are trying to answer some of the questions raised here and elsewhere. [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5244&amp;subd=rjlipton&amp;ref=&amp;feed=1&quot; /&gt;
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/08/10/update-on-deolalikars-proof-that-p%e2%89%a0np/" start="Wed Aug 11 2010 03:47:13 GMT+0000" title="Update on Deolalikar&amp;#8217;s Proof that P&amp;#8800;NP">
-    An update on the P not equal to NP proof Timothy Gowers, Gil Kalai, Ken Regan, Terence Tao, and Suresh Venkatasubramanian are some of the top people who are trying to unravel what is up with the claimed proof that P NP. I will call them the Group today&amp;#8212;I hope that is okay. It&amp;#8217;s a [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5221&amp;subd=rjlipton&amp;ref=&amp;feed=1&quot; /&gt;
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/08/09/issues-in-the-proof-that-p%e2%89%a0np/" start="Tue Aug 10 2010 01:20:45 GMT+0000" title="Issues In The Proof That P&amp;#8800;NP">
-    Some further comments into the recent claimed proof that P is not equal to NP Yuri Manin is a great mathematician, who has worked in various areas and also is well known for his many excellent expository works. He has won numerous prizes, including the Nemmers Prize in Mathematics in 1994, and the Cantor Medal [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5206&amp;subd=rjlipton&amp;ref=&amp;feed=1&quot; /&gt;
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/08/08/a-proof-that-p-is-not-equal-to-np/" start="Mon Aug 09 2010 01:11:24 GMT+0000" title="A Proof That P Is Not Equal To NP?">
-    A serious proof that claims to have resolved the P=NP question. Vinay Deolalikar is a Principal Research Scientist at HP Labs who has done important research in various areas of networks. He also has worked on complexity theory, including previous work on infinite versions of the P=NP question. He has just claimed that he has [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5195&amp;subd=rjlipton&amp;ref=&amp;feed=1&quot; /&gt;
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/08/07/hilberts-tenth-over-the-rationals/" start="Sat Aug 07 2010 18:31:12 GMT+0000" title="Hilbert&amp;#8217;s Tenth Over The Rationals">
-    Can there be a bijection from pairs of rationals to rationals? Bjorn Poonen is a famous number theorist at MIT, who has many honors and awards, including being one of the few four time Putnam Exam winners. He has worked on many aspects of number theory, including the &amp;#8220;dark side&amp;#8221; as he calls it. The [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5182&amp;subd=rjlipton&amp;ref=&amp;feed=1&quot; /&gt;
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/08/02/quantum-algorithms-via-linear-algebra/" start="Mon Aug 02 2010 12:38:47 GMT+0000" title="Quantum Algorithms A Different View">
-    Quantum algorithms explained in a different way, not the usual way. David Deutsch is one of the founders of quantum computing. His 1985 paper described quantum Turing machines, and his famous algorithm is considered the first quantum algorithm ever. Today I plan to try to explain the power of quantum algorithms, but do it in [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5166&amp;subd=rjlipton&amp;ref=&amp;feed=1&quot; /&gt;
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/07/30/paranoia-and-selling-a-laptop/" start="Fri Jul 30 2010 15:56:30 GMT+0000" title="Paranoia And Selling A Laptop">
-    How do we erase a hard disk, for sure? Richard Nixon was not a theorist&amp;#8212;of course. He was the President of the United States from 1969 until 1974, when he resigned in disgrace. He was also an &amp;#8220;un-indicted co-conspirator&amp;#8221; and probably one of the most paranoid presidents ever. Today I want to talk about paranoia. [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5154&amp;subd=rjlipton&amp;ref=&amp;feed=1&quot; /&gt;
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/07/25/hedy-lamarr-the-inventor/" start="Sun Jul 25 2010 14:10:51 GMT+0000" title="Hedy Lamarr the Inventor">
-    The invention of spread spectrum and the role of amateurs in science and technology Hedy Lamarr was, of course, not a mathematician nor a complexity theorist. She was one of the great movie actresses of all time, and was once &amp;#8220;voted&amp;#8221; the most beautiful woman in the world. She was also an inventor. Today I [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5142&amp;subd=rjlipton&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://twitter.com/favicon.ico" link="http://twitter.com/p6/statuses/21325513289" start="Mon Aug 16 2010 15:58:13 GMT+0000" title="LOLWAT: http://is.gd/eki8P #PvsNP #WTF."/>
-    <event icon="http://twitter.com/favicon.ico" link="http://twitter.com/exhibitiontiz/statuses/21241954360" start="Sun Aug 15 2010 16:13:10 GMT+0000" title="#PvsNP running into trouble? - New Scientist http://t.co/78XxQ85"/>
-    <event icon="http://twitter.com/favicon.ico" link="http://twitter.com/rcvnglst/statuses/21231066765" start="Sun Aug 15 2010 13:35:59 GMT+0000" title="Proof for P!=NP? http://bit.ly/aI8yMf #compsci #math #in #pvsnp"/>
-    <event icon="http://twitter.com/favicon.ico" link="http://twitter.com/M_Turkistany/statuses/21189271271" start="Sun Aug 15 2010 00:29:01 GMT+0000" title="We must give up the tradition of equating efficient computation with polynomial running time #pnp #pvsnp"/>
-    <event icon="http://twitter.com/favicon.ico" link="http://twitter.com/M_Turkistany/statuses/21189117873" start="Sun Aug 15 2010 00:26:17 GMT+0000" title="To resolve P vs. NP Problem, we need to redefine concepts (discover alternatives) like tractable problem and efficient algorithm #pnp #pvsnp"/>
-    <event icon="http://twitter.com/favicon.ico" link="http://twitter.com/Vallenain/statuses/21129826237" start="Sat Aug 14 2010 06:19:36 GMT+0000" title="RT @4ugeek: La solution &amp;#224; 1 million de $ de la plus grande &amp;#233;nigme informatique contrevers&amp;#233;e http://0z.fr/k4Tqp #science #math #PvsNP"/>
-    <event icon="http://twitter.com/favicon.ico" link="http://twitter.com/bigdougschmidt/statuses/21126179450" start="Sat Aug 14 2010 05:10:48 GMT+0000" title="If you could recognize the right solution to a problem if it was given to you does it mean the solution could be found automatically? #PvsNP"/>
-    <event icon="http://twitter.com/favicon.ico" link="http://twitter.com/4ugeek/statuses/21099539564" start="Fri Aug 13 2010 22:11:32 GMT+0000" title="La solution &amp;#224; 1 million de $ de la plus grande &amp;#233;nigme informatique contrevers&amp;#233;e http://0z.fr/k4Tqp #science #math #PvsNP"/>
-    <event icon="http://twitter.com/favicon.ico" link="http://twitter.com/GISDJL/statuses/21085158454" start="Fri Aug 13 2010 18:17:13 GMT+0000" title="A speculative effort at writing about something I don't really understand: http://bit.ly/9UAyAP #P!=NP #PvsNP #geography"/>
-    <event icon="http://twitter.com/favicon.ico" link="http://twitter.com/ImaginaryTime/statuses/21048517347" start="Fri Aug 13 2010 08:43:15 GMT+0000" title="P=NP explained using #Minesweeper as an analogy. http://bit.ly/aTnsj3 #mathematics #PvsNP"/>
-    <event icon="http://twitter.com/favicon.ico" link="http://twitter.com/nishkarshs/statuses/21046297186" start="Fri Aug 13 2010 07:53:52 GMT+0000" title="RT: @cowmaaa: P=NP Solved? Indian scientist thinks so. http://alturl.com/zvzzg #pvsnp #vinaydeolalikar"/>
-    <event icon="http://twitter.com/favicon.ico" link="http://twitter.com/sc_k/statuses/20945767212" start="Thu Aug 12 2010 04:14:56 GMT+0000" title="&quot;The ethics of scientific betting&quot; http://scottaaronson.com/blog/?p=457 with new and improved hilarious comment thread! #pvsnp"/>
-    <event icon="http://twitter.com/favicon.ico" link="http://twitter.com/Crayolasaurus/statuses/20927114621" start="Wed Aug 11 2010 23:47:17 GMT+0000" title="P vs NP math problem possibly solved, your online credit card info is safe http://bit.ly/cKZDal #pvsnp #math"/>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8235#Comment_8235" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Greg Kuperberg comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8235)">
-    I think that it's fine to ask once for references that discuss an attempted proof of a big conjecture.  There have now been several questions posted on this theme, generally in a dubious form such as ...
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8226#Comment_8226" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Kaveh comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8226)">
-    &quot;Ramping up the hype&quot; was not my intension, but I drop my position as there is now a polymath like attempt to discuss the paper.
-
-Now any suggestion about what to do with the question I ...
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8221#Comment_8221" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Pete L. Clark comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8221)">
-    @Felipe: The question is now deleted.
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8219#Comment_8219" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="voloch comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8219)">
-    @Pete I just clicked on &amp;quot;delete&amp;quot;. I am not sure whether it's done anything.
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8217#Comment_8217" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Pete L. Clark comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8217)">
-    @Felipe Voloch: I think I agree with you that my question on the Datta paper and your answer are both somewhat regrettable. I have just unaccepted your answer.  I believe you have to delete your ...
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8213#Comment_8213" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Scott Carnahan comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8213)">
-    I am inclined to close the newer question, because I think blogs are a better format than StackExchange for asking about and disseminating breaking news and rumors.  If you're wondering what ...
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8212#Comment_8212" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Kaveh comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8212)">
-    voloch: &quot;Stop bumping old complexity theory questions. I think, despite you appearing to want to follow MO rules, you are really just trying to stir the pot&quot;
-
-Sorry if I have done anything ...
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8211#Comment_8211" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="voloch comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8211)">
-    @Kaveh Stop bumping old complexity theory questions. I think, despite you appearing to want to follow MO rules, you are really just trying to stir the pot. I haven't yet voted to close your question ...
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8210#Comment_8210" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Bill Dubuque comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8210)">
-    @VP I don't find your argument persuasive. There are plently of unpublished results that are discussed on MO and many other mathematical forums. Some may never be published - for a variety of ...
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8209#Comment_8209" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="SimonPL comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8209)">
-    I think that algori's reformulation &amp;quot;What are, in your opinion, potential difficulties with the strategy proposed in this paper?&amp;quot; is a good way to transfer knowledge between people ...
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8208#Comment_8208" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Olivier comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8208)">
-    @Kaveh Forgive me, I guess this is because I am a junior researcher, but I don't understand this, can someone be more specific?@algori I'm not sure I understand the proof of lemma 15.3, specifically ...
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8207#Comment_8207" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="algori comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8207)">
-    VP, Kevin -- I think there may be a reasonable question along these lines. Example: &amp;quot;What are, in your opinion, potential difficulties with the strategy proposed in this paper?&amp;quot; ...
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8206#Comment_8206" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Kaveh comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8206)">
-    VP (Vicktor  Protsak?): &quot;In particular, reviews are not openly disseminated until the paper has been published in a journal. There are very, very good reasons not to deviate from these ...
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8205#Comment_8205" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Olivier comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8205)">
-    @Suresh and more generally&amp;quot;To follow up on Kaveh, I have been reading the paper and have an even more specific question relating to background knowledge about phase transitions for random ...
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8204#Comment_8204" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Suresh Venkat comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8204)">
-    I was referring to Kaveh's question, not the original post ....
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-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8203#Comment_8203" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Kevin Buzzard comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8203)">
-    @Suresh: &amp;quot;...an even more specific question&amp;quot;. The original question could barely be less specific! &amp;quot;Is there a problem in these 100 pages of mathematics?&amp;quot;. I agree ...
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8201#Comment_8201" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Suresh Venkat comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8201)">
-    To follow up on Kaveh, I have been reading the paper and have an even more specific question relating to background knowledge about phase transitions for random k-SAT. Would that be admissible ?
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8200#Comment_8200" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Kaveh comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8200)">
-    Is it OK if I ask a more specific question like:
-
-
-  does his approach avoid being a natural proof?
-
-
-By the way, here is Dick Lipton's related ...
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-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8194#Comment_8194" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Andy Putman comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8194)">
-    I agree strongly with the points VP made and just cast the deciding vote to close.
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8193#Comment_8193" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Qiaochu Yuan comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8193)">
-    @VP:  Good points.  I defer to your expertise.
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8192#Comment_8192" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="VP comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8192)">
-    Qiaochu, are you suggesting that MO be used as a repository of expert opinions on unpublished papers?  Even if &quot;there are experts who might be able to evaluate the content&quot;, there are ...
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8191#Comment_8191" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Gjergji comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8191)">
-    I really want to know the answer to the question, however I don't see what purpose it serves on MO at this point. The specialists are probably already aware of this paper and so it seems like a good ...
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8190#Comment_8190" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="Qiaochu Yuan comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8190)">
-    I'm torn.  On the one hand, there are several experts here who might be capable of evaluating the content of this paper, and MO would be a valuable place to store those evaluations.  On the other ...
-    </event>
-    <event icon="http://stackexchange.com/img/favicon.png" link="http://meta.mathoverflow.net/discussion/590/whats-wrong-with-this-proof/?Focus=8189#Comment_8189" start="Tue Aug 17 2010 01:02:48 GMT+0000" title="theojf comments on &quot;&quot;What&amp;#8217;s wrong with this proof?&quot;&quot; (8189)">
-    As written, http://mathoverflow.net/questions/34953/whats-wrong-with-this-proof is an awful question.  The title has almost no information, and sounds crankier than it is, and the question is not a ...
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/08/14/a-possible-answer-to-my-objection/" start="Sat Aug 14 2010 07:57:37 GMT+0000" title="A possible answer to my objection">
-    It seems that the informal argument in my previous post may have been aiming off target. Here is a sort of counter to it, by Jun Tarui. (See also this earlier comment of his, and the ensuing discussion.) If I&amp;#8217;ve understood various comments correctly, Deolalikar is making a move that I, for one, had not [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1920&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/08/13/could-anything-like-deolalikars-strategy-work/" start="Fri Aug 13 2010 08:31:06 GMT+0000" title="Could anything like Deolalikar&amp;#8217;s strategy work?">
-    This post comes with the same warning as my previous one: my acquaintance with the ideas of Deolalikar&amp;#8217;s paper is derived not from the paper itself (apart from a glance through it to get some kind of general impression) but from some of the excellent commentary that has been taking place on the blogs of [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1914&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/08/11/my-pennyworth-about-deolalikar/" start="Wed Aug 11 2010 14:16:31 GMT+0000" title="My pennyworth about Deolalikar">
-    Given that I have expressed rather unambiguously an interest in the P/NP problem on this blog, I feel duty bound to make some kind of remark about Deolalikar&amp;#8217;s recent attempt to prove the theorem. (If by some extraordinary chance you don&amp;#8217;t know what I am talking about, then an excellent starting point is Dick Lipton&amp;#8217;s [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1903&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/07/18/edp17-are-we-nearly-there/" start="Sun Jul 18 2010 21:53:06 GMT+0000" title="EDP17 &amp;#8212; are we nearly there?">
-    Apologies for the attention-seeking title, but that really is the purpose of this post. I want to draw attention to some ideas that are buried in more comments that most people are likely to want to read, because I think there is a chance that all that stands between where we are now and a [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1839&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/07/13/polymath-news/" start="Tue Jul 13 2010 16:40:32 GMT+0000" title="Polymath news">
-    If you haven&amp;#8217;t already spotted this, you might like to know that Scott Aaronson has just posed a very interesting unsolved combinatorial problem and invited Polymath-style thoughts on it. I&amp;#8217;ll give a quick and I hope appetite-whetting account of the problem here, but you could skip it if you want and jump straight to his [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1832&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/07/04/edp16-from-ap-discrepancy-to-hap-discrepancy/" start="Sun Jul 04 2010 09:27:07 GMT+0000" title="EDP16 &amp;#8212; from AP-discrepancy to HAP-discrepancy?">
-    In this post I want to elaborate on a strategy for proving EDP that I discussed in EDP15. Briefly, the idea is to take a representation-of-identity proof of Roth&amp;#8217;s AP-discrepancy result and modify it so that it becomes a representation-of-diagonal proof of unbounded HAP-discrepancy. The first step of this programme was an obvious one: obtain [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1780&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/06/24/a-mathematician-watches-tennis-ii/" start="Thu Jun 24 2010 09:00:38 GMT+0000" title="A mathematician watches tennis II">
-    This has been a year to remember for anybody whose interest in tennis is more that of a nerd than that of a tennis player (which, given the uselessness of my serve, very much applies to me), in that it has given us two records that may well never be beaten. First we have Roger [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1768&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/06/21/edp15-finding-a-diagonal-matrix/" start="Mon Jun 21 2010 11:09:16 GMT+0000" title="EDP15 &amp;#8212; finding a diagonal matrix">
-    It is not at all clear to me what should now happen with the project to solve the Erd&amp;#337;s discrepancy problem. A little while ago, it seems that either the project ran out of steam, or else several people, at roughly the same time, decided that other things they were doing should take higher priority [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1725&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/05/09/if-politicians-were-mathematicians/" start="Sun May 09 2010 15:26:36 GMT+0000" title="If politicians were mathematicians">
-    Before I start, let me get one thing over and done with: I fully admit that professional mathematicians are as capable as anyone else of making stupid collective decisions. But I don&amp;#8217;t want to imagine what the world would be like if it were run by mathematical researchers. I just wonder how much difference it [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1715&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/05/08/a-little-experiment-iv/" start="Sat May 08 2010 12:25:42 GMT+0000" title="A little experiment IV">
-    I have another experiment that may not add all that much to the previous one, but I&amp;#8217;m posting it anyway because I don&amp;#8217;t want to waste the (admittedly not huge) effort it has just taken me to follow Jason Dyer&amp;#8217;s suggestion and create a presentation using Prezi. If you follow this link, you will be [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1703&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/05/07/a-little-experiment-iii/" start="Fri May 07 2010 09:33:45 GMT+0000" title="A little experiment III">
-    This one is a little bit different, and the associated poll questions are rather vague. I am curious to know what effect it would have on our ability to do routine manipulations if we could look at only one symbol at a time. Maybe at some point after the experiment I will say what my [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1680&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/05/03/is-the-british-voting-system-fair/" start="Sun May 02 2010 23:54:54 GMT+0000" title="Is the British voting system fair?">
-    At the last general election, the percentages of votes and numbers of seats in parliament for the three main parties in Britain were as follows: Labour, 35.3% of votes, 356 seats; Conservatives, 32.3% of votes, 198 seats; Liberal Democrats, 22.1% of votes, 62 seats. In the election coming up on the 6th May, there is [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1651&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/05/01/a-little-experiment-ii/" start="Sat May 01 2010 14:57:19 GMT+0000" title="A little experiment II">
-    A few people felt that my experiment about solving the equation would have been more interesting if the equation had been more interesting. I&amp;#8217;m not sure I agree with that: part of my intention was to get some evidence about what we do in a very simple situation. However, I now have another example that [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1656&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/04/25/edp14-strategic-questions/" start="Sun Apr 25 2010 21:39:55 GMT+0000" title="EDP14 &amp;#8212; strategic questions">
-    In this post I want to take the following attitude. Although there are several promising approaches to solving EDP, I am going to concentrate just on the representation-of-diagonals idea and pretend that that is the problem. That is, I want to pretend that the main problem we are trying to solve is not a problem [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1643&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/04/22/a-little-experiment/" start="Thu Apr 22 2010 17:18:22 GMT+0000" title="A little experiment">
-    For a long time, a side interest of mine has been how people think when they are doing mathematics. Two difficulties in investigating this general question are (i) that it is quite hard to examine one&amp;#8217;s own thought processes reliably (since very often what one remembers of these processes after solving a problem is a [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1620&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/04/02/use-of-mathematics-ii/" start="Fri Apr 02 2010 11:57:51 GMT+0000" title="Use of mathematics II">
-    Today I had an experience that I have had many times before, and so, I imagine, has almost everybody (at least if they are old enough to be the kind of person who might conceivably read this blog post). I was in a queue in a chemist (=pharmacy=drugstore), and I knew that my particular item [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1612&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/03/28/when-is-proof-by-contradiction-necessary/" start="Sun Mar 28 2010 14:59:10 GMT+0000" title="When is proof by contradiction necessary?">
-    It&amp;#8217;s been a while since I have written a post in the &amp;#8220;somewhat philosophical&amp;#8221; category, which is where I put questions like &amp;#8220;How can one statement be stronger than an another, equivalent, statement?&amp;#8221; This post is about a question that I&amp;#8217;ve intended for a long time to sort out in my mind but have found [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1595&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/03/23/edp13-quick-summary/" start="Tue Mar 23 2010 22:06:00 GMT+0000" title="EDP13 &amp;#8212; quick summary">
-    I don&amp;#8217;t have any major new themes to talk about in this post. We are continuing to look for ways of decomposing matrices that would give rise to proofs of strengthenings of EDP, but I would say that at the moment we are still at the stage of trying to get a feel for the [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1589&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/03/20/hello-world-2/" start="Sat Mar 20 2010 12:49:47 GMT+0000" title="&amp;#8220;Hello World!&amp;#8221;">
-    As anyone who has been following the discussion of EDP will be aware, I am interested in programming, but have until very recently favoured an extremely high-level language called BLOG. The principle of this language is that you describe in English in a blog post or comment what you want an algorithm to do, after [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1581&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/03/13/edp12-representing-diagonal-maps/" start="Sat Mar 13 2010 22:57:44 GMT+0000" title="EDP12 &amp;#8212; representing diagonal maps">
-    In this post I want to use my unfair advantage as host of this blog to push an approach that is the one that I am currently most interested in. So let me precede it with the qualification that although I shall present the approach as favourably as I can, it may well be that [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1565&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/03/07/edp11-the-search-continues/" start="Sun Mar 07 2010 11:27:02 GMT+0000" title="EDP11 &amp;#8212; the search continues">
-    I do not have too much to say about the new programme to use SDP to solve EDP, other than that it is still being actively pursued. One remark I would make is that I have rather forgotten recently about keeping the wiki updated, and SDP is a major omission from it. It would be [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1561&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
-    </event>
-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/03/02/edp10-a-new-and-very-promising-approach/" start="Tue Mar 02 2010 17:40:07 GMT+0000" title="EDP10 &amp;#8212; a new and very promising approach">
-    From time to time, there has been an input into this project that has given rise to a burst of optimism (on my part anyway). Perhaps the first was the rapid discovery of very long sequences with low discrepancy, and especially the fact that these sequences had interesting structure to them. (The length led me [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1554&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/02/24/edp9-a-change-of-focus/" start="Wed Feb 24 2010 11:49:09 GMT+0000" title="EDP9 &amp;#8212; a change of focus">
-    The discussion in the last thread has noticeably moved on to new topics. In particular, multiplicative functions have been much less in the spotlight. Some progress has been made on the question of whether the Fourier transform of a sequence of bounded discrepancy must be very large somewhere, though the question is far from answered, [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1542&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/02/19/edp8-what-next/" start="Fri Feb 19 2010 08:10:17 GMT+0000" title="EDP8 &amp;#8212; what next?">
-    It&amp;#8217;s taken noticeably longer than usual for the number of comments on the previous EDP post to reach 100, so this is perhaps a good moment to think strategically about what we should do. Individual researchers continually have a choice &amp;#8212; whether to take a break from the problem and work on other, possibly more [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1533&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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-    <event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gowers.wordpress.com/2010/02/17/dhj-latest/" start="Wed Feb 17 2010 00:14:17 GMT+0000" title="DHJ latest">
-    A quick post to say that earlier today I put a new version of the write-up of Polymath1&amp;#8242;s proof of the density Hales-Jewett theorem on the arXiv. Very soon it will be submitted for publication. I will not say more at this stage (since I don&amp;#8217;t want the journal to evaluate the paper in the [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=gowers.wordpress.com&amp;blog=1659011&amp;post=1531&amp;subd=gowers&amp;ref=&amp;feed=1&quot; /&gt;
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+<data><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/08/07/p-n-np/" start="Sun Aug 08 2010 03:21:15 GMT+0000" title="P &#8800; NP">An email I was recently forwarded (a couple of steps removed) from Vinay Deolalikar from HP Labs: Dear Fellow Researchers, I am pleased to announce a proof that P is not equal to NP, which is attached in 10pt and 12pt fonts. The proof required the piecing together of principles from multiple areas within mathematics. [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/06/24/everything-i-know-about-databases-is-wrong-also-right/" start="Thu Jun 24 2010 19:48:44 GMT+0000" title="Everything I know about databases is wrong.  Also, right.">I have been teaching CMPT 470 for six years now, with my 13th offering going on right now. Anybody doing that is going to pick up a thing or two about web systems. I was there for the rise of the MVC frameworks and greeted them with open arms. I watched Web 2.0 proclaim &amp;#8220;screw [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/06/10/computer-woes/" start="Thu Jun 10 2010 17:32:46 GMT+0000" title="Computer Woes">My computer at home has been locking up occasionally for the last few weeks. This has been happening since my upgrade to Ubuntu 10.04/Lucid, but I suspect this is a coincidence. (1) The lockups are hard: even the SysRq magic doesn&amp;#8217;t do anything, so I deduce that the problem is in the kernel or below. [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/06/03/future-epic-culinary-journeys/" start="Fri Jun 04 2010 06:46:31 GMT+0000" title="Future epic culinary journeys?">Now that me fried rice journey has ended, I find myself contemplating the next epic food journey. Having gone through one, I think I can handle another. The key is to only ask individual people, namely either my Aunt Daisy or my mom, very specific questions about whatever dish I am trying to make. I [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/06/01/how-fried-rice-drove-me-insane-part-3/" start="Wed Jun 02 2010 04:55:58 GMT+0000" title="How fried rice drove me insane (Part 3)">Following Saturday&amp;#8217;s soul-crushing family conversation about my Ama&amp;#8217;s fried rice, during which I was informed of additonal, never-before-mentioned fried rice ingredients and cooking techniques, I have to say that I was at my wit&amp;#8217;s end. There may have been a few instances that I sat in a ball and ricked back and forth. However, Sunday [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/05/30/how-fried-rice-drove-me-insane-part-2/" start="Sun May 30 2010 18:20:30 GMT+0000" title="How fried rice drove me insane (Part 2)">My mom is in town, so Greg and I went to have dinner at my Ama&amp;#8217;s house. Inevitably the conversation turned to food, which led me to ask about my Ama&amp;#8217;s fried rice for the 100th time. This time there were more people around, and everyone had an opinion about what went into Ama&amp;#8217;s fried [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/05/29/how-fried-rice-drove-me-insane-part-1/" start="Sun May 30 2010 05:32:09 GMT+0000" title="How fried rice drove me insane (Part 1)">For the last ~5+ years I have been attempting to make fried rice. Not just any fried rice though, my Ama&amp;#8217;s fried rice. I have wonderful memories of visiting Vancouver in the summer and for Christmas and having my Ama&amp;#8217;s fried rice for dinner, usually with an accompanying soup of some sort. The combination of [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/05/28/how-to-not-attend-a-lecture/" start="Fri May 28 2010 07:06:10 GMT+0000" title="How to not attend a lecture">I teach at a university. That comes with certain parameters: most of my students are in their late teens or early twenties, the average student is reasonably bright but occasionally unmotivated, and I don&amp;#8217;t really have any way to compel students to come to lectures. I do my best to give interesting, informative, and entertaining [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/04/19/cmpt-383-for-real-this-time/" start="Tue Apr 20 2010 01:09:07 GMT+0000" title="CMPT 383: for real this time">I have mentioned here before that I was planning to teach CMPT 383. It ended up being a no-go this semester because of a very productive capstone project team (more on that later). But, I&amp;#8217;m on-deck to teach it in the summer. The class is full; the waiting list is full; must be time to [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://gregbaker.ca/blog/2010/03/19/the-history-of-html/" start="Fri Mar 19 2010 15:46:21 GMT+0000" title="The History of HTML">After a simple query from a colleague about the differences between HTML versions, I wrote this. I thought I might as well post it. Everything was from-memory, so there may be some minor errors. HTML 1 never existed (it was the informal &amp;#8220;standard&amp;#8221; that the first documentation implied). HTML 2 was a really minimal initial [...]</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://gregbaker.ca/blog/2010/08/07/p-n-np/comment-page-1/#comment-15422" start="Tue Aug 10 2010 20:33:53 GMT+0000" title="By: HP Labs researcher Vinay Deolalikar proves P != NP (possibly) | LogicHP - HP Laptop Coupons, Deals, Reviews, News, and Forum">[...] given the $1M prize associated with solving one of the Millennium Problems.Via Slashdot via Greg Baker&#8217;s BlogRelated posts:New photo editing programs coming from HP LabsHP Labs to become more cost effective [...]</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://gregbaker.ca/blog/2010/08/07/p-n-np/comment-page-1/#comment-15421" start="Tue Aug 10 2010 18:39:18 GMT+0000" title="By: P &#8800; NP? &#171; The Math Less Traveled">[...] draft of a paper entitled &#8220;P &#8800; NP&#8221;. The mathematics and computer science communities immediately erupted in a frenzy of excitement and [...]</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://gregbaker.ca/blog/2010/08/07/p-n-np/comment-page-1/#comment-15419" start="Tue Aug 10 2010 15:24:01 GMT+0000" title="By: Opposed Systems Design :: P != NP :: August :: 2010">[...] looks like a serious proof for one of the biggest unsolved problems in mathematics has been presented. It is not yet peer-reviewed, so we&#8217;ll see how this turns out, but this is a big [...]</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://gregbaker.ca/blog/2010/08/07/p-n-np/comment-page-1/#comment-15418" start="Tue Aug 10 2010 15:03:04 GMT+0000" title="By: P &#8800; NP? | Elias Diab log">[...] was on holidays and when I came back I came&#160;across this blog post in slashdot which introduced me to&#160;this paper-draft by Vinay Deolalikar, where he claims to have [...]</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://gregbaker.ca/blog/2010/08/07/p-n-np/comment-page-1/#comment-15416" start="Tue Aug 10 2010 13:21:43 GMT+0000" title="By: Science in the Open &#187; Blog Archive &#187; P &#8800; NP and the future of peer review">[...] S. Cook: &#8220;This appears to be a relatively serious claim to have solved P vs NP.&#8221; (gregbaker.ca) [...]</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://gregbaker.ca/blog/2010/08/07/p-n-np/comment-page-1/#comment-15415" start="Tue Aug 10 2010 12:29:27 GMT+0000" title="By: HP engineer claims to have proven P &#8800; NP">[...] engineer claims to have proven P &#8800; NP     Millennial Prize if it holds up, and solves one of the biggest open compsci math problems existent.  If you use an ATM card or engage in E-commerce or basically have any internet passwords....this is [...]</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://gregbaker.ca/blog/2010/08/07/p-n-np/comment-page-1/#comment-15414" start="Tue Aug 10 2010 11:56:28 GMT+0000" title="By: Pesquisador da HP demonstra poss&#237;vel prova de que P != NP | Pablo Ximenes">[...] http://gregbaker.ca/blog/2010/08/07/p-n-np/   This entry was posted in Ensino, Outros and tagged algoritmos, ci&#234;ncia da computa&#231;&#227;o, Cook, P [...]</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://gregbaker.ca/blog/2010/08/07/p-n-np/comment-page-1/#comment-15413" start="Tue Aug 10 2010 10:34:37 GMT+0000" title="By: Chris Alexander - Possible proof emerges that P != NP">[...] Yesterday news emerged of an HP researcher who has claimed that he has proved that P != NP. [...]</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://gregbaker.ca/blog/2010/08/07/p-n-np/comment-page-1/#comment-15412" start="Tue Aug 10 2010 09:11:33 GMT+0000" title="By: &#191;Una prueba de que P&#8800;NP? &#171; :: ZTFNews.org">[...] Greg Baker, P&#8800;NP [7 de agosto de 2010] [...]</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://gregbaker.ca/blog/2010/08/07/p-n-np/comment-page-1/#comment-15411" start="Tue Aug 10 2010 07:37:56 GMT+0000" title="By: P&#8800;NP? | Q">[...] (no en el sentit de que sigui correcte o no, encara, sin&#243; en el que mereix una lectura s&#232;ria): Greg Baker, Dick Lipton, David Bacon i Scott Aaronson. Si &#233;s correcte, el premi &#233;s d&#8217;un mili&#243; de [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/" start="Sun Aug 15 2010 18:30:20 GMT+0000" title="The P&#8800;NP &#8220;Proof&#8221; Is One Week Old">Another update on progress understanding the proof of Deolalikar Terence Tao is leading, as often is the case, with some very insightful comments on the status of Vinay Deolalikar&amp;#8217;s proof. Today Ken Regan and I want to try to summarize the main issues with Deolalikar&amp;#8217;s proof of PNP raised by Tao and many many others. [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5273&amp;subd=rjlipton&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/08/12/fatal-flaws-in-deolalikars-proof/" start="Fri Aug 13 2010 04:15:21 GMT+0000" title="Fatal Flaws in Deolalikar&#8217;s Proof?">Possible fatal flaws in the finite model part of Deolalikar&amp;#8217;s proof Neil Immerman is one of the world&amp;#8217;s experts on Finite Model Theory. He used insights from this area to co-discover the great result that is closed under complement. Today I had planned not to discuss the proof, but I just received a note from [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5260&amp;subd=rjlipton&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/08/11/deolalikar-responds-to-issues-about-his-p%e2%89%a0np-proof/" start="Thu Aug 12 2010 02:14:13 GMT+0000" title="Deolalikar Responds To Issues About His P&#8800;NP Proof">A principle for checking proofs with an application to the proof Vinay Deolalikar is standing by his claim and proof. He and I have been exchanging e-mails, and as noted in the several comments, he has updated his paper several times. The updates are trying to answer some of the questions raised here and elsewhere. [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5244&amp;subd=rjlipton&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/08/10/update-on-deolalikars-proof-that-p%e2%89%a0np/" start="Wed Aug 11 2010 03:47:13 GMT+0000" title="Update on Deolalikar&#8217;s Proof that P&#8800;NP">An update on the P not equal to NP proof Timothy Gowers, Gil Kalai, Ken Regan, Terence Tao, and Suresh Venkatasubramanian are some of the top people who are trying to unravel what is up with the claimed proof that P NP. I will call them the Group today&amp;#8212;I hope that is okay. It&amp;#8217;s a [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5221&amp;subd=rjlipton&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/08/09/issues-in-the-proof-that-p%e2%89%a0np/" start="Tue Aug 10 2010 01:20:45 GMT+0000" title="Issues In The Proof That P&#8800;NP">Some further comments into the recent claimed proof that P is not equal to NP Yuri Manin is a great mathematician, who has worked in various areas and also is well known for his many excellent expository works. He has won numerous prizes, including the Nemmers Prize in Mathematics in 1994, and the Cantor Medal [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5206&amp;subd=rjlipton&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/08/08/a-proof-that-p-is-not-equal-to-np/" start="Mon Aug 09 2010 01:11:24 GMT+0000" title="A Proof That P Is Not Equal To NP?">A serious proof that claims to have resolved the P=NP question. Vinay Deolalikar is a Principal Research Scientist at HP Labs who has done important research in various areas of networks. He also has worked on complexity theory, including previous work on infinite versions of the P=NP question. He has just claimed that he has [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5195&amp;subd=rjlipton&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/08/07/hilberts-tenth-over-the-rationals/" start="Sat Aug 07 2010 18:31:12 GMT+0000" title="Hilbert&#8217;s Tenth Over The Rationals">Can there be a bijection from pairs of rationals to rationals? Bjorn Poonen is a famous number theorist at MIT, who has many honors and awards, including being one of the few four time Putnam Exam winners. He has worked on many aspects of number theory, including the &amp;#8220;dark side&amp;#8221; as he calls it. The [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5182&amp;subd=rjlipton&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/08/02/quantum-algorithms-via-linear-algebra/" start="Mon Aug 02 2010 12:38:47 GMT+0000" title="Quantum Algorithms A Different View">Quantum algorithms explained in a different way, not the usual way. David Deutsch is one of the founders of quantum computing. His 1985 paper described quantum Turing machines, and his famous algorithm is considered the first quantum algorithm ever. Today I plan to try to explain the power of quantum algorithms, but do it in [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5166&amp;subd=rjlipton&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/07/30/paranoia-and-selling-a-laptop/" start="Fri Jul 30 2010 15:56:30 GMT+0000" title="Paranoia And Selling A Laptop">How do we erase a hard disk, for sure? Richard Nixon was not a theorist&amp;#8212;of course. He was the President of the United States from 1969 until 1974, when he resigned in disgrace. He was also an &amp;#8220;un-indicted co-conspirator&amp;#8221; and probably one of the most paranoid presidents ever. Today I want to talk about paranoia. [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5154&amp;subd=rjlipton&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://rjlipton.wordpress.com/2010/07/25/hedy-lamarr-the-inventor/" start="Sun Jul 25 2010 14:10:51 GMT+0000" title="Hedy Lamarr the Inventor">The invention of spread spectrum and the role of amateurs in science and technology Hedy Lamarr was, of course, not a mathematician nor a complexity theorist. She was one of the great movie actresses of all time, and was once &amp;#8220;voted&amp;#8221; the most beautiful woman in the world. She was also an inventor. Today I [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=rjlipton.wordpress.com&amp;blog=6472207&amp;post=5142&amp;subd=rjlipton&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/#comment-5860" start="Tue Aug 17 2010 21:33:33 GMT+0000" title="Comment on The P&#8800;NP &#8220;Proof&#8221; Is One Week Old by Anonymous">Prof. Lipton asked in one of his earlier blogs
+
+"is it possible for an amateur to solve the P/NP problem?".
+
+
+On a second thought - the question is almost equivalent to P/NP problem itself  - 
+i.e., 
+1. you consider an amateur to be a machine with smaller resources 
+2. the question at hand sounds analogus to a NP problem in the sense that - given a proof, one (or the community) can verify it in polynomial time ( 1 week in this case)
+
+
+:)</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/#comment-5859" start="Tue Aug 17 2010 21:14:59 GMT+0000" title="Comment on The P&#8800;NP &#8220;Proof&#8221; Is One Week Old by Harvey Friedman">It is of course *possible* that logical issues are what is holding up progress on the massive number of critical lower bound issues throughout theoretical computer science. E.g, that one has to use more than ZFC in order to make progress on these problems. 
+
+Unfortunately, I cannot say that I have any kind of viable research program for showing that ZFC is inadequate for these crucial problems. 
+
+Let me also add that if I did have such a viable research program for showing this, I would probably write this posting.</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/#comment-5858" start="Tue Aug 17 2010 21:14:33 GMT+0000" title="Comment on The P&#8800;NP &#8220;Proof&#8221; Is One Week Old by anonymous">&lt;i&gt;do you mean something like applying the relevant f for each instance separately, and then look on the structure of the entire solution space? would that still be considered &#8220;the same structure&#8221; to the solution space of the original problem?&lt;/i&gt;
+
+yes.... the map is not supposed to be uniform... i think the point was that the structure of the solution space has nothing to do with the complexity of the problem...</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/#comment-5857" start="Tue Aug 17 2010 21:04:08 GMT+0000" title="Comment on The P&#8800;NP &#8220;Proof&#8221; Is One Week Old by CS grad">For a single instance - there is no difference between the two, since an instance (\phi,a') [with solution a] already includes the number of variables to complete m=|a|. There's no notion of Kolmogorov complexity here, since we only discuss a single string.
+
+I understand that for a *set* of instances (i.e. \phi with different assignments a_1,a_2,... ; or different formulas \phi with increasing size) there is a difference. 
+However (that's point 2.), different instances (even of the same formula) require different mapping functions f. So now you have a polynomial algorithm, but I'm not sure it solves a similar problem.
+
+It is not the same to claim that f_{\phi,a} preserves the structure of solution space of \phi (which is obviously true), and that separately mapping each instance of a certain set (or a probability distribution over such set) , preserves the structure of the entire space of solutions to all instances. The latter seems like a much stronger claim, and not necessarily true.
+
+As a metaphor, consider a linear transformation carried on, say, a tree in R^2. Clearly all properties of the tree remain. Now suppose you perform a (different) linear transformation on each of a large set of distinct trees, so that all their root now coincide. This is far from being the same - (non)connectivity for example is not kept.</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/#comment-5856" start="Tue Aug 17 2010 20:57:46 GMT+0000" title="Comment on The P&#8800;NP &#8220;Proof&#8221; Is One Week Old by Ryan Williams">Thanks vloodin!  It looks like you have given a more direct argument than my half-baked intuitions about SAT vs SAT0. I am not sure you disagree with me as much as you think you do! 
+
+Sorry if my comments were sometimes cryptic, but the last few days have been extremely busy for me, and I could only comment sporadically. But I have enjoyed catching up on the discussion over the last hour, especially the "true" meaning of parameters.</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/#comment-5855" start="Tue Aug 17 2010 20:50:28 GMT+0000" title="Comment on The P&#8800;NP &#8220;Proof&#8221; Is One Week Old by vloodin">He should get a quick review. Things are pretty clear by now.
+
+I cannot check now his "ampleness" assumption (I did not save his draft 3), but if Tao is right, and he relied on supports of ppp distributions, than there is no chance he has proved anything. He might have fixed (a) ,(b) and (c), but his proof is dead due to pp vs ppp issue. I guess he is just trying to save face, it doesn't seem to be easy for him to admit defeat. 
+
+A retraction would have been a much wiser move.</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/#comment-5853" start="Tue Aug 17 2010 20:39:59 GMT+0000" title="Comment on The P&#8800;NP &#8220;Proof&#8221; Is One Week Old by NK">Also, only the synopsis of the proof is now available both from his web page as well as from his paper web folder.</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/#comment-5851" start="Tue Aug 17 2010 20:06:41 GMT+0000" title="Comment on The P&#8800;NP &#8220;Proof&#8221; Is One Week Old by Terence Tao">Ah, I see.  So ppp is an even wider class than I thought, it that it contains distributions supported on SAT0 solution spaces.  The same argument also gives a (non-uniform) way to create ppp distributions on any non-empty SAT solution space (or indeed, any non-empty solution space of a function in P), by using some (non-uniformly selected) specific solution (a_1,...,a_n) to the SAT problem as the "default".
+
+If so, then it appears that there is in fact no separation of P and NP going on here, even at the conjectural level; regardless of whether P equals NP or not, the k-SAT solution space supports a ppp distribution.   So the ppp concept is of no actual use for this problem.</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/#comment-5850" start="Tue Aug 17 2010 19:56:52 GMT+0000" title="Comment on The P&#8800;NP &#8220;Proof&#8221; Is One Week Old by Interested Amateur">My thanks for the time you've spent on your illumination Mr Regan. 
+It helps.
+
+I'm beginning to wonder if just maybe there is a duality involved?  Sometimes P will = NP and sometimes it won't - and determining &gt;&gt;when&lt;&lt;  it is and when it isn't is another "Uncertainty Principle",  spoiling the Order we'd perhaps prefer!
+
+I will follow this discussion with added interest (and a little more enlightened).</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/#comment-5849" start="Tue Aug 17 2010 19:56:27 GMT+0000" title="Comment on The P&#8800;NP &#8220;Proof&#8221; Is One Week Old by NK">Vinay D. has updated his web page as follows:
+"The preliminary version was meant to solicit feedback from a few researchers as is customarily done. I have fixed all the issues that were raised about the preliminary version in a revised manuscript; clarified some concepts; and obtained simpler proofs of several claims. This revised manuscript has been sent to a small number of researchers.  I will send the manuscript to journal review this week. Once I hear back from the journal, I will put up the final version on this website."</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://scottaaronson.com/blog/?p=460" start="Sun Aug 15 2010 21:48:04 GMT+0000" title="And now a word from our sponsors">Today I interrupt your regularly-scheduled P vs. NP programming to bring you a special message from Dmitry Maslov, the program director at NSF Computing and Communication Foundations (CCF) who handles quantum information science.&#160; Besides paying for my CAREER grant (and thus, arguably, in an indirect sense, for this blog), Dmitry also happens to be one [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://scottaaronson.com/blog/?p=459" start="Sun Aug 15 2010 15:39:05 GMT+0000" title="P vs. NP for Dummies">A reader named Darren commented on my last post:
+I have this feeling that this whole P and NP thing is not only a  profound problem that needs solving, but something that can be  infinitely curious to try and wrap your mind around&#8230;
+Thing is- there&#8217;s a whole world of great minded, genius hackers out [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://scottaaronson.com/blog/?p=458" start="Sat Aug 14 2010 01:08:28 GMT+0000" title="Eight Signs A Claimed P&#8800;NP Proof Is Wrong">As of this writing, Vinay Deolalikar still hasn&amp;#8217;t retracted his P&#8800;NP claim, but a clear consensus has emerged that the proof, as it stands, is fatally flawed.&#160; The first reason is that we&amp;#8217;re not going to separate k-SAT from much easier problems purely by looking at the structure of the solution space: see for example [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://scottaaronson.com/blog/?p=457" start="Thu Aug 12 2010 01:03:07 GMT+0000" title="The ethics of scientific betting">Throughout the day, Dick Lipton&amp;#8217;s blog has hosted a phenomenal discussion of Vinay Deolalikar&amp;#8217;s attempted proof of P&#8800;NP (of which a new draft appeared as this blog entry was going to press).&#160; As of this writing, the discussion seems to have led to the following two conclusions:
+
+Deolalikar deserves our gratitude; he did a wonderful thing [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://scottaaronson.com/blog/?p=456" start="Mon Aug 09 2010 06:53:58 GMT+0000" title="Putting my money where my mouth isn&#8217;t">A few days ago, Vinay Deolalikar of HP Labs started circulating a claimed proof of P&#8800;NP.&#160; As anyone could predict, the alleged proof has already been Slashdotted (see also Lipton&amp;#8217;s blog and Bacon&amp;#8217;s blog), and my own inbox has been filling up faster than the Gulf of Mexico.
+Alas, a simple &amp;#8220;top kill&amp;#8221; seems unlikely to [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://scottaaronson.com/blog/?p=455" start="Sat Jul 31 2010 21:20:51 GMT+0000" title="Going into deep freeze">I&amp;#8217;m leaving tomorrow for a grand tour of Banff, then Israel, then Greece, then Princeton.&#160; Blogging may be even lighter than usual.
+In the meantime, my friend Michael Vassar has asked me to advertise the 2010 Singularity Summit, to be held August 14-15 in San Francisco.&#160; Register now, because the summit is approaching so rapidly that [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://scottaaronson.com/blog/?p=454" start="Sat Jul 24 2010 16:54:50 GMT+0000" title="My diavlog with Anthony Aguirre">Bloggingheads has&#160;just posted an hour-long diavlog&#160;between the cosmologist Anthony Aguirre and your humble blogger. &#160;Topics discussed include: the anthropic principle; how to do quantum mechanics if the universe is so large that there could be multiple copies of you; Nick Bostrom&amp;#8217;s &amp;#8220;God&amp;#8217;s Coin Toss&amp;#8221; thought experiment; the cosmological constant; the total amount of computation in [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://scottaaronson.com/blog/?p=453" start="Tue Jul 13 2010 13:48:31 GMT+0000" title="My philomath project: Sensitivity versus block-sensitivity">If you like math, and you don&amp;#8217;t yet have a Math Overflow account, stop reading this post now (not right now, but by the end of the sentence) and set one up, before returning here to finish reading the post.&#160; Math Overflow is the real deal: something that I&amp;#8217;ve missed, dreamed about, and told my [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://scottaaronson.com/blog/?p=452" start="Sun Jul 11 2010 05:02:52 GMT+0000" title="The Generalized Linial-Nisan Conjecture is false">In a post a year and a half ago, I offered a prize of $200 for proving something called the Generalized Linial-Nisan Conjecture, which basically said that almost k-wise independent distributions fool AC0 circuits.&#160; (Go over to that post if you want to know what that means and why I cared about it.)
+Well, I&amp;#8217;m pleased [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://scottaaronson.com/blog/?p=451" start="Mon Jul 05 2010 21:53:49 GMT+0000" title="Doing my oracle duty">I promised myself I&amp;#8217;d stop blogging about controversial issues whose mere mention could instigate a flamewar and permanently get me in trouble.&#160; Well, today I&amp;#8217;m going to violate that rule, by blogging about the difference relativized and unrelativized complexity classes.
+Recently a colleague of mine, who works in the foundations of quantum mechanics, sent me a [...]</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://scottaaronson.com/blog/?p=457#comment-45925" start="Tue Aug 17 2010 21:30:26 GMT+0000" title="Comment on The ethics of scientific betting by John Sidles">Quantum Tennis Referee Says:   &lt;i&gt;@Sidles: ... The "QM as OS" quote is fantastic.&lt;/i&gt;
+
+The remark "quantum mechanics is the operating system that other physical theories run on as application software" can be found in Scott's &lt;a href="http://www.scottaaronson.com/democritus/lec9.html" rel="nofollow"&gt;Quantum computing since Democritus #9&lt;/a&gt; ... a lecture series that I sincerely admire and recommend (even though I don't agree with everything in it).
+
+QTR, you can take the work "sincerely" for granted, because writing "funny" is completely beyond my talents.  
+
+Moreover, during the past week, the Deolalikar episode has amply demonstrated that writing high-quality sardonicism is even tougher than writing "funny" ... requiring Joseph-Heller levels of narrative talent (at a minimum).
+
+IMHO no-one to date has successfully written sardonically about the Deolalikar's claimed proof (despite numerous attempts) ... in fact, no attempt has even come close.  
+
+Perhaps more folks need to keep in-mind that (IMHO) Heller's &lt;i&gt;Catch-22&lt;/i&gt; worked as a masterpiece of sardonicism only because at the end, Orr *did* get to Sweden.</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://scottaaronson.com/blog/?p=460#comment-45923" start="Tue Aug 17 2010 20:21:16 GMT+0000" title="Comment on And now a word from our sponsors by Joshua Zelinsky">Greg, sorry yes, that makes more clear what you are trying to do.</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://scottaaronson.com/blog/?p=457#comment-45919" start="Tue Aug 17 2010 19:42:27 GMT+0000" title="Comment on The ethics of scientific betting by Matthias Gall&#233;">Just found this quote from E Rutherford: &quot;&#8220;You should never bet against anything in science at odds of more than about 10-12 to 1 against&#8221;. Scott, now you share something with Einstein (http://thinkexist.com/quotation/anyone_who_expects_a_source_of_power_from_the/191140.html)
+
+More funny yet (because some starts to think in terms of war of bands) this quote can be found on rjliptons webpage :D</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://scottaaronson.com/blog/?p=458#comment-45916" start="Tue Aug 17 2010 18:40:34 GMT+0000" title="Comment on Eight Signs A Claimed P&#8800;NP Proof Is Wrong by Quantum Tennis Receiver">??? I don't understand where these comments are coming from.
+
+My question is independent of the Deolalikar affair.
+
+I want Scott to write a blog post about reading mathematics. This has absolutely nothing to do with P/NP or Deolalikar or proofs of famous problems.
+
+I am talking about simple STOC/FOCS/journal level papers and tips on reading these. Again, nothing to do with verifying them ... simply tips Scott might have for students.</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://scottaaronson.com/blog/?p=459#comment-45909" start="Tue Aug 17 2010 17:43:16 GMT+0000" title="Comment on P vs. NP for Dummies by Peter Drubetskoy">Anybody able to throw some light on the assertion (such as voiced by Wigderson &lt;a href="http://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/W06/w06.pdf" rel="nofollow"&gt;here&lt;/a&gt; and elsewhere), that pseudorandom generators are used to derandomize BPP algorithms? I understand how they can take the second &lt;b&gt;P&lt;/b&gt; out of &lt;b&gt;BPP&lt;/b&gt; (i.e., by fooling the algorithm with pseudorandom numbers you get a result that is no longer dependent on truly random numbers) but how do you get rid of the &lt;b&gt;B&lt;/b&gt;, i.e., the fact that the resulting algorithm, while deterministic, is still bounded-error? Maybe for all practical intents and purposes there is no diffrence between a polynomial algo that returns correct result with prob 100% and the one that does so with prob 100%-epsilon, but theoretically there certainly is. 
+And how does one see that the AKS algo for PRIMES is an example of derandomization  of this type (as Wigderson says in the above paper: &lt;i&gt;Indeed, careful analysis
+of some important probabilistic algorithms, and the way they use their randomness, has enabled making them deterministic via tailor-made generators. These success stories (of which the most dramatic is the recent deterministic primality test of [3]) actually suggest the route of probabilistic algorithms and then derandomization as a paradigm for deterministic algorithm design.&lt;/i&gt;)</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://scottaaronson.com/blog/?p=459#comment-45908" start="Tue Aug 17 2010 16:53:47 GMT+0000" title="Comment on P vs. NP for Dummies by Rahul">Raoul: I was not making a judgment about which question is harder or more interesting, just saying that they might be somewhat orthogonal.</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://scottaaronson.com/blog/?p=460#comment-45903" start="Tue Aug 17 2010 16:07:00 GMT+0000" title="Comment on And now a word from our sponsors by Greg Kuperberg">Joshua, certainly L is an interesting smaller class and probably should be included.  There is a technical reason that smaller classes work less well in Complexity Zoology:  On the one hand, I have to use oracle separations to make sense of the map of complexity classes.  On the other hand, the oracle model begins to break down for classes that are too small (or too large), because it is tricky to define a natural level of oracle access.  Nonetheless, for L it's still okay.
+
+The sense of my question, though, is that I don't need help with individual classes; I have plenty of them.  The problem is an excessive proliferation of *pairs* of classes.  To see what I'm talking about, take a look at the &lt;a href=&quot;http://www.math.ucdavis.edu/~greg/zoology/&quot; rel=&quot;nofollow&quot;&gt;Complexity Zoology project&lt;/a&gt; where I last left it off.
+My request is for a large clique of interesting pairs of classes.  I think this much at least is safe:
+
+L, P, NP, RP, BPP, BQP, MA, AM, SZK, CZK, IP, QIP, &#931;_2P, &#931;_3P, PP, PH, BQP, QMA, S_2P, BPP_path, MP, RG, QRG, +P, PSPACE, EXP, NEXP
+
+plus the polynomial Turing closure (A -&gt; P^A) and the polynomial advice closure (A -&gt; A/poly) of some of these classes.  (But I am not sure which ones should be so extended.)  In fact, this list seems too conservative.  The disease that eventually infected the project was comparisons like DQP vs ModP, where understandably no one cares to even call the relation an open problem.</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://scottaaronson.com/blog/?p=460#comment-45899" start="Tue Aug 17 2010 15:26:42 GMT+0000" title="Comment on And now a word from our sponsors by Joshua Zelinsky">Greg, it seems like you primarily have classes which are comparatively large.  Maybe looking at small classes might help? L might be an example (if I'm not mistaken whether L=P is open as is whether L = RL but I'm not aware of any interesting connections between L and comparatively large classes like PP and NP. )</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://scottaaronson.com/blog/?p=460#comment-45896" start="Tue Aug 17 2010 15:07:33 GMT+0000" title="Comment on And now a word from our sponsors by Greg Kuperberg">Andras:  Certainly you should include BPP and P^#P.  (You can't add #P itself because it is not Boolean.)  In all I would want at least 50 classes (expanded to more with complementation and intersection with complement, maybe also with the class operator A -&gt; P^A).</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://scottaaronson.com/blog/?p=459#comment-45894" start="Tue Aug 17 2010 14:45:22 GMT+0000" title="Comment on P vs. NP for Dummies by Raoul Ohio">Rahul (#70): Not sure I agree with you there. "making a taxonomy of polynomial-time algorithms" should be doable and useful, just a lot of hard work. On the other hand, "P vs. NP" is likely to never be decided, has little if any practical use, and is absorbing huge numbers of smart people hours.
+
+
+The TCS community should beware of the example of Point Set Topology. Several decades ago PST was one of the most important areas in math. They have veered off into recreational math areas such as "topology of transfinite cardinals". It might be the case that PST has fully explored everything worth exploring, and this is the best they can do. Actually, a lot of PST people switched to TCS, which probably explains a lot.</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/TheGeomblog/~3/iQW505GA56A/cstheory-q-site-now-open.html" start="Mon Aug 16 2010 15:53:25 GMT+0000" title="cstheory q&amp;a site now open !" /><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/TheGeomblog/~3/1l0UDLuVLI0/social-theorycsnow-98-better.html" start="Sat Aug 14 2010 02:12:55 GMT+0000" title="Social theoryCS...now 98% better..." /><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/TheGeomblog/~3/Q66fRhPA8bE/p-vs-np-what-ive-learnt-so-far.html" start="Wed Aug 11 2010 11:39:59 GMT+0000" title="P vs NP: What I&apos;ve learnt so far..." /><event 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icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/TheGeomblog/~3/pP7x9ZzmPvk/case-study-in-large-theory-conferences.html" start="Wed Jun 23 2010 00:59:31 GMT+0000" title="Case Study in Large Theory Conferences: ICALP" /><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/TheGeomblog/~3/2NxFKfrwJ30/on-acceptance-rates-and-flagship.html" start="Mon Jun 21 2010 10:09:48 GMT+0000" title="On acceptance rates and flagship conferences" /><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/TheGeomblog/~3/X4B38aEUknE/shape-of-shape-analysis-research-part_19.html" start="Sat Jun 19 2010 01:48:52 GMT+0000" title="The Shape of Shape Analysis Research, Part III" /><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/TheGeomblog/~3/QnVfc9knPNU/rebooting-how-we-publish-in-cs.html" start="Thu Jun 17 2010 11:42:48 GMT+0000" title="Rebooting how we publish in CS." /><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/TheGeomblog/~3/CiTZZyXQGcM/its-over.html" start="Thu Jun 17 2010 11:38:42 GMT+0000" title="It&apos;s over !!" /><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/TheGeomblog/~3/lLTp1DcKmpk/shape-of-shape-analysis-research-part.html" start="Tue Jun 15 2010 02:34:38 GMT+0000" title="The Shape of Shape Analysis Research: Part II" /><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/TheGeomblog/~3/v2PIz_BRGm4/active-learning-modules-for-grad.html" start="Tue Jun 08 2010 12:16:25 GMT+0000" title="Active learning modules for grad algorithms ?" /><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/TheGeomblog/~3/R6o8YT9utvo/why-double-blind-review-occasionally.html" start="Mon Jun 07 2010 23:02:18 GMT+0000" title="Why double blind review occasionally annoys me." /><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/TheGeomblog/~3/d8hYjum9vzQ/bibtex-style-question.html" start="Sat Jun 05 2010 00:37:41 GMT+0000" title="bibtex style question" /><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/TheGeomblog/~3/xh8dD1Vre_k/minor-conundrum-when-citing-related.html" start="Wed Jun 02 2010 06:19:30 GMT+0000" title="A (minor) conundrum when citing related work" /><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/TheGeomblog/~3/fgCOWZcI2eU/avner-magen.html" start="Tue Jun 01 2010 11:17:15 GMT+0000" title="Avner Magen" /><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/TheGeomblog/~3/-L6KO7EE9SI/summer-seminars.html" start="Mon May 31 2010 14:53:15 GMT+0000" title="Summer Seminars" /><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Polylog_parameterizability&amp;diff=3615&amp;oldid=prev" start="Tue Aug 17 2010 20:47:33 GMT+0000" title="Polylog parameterizability">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;pp vs ppp:&amp;#32;&lt;/span&gt; vloodin's example that the solution space of any k-SAT problem with at least one solution is ppp&lt;/p&gt;
+
+			&lt;table&gt;
+			&lt;col class=&quot;diff-marker&quot; /&gt;
+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;col class=&quot;diff-marker&quot; /&gt;
+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;tr valign=&quot;top&quot;&gt;
+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 20:47, 17 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 103:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 103:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;However this does not allow us to conclude that P!=NP, because of the substantial gap between pp and ppp.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;However this does not allow us to conclude that P!=NP, because of the substantial gap between pp and ppp.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;Furthermore, it is in fact true that the solution space of any k-SAT problem with at least one solution supports a ppp distribution regardless of whether P=NP, though the ppp distribution is not uniform in the sense that it depends on knowing the specific solution.&amp;nbsp; Specifically, one randomly selects an element of {0,1}^n, and tests whether it solves the k-SAT problem.&amp;nbsp; If it does, output that element; otherwise output the specific solution.&amp;nbsp; It is easy to see that this distribution is ppp and is supported exactly on the solution space of k-SAT.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:09 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper&amp;diff=3614&amp;oldid=prev" start="Tue Aug 17 2010 20:43:03 GMT+0000" title="Deolalikar P vs NP paper">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;Theory blogs:&amp;#32;&lt;/span&gt; &lt;/p&gt;
+
+			&lt;table&gt;
+			&lt;col class=&quot;diff-marker&quot; /&gt;
+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;col class=&quot;diff-marker&quot; /&gt;
+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;tr valign=&quot;top&quot;&gt;
+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 20:43, 17 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;4&quot; align=&quot;center&quot; class=&quot;diff-multi&quot;&gt;(6 intermediate revisions not shown.)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 25:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 25:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;These links are taken from [http://www.hpl.hp.com/personal/Vinay_Deolalikar/ Vinay Deolalikar's web page].&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;These links are taken from [http://www.hpl.hp.com/personal/Vinay_Deolalikar/ Vinay Deolalikar's web page].&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://www.hpl.hp.com/personal/Vinay_Deolalikar/Papers/pnp12pt.pdf First draft], Aug 6, 2010&amp;nbsp; '''&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;Has since been replaced with the most recent version.&lt;/del&gt;'''&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://www.hpl.hp.com/personal/Vinay_Deolalikar/Papers/pnp12pt.pdf First draft], Aug 6, 2010&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;. &lt;/ins&gt; '''&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;File overwritten several times and then finally removed,&lt;/ins&gt;''' &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;Aug 17 2010.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://www.hpl.hp.com/personal/Vinay_Deolalikar/Papers/pnp_updated.pdf Second draft] Aug 9, 2010.&amp;nbsp; '''File removed,''' Aug 10 2010. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://www.hpl.hp.com/personal/Vinay_Deolalikar/Papers/pnp_updated.pdf Second draft] Aug 9, 2010.&amp;nbsp; '''File removed,''' Aug 10 2010. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://www.hpl.hp.com/personal/Vinay_Deolalikar/Papers/pnp_updated-mjr.pdf draft 2 + &amp;amp;epsilon;], Aug 9 2010. '''File removed,''' Aug 11 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://www.hpl.hp.com/personal/Vinay_Deolalikar/Papers/pnp_updated-mjr.pdf draft 2 + &amp;amp;epsilon;], Aug 9 2010. '''File removed,''' Aug 11 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://www.hpl.hp.com/personal/Vinay_Deolalikar/Papers/pnp_8_11.pdf Third draft], Aug 11 2010. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://www.hpl.hp.com/personal/Vinay_Deolalikar/Papers/pnp_8_11.pdf Third draft], Aug 11 &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;2010.&amp;nbsp;  '''File removed,''' Aug 17 &lt;/ins&gt;2010. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://www.hpl.hp.com/personal/Vinay_Deolalikar/Papers/pnp_synopsis.pdf Synopsis of proof], Aug 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://www.hpl.hp.com/personal/Vinay_Deolalikar/Papers/pnp_synopsis.pdf Synopsis of proof], Aug 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 291:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 291:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/10/update-on-deolalikars-proof-that-p%E2%89%A0np/ Update on Deolalikar's Proof that P&#8800;NP], Richard Lipton and Ken Regan, G&#246;del&#8217;s lost letter and P=NP, August 10 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/10/update-on-deolalikars-proof-that-p%E2%89%A0np/ Update on Deolalikar's Proof that P&#8800;NP], Richard Lipton and Ken Regan, G&#246;del&#8217;s lost letter and P=NP, August 10 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://dabacon.org/pontiff/?p=4292 P&amp;lt;&amp;gt;NP Hype], Dave Bacon, The Quantum Pontiff, August 10 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://dabacon.org/pontiff/?p=4292 P&amp;lt;&amp;gt;NP Hype], Dave Bacon, The Quantum Pontiff, August 10 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/11/deolalikar-responds-to-issues-about-his-p%e2%89%a0np-proof/ Deolalikar Responds To Issues About His P&#8800;NP Proof] Richard Lipton, G&#246;del&#8217;s lost letter and P=NP, August 11, 2010&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/11/deolalikar-responds-to-issues-about-his-p%e2%89%a0np-proof/ Deolalikar Responds To Issues About His P&#8800;NP Proof] Richard Lipton &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;and Ken Regan&lt;/ins&gt;, G&#246;del&#8217;s lost letter and P=NP, August 11, 2010&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://scottaaronson.com/blog/?p=457 The ethics of scientific betting], Scott Aaronson, Shtetl-Optimized, August 11 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://scottaaronson.com/blog/?p=457 The ethics of scientific betting], Scott Aaronson, Shtetl-Optimized, August 11 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://geomblog.blogspot.com/2010/08/p-vs-np-what-ive-learnt-so-far.html P vs NP: What I've learnt so far...], Suresh Venkatasubramanian, The Geomblog, August 11 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://geomblog.blogspot.com/2010/08/p-vs-np-what-ive-learnt-so-far.html P vs NP: What I've learnt so far...], Suresh Venkatasubramanian, The Geomblog, August 11 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://gowers.wordpress.com/2010/08/11/my-pennyworth-about-deolalikar/ My pennyworth about Deolalikar], Timothy Gowers, Aug 11 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://gowers.wordpress.com/2010/08/11/my-pennyworth-about-deolalikar/ My pennyworth about Deolalikar], Timothy Gowers, Aug 11 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/12/fatal-flaws-in-deolalikars-proof Fatal Flaws in Deolalikar&#8217;s Proof?], Richard Lipton, G&#246;del&#8217;s lost letter and P=NP, August 12 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/12/fatal-flaws-in-deolalikars-proof Fatal Flaws in Deolalikar&#8217;s Proof?], Richard Lipton &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;and Ken Regan&lt;/ins&gt;, G&#246;del&#8217;s lost letter and P=NP, August 12 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://gowers.wordpress.com/2010/08/13/could-anything-like-deolalikars-strategy-work/ Could anything like Deolalikar&#8217;s strategy work?], Timothy Gowers, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://gowers.wordpress.com/2010/08/13/could-anything-like-deolalikars-strategy-work/ Could anything like Deolalikar&#8217;s strategy work?], Timothy Gowers, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://scottaaronson.com/blog/?p=458 Eight Signs A Claimed P&#8800;NP Proof Is Wrong], Scott Aaronson, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://scottaaronson.com/blog/?p=458 Eight Signs A Claimed P&#8800;NP Proof Is Wrong], Scott Aaronson, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://gowers.wordpress.com/2010/08/14/a-possible-answer-to-my-objection/ A possible answer to my objection], Tim Gowers, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://gowers.wordpress.com/2010/08/14/a-possible-answer-to-my-objection/ A possible answer to my objection], Tim Gowers, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://constraints.wordpress.com/2010/08/13/p-not-equal-to-np-consequences/ &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;(&lt;/del&gt;P not equal to NP (P&#8800;NP) Consequences], Andr&#225;s Salamon, August 13, 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://constraints.wordpress.com/2010/08/13/p-not-equal-to-np-consequences/ P not equal to NP (P&#8800;NP) Consequences], Andr&#225;s Salamon, August 13, 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/ The P&#8800;NP &amp;quot;Proof&amp;quot; Is One Week Old], Richard Lipton, G&#246;del&#8217;s lost letter and P=NP, August 15 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/ The P&#8800;NP &amp;quot;Proof&amp;quot; Is One Week Old], Richard Lipton &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;and Ken Regan&lt;/ins&gt;, G&#246;del&#8217;s lost letter and P=NP, August 15 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://constraints.wordpress.com/2010/08/15/a-simple-reduction/#more-391 A simple reduction]. Andr&#225;s Salamon, August 15, 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://constraints.wordpress.com/2010/08/15/a-simple-reduction/#more-391 A simple reduction]. Andr&#225;s Salamon, August 15, 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://blog.computationalcomplexity.org/2010/08/but-this-one-is-different.html But this one is different], Lance Fortnow, August 16 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://blog.computationalcomplexity.org/2010/08/but-this-one-is-different.html But this one is different], Lance Fortnow, August 16 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 394:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 394:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* August 12: Neil Immerman [[Immerman's letter|describes what appear to be fatal flaws in the argument]].&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* August 12: Neil Immerman [[Immerman's letter|describes what appear to be fatal flaws in the argument]].&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* August 13: A three-page [http://www.hpl.hp.com/personal/Vinay_Deolalikar/Papers/pnp_synopsis.pdf synopsis of the proof] is posted on Deolalikar's [http://www.hpl.hp.com/personal/Vinay_Deolalikar/ home page].&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* August 13: A three-page [http://www.hpl.hp.com/personal/Vinay_Deolalikar/Papers/pnp_synopsis.pdf synopsis of the proof] is posted on Deolalikar's [http://www.hpl.hp.com/personal/Vinay_Deolalikar/ home page].&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;* August 15: Further difficulties in the proof are summarized in another [http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/ post of Dick Lipton and Ken Regan].&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;* August 17: All drafts of the paper are now removed from Deolalikar's [http://www.hpl.hp.com/personal/Vinay_Deolalikar/ home page], leaving only the synopsis and the comment &amp;quot;The preliminary version was meant to solicit feedback from a few researchers as is customarily done. It illustrated the interplay of principles from various areas, which was the major effort in constructing the proof. I have fixed all the issues that were raised about the preliminary version in a revised manuscript (126 pages); clarified some concepts; and obtained simpler proofs of several claims. This revised manuscript has been sent to a small number of researchers.&amp;nbsp; I will send the manuscript to journal review this week. Once I hear back from the journal as part of due process, I will put up the final version on this website.&amp;quot;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;== Bibliography ==&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;== Bibliography ==&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:10 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Polylog_parameterizability&amp;diff=3607&amp;oldid=prev" start="Tue Aug 17 2010 15:44:31 GMT+0000" title="Polylog parameterizability">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;pp vs ppp:&amp;#32;&lt;/span&gt; one way function example&lt;/p&gt;
+
+			&lt;table&gt;
+			&lt;col class=&quot;diff-marker&quot; /&gt;
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+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;tr valign=&quot;top&quot;&gt;
+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 15:44, 17 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;4&quot; align=&quot;center&quot; class=&quot;diff-multi&quot;&gt;(One intermediate revision not shown.)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 92:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 92:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;Thus, for instance, the above discussion shows that the graph of a straight line program of polynomial length is in ppp, and if P=NP then the solution space of every NP problem has a ppp distribution supported on it.&amp;nbsp; The sets EVEN and &amp;lt;math&amp;gt;\{0,1\}^n \backslash (0,\ldots,0)&amp;lt;/math&amp;gt; can also be easily seen to support ppp distributions (e.g. the uniform distribution on either is ppp), but neither of these sets supports pp distributions.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;Thus, for instance, the above discussion shows that the graph of a straight line program of polynomial length is in ppp, and if P=NP then the solution space of every NP problem has a ppp distribution supported on it.&amp;nbsp; The sets EVEN and &amp;lt;math&amp;gt;\{0,1\}^n \backslash (0,\ldots,0)&amp;lt;/math&amp;gt; can also be easily seen to support ppp distributions (e.g. the uniform distribution on either is ppp), but neither of these sets supports pp distributions.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;As another example, consider the space of collections of k-clauses on n-variables for which the corresponding SAT problem is solvable, where k is fixed (e.g. k=10) and n is large. (This is a subset of {0,1}^C, where C is the space of k-clauses on n variables.) [Note that this space is NOT the same as the solution space of a single k-SAT instance; the former is the &amp;quot;base&amp;quot; of the k-SAT relation, while the latter is a &amp;quot;fiber&amp;quot;.]&amp;nbsp; On the one hand, if P!=NP, we know that this space is not in P. On the other hand, it supports a ppp distribution, because one can randomly select a solution &amp;lt;math&amp;gt;(x_1,\ldots,x_n)&amp;lt;/math&amp;gt; ''first'', and then select a random family of the clauses that are obeyed by this solution. Thus we see that ppp supports can actually be quite &amp;quot;hard&amp;quot; in nature.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;As a further example of a set supporting a ppp distribution, consider the image of {0,1}^n via a hash function or some other computable &amp;quot;one-way function&amp;quot;.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;The pp vs ppp distinction appears to be a crucial issue in the Deolalikar argument.&amp;nbsp; As far as can be discerned from the manuscript, the paper establishes the following facts:&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;The pp vs ppp distinction appears to be a crucial issue in the Deolalikar argument.&amp;nbsp; As far as can be discerned from the manuscript, the paper establishes the following facts:&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:10 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper&amp;diff=3605&amp;oldid=prev" start="Tue Aug 17 2010 15:10:29 GMT+0000" title="Deolalikar P vs NP paper">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;Online reactions:&amp;#32;&lt;/span&gt; Gasarch&lt;/p&gt;
+
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+			&lt;tr valign=&quot;top&quot;&gt;
+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 15:10, 17 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 303:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 303:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://constraints.wordpress.com/2010/08/15/a-simple-reduction/#more-391 A simple reduction]. Andr&#225;s Salamon, August 15, 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://constraints.wordpress.com/2010/08/15/a-simple-reduction/#more-391 A simple reduction]. Andr&#225;s Salamon, August 15, 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://blog.computationalcomplexity.org/2010/08/but-this-one-is-different.html But this one is different], Lance Fortnow, August 16 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://blog.computationalcomplexity.org/2010/08/but-this-one-is-different.html But this one is different], Lance Fortnow, August 16 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;www&lt;/del&gt;.&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;google&lt;/del&gt;.&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;com&lt;/del&gt;/&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;buzz&lt;/del&gt;/&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;114134834346472219368&lt;/del&gt;/&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;WJkEENg19Sz/Formally&lt;/del&gt;-&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;a&lt;/del&gt;-&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;mathematical&lt;/del&gt;-&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;proof&lt;/del&gt;-&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;consists&lt;/del&gt;-&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;of&lt;/del&gt;-&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;a Formally a mathematical proof&lt;/del&gt;], &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;Terence Tao, Buzz&lt;/del&gt;, August &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;16 &lt;/del&gt;2010&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;blog&lt;/ins&gt;.&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;computationalcomplexity&lt;/ins&gt;.&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;org&lt;/ins&gt;/&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;2010&lt;/ins&gt;/&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;08&lt;/ins&gt;/&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;my&lt;/ins&gt;-&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;last&lt;/ins&gt;-&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;post&lt;/ins&gt;-&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;on&lt;/ins&gt;-&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;alleged&lt;/ins&gt;-&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;p&lt;/ins&gt;-&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;ne-np-paper.html My last post on the alleged P NE NP paper&lt;/ins&gt;], &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;Bill Gasarch&lt;/ins&gt;, August &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;17 &lt;/ins&gt;2010&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Media and aggregators ===&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Media and aggregators ===&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 375:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 375:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://computacioncuantica.blogspot.com/2010/08/una-prueba-de-que-p-np.html Una prueba de que P&#8800;NP? (Spanish)], Alejandro D&#237;az-Caro, Computaci&#243;n Cu&#225;ntica, Aug 10 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://computacioncuantica.blogspot.com/2010/08/una-prueba-de-que-p-np.html Una prueba de que P&#8800;NP? (Spanish)], Alejandro D&#237;az-Caro, Computaci&#243;n Cu&#225;ntica, Aug 10 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://www.jtolds.com/newsletter/2010/8/11/watching-great-mathematics-unfold Watching great mathematics unfold], JT Olds, Aug 10, 2010. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://www.jtolds.com/newsletter/2010/8/11/watching-great-mathematics-unfold Watching great mathematics unfold], JT Olds, Aug 10, 2010. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;del&gt;&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://blog.oddhead.com/2010/08/11/betting-on-p-equals-np/ Where is the betting market for P=NP when you need it?], David Pennock, Oddhead Blog, August 11 2010&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://blog.oddhead.com/2010/08/11/betting-on-p-equals-np/ Where is the betting market for P=NP when you need it?], David Pennock, Oddhead Blog, August 11 2010&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;* [http://www.google.com/buzz/114134834346472219368/WJkEENg19Sz/Formally-a-mathematical-proof-consists-of-a Formally a mathematical proof], Terence Tao, Buzz, August 16 2010&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:10 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper&amp;diff=3604&amp;oldid=prev" start="Tue Aug 17 2010 06:37:20 GMT+0000" title="Deolalikar P vs NP paper">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;Issues with random k-SAT:&amp;#32;&lt;/span&gt; &lt;/p&gt;
+
+			&lt;table&gt;
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+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 06:37, 17 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 165:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 165:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;'''Complex solution spaces are uncorrelated with time complexity'''. (The below is a greatly expanded version of a series of twitter comments by Ryan Williams, on [http://twitter.com/rrwilliams/status/20741046788 twitter]) The author tries to use the fact that for certain distributions of random k-SAT, the solution space has a &amp;quot;hard structure&amp;quot;. For certain parameterizations, the space of satisfying assignments to a random k-SAT instance has some intriguing structure. If SAT is in P, then SAT can be captured in a certain logic (equivalent to P in some sense). The author claims that anything captured in this logic can't have a solution space with this intriguing structure. There are two &amp;quot;meta&amp;quot; objections to this. One is that &amp;quot;intriguing structure in the solution space is not sufficient for NP hardness&amp;quot;. The second is that &amp;quot;intriguing structure is not necessary for NP hardness&amp;quot;. They don't actually point to a place where the proof is wrong. But they do appear to give an obstacle to the general proof method. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;'''Complex solution spaces are uncorrelated with time complexity'''. (The below is a greatly expanded version of a series of twitter comments by Ryan Williams, on [http://twitter.com/rrwilliams/status/20741046788 twitter]) The author tries to use the fact that for certain distributions of random k-SAT, the solution space has a &amp;quot;hard structure&amp;quot;. For certain parameterizations, the space of satisfying assignments to a random k-SAT instance has some intriguing structure. If SAT is in P, then SAT can be captured in a certain logic (equivalent to P in some sense). The author claims that anything captured in this logic can't have a solution space with this intriguing structure. There are two &amp;quot;meta&amp;quot; objections to this. One is that &amp;quot;intriguing structure in the solution space is not sufficient for NP hardness&amp;quot;. The second is that &amp;quot;intriguing structure is not necessary for NP hardness&amp;quot;. They don't actually point to a place where the proof is wrong. But they do appear to give an obstacle to the general proof method. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;::'''How to map satisfiable formulas to trivial problems with exactly the same solution space structure.''' Very easy problems can also have complicated solution distributions. The following arose from discussions with Jeremiah Blocki. Here is &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;an &lt;/del&gt;easy way to (non-uniformly!) map any infinite collection of satisfiable formulas into &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt;, the set of formulas satisfied by the all-zeroes assignment. The map completely preserves the number of variables, the number of satisfying assignments, the distances between assignments, the clusters--as far as I can tell it preserves '''every property''' that has been studied about random k-SAT. Observe &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt; is trivially decidable in polynomial time. This should be a barrier to any proof of P vs NP that attempts to argue that the solution space of an NP-hard problem is &amp;quot;too complex&amp;quot; for P. (It even blocks any proof of AC0 not equal NP with this pattern!) This is the objection which is most germane to the current proposed proof: it opposes the claim that &amp;quot;anything in P can't have a solution space with complicated structure, but SAT can&amp;quot;. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;::'''How to map satisfiable formulas to trivial problems with exactly the same solution space structure.''' Very easy problems can also have complicated solution distributions. The following arose from discussions with Jeremiah Blocki. Here is &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;one &lt;/ins&gt;easy way to (non-uniformly!) map any infinite collection of satisfiable formulas into &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt;, the set of formulas satisfied by the all-zeroes assignment. The map completely preserves the number of variables, the number of satisfying assignments, the distances between assignments, the clusters--as far as I can tell it preserves '''every property''' that has been studied about random k-SAT. Observe &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt; is trivially decidable in polynomial time. This should be a barrier to any proof of P vs NP that attempts to argue that the solution space of an NP-hard problem is &amp;quot;too complex&amp;quot; for P. (It even blocks any proof of AC0 not equal NP with this pattern!) This is the objection which is most germane to the current proposed proof: it opposes the claim that &amp;quot;anything in P can't have a solution space with complicated structure, but SAT can&amp;quot;. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;::The map is simple: for a satisfiable formula F on n variables, let &amp;lt;math&amp;gt;(A_1,\ldots,A_n)&amp;lt;/math&amp;gt; be a satisfying assignment for it, and define F' to be the formula obtained by the following procedure: ''flip the signs of all literals involving &amp;lt;math&amp;gt;x_i&amp;lt;/math&amp;gt; in F iff &amp;lt;math&amp;gt;A_i = 1&amp;lt;/math&amp;gt;''. Now the space of satisfying assignments to F is transformed by the map &amp;lt;math&amp;gt;\phi_A(x_1,\ldots,x_n) = (x_1 \oplus A_1,\ldots,x_n \oplus A_n)&amp;lt;/math&amp;gt;. Observe this map completely preserves all distances between assignments, clusters, etc., and that the all-zeroes assignment satisfies F'. So for any infinite collection &amp;lt;math&amp;gt;{C}&amp;lt;/math&amp;gt; of satisfiable formulas with &amp;quot;hard solution structure&amp;quot; there is also an infinite collection of formulas &amp;lt;math&amp;gt;{C'}&amp;lt;/math&amp;gt; with analogous solution structure, where &amp;lt;math&amp;gt;C'&amp;lt;/math&amp;gt; is a subset of &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt;. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;::The map is simple: for a satisfiable formula &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;math&amp;gt;&lt;/ins&gt;F&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;/math&amp;gt; &lt;/ins&gt;on &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;math&amp;gt;&lt;/ins&gt;n&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;/math&amp;gt; &lt;/ins&gt;variables, let &amp;lt;math&amp;gt;(A_1,\ldots,A_n)&amp;lt;/math&amp;gt; be a satisfying assignment for it, and define F' to be the formula obtained by the following procedure: ''flip the signs of all literals involving &amp;lt;math&amp;gt;x_i&amp;lt;/math&amp;gt; in &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;math&amp;gt;&lt;/ins&gt;F&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;/math&amp;gt; &lt;/ins&gt;iff &amp;lt;math&amp;gt;A_i = 1&amp;lt;/math&amp;gt;''. Now the space of satisfying assignments to &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;math&amp;gt;&lt;/ins&gt;F&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;/math&amp;gt; &lt;/ins&gt;is transformed by the map &amp;lt;math&amp;gt;\phi_A(x_1,\ldots,x_n) = (x_1 \oplus A_1,\ldots,x_n \oplus A_n)&amp;lt;/math&amp;gt;. Observe this map completely preserves all distances between assignments, clusters, etc., and that the all-zeroes assignment satisfies &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;math&amp;gt;&lt;/ins&gt;F'&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;/math&amp;gt;&lt;/ins&gt;. So for any infinite collection &amp;lt;math&amp;gt;{C}&amp;lt;/math&amp;gt; of satisfiable formulas with &amp;quot;hard solution structure&amp;quot; there is also an infinite collection of formulas &amp;lt;math&amp;gt;{C'}&amp;lt;/math&amp;gt; with analogous solution structure, where &amp;lt;math&amp;gt;C'&amp;lt;/math&amp;gt; is a subset of &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt;. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;::Furthermore, the map easily extends to the generalization of k-SAT which consists of pairs &amp;lt;math&amp;gt;(F,A)&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; is a k-SAT formula, &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt; is a partial assignment to some variables, and we wish to know if it is possible to complete the partial assignment into a full satisfying assignment. By flipping literals appropriately in &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt;, there is always an instance &amp;lt;math&amp;gt;(G,A)&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt; can be satisfied by setting the rest of the variables to zero, and yet &amp;lt;math&amp;gt;G|_{A}&amp;lt;/math&amp;gt; has an isomorphic solution space to &amp;lt;math&amp;gt;F|_{A}&amp;lt;/math&amp;gt;. So for every instance of &amp;quot;k-SAT-Partial-Assignment&amp;quot; we can find an instance of &amp;quot;SAT0-Partial-Assignment&amp;quot; (which is again a trivial problem) with the same solution space structure. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;::Furthermore, the map easily extends to the generalization of k-SAT which consists of pairs &amp;lt;math&amp;gt;(F,A)&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; is a k-SAT formula, &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt; is a partial assignment to some variables, and we wish to know if it is possible to complete the partial assignment into a full satisfying assignment. By flipping literals appropriately in &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt;, there is always an instance &amp;lt;math&amp;gt;(G,A)&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt; can be satisfied by setting the rest of the variables to zero, and yet &amp;lt;math&amp;gt;G|_{A}&amp;lt;/math&amp;gt; has an isomorphic solution space to &amp;lt;math&amp;gt;F|_{A}&amp;lt;/math&amp;gt;. So for every instance of &amp;quot;k-SAT-Partial-Assignment&amp;quot; we can find an instance of &amp;quot;SAT0-Partial-Assignment&amp;quot; (which is again a trivial problem) with the same solution space structure. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;::The above looks like a cheat. Of course it is! We are exploiting the fact that, for arbitrary problems in NP (and therefore P as well), ''the solution space of that problem is not uniquely well-defined in general''. (Hubie Chen raised a similar point in email correspondence.) A solution space for an instance can only be defined with respect to some polynomial time verifier for the problem: this space is the set of witnesses that make the verifier accept on the instance. ''If you change the verifier, you change the solution space.''&amp;nbsp; The usual solution space for SAT comes from the verifier (call it &amp;lt;math&amp;gt;V&amp;lt;/math&amp;gt;) that checks a candidate assignment. If &amp;lt;math&amp;gt;V&amp;lt;/math&amp;gt; is used on SAT0, we get a complex solution space. If we use a sane verifier for SAT0 instead (the verifier that checks all-zeroes), the solution space becomes trivial. However, if P=NP, then there's also a verifier such that ''SAT has a trivial solution space'', namely the verifier which ignores its witness and just runs a polynomial time algorithm for SAT. The above argument only arises because the notion of solution space was forced to be verifier-independent over P and NP problems (which looks critical to the P vs NP paper).&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;::The above looks like a cheat. Of course it is! We are exploiting the fact that, for arbitrary problems in NP (and therefore P as well), ''the solution space of that problem is not uniquely well-defined in general''. (Hubie Chen raised a similar point in email correspondence.) A solution space for an instance can only be defined with respect to some polynomial time verifier for the problem: this space is the set of witnesses that make the verifier accept on the instance. ''If you change the verifier, you change the solution space.''&amp;nbsp; The usual solution space for SAT comes from the verifier (call it &amp;lt;math&amp;gt;V&amp;lt;/math&amp;gt;) that checks a candidate assignment. If &amp;lt;math&amp;gt;V&amp;lt;/math&amp;gt; is used on &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;math&amp;gt;&lt;/ins&gt;SAT0&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;/math&amp;gt;&lt;/ins&gt;, we get a complex solution space. If we use a sane verifier for &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;math&amp;gt;&lt;/ins&gt;SAT0&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;/math&amp;gt; &lt;/ins&gt;instead (the verifier that checks all-zeroes), the solution space becomes trivial. However, if &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;math&amp;gt;&lt;/ins&gt;P=NP&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;/math&amp;gt;&lt;/ins&gt;, then there's also a verifier such that ''SAT has a trivial solution space'', namely the verifier which ignores its witness and just runs a polynomial time algorithm for SAT. The above argument only arises because the notion of solution space was forced to be verifier-independent over P and NP problems (which looks critical to the P vs NP paper).&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;::Note there are also ways to construct infinitely many 2-CNF formulas and XOR-SAT formulas with &amp;quot;complex&amp;quot; distributions, in that the solution space (with respect to the verifier &amp;lt;math&amp;gt;V&amp;lt;/math&amp;gt; above&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;!&lt;/del&gt;) has many clusters, large distances between clusters, frozen variables--the kind of properties one finds with random k-SAT. (See also &amp;quot;The XOR-SAT Objection&amp;quot; above.)&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;::Note there are also ways to construct infinitely many 2-CNF formulas and XOR-SAT formulas with &amp;quot;complex&amp;quot; distributions, in that the solution space (with respect to the verifier &amp;lt;math&amp;gt;V&amp;lt;/math&amp;gt; above) has many clusters, large distances between clusters, frozen variables--the kind of properties one finds with random k-SAT&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;. For example, for any &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; one can easily construct 2-CNF formulas on &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; variables with &amp;lt;math&amp;gt;2^k&amp;lt;/math&amp;gt; satisfying assignments, each of which have &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; frozen variables and every pair has hamming distance at least &amp;lt;math&amp;gt;n/k&amp;lt;/math&amp;gt;&lt;/ins&gt;. (See also &amp;quot;The XOR-SAT Objection&amp;quot; above.)&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;::'''Satisfiable formulas with 'complex' solution spaces can be efficiently mapped to satisfiable formulas with 'simple' solution spaces.''' A hard distribution of solutions is not ''necessary'' for NP-hardness, either. A weird distribution is ''not'' what makes a problem hard, it's the representation of that solution space (e.g., a 3-CNF formula, a 2-CNF formula, etc.). The &amp;quot;hard&amp;quot; case of 3-SAT is the case where there is ''at most one'' satisfying assignment, since there is a randomized reduction from 3-SAT to 3-SAT with at most one satisfying assignment ([http://en.wikipedia.org/wiki/Valiant%E2%80%93Vazirani_theorem Valiant-Vazirani]). This reduction increases the number of clauses and the number of variables, but that doesn't really matter. The point is that you can always reduce 3-SAT with a &amp;quot;complex&amp;quot; solution space to one with an &amp;quot;easy&amp;quot; solution space, so how can a proof separating P from NP rely on the former? (Note that, if plausible circuit lower bounds hold up, then Valiant-Vazirani can be derandomized to run in deterministic polynomial time.) &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;To summarize, there is essentially no correlation between the &amp;quot;hard structure&amp;quot; of the solution space for instances of some problem, and the NP-hardness of that problem.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;::'''Satisfiable formulas with 'complex' solution spaces can be efficiently mapped to satisfiable formulas with 'simple' solution spaces.''' A hard distribution of solutions is not ''necessary'' for NP-hardness, either. A weird distribution is ''not'' what makes a problem hard, it's the representation of that solution space (e.g., a 3-CNF formula, a 2-CNF formula, etc.). The &amp;quot;hard&amp;quot; case of 3-SAT is the case where there is ''at most one'' satisfying assignment, since there is a randomized reduction from 3-SAT to 3-SAT with at most one satisfying assignment ([http://en.wikipedia.org/wiki/Valiant%E2%80%93Vazirani_theorem Valiant-Vazirani]). This reduction increases the number of clauses and the number of variables, but that doesn't really matter. The point is that you can always reduce 3-SAT with a &amp;quot;complex&amp;quot; solution space to one with an &amp;quot;easy&amp;quot; solution space, so how can a proof separating P from NP rely on the former? (Note that, if plausible circuit lower bounds hold up, then Valiant-Vazirani can be derandomized to run in deterministic polynomial time.)&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;::Of course, this point on its own does not invalidate Deolalikar's approach. To prove just one NP-complete problem has a complex solution space would be enough ''if'' it was also proved that all P problems have simple solution spaces. But it is hard to make sense of what this proposition &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;even means&lt;/del&gt;, in light of the above.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;::Of course, this point on its own does not invalidate Deolalikar's approach. To prove just one NP-complete problem has a complex solution space would be enough ''if'' it was also proved that all P problems have simple solution spaces. But it is hard to make sense of what this proposition &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;could really mean&lt;/ins&gt;, in light of the above&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;#160;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;:To summarize, there is little correlation between the &amp;quot;hard structure&amp;quot; of the solution space for instances of some problem, and the NP-hardness of that problem&lt;/ins&gt;.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Uniformity issues ===&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Uniformity issues ===&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:10 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper&amp;diff=3603&amp;oldid=prev" start="Tue Aug 17 2010 00:02:28 GMT+0000" title="Deolalikar P vs NP paper">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;Bibliography:&amp;#32;&lt;/span&gt; &lt;/p&gt;
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+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 00:02, 17 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;4&quot; align=&quot;center&quot; class=&quot;diff-multi&quot;&gt;(2 intermediate revisions not shown.)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 301:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 301:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://constraints.wordpress.com/2010/08/15/a-simple-reduction/#more-391 A simple reduction]. Andr&#225;s Salamon, August 15, 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://constraints.wordpress.com/2010/08/15/a-simple-reduction/#more-391 A simple reduction]. Andr&#225;s Salamon, August 15, 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://blog.computationalcomplexity.org/2010/08/but-this-one-is-different.html But this one is different], Lance Fortnow, August 16 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://blog.computationalcomplexity.org/2010/08/but-this-one-is-different.html But this one is different], Lance Fortnow, August 16 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;* [http://www.google.com/buzz/114134834346472219368/WJkEENg19Sz/Formally-a-mathematical-proof-consists-of-a Formally a mathematical proof], Terence Tao, Buzz, August 16 2010&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Media and aggregators ===&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Media and aggregators ===&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 345:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 346:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://techcrunch.com/2010/08/12/fuzzy-math/ Attempt At P &#8800; NP Proof Gets Torn Apart Online], Alexia Totsis, Techcrunch, August 12, 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://techcrunch.com/2010/08/12/fuzzy-math/ Attempt At P &#8800; NP Proof Gets Torn Apart Online], Alexia Totsis, Techcrunch, August 12, 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;====13th August ====&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;====13th August====&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://www.newscientist.com/article/dn19313-tide-turns-against-milliondollar-maths-proof.html Tide turns against million-dollar maths proof], Jacob Aron, New Scientist, August 13, 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://www.newscientist.com/article/dn19313-tide-turns-against-milliondollar-maths-proof.html Tide turns against million-dollar maths proof], Jacob Aron, New Scientist, August 13, 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;====&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;15 &lt;/del&gt;August ====&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;====&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;15th &lt;/ins&gt;August====&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://cacm.acm.org/blogs/blog-cacm/97587-a-tale-of-a-serious-attempt-at-p%E2%89%A0np/fulltext A Tale of A Serious Attempt At P&#8800;NP], Richard J. Lipton, Comm. ACM blog, August 15 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://cacm.acm.org/blogs/blog-cacm/97587-a-tale-of-a-serious-attempt-at-p%E2%89%A0np/fulltext A Tale of A Serious Attempt At P&#8800;NP], Richard J. Lipton, Comm. ACM blog, August 15 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;====16th August====&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;* [http://www.nytimes.com/2010/08/17/science/17proof.html Step 1: Post Elusive Proof. Step 2: Watch Fireworks.], John Markoff, New York Times, August 16 2010 online, August 17 in print&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Real-time searches ===&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Real-time searches ===&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 390:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 395:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;== Bibliography ==&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;== Bibliography ==&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;===Papers===&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [AVV1997] S. Abiteboul, M. Y. Yardi, V. Vianu, &amp;quot;[http://portal.acm.org/citation.cfm?id=256295 Fixpoint logics, relational machines, and computational complexity]&amp;quot;, Journal of the ACM (JACM) Volume 44,&amp;nbsp; Issue 1&amp;nbsp; (January 1997), 30-56.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [AVV1997] S. Abiteboul, M. Y. Yardi, V. Vianu, &amp;quot;[http://portal.acm.org/citation.cfm?id=256295 Fixpoint logics, relational machines, and computational complexity]&amp;quot;, Journal of the ACM (JACM) Volume 44,&amp;nbsp; Issue 1&amp;nbsp; (January 1997), 30-56.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [AM2003] D. Achlioptas, C. Moore, &amp;quot;[http://www.sciencedirect.com/science?_ob=ArticleURL&amp;amp;_udi=B6WJ0-49M05RK-3&amp;amp;_user=10&amp;amp;_coverDate=09%2F30%2F2003&amp;amp;_rdoc=1&amp;amp;_fmt=high&amp;amp;_orig=search&amp;amp;_sort=d&amp;amp;_docanchor=&amp;amp;view=c&amp;amp;_acct=C000050221&amp;amp;_version=1&amp;amp;_urlVersion=0&amp;amp;_userid=10&amp;amp;md5=13eae49445b87797b1f90aa42e54b5a5 Almost all graphs with average degree 4 are 3-colorable]&amp;quot;, Journal of Computer and System Sciences 67, Issue 2, September 2003, 441-471.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [AM2003] D. Achlioptas, C. Moore, &amp;quot;[http://www.sciencedirect.com/science?_ob=ArticleURL&amp;amp;_udi=B6WJ0-49M05RK-3&amp;amp;_user=10&amp;amp;_coverDate=09%2F30%2F2003&amp;amp;_rdoc=1&amp;amp;_fmt=high&amp;amp;_orig=search&amp;amp;_sort=d&amp;amp;_docanchor=&amp;amp;view=c&amp;amp;_acct=C000050221&amp;amp;_version=1&amp;amp;_urlVersion=0&amp;amp;_userid=10&amp;amp;md5=13eae49445b87797b1f90aa42e54b5a5 Almost all graphs with average degree 4 are 3-colorable]&amp;quot;, Journal of Computer and System Sciences 67, Issue 2, September 2003, 441-471.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 395:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 401:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [VV1986] L. G. Valiant, V. V. Vazirani, &amp;quot;[http://www.cs.princeton.edu/courses/archive/fall05/cos528/handouts/NP_is_as.pdf NP is as easy as detecting unique solutions]&amp;quot;, Theoretical Computer Science (North-Holland) 47: 85&#8211;93 (1986). doi:10.1016/0304-3975(86)90135-0.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [VV1986] L. G. Valiant, V. V. Vazirani, &amp;quot;[http://www.cs.princeton.edu/courses/archive/fall05/cos528/handouts/NP_is_as.pdf NP is as easy as detecting unique solutions]&amp;quot;, Theoretical Computer Science (North-Holland) 47: 85&#8211;93 (1986). doi:10.1016/0304-3975(86)90135-0.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [V1982] M. Vardi, &amp;quot;[http://www.dcc.uchile.cl/~cgutierr/cursos/CC/vardi-complex-bd.pdf Complexity of Relational Query Languages]&amp;quot;, 14th Symposium on Theory of Computation (1982), 137-146.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [V1982] M. Vardi, &amp;quot;[http://www.dcc.uchile.cl/~cgutierr/cursos/CC/vardi-complex-bd.pdf Complexity of Relational Query Languages]&amp;quot;, 14th Symposium on Theory of Computation (1982), 137-146.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;===Books===&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;* [http://books.google.com/books?id=jhCM7i0a6UUC Information, Physics, and Computation], Marc M&#233;zard, Andrea Montanari, Oxford University Press, 2009, ISBN-13: 978-0198570837, - the interface between statistical physics, theoretical computer science/discrete mathematics, and coding/information theory.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;== Other links ==&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;== Other links ==&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:10 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper&amp;diff=3600&amp;oldid=prev" start="Mon Aug 16 2010 22:43:07 GMT+0000" title="Deolalikar P vs NP paper">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;Theory blogs:&amp;#32;&lt;/span&gt; &lt;/p&gt;
+
+			&lt;table&gt;
+			&lt;col class=&quot;diff-marker&quot; /&gt;
+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;col class=&quot;diff-marker&quot; /&gt;
+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;tr valign=&quot;top&quot;&gt;
+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 22:43, 16 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;4&quot; align=&quot;center&quot; class=&quot;diff-multi&quot;&gt;(One intermediate revision not shown.)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 297:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 297:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://scottaaronson.com/blog/?p=458 Eight Signs A Claimed P&#8800;NP Proof Is Wrong], Scott Aaronson, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://scottaaronson.com/blog/?p=458 Eight Signs A Claimed P&#8800;NP Proof Is Wrong], Scott Aaronson, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://gowers.wordpress.com/2010/08/14/a-possible-answer-to-my-objection/ A possible answer to my objection], Tim Gowers, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://gowers.wordpress.com/2010/08/14/a-possible-answer-to-my-objection/ A possible answer to my objection], Tim Gowers, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;* [http://constraints.wordpress.com/2010/08/13/p-not-equal-to-np-consequences/ (P not equal to NP (P&#8800;NP) Consequences], Andr&#225;s Salamon, August 13, 2010.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/ The P&#8800;NP &amp;quot;Proof&amp;quot; Is One Week Old], Richard Lipton, G&#246;del&#8217;s lost letter and P=NP, August 15 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/ The P&#8800;NP &amp;quot;Proof&amp;quot; Is One Week Old], Richard Lipton, G&#246;del&#8217;s lost letter and P=NP, August 15 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;* [http://constraints.wordpress.com/2010/08/15/a-simple-reduction/#more-391 A simple reduction]. Andr&#225;s Salamon, August 15, 2010.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://blog.computationalcomplexity.org/2010/08/but-this-one-is-different.html But this one is different], Lance Fortnow, August 16 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://blog.computationalcomplexity.org/2010/08/but-this-one-is-different.html But this one is different], Lance Fortnow, August 16 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:10 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper&amp;diff=3598&amp;oldid=prev" start="Mon Aug 16 2010 22:30:43 GMT+0000" title="Deolalikar P vs NP paper">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;15 August:&amp;#32;&lt;/span&gt; timestamp&lt;/p&gt;
+
+			&lt;table&gt;
+			&lt;col class=&quot;diff-marker&quot; /&gt;
+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;col class=&quot;diff-marker&quot; /&gt;
+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;tr valign=&quot;top&quot;&gt;
+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 22:30, 16 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;4&quot; align=&quot;center&quot; class=&quot;diff-multi&quot;&gt;(3 intermediate revisions not shown.)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 93:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 93:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;:: We have embedded our original set of variates into a &amp;lt;I&amp;gt;polynomially larger&amp;lt;/I&amp;gt; product space, and obtained a directed graphical model on this larger space. This product space has a nice factorization due to the directed graph structure. This is what we will exploit.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;:: We have embedded our original set of variates into a &amp;lt;I&amp;gt;polynomially larger&amp;lt;/I&amp;gt; product space, and obtained a directed graphical model on this larger space. This product space has a nice factorization due to the directed graph structure. This is what we will exploit.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;It is clear here that the problem is being lifted from the original space &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt; to a larger space &amp;lt;math&amp;gt;\{0,1\}^N&amp;lt;/math&amp;gt;, but then the solution space of (say) k-SAT on &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt; also gets lifted to a solution space which could have a completely different geometry.&amp;nbsp; Given that the proof sketch of P != NP (pages 99-100, third draft) is based on the cluster geometry of the solution space of k-SAT itself, rather than a lift thereof, this is a significant gap in the argument.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;It is clear here that the problem is being lifted from the original space &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt; to a larger space &amp;lt;math&amp;gt;\{0,1\}^N&amp;lt;/math&amp;gt;, but then the solution space of (say) k-SAT on &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt; also gets lifted to a solution space which could have a completely different geometry.&amp;nbsp; Given that the proof sketch of P != NP (pages 99-100, third draft) is based on the cluster geometry of the solution space of k-SAT itself, rather than a lift thereof, this is a significant gap in the argument&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;.&amp;nbsp; (Specifically, the variables &amp;lt;math&amp;gt;\alpha,\beta,\gamma&amp;lt;/math&amp;gt; etc. appearing on page 100 are implicitly assumed to belong to the original set of n literals &amp;lt;math&amp;gt;x_1,\ldots,x_n&amp;lt;/math&amp;gt;, but due to the embedding into the polynomially larger space, they are in fact likely to instead be scattered among a much larger set &amp;lt;math&amp;gt;x_1,\ldots,x_N&amp;lt;/math&amp;gt; of literals&lt;/ins&gt;.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;nbsp; &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;nbsp; &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 132:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 132:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;:: [http://www.cs.umass.edu/~immerman/ Neil Immerman] has written a very interesting letter to Deolalikar critiquing his proof. With Immerman's permission, this letter is reproduced on the following page: [[Immerman's letter]]. Immerman also raises the problem of order-invariance mentioned by Anuj Dawar above. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;:: [http://www.cs.umass.edu/~immerman/ Neil Immerman] has written a very interesting letter to Deolalikar critiquing his proof. With Immerman's permission, this letter is reproduced on the following page: [[Immerman's letter]]. Immerman also raises the problem of order-invariance mentioned by Anuj Dawar above. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;* ''&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;Placeholder for specific critique based &lt;/del&gt;on the &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;structure &lt;/del&gt;of k-XORSAT - &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;it seems like we're ready &lt;/del&gt;to &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;phrase &lt;/del&gt;this &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;one&lt;/del&gt;.&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;''&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;* ''&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;'The XORSAT objection'''&amp;nbsp; The claimed proof of P!=NP &lt;/ins&gt;on &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;pages 99-100 of the third draft assumes for contradiction that the extension problem for k-SAT is in P, concludes that &lt;/ins&gt;the &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;solution space &lt;/ins&gt;of &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;a random k-SAT problem is polylog parameterizable (in the ppp sense), and then shows that this is in contradiction with the known cluster geometry of that solution space at the hard phase, so that the k-SAT extension problem is in fact not in P.&amp;nbsp; However, if this argument were valid, one could substitute k-SAT for &lt;/ins&gt;k-XORSAT &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;throughout, and obtain the false conclusion that the extension problem for k&lt;/ins&gt;-&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;XORSAT were similarly not in P.&amp;nbsp; Thus, if the argument were valid, there must be a specific step in the argument which worked for k-SAT but not for k-XORSAT; but no such specific step has been identified.&amp;nbsp; (It is indeed true that k-XORSAT is in ppp, while k-SAT is not believed &lt;/ins&gt;to &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;be in ppp, but &lt;/ins&gt;this &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;is not a specific step in the argument.)&amp;nbsp; Indeed, it appears that the argument fails for both k-XORSAT and k-SAT at the same place, namely the projection issue mentioned earlier in which the variables &amp;lt;math&amp;gt;\alpha,\beta,\gamma&amp;lt;/math&amp;gt; of the ENSP model are erroneously assumed to belong to the initial set of n variables, rather than the polynomially expanded set&lt;/ins&gt;.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;== General issues ==&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;== General issues ==&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 347:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 347:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;====15 August ====&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;====15 August ====&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://cacm.acm.org/blogs/blog-cacm/97587-a-tale-of-a-serious-attempt-at-p%E2%89%A0np/fulltext A Tale of A Serious Attempt At P&#8800;NP], Richard J. Lipton, Comm. ACM blog,&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://cacm.acm.org/blogs/blog-cacm/97587-a-tale-of-a-serious-attempt-at-p%E2%89%A0np/fulltext A Tale of A Serious Attempt At P&#8800;NP], Richard J. Lipton, Comm. ACM blog, &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;August 15 2010.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Real-time searches ===&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Real-time searches ===&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:10 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Polylog_parameterizability&amp;diff=3594&amp;oldid=prev" start="Mon Aug 16 2010 20:28:43 GMT+0000" title="Polylog parameterizability">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;Counterexamples:&amp;#32;&lt;/span&gt; Discussion of pp vs ppp&lt;/p&gt;
+
+			&lt;table&gt;
+			&lt;col class=&quot;diff-marker&quot; /&gt;
+			&lt;col class=&quot;diff-content&quot; /&gt;
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+			&lt;tr valign=&quot;top&quot;&gt;
+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 20:28, 16 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 82:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 82:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;On the other hand EVEN can be polylog parametrized if we add a number of auxilary variables, placeholders for partial sums, so that we never add more than polylog elements (we can always add a fixed number), and similarly for #2.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;On the other hand EVEN can be polylog parametrized if we add a number of auxilary variables, placeholders for partial sums, so that we never add more than polylog elements (we can always add a fixed number), and similarly for #2.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;== pp vs ppp ==&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;For the purposes of discussion, the following abbreviations have been adopted:&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;* A probability distribution on &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt; is ''pp'' (polylog parameterizable) if it is polylog parameterizable in the above sense, WITHOUT the addition of any additional variables beyond the existing n literals &amp;lt;math&amp;gt;x_1,\ldots,x_n&amp;lt;/math&amp;gt;.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;* A probability distribution on &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt; is ''ppp'' (projected polylog parameterizable) if it is the ''projection'' of a pp distribution on &amp;lt;math&amp;gt;\{0,1\}^N&amp;lt;/math&amp;gt; for some polynomial size &amp;lt;math&amp;gt;N=n^{O(1)}&amp;lt;/math&amp;gt; under a map that keeps n of the N literals and discards the rest.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;Thus, for instance, the above discussion shows that the graph of a straight line program of polynomial length is in ppp, and if P=NP then the solution space of every NP problem has a ppp distribution supported on it.&amp;nbsp; The sets EVEN and &amp;lt;math&amp;gt;\{0,1\}^n \backslash (0,\ldots,0)&amp;lt;/math&amp;gt; can also be easily seen to support ppp distributions (e.g. the uniform distribution on either is ppp), but neither of these sets supports pp distributions.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;The pp vs ppp distinction appears to be a crucial issue in the Deolalikar argument.&amp;nbsp; As far as can be discerned from the manuscript, the paper establishes the following facts:&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;* If P=NP, then the solution space of any k-SAT problem with at least one solution, supports a ppp distribution.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;* The hard phase of k-SAT does not support any pp distribution.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;However this does not allow us to conclude that P!=NP, because of the substantial gap between pp and ppp.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:11 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=User:Jennie&amp;diff=0&amp;oldid=prev" start="Mon Aug 16 2010 20:05:14 GMT+0000" title="User:Jennie">&lt;p&gt;blocked [[&lt;a href="http://michaelnielsen.org/polymath1/index.php?title=User:Jennie&amp;action=edit&amp;redlink=1" class="new" title="User:Jennie (not yet written)"&gt;Jennie&lt;/a&gt; (&lt;a href="http://michaelnielsen.org/polymath1/index.php?title=User_talk:Jennie&amp;action=edit&amp;redlink=1" class="new" title="User talk:Jennie (not yet written)"&gt;Talk&lt;/a&gt; | &lt;a href="http://michaelnielsen.org/polymath1/index.php?title=Special:Contributions/Jennie" title="Special:Contributions/Jennie"&gt;contribs&lt;/a&gt;)]] with an expiry time of &lt;span title="infinite"&gt;infinite&lt;/span&gt; (account creation disabled) Spamming links to external sites&lt;/p&gt;
+&lt;p&gt;&lt;b&gt;New page&lt;/b&gt;&lt;/p&gt;&lt;div&gt;&lt;/div&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper&amp;diff=3593&amp;oldid=prev" start="Mon Aug 16 2010 19:36:21 GMT+0000" title="Deolalikar P vs NP paper">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;Issues with random k-SAT:&amp;#32;&lt;/span&gt; &lt;/p&gt;
+
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+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 19:36, 16 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 177:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 177:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;::'''Satisfiable formulas with 'complex' solution spaces can be efficiently mapped to satisfiable formulas with 'simple' solution spaces.''' A hard distribution of solutions is not ''necessary'' for NP-hardness, either. A weird distribution is ''not'' what makes a problem hard, it's the representation of that solution space (e.g., a 3-CNF formula, a 2-CNF formula, etc.). The &amp;quot;hard&amp;quot; case of 3-SAT is the case where there is ''at most one'' satisfying assignment, since there is a randomized reduction from 3-SAT to 3-SAT with at most one satisfying assignment ([http://en.wikipedia.org/wiki/Valiant%E2%80%93Vazirani_theorem Valiant-Vazirani]). This reduction increases the number of clauses and the number of variables, but that doesn't really matter. The point is that you can always reduce 3-SAT with a &amp;quot;complex&amp;quot; solution space to one with an &amp;quot;easy&amp;quot; solution space, so how can a proof separating P from NP rely on the former? (Note that, if plausible circuit lower bounds hold up, then Valiant-Vazirani can be derandomized to run in deterministic polynomial time.) To summarize, there is essentially no correlation between the &amp;quot;hard structure&amp;quot; of the solution space for instances of some problem, and the NP-hardness of that problem.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;::'''Satisfiable formulas with 'complex' solution spaces can be efficiently mapped to satisfiable formulas with 'simple' solution spaces.''' A hard distribution of solutions is not ''necessary'' for NP-hardness, either. A weird distribution is ''not'' what makes a problem hard, it's the representation of that solution space (e.g., a 3-CNF formula, a 2-CNF formula, etc.). The &amp;quot;hard&amp;quot; case of 3-SAT is the case where there is ''at most one'' satisfying assignment, since there is a randomized reduction from 3-SAT to 3-SAT with at most one satisfying assignment ([http://en.wikipedia.org/wiki/Valiant%E2%80%93Vazirani_theorem Valiant-Vazirani]). This reduction increases the number of clauses and the number of variables, but that doesn't really matter. The point is that you can always reduce 3-SAT with a &amp;quot;complex&amp;quot; solution space to one with an &amp;quot;easy&amp;quot; solution space, so how can a proof separating P from NP rely on the former? (Note that, if plausible circuit lower bounds hold up, then Valiant-Vazirani can be derandomized to run in deterministic polynomial time.) To summarize, there is essentially no correlation between the &amp;quot;hard structure&amp;quot; of the solution space for instances of some problem, and the NP-hardness of that problem.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;::Of course, this point on its own does not &lt;del class=&quot;diffchange diffchange-inline&quot;&gt; &lt;/del&gt;invalidate Deolalikar's approach. &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;&amp;nbsp; In principle it's possible that some NP-complete problems have complex solutions spaces and some do not.&amp;nbsp; &amp;nbsp; &lt;/del&gt;To prove just one NP-complete problem has a complex solution space would be enough ''if'' it was also proved that all P problems have simple solution spaces. &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;&amp;nbsp; For &lt;/del&gt;this &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;to work&lt;/del&gt;, &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;it would clearly be necessary that reductions between NP problems need not preserve complexity &lt;/del&gt;of &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;solution space, but &lt;/del&gt;the &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;examples given &lt;/del&gt;above &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;indicate this is the case&lt;/del&gt;.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;::Of course, this point on its own does not invalidate Deolalikar's approach. To prove just one NP-complete problem has a complex solution space would be enough ''if'' it was also proved that all P problems have simple solution spaces. &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;But it is hard to make sense of what &lt;/ins&gt;this &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;proposition even means&lt;/ins&gt;, &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;in light &lt;/ins&gt;of the above.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Uniformity issues ===&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Uniformity issues ===&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:11 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper&amp;diff=3592&amp;oldid=prev" start="Mon Aug 16 2010 14:36:33 GMT+0000" title="Deolalikar P vs NP paper">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;Theory blogs:&amp;#32;&lt;/span&gt; Fortnow&lt;/p&gt;
+
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+			&lt;tr valign=&quot;top&quot;&gt;
+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 14:36, 16 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 297:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 297:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://scottaaronson.com/blog/?p=458 Eight Signs A Claimed P&#8800;NP Proof Is Wrong], Scott Aaronson, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://scottaaronson.com/blog/?p=458 Eight Signs A Claimed P&#8800;NP Proof Is Wrong], Scott Aaronson, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://gowers.wordpress.com/2010/08/14/a-possible-answer-to-my-objection/ A possible answer to my objection], Tim Gowers, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://gowers.wordpress.com/2010/08/14/a-possible-answer-to-my-objection/ A possible answer to my objection], Tim Gowers, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/ The P&#8800;NP &amp;quot;Proof&amp;quot; Is One Week Old], Richard Lipton, G&#246;del&#8217;s lost letter and P=NP, August 15, 2010&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/ The P&#8800;NP &amp;quot;Proof&amp;quot; Is One Week Old], Richard Lipton, G&#246;del&#8217;s lost letter and P=NP, August 15 &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;2010.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;* [http://blog.computationalcomplexity.org/2010/08/but-this-one-is-different.html But this one is different], Lance Fortnow&lt;/ins&gt;, &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;August 16 &lt;/ins&gt;2010&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Media and aggregators ===&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Media and aggregators ===&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:11 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper&amp;diff=3591&amp;oldid=prev" start="Mon Aug 16 2010 08:12:47 GMT+0000" title="Deolalikar P vs NP paper">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;13th August:&amp;#32;&lt;/span&gt; &lt;/p&gt;
+
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+			&lt;tr valign=&quot;top&quot;&gt;
+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 08:12, 16 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 344:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 344:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;====13th August ====&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;====13th August ====&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://www.newscientist.com/article/dn19313-tide-turns-against-milliondollar-maths-proof.html Tide turns against million-dollar maths proof], Jacob Aron, New Scientist, August 13, 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://www.newscientist.com/article/dn19313-tide-turns-against-milliondollar-maths-proof.html Tide turns against million-dollar maths proof], Jacob Aron, New Scientist, August 13, 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;====15 August ====&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;* [http://cacm.acm.org/blogs/blog-cacm/97587-a-tale-of-a-serious-attempt-at-p%E2%89%A0np/fulltext A Tale of A Serious Attempt At P&#8800;NP], Richard J. Lipton, Comm. ACM blog,&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Real-time searches ===&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Real-time searches ===&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:11 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper&amp;diff=3590&amp;oldid=prev" start="Mon Aug 16 2010 05:10:16 GMT+0000" title="Deolalikar P vs NP paper">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;Issues with random k-SAT:&amp;#32;&lt;/span&gt; &lt;/p&gt;
+
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+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 05:10, 16 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 165:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 165:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;'''Complex solution spaces are uncorrelated with time complexity'''. (The below is a greatly expanded version of a series of twitter comments by Ryan Williams, on [http://twitter.com/rrwilliams/status/20741046788 twitter]) The author tries to use the fact that for certain distributions of random k-SAT, the solution space has a &amp;quot;hard structure&amp;quot;. For certain parameterizations, the space of satisfying assignments to a random k-SAT instance has some intriguing structure. If SAT is in P, then SAT can be captured in a certain logic (equivalent to P in some sense). The author claims that anything captured in this logic can't have a solution space with this intriguing structure. There are two &amp;quot;meta&amp;quot; objections to this. One is that &amp;quot;intriguing structure in the solution space is not sufficient for NP hardness&amp;quot;. The second is that &amp;quot;intriguing structure is not necessary for NP hardness&amp;quot;. They don't actually point to a place where the proof is wrong. But they do appear to give an obstacle to the general proof method. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;'''Complex solution spaces are uncorrelated with time complexity'''. (The below is a greatly expanded version of a series of twitter comments by Ryan Williams, on [http://twitter.com/rrwilliams/status/20741046788 twitter]) The author tries to use the fact that for certain distributions of random k-SAT, the solution space has a &amp;quot;hard structure&amp;quot;. For certain parameterizations, the space of satisfying assignments to a random k-SAT instance has some intriguing structure. If SAT is in P, then SAT can be captured in a certain logic (equivalent to P in some sense). The author claims that anything captured in this logic can't have a solution space with this intriguing structure. There are two &amp;quot;meta&amp;quot; objections to this. One is that &amp;quot;intriguing structure in the solution space is not sufficient for NP hardness&amp;quot;. The second is that &amp;quot;intriguing structure is not necessary for NP hardness&amp;quot;. They don't actually point to a place where the proof is wrong. But they do appear to give an obstacle to the general proof method. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;::'''How to map satisfiable formulas to trivial problems with exactly the same solution space structure.''' Very easy problems can also have complicated solution distributions. The following arose from discussions with Jeremiah Blocki. Here is an easy way to (non-uniformly!) map any infinite collection of satisfiable formulas into &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt;, the set of formulas satisfied by the all-zeroes assignment. The map completely preserves the number of variables, the number of satisfying assignments, the distances between assignments, the clusters--as far as I can tell it preserves '''every property''' that has been studied about random k-SAT. Observe &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt; is trivially decidable in polynomial time. This should be a barrier to any proof of P vs NP that attempts to argue that the solution space of an NP-hard problem is &amp;quot;too complex&amp;quot; for P. (It even blocks any proof of AC0 not equal NP with this pattern!) This is the objection which is most germane to the current proposed proof: it opposes the claim that &amp;quot;anything in P can't have a solution space with complicated structure&amp;quot;. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;::'''How to map satisfiable formulas to trivial problems with exactly the same solution space structure.''' Very easy problems can also have complicated solution distributions. The following arose from discussions with Jeremiah Blocki. Here is an easy way to (non-uniformly!) map any infinite collection of satisfiable formulas into &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt;, the set of formulas satisfied by the all-zeroes assignment. The map completely preserves the number of variables, the number of satisfying assignments, the distances between assignments, the clusters--as far as I can tell it preserves '''every property''' that has been studied about random k-SAT. Observe &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt; is trivially decidable in polynomial time. This should be a barrier to any proof of P vs NP that attempts to argue that the solution space of an NP-hard problem is &amp;quot;too complex&amp;quot; for P. (It even blocks any proof of AC0 not equal NP with this pattern!) This is the objection which is most germane to the current proposed proof: it opposes the claim that &amp;quot;anything in P can't have a solution space with complicated structure&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;, but SAT can&lt;/ins&gt;&amp;quot;. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;::The map is simple: for a satisfiable formula F on n variables, let &amp;lt;math&amp;gt;(A_1,\ldots,A_n)&amp;lt;/math&amp;gt; be a satisfying assignment for it, and define F' to be the formula obtained by the following procedure: ''flip the signs of all literals involving &amp;lt;math&amp;gt;x_i&amp;lt;/math&amp;gt; in F iff &amp;lt;math&amp;gt;A_i = 1&amp;lt;/math&amp;gt;''. Now the space of satisfying assignments to F is transformed by the map &amp;lt;math&amp;gt;\phi_A(x_1,\ldots,x_n) = (x_1 \oplus A_1,\ldots,x_n \oplus A_n)&amp;lt;/math&amp;gt;. Observe this map completely preserves all distances between assignments, clusters, etc., and that the all-zeroes assignment satisfies F'. So for any infinite collection &amp;lt;math&amp;gt;{C}&amp;lt;/math&amp;gt; of satisfiable formulas with &amp;quot;hard solution structure&amp;quot; there is also an infinite collection of formulas &amp;lt;math&amp;gt;{C'}&amp;lt;/math&amp;gt; with analogous solution structure, where &amp;lt;math&amp;gt;C'&amp;lt;/math&amp;gt; is a subset of &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt;. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;::The map is simple: for a satisfiable formula F on n variables, let &amp;lt;math&amp;gt;(A_1,\ldots,A_n)&amp;lt;/math&amp;gt; be a satisfying assignment for it, and define F' to be the formula obtained by the following procedure: ''flip the signs of all literals involving &amp;lt;math&amp;gt;x_i&amp;lt;/math&amp;gt; in F iff &amp;lt;math&amp;gt;A_i = 1&amp;lt;/math&amp;gt;''. Now the space of satisfying assignments to F is transformed by the map &amp;lt;math&amp;gt;\phi_A(x_1,\ldots,x_n) = (x_1 \oplus A_1,\ldots,x_n \oplus A_n)&amp;lt;/math&amp;gt;. Observe this map completely preserves all distances between assignments, clusters, etc., and that the all-zeroes assignment satisfies F'. So for any infinite collection &amp;lt;math&amp;gt;{C}&amp;lt;/math&amp;gt; of satisfiable formulas with &amp;quot;hard solution structure&amp;quot; there is also an infinite collection of formulas &amp;lt;math&amp;gt;{C'}&amp;lt;/math&amp;gt; with analogous solution structure, where &amp;lt;math&amp;gt;C'&amp;lt;/math&amp;gt; is a subset of &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt;. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:11 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper&amp;diff=3589&amp;oldid=prev" start="Sun Aug 15 2010 19:59:56 GMT+0000" title="Deolalikar P vs NP paper">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;Theory blogs:&amp;#32;&lt;/span&gt; &lt;/p&gt;
+
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+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 19:59, 15 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;4&quot; align=&quot;center&quot; class=&quot;diff-multi&quot;&gt;(2 intermediate revisions not shown.)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 19:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 19:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/11/deolalikar-responds-to-issues-about-his-p%e2%89%a0np-proof/ Deolalikar Responds To Issues About His P&#8800;NP Proof] August 11, 2010 (Inactive)&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/11/deolalikar-responds-to-issues-about-his-p%e2%89%a0np-proof/ Deolalikar Responds To Issues About His P&#8800;NP Proof] August 11, 2010 (Inactive)&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/12/fatal-flaws-in-deolalikars-proof Fatal Flaws in Deolalikar&#8217;s Proof?] August 12, 2010 (Inactive)&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/12/fatal-flaws-in-deolalikars-proof Fatal Flaws in Deolalikar&#8217;s Proof?] August 12, 2010 (Inactive)&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/ The P&#8800;NP Proof Is One Week Old] August 15, 2010 ('''Active''')&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/ The P&#8800;NP &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;quot;&lt;/ins&gt;Proof&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;quot; &lt;/ins&gt;Is One Week Old] August 15, 2010 ('''Active''')&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;== The paper ==&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;== The paper ==&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 87:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 87:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;== Specific issues ==&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;== Specific issues ==&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;* '''P=NP only gives polylog parameterizability property of a lift of the k-SAT solution space, rather than that solution space itself'''&amp;nbsp; If P=NP, then the procedure in Section 8.3 of the third draft does not create a polylog parameterizable distribution on the solution space of a hard phase k-SAT problem itself, but rather on a &amp;lt;I&amp;gt;lift&amp;lt;/I&amp;gt; of that solution space in which many additional variables are added, so that one is now working in &amp;lt;math&amp;gt;\{0,1\}^N&amp;lt;/math&amp;gt; for some &amp;lt;math&amp;gt;N=n^{O(1)}&amp;lt;/math&amp;gt; rather than in &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt;.&amp;nbsp; Of course, one can forget all but the original n variables and return to a distribution on the solution space by projecting, but it is not clear that the polylog parameterizable property is preserved by this (there is no reason why the directed acyclic graph on the N literals has any meaningful projection onto the original n literals).&amp;nbsp; In particular, the projected distribution is not in a recursively factorizable form, and the cluster geometry of the lifted solution space could be quite different from that of the original k-SAT problem (e.g. the set of frozen variables could change, as could the definition of a cluster, and number of flips required to get from one cluster to the next).&amp;nbsp;  This issue does not seem to be discussed at all in the paper.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;* '''P=NP only gives polylog parameterizability property of a lift of the k-SAT solution space, rather than that solution space itself'''&amp;nbsp; If P=NP, then the procedure in Section 8.3 of the third draft does not create a &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;[[polylog parameterizability|&lt;/ins&gt;polylog parameterizable&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;]] &lt;/ins&gt;distribution on the solution space of a hard phase k-SAT problem itself, but rather on a &amp;lt;I&amp;gt;lift&amp;lt;/I&amp;gt; of that solution space in which many additional variables are added, so that one is now working in &amp;lt;math&amp;gt;\{0,1\}^N&amp;lt;/math&amp;gt; for some &amp;lt;math&amp;gt;N=n^{O(1)}&amp;lt;/math&amp;gt; rather than in &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt;.&amp;nbsp; Of course, one can forget all but the original n variables and return to a distribution on the solution space by projecting, but it is not clear that the polylog parameterizable property is preserved by this (there is no reason why the directed acyclic graph on the N literals has any meaningful projection onto the original n literals).&amp;nbsp; In particular, the projected distribution is not in a recursively factorizable form, and the cluster geometry of the lifted solution space could be quite different from that of the original k-SAT problem (e.g. the set of frozen variables could change, as could the definition of a cluster, and number of flips required to get from one cluster to the next).&amp;nbsp;  This issue does not seem to be discussed at all in the paper.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;To phrase it another way, consider the following excerpt from the paper describing part of the strategy (Page 92, third draft):&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;To phrase it another way, consider the following excerpt from the paper describing part of the strategy (Page 92, third draft):&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 297:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 297:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://scottaaronson.com/blog/?p=458 Eight Signs A Claimed P&#8800;NP Proof Is Wrong], Scott Aaronson, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://scottaaronson.com/blog/?p=458 Eight Signs A Claimed P&#8800;NP Proof Is Wrong], Scott Aaronson, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://gowers.wordpress.com/2010/08/14/a-possible-answer-to-my-objection/ A possible answer to my objection], Tim Gowers, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://gowers.wordpress.com/2010/08/14/a-possible-answer-to-my-objection/ A possible answer to my objection], Tim Gowers, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/ The P&#8800;NP Proof Is One Week Old], Richard Lipton, G&#246;del&#8217;s lost letter and P=NP, August 15, 2010&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/ The P&#8800;NP &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;quot;&lt;/ins&gt;Proof&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;quot; &lt;/ins&gt;Is One Week Old], Richard Lipton, G&#246;del&#8217;s lost letter and P=NP, August 15, 2010&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Media and aggregators ===&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Media and aggregators ===&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:11 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper&amp;diff=3586&amp;oldid=prev" start="Sun Aug 15 2010 19:26:29 GMT+0000" title="Deolalikar P vs NP paper">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;Theory blogs:&amp;#32;&lt;/span&gt; &lt;/p&gt;
+
+			&lt;table&gt;
+			&lt;col class=&quot;diff-marker&quot; /&gt;
+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;col class=&quot;diff-marker&quot; /&gt;
+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;tr valign=&quot;top&quot;&gt;
+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 19:26, 15 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;4&quot; align=&quot;center&quot; class=&quot;diff-multi&quot;&gt;(One intermediate revision not shown.)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 18:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 18:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/10/update-on-deolalikars-proof-that-p%E2%89%A0np/ Update on Deolalikar's Proof that P&#8800;NP], August 10 2010. (Inactive)&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/10/update-on-deolalikars-proof-that-p%E2%89%A0np/ Update on Deolalikar's Proof that P&#8800;NP], August 10 2010. (Inactive)&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/11/deolalikar-responds-to-issues-about-his-p%e2%89%a0np-proof/ Deolalikar Responds To Issues About His P&#8800;NP Proof] August 11, 2010 (Inactive)&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/11/deolalikar-responds-to-issues-about-his-p%e2%89%a0np-proof/ Deolalikar Responds To Issues About His P&#8800;NP Proof] August 11, 2010 (Inactive)&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/12/fatal-flaws-in-deolalikars-proof Fatal Flaws in Deolalikar&#8217;s Proof?] August 12, 2010 ('''Active''')&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://rjlipton.wordpress.com/2010/08/12/fatal-flaws-in-deolalikars-proof Fatal Flaws in Deolalikar&#8217;s Proof?] August 12&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;, 2010 (Inactive)&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;* [http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/ The P&#8800;NP Proof Is One Week Old] August 15&lt;/ins&gt;, 2010 ('''Active''')&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;== The paper ==&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;== The paper ==&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 296:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 297:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://scottaaronson.com/blog/?p=458 Eight Signs A Claimed P&#8800;NP Proof Is Wrong], Scott Aaronson, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://scottaaronson.com/blog/?p=458 Eight Signs A Claimed P&#8800;NP Proof Is Wrong], Scott Aaronson, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://gowers.wordpress.com/2010/08/14/a-possible-answer-to-my-objection/ A possible answer to my objection], Tim Gowers, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://gowers.wordpress.com/2010/08/14/a-possible-answer-to-my-objection/ A possible answer to my objection], Tim Gowers, August 13 2010.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;* [http://rjlipton.wordpress.com/2010/08/15/the-p%e2%89%a0np-proof-is-one-week-old/ The P&#8800;NP Proof Is One Week Old], Richard Lipton, G&#246;del&#8217;s lost letter and P=NP, August 15, 2010&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Media and aggregators ===&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Media and aggregators ===&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:12 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Random_k-SAT&amp;diff=3584&amp;oldid=prev" start="Sun Aug 15 2010 17:29:42 GMT+0000" title="Random k-SAT">&lt;p&gt;&lt;/p&gt;
+
+			&lt;table&gt;
+			&lt;col class=&quot;diff-marker&quot; /&gt;
+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;col class=&quot;diff-marker&quot; /&gt;
+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;tr valign=&quot;top&quot;&gt;
+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 17:29, 15 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;4&quot; align=&quot;center&quot; class=&quot;diff-multi&quot;&gt;(One intermediate revision not shown.)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 80:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 80:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Random k-XOR-SAT ===&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Random k-XOR-SAT ===&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;Finally, what about random k-XOR-SAT ? &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;Finally, what about random k-XOR-SAT ? &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;Random k-XOR-SAT has been solved in [MRZ03] and there the existence&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;of clusters, frozen variables, etc... have been proved rigorously.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* First, a technical point. If one does indeed generate instances by selecting the variables in each constraint uniformly at random (without insisting that each variable has degree at least 2 as one does in LDPC codes), then there will be variables which appears in 0 or 1 constraints. Clearly, such variables can be &amp;quot;repeatedly removed&amp;quot; leaving a &amp;quot;core&amp;quot; structure.&amp;nbsp; It is clear that if one finds a solution to the core, then this solution can be trivially extended to the rest. OK, what about solutions to this core? &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* First, a technical point. If one does indeed generate instances by selecting the variables in each constraint uniformly at random (without insisting that each variable has degree at least 2 as one does in LDPC codes), then there will be variables which appears in 0 or 1 constraints. Clearly, such variables can be &amp;quot;repeatedly removed&amp;quot; leaving a &amp;quot;core&amp;quot; structure.&amp;nbsp; It is clear that if one finds a solution to the core, then this solution can be trivially extended to the rest. OK, what about solutions to this core? &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 93:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 96:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [KMR+07] [http://www.pnas.org/content/104/25/10318.abstract Gibbs states and the set of solutions of random constraint satisfaction problems],&amp;nbsp; &amp;nbsp; F. Krzakala, A. Montanari, F. Ricci-Tersenghi, G. Semerjian and L. Zdeborova, PNAS&amp;nbsp; 2007, 104, 10318-10323.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [KMR+07] [http://www.pnas.org/content/104/25/10318.abstract Gibbs states and the set of solutions of random constraint satisfaction problems],&amp;nbsp; &amp;nbsp; F. Krzakala, A. Montanari, F. Ricci-Tersenghi, G. Semerjian and L. Zdeborova, PNAS&amp;nbsp; 2007, 104, 10318-10323.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [MPZ02] [http://www.sciencemag.org/cgi/content/abstract/297/5582/812?ijkey=bb692e28fb17ceba785ae57d9c5db2cf21a5fa62&amp;amp;keytype2=tf_ipsecsha Analytic and Algorithmic Solution of Random Satisfiability Problems ], M. Mezard M, G. Parisi and R.&amp;nbsp; Zecchina, 2002 Science 297:812&#8211;815.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [MPZ02] [http://www.sciencemag.org/cgi/content/abstract/297/5582/812?ijkey=bb692e28fb17ceba785ae57d9c5db2cf21a5fa62&amp;amp;keytype2=tf_ipsecsha Analytic and Algorithmic Solution of Random Satisfiability Problems ], M. Mezard M, G. Parisi and R.&amp;nbsp; Zecchina, 2002 Science 297:812&#8211;815.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;* [MRZ03] Two Solutions to Diluted p-Spin Models and XORSAT Problems, M. M&#233;zard, F. Ricci-Tersenghi and R. Zecchina, J. Stat. Phys. 111 (2003) 505.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [S07] [http://www.springerlink.com/content/n710514276010144/&amp;nbsp; On the Freezing of Variables in Random Constraint Satisfaction Problems], G. Semerjian, J. Stat. Phys. 130, 251 (2008).&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [S07] [http://www.springerlink.com/content/n710514276010144/&amp;nbsp; On the Freezing of Variables in Random Constraint Satisfaction Problems], G. Semerjian, J. Stat. Phys. 130, 251 (2008).&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [ZK07] [http://pre.aps.org/abstract/PRE/v76/i3/e031131 Phase transitions in the coloring of random graphs], L. Zdeborova and F. Krzakala, Phys. Rev. E 76, 031131 (2007)&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [ZK07] [http://pre.aps.org/abstract/PRE/v76/i3/e031131 Phase transitions in the coloring of random graphs], L. Zdeborova and F. Krzakala, Phys. Rev. E 76, 031131 (2007)&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [ZM08] [http://iopscience.iop.org/1742-5468/2008/12/P12004&amp;nbsp; Constraint satisfaction problems with isolated solutions are hard], L. Zdeborova and M. Mezard, J. Stat. Mech. (2008) P12004.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [ZM08] [http://iopscience.iop.org/1742-5468/2008/12/P12004&amp;nbsp; Constraint satisfaction problems with isolated solutions are hard], L. Zdeborova and M. Mezard, J. Stat. Mech. (2008) P12004.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:12 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper&amp;diff=3582&amp;oldid=prev" start="Sun Aug 15 2010 17:04:46 GMT+0000" title="Deolalikar P vs NP paper">&lt;p&gt;&lt;/p&gt;
+
+			&lt;table&gt;
+			&lt;col class=&quot;diff-marker&quot; /&gt;
+			&lt;col class=&quot;diff-content&quot; /&gt;
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+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;tr valign=&quot;top&quot;&gt;
+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 17:04, 15 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 105:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 105:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;::As phrased by [http://rjlipton.wordpress.com/2010/08/11/deolalikar-responds-to-issues-about-his-p%e2%89%a0np-proof#comment-5138 Albert Atserias] and [[Lindell's Critique|Steven Lindell]]:&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;::As phrased by [http://rjlipton.wordpress.com/2010/08/11/deolalikar-responds-to-issues-about-his-p%e2%89%a0np-proof#comment-5138 Albert Atserias] and [[Lindell's Critique|Steven Lindell]]:&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;lt;blockquote&amp;gt;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;lt;blockquote&amp;gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;Let &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;&#8220;polylog&lt;/del&gt;-&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;parametrizable&#8221; &lt;/del&gt;be a property of solution spaces yet to be defined &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;(see [[Polylog parameterizability]]&lt;/del&gt;.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;Let &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&#8220;[[Polylog parameterizability|polylog&lt;/ins&gt;-&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;parametrizable]]&#8221; &lt;/ins&gt;be a property of solution spaces yet to be defined.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* Claim 1: The solution spaces of problems in P are polylog-parametrizable.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* Claim 1: The solution spaces of problems in P are polylog-parametrizable.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:12 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Polylog_parameterizability&amp;diff=3581&amp;oldid=prev" start="Sun Aug 15 2010 17:03:13 GMT+0000" title="Polylog parameterizability">&lt;p&gt;&lt;/p&gt;
+
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+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 17:03, 15 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;4&quot; align=&quot;center&quot; class=&quot;diff-multi&quot;&gt;(One intermediate revision not shown.)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 15:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 15:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;: Claim. If a solution space in &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt; can be &amp;quot;explored by a P algorithm&amp;quot; in a suitable sense, then the uniform distribution &amp;lt;math&amp;gt;\mu&amp;lt;/math&amp;gt; of that space is the _projection_ of a polylog-parameterisable measure &amp;lt;math&amp;gt;\mu'&amp;lt;/math&amp;gt; on a subset of a polynomially larger space &amp;lt;math&amp;gt;\{0,1\}^{n^{O(1)}}&amp;lt;/math&amp;gt;.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;: Claim. If a solution space in &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt; can be &amp;quot;explored by a P algorithm&amp;quot; in a suitable sense, then the uniform distribution &amp;lt;math&amp;gt;\mu&amp;lt;/math&amp;gt; of that space is the _projection_ of a polylog-parameterisable measure &amp;lt;math&amp;gt;\mu'&amp;lt;/math&amp;gt; on a subset of a polynomially larger space &amp;lt;math&amp;gt;\{0,1\}^{n^{O(1)}}&amp;lt;/math&amp;gt;.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;However, a key point that does not seem to be sufficiently addressed in the paper is whether the projection of a polylog-parameterisable measure remains polylog-parameterisable, and which I believe could in fact be a serious problem with the proof.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;#160;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;The following can be easily shown (see [http://rjlipton.wordpress.com/2010/08/12/fatal-flaws-in-deolalikars-proof/#comment-5566 proof]):&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;#160;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;A probability distribution on &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt; that can be computed by a polynomial time run on a probabilistic Turing machine is a projection of a polylog parametrizable distribution.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;#160;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;However, a key point that does not seem to be sufficiently addressed in the paper is whether the projection of a polylog-parameterisable measure remains polylog-parameterisable, and which I believe could in fact be a serious problem with the proof&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;. In fact, it is not the case, shown by counterexamples below&lt;/ins&gt;.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;== Examples ==&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;== Examples ==&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:12 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper&amp;diff=3579&amp;oldid=prev" start="Sun Aug 15 2010 16:53:22 GMT+0000" title="Deolalikar P vs NP paper">&lt;p&gt;&lt;/p&gt;
+
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+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;tr valign=&quot;top&quot;&gt;
+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 16:53, 15 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 105:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 105:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;::As phrased by [http://rjlipton.wordpress.com/2010/08/11/deolalikar-responds-to-issues-about-his-p%e2%89%a0np-proof#comment-5138 Albert Atserias] and [[Lindell's Critique|Steven Lindell]]:&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;::As phrased by [http://rjlipton.wordpress.com/2010/08/11/deolalikar-responds-to-issues-about-his-p%e2%89%a0np-proof#comment-5138 Albert Atserias] and [[Lindell's Critique|Steven Lindell]]:&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;lt;blockquote&amp;gt;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;lt;blockquote&amp;gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;Let &#8220;polylog-parametrizable&#8221; be a property of solution spaces yet to be defined.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;Let &#8220;polylog-parametrizable&#8221; be a property of solution spaces yet to be defined &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;(see [[Polylog parameterizability]]&lt;/ins&gt;.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* Claim 1: The solution spaces of problems in P are polylog-parametrizable.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* Claim 1: The solution spaces of problems in P are polylog-parametrizable.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:12 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper&amp;diff=3578&amp;oldid=prev" start="Sun Aug 15 2010 16:43:49 GMT+0000" title="Deolalikar P vs NP paper">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;Further reading:&amp;#32;&lt;/span&gt; Aaronson post&lt;/p&gt;
+
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+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;tr valign=&quot;top&quot;&gt;
+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 16:43, 15 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;4&quot; align=&quot;center&quot; class=&quot;diff-multi&quot;&gt;(2 intermediate revisions not shown.)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 87:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 87:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* '''P=NP only gives polylog parameterizability property of a lift of the k-SAT solution space, rather than that solution space itself'''&amp;nbsp; If P=NP, then the procedure in Section 8.3 of the third draft does not create a polylog parameterizable distribution on the solution space of a hard phase k-SAT problem itself, but rather on a &amp;lt;I&amp;gt;lift&amp;lt;/I&amp;gt; of that solution space in which many additional variables are added, so that one is now working in &amp;lt;math&amp;gt;\{0,1\}^N&amp;lt;/math&amp;gt; for some &amp;lt;math&amp;gt;N=n^{O(1)}&amp;lt;/math&amp;gt; rather than in &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt;.&amp;nbsp; Of course, one can forget all but the original n variables and return to a distribution on the solution space by projecting, but it is not clear that the polylog parameterizable property is preserved by this (there is no reason why the directed acyclic graph on the N literals has any meaningful projection onto the original n literals).&amp;nbsp; In particular, the projected distribution is not in a recursively factorizable form, and the cluster geometry of the lifted solution space could be quite different from that of the original k-SAT problem (e.g. the set of frozen variables could change, as could the definition of a cluster, and number of flips required to get from one cluster to the next).&amp;nbsp;  This issue does not seem to be discussed at all in the paper.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* '''P=NP only gives polylog parameterizability property of a lift of the k-SAT solution space, rather than that solution space itself'''&amp;nbsp; If P=NP, then the procedure in Section 8.3 of the third draft does not create a polylog parameterizable distribution on the solution space of a hard phase k-SAT problem itself, but rather on a &amp;lt;I&amp;gt;lift&amp;lt;/I&amp;gt; of that solution space in which many additional variables are added, so that one is now working in &amp;lt;math&amp;gt;\{0,1\}^N&amp;lt;/math&amp;gt; for some &amp;lt;math&amp;gt;N=n^{O(1)}&amp;lt;/math&amp;gt; rather than in &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt;.&amp;nbsp; Of course, one can forget all but the original n variables and return to a distribution on the solution space by projecting, but it is not clear that the polylog parameterizable property is preserved by this (there is no reason why the directed acyclic graph on the N literals has any meaningful projection onto the original n literals).&amp;nbsp; In particular, the projected distribution is not in a recursively factorizable form, and the cluster geometry of the lifted solution space could be quite different from that of the original k-SAT problem (e.g. the set of frozen variables could change, as could the definition of a cluster, and number of flips required to get from one cluster to the next).&amp;nbsp;  This issue does not seem to be discussed at all in the paper.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;To phrase it another way, consider the following excerpt from the paper describing part of the strategy (Page 92, third draft):&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;:: We have embedded our original set of variates into a &amp;lt;I&amp;gt;polynomially larger&amp;lt;/I&amp;gt; product space, and obtained a directed graphical model on this larger space. This product space has a nice factorization due to the directed graph structure. This is what we will exploit.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;It is clear here that the problem is being lifted from the original space &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt; to a larger space &amp;lt;math&amp;gt;\{0,1\}^N&amp;lt;/math&amp;gt;, but then the solution space of (say) k-SAT on &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt; also gets lifted to a solution space which could have a completely different geometry.&amp;nbsp; Given that the proof sketch of P != NP (pages 99-100, third draft) is based on the cluster geometry of the solution space of k-SAT itself, rather than a lift thereof, this is a significant gap in the argument.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;nbsp; &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;nbsp; &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* '''The problem in the LFP section of the proof:''' &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* '''The problem in the LFP section of the proof:''' &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 398:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 405:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://kintali.wordpress.com/2009/09/01/relativization-barrier/ Relativization Barrier]&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://kintali.wordpress.com/2009/09/01/relativization-barrier/ Relativization Barrier]&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://terrytao.wordpress.com/2009/08/01/pnp-relativisation-and-multiple-choice-exams P=NP, relativisation, and multiple choice exams], Terence Tao&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* [http://terrytao.wordpress.com/2009/08/01/pnp-relativisation-and-multiple-choice-exams P=NP, relativisation, and multiple choice exams], Terence Tao&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;* [http://scottaaronson.com/blog/?p=459 P=NP for dummies], Scott Aaronson, August 15 2010&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* There are many other complexity classes besides P and NP. See the [http://qwiki.stanford.edu/wiki/Complexity_Zoo Complexity Zoo]&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;* There are many other complexity classes besides P and NP. See the [http://qwiki.stanford.edu/wiki/Complexity_Zoo Complexity Zoo]&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:12 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Polylog_parameterizability&amp;diff=3575&amp;oldid=prev" start="Sun Aug 15 2010 15:16:05 GMT+0000" title="Polylog parameterizability">&lt;p&gt;Adding examples of Tarui and vloodin&lt;/p&gt;
+
+			&lt;table&gt;
+			&lt;col class=&quot;diff-marker&quot; /&gt;
+			&lt;col class=&quot;diff-content&quot; /&gt;
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+			&lt;col class=&quot;diff-content&quot; /&gt;
+			&lt;tr valign=&quot;top&quot;&gt;
+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 15:16, 15 August 2010&lt;/td&gt;
+			&lt;/tr&gt;
+		&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;In this post I will try to describe what I believe Deolilakar means by &amp;quot;polylog parameterizability&amp;quot;, a key concept in his work.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;In this post I will try to describe what I believe Deolilakar means by &amp;quot;polylog parameterizability&amp;quot;, a key concept in his work. &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt; (A more accurate term here would be &amp;quot;polylog recursive factorizability&amp;quot;.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;#160;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;== Definition ==&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;Let &amp;lt;math&amp;gt;\mu&amp;lt;/math&amp;gt; be a probability distribution on the cube &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt;.&amp;nbsp; Then to describe this distribution fully, one needs &amp;lt;math&amp;gt;2^n-1&amp;lt;/math&amp;gt; real parameters; one real number &amp;lt;math&amp;gt;\mu(\{x\})&amp;lt;/math&amp;gt; for each of the &amp;lt;math&amp;gt;2^n&amp;lt;/math&amp;gt; elements of the cube, minus one for the constraint &amp;lt;math&amp;gt;\sum_{x \in \{0,1\}^n} \mu(\{x\}) = 1&amp;lt;/math&amp;gt;.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;Let &amp;lt;math&amp;gt;\mu&amp;lt;/math&amp;gt; be a probability distribution on the cube &amp;lt;math&amp;gt;\{0,1\}^n&amp;lt;/math&amp;gt;.&amp;nbsp; Then to describe this distribution fully, one needs &amp;lt;math&amp;gt;2^n-1&amp;lt;/math&amp;gt; real parameters; one real number &amp;lt;math&amp;gt;\mu(\{x\})&amp;lt;/math&amp;gt; for each of the &amp;lt;math&amp;gt;2^n&amp;lt;/math&amp;gt; elements of the cube, minus one for the constraint &amp;lt;math&amp;gt;\sum_{x \in \{0,1\}^n} \mu(\{x\}) = 1&amp;lt;/math&amp;gt;.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 14:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 16:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;However, a key point that does not seem to be sufficiently addressed in the paper is whether the projection of a polylog-parameterisable measure remains polylog-parameterisable, and which I believe could in fact be a serious problem with the proof.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;However, a key point that does not seem to be sufficiently addressed in the paper is whether the projection of a polylog-parameterisable measure remains polylog-parameterisable, and which I believe could in fact be a serious problem with the proof.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins&gt;== Examples ==&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;Let me illustrate the above Claim with a simple example, that of a straight line program.&amp;nbsp; Let &amp;lt;math&amp;gt;N := n^{100}&amp;lt;/math&amp;gt;, and consider a boolean function &amp;lt;math&amp;gt;f: \{0,1\}^n \to \{0,1\}&amp;lt;/math&amp;gt; obtained by the formula&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;Let me illustrate the above Claim with a simple example, that of a straight line program.&amp;nbsp; Let &amp;lt;math&amp;gt;N := n^{100}&amp;lt;/math&amp;gt;, and consider a boolean function &amp;lt;math&amp;gt;f: \{0,1\}^n \to \{0,1\}&amp;lt;/math&amp;gt; obtained by the formula&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 52:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 56:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;where &amp;lt;math&amp;gt;p_i(\cdot; a_0, a_1)&amp;lt;/math&amp;gt; is the delta distribution &amp;lt;math&amp;gt;\delta_0&amp;lt;/math&amp;gt; if &amp;lt;math&amp;gt;a_0=1, a_1 = 0&amp;lt;/math&amp;gt;, the delta distribution &amp;lt;math&amp;gt;\delta_1&amp;lt;/math&amp;gt; if &amp;lt;math&amp;gt;a_0=0,a_1=1&amp;lt;/math&amp;gt;, the uniform distribution &amp;lt;math&amp;gt;\frac{1}{2}\delta_0+\frac{1}{2}\delta_1&amp;lt;/math&amp;gt; if &amp;lt;math&amp;gt;a_0=a_1=1&amp;lt;/math&amp;gt;, and some arbitrary distribution (e.g. &amp;lt;math&amp;gt;\delta_0&amp;lt;/math&amp;gt; for concreteness) in the (unattainable) case &amp;lt;math&amp;gt;a_0=a_1=1&amp;lt;/math&amp;gt;.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;where &amp;lt;math&amp;gt;p_i(\cdot; a_0, a_1)&amp;lt;/math&amp;gt; is the delta distribution &amp;lt;math&amp;gt;\delta_0&amp;lt;/math&amp;gt; if &amp;lt;math&amp;gt;a_0=1, a_1 = 0&amp;lt;/math&amp;gt;, the delta distribution &amp;lt;math&amp;gt;\delta_1&amp;lt;/math&amp;gt; if &amp;lt;math&amp;gt;a_0=0,a_1=1&amp;lt;/math&amp;gt;, the uniform distribution &amp;lt;math&amp;gt;\frac{1}{2}\delta_0+\frac{1}{2}\delta_1&amp;lt;/math&amp;gt; if &amp;lt;math&amp;gt;a_0=a_1=1&amp;lt;/math&amp;gt;, and some arbitrary distribution (e.g. &amp;lt;math&amp;gt;\delta_0&amp;lt;/math&amp;gt; for concreteness) in the (unattainable) case &amp;lt;math&amp;gt;a_0=a_1=1&amp;lt;/math&amp;gt;.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;This is not yet of a polylog &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;parameterisable &lt;/del&gt;form (each i has i-1 ancestors rather than polylog ancestors).&amp;nbsp; &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;This is not yet of a polylog &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;parameterizable &lt;/ins&gt;form (each i has i-1 ancestors rather than polylog ancestors).&amp;nbsp; &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;#160;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;However, if one expands all the straight line programs &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; and adds all the literals required for computing such programs, then we can lift &amp;lt;math&amp;gt;\nu&amp;lt;/math&amp;gt; to a polylog parameterizable measure &amp;lt;math&amp;gt;\nu'&amp;lt;/math&amp;gt; similarly to before.&amp;nbsp; However &amp;lt;math&amp;gt;\nu&amp;lt;/math&amp;gt; itself is not obviously a polylog parameterizable measure, merely the projection of one that involves additional &amp;quot;hidden variables&amp;quot;.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;#160;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;Another remark: since NP is in EXP, I think that the solution space to a k-SAT problem, if it is non-empty admits, the projection of a degree O(1) recursively factorizable measure on exponentially many variables.&amp;nbsp; This already illustrates that the projection operation is powerful enough to convert even the simplest measure into an extremely general one, at least if one is allowed to discard an exponential number of variables.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;#160;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;== Counterexamples ==&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;#160;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;The following measures are not polylog parameterisable, though they are projections of polylog parameterisable measures in a higher dimensional space:&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;#160;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;# uniform (or any positive) measure on &amp;lt;math&amp;gt;EVEN:=\{x \in \{0,1\}^n: x_1+\ldots+x_n =0 (\mod 2)\}&amp;lt;/math&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;# uniform (or any positive) measure on &amp;lt;math&amp;gt;\{0,1\}^n \backslash (0,\ldots,0)&amp;lt;/math&amp;gt;.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;#160;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;To see the non polylog parameterisability, we use&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;#160;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;: Claim: Let &amp;lt;math&amp;gt;S \subset \{0,1\}^n&amp;lt;/math&amp;gt;, let &amp;lt;math&amp;gt;\Chi_S&amp;lt;/math&amp;gt; be the characteristic function of S, and let &amp;lt;math&amp;gt;\mu&amp;lt;/math&amp;gt; be a probability measure whose support is exactly S (i.e. &amp;lt;math&amp;gt;\mu(x) &amp;gt; 0&amp;lt;/math&amp;gt; if and only if x is in S. Assume that &amp;lt;math&amp;gt;\mu&amp;lt;/math&amp;gt; is polylog parametriable. Then, there is some permutation (reordering) of [n] such that Boolean function &amp;lt;math&amp;gt;\Chi_S(x_1,\ldots,x_n)&amp;lt;/math&amp;gt; can be expressed/computed as follows:&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;: &amp;lt;math&amp;gt;\Chi_S(x_1\ldots,x_n) = \prod_{i=1}^n g(x_i; x_j&#8217;s)&amp;lt;/math&amp;gt;, where each g is a Boolean function of &amp;lt;math&amp;gt;x_i&amp;lt;/math&amp;gt; and polylog-many &amp;lt;math&amp;gt;x_j&amp;lt;/math&amp;gt;&#8217;s with j &amp;lt; i.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;#160;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;We can see the claim by focusing on measure being zero or nonzero in the definition of polylog paramateizability. Now consider #1. With respect to any ordering (permutation) of [n], the membership in EVEN cannot be expressed by such an AND as in the claim because the last factor &amp;lt;math&amp;gt;g(x_n; x_j's)&amp;lt;/math&amp;gt; only depend on *fixed* polylog many &amp;lt;math&amp;gt;x_j&amp;lt;/math&amp;gt;'s. By the same reason, #2 is not polylog parametrizable. (To express #1 and #2 in such a form, the last g(xn; xj's) have to depend on all n variables.)&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;However, if one expands all &lt;/del&gt;the &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;straight line programs &amp;lt;math&amp;gt;f_i&amp;lt;/math&amp;gt; and adds all the literals required &lt;/del&gt;for &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;computing such programs&lt;/del&gt;, &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;then &lt;/del&gt;we can &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;lift &amp;lt;math&amp;gt;\nu&amp;lt;/math&amp;gt; to &lt;/del&gt;a &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;polylog parameterisable measure &amp;lt;math&amp;gt;\nu'&amp;lt;/math&amp;gt; similarly to before.&amp;nbsp; However &amp;lt;math&amp;gt;\nu&amp;lt;/math&amp;gt; itself is not obviously a polylog parameterisable measure&lt;/del&gt;, &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;merely the projection of one that involves additional &amp;quot;hidden variables&amp;quot;&lt;/del&gt;.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;On &lt;/ins&gt;the &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;other hand EVEN can be polylog parametrized if we add a number of auxilary variables, placeholders &lt;/ins&gt;for &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;partial sums&lt;/ins&gt;, &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;so that we never add more than polylog elements (&lt;/ins&gt;we can &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;always add &lt;/ins&gt;a &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;fixed number)&lt;/ins&gt;, &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;and similarly for #2&lt;/ins&gt;.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:12 --&gt;
+&lt;/table&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://michaelnielsen.org/polymath1/index.php?title=Deolalikar_P_vs_NP_paper&amp;diff=3574&amp;oldid=prev" start="Sun Aug 15 2010 06:18:12 GMT+0000" title="Deolalikar P vs NP paper">&lt;p&gt;&lt;span class=&quot;autocomment&quot;&gt;Issues with random k-SAT:&amp;#32;&lt;/span&gt; &lt;/p&gt;
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+				&lt;td colspan=&quot;2&quot;&gt;&#8592;Older revision&lt;/td&gt;
+				&lt;td colspan=&quot;2&quot;&gt;Revision as of 06:18, 15 August 2010&lt;/td&gt;
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+		&lt;tr&gt;&lt;td colspan=&quot;4&quot; align=&quot;center&quot; class=&quot;diff-multi&quot;&gt;(7 intermediate revisions not shown.)&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 155:&lt;/td&gt;
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+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;===Issues with random k-SAT===&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;===Issues with random k-SAT===&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;'''Complex solution spaces are uncorrelated with time complexity'''. (The below is a greatly expanded version of a series of twitter comments by Ryan Williams, on [http://twitter.com/rrwilliams/status/20741046788 twitter]) The author tries to use the fact that for certain distributions of random k-SAT, the solution space has a &amp;quot;hard structure&amp;quot;. For certain parameterizations, the space of satisfying assignments to a random k-SAT instance has some intriguing structure. If SAT is in P, then SAT can be captured in a certain logic (equivalent to P in some sense). The author claims that anything captured in this logic can't have a solution space with this intriguing structure. There are two &amp;quot;meta&amp;quot; objections to this. One is that &amp;quot;intriguing structure in the solution space is not sufficient for NP hardness&amp;quot;. The second is that &amp;quot;intriguing structure is not necessary for NP hardness&amp;quot;. They don't actually point to a place where the proof is wrong. But they do appear to give an obstacle to the general proof method&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;. They exploit the fact that the concept of a &amp;quot;solution space&amp;quot; cannot be coherently defined for NP problems, without being verifier-dependent&lt;/del&gt;.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;'''Complex solution spaces are uncorrelated with time complexity'''. (The below is a greatly expanded version of a series of twitter comments by Ryan Williams, on [http://twitter.com/rrwilliams/status/20741046788 twitter]) The author tries to use the fact that for certain distributions of random k-SAT, the solution space has a &amp;quot;hard structure&amp;quot;. For certain parameterizations, the space of satisfying assignments to a random k-SAT instance has some intriguing structure. If SAT is in P, then SAT can be captured in a certain logic (equivalent to P in some sense). The author claims that anything captured in this logic can't have a solution space with this intriguing structure. There are two &amp;quot;meta&amp;quot; objections to this. One is that &amp;quot;intriguing structure in the solution space is not sufficient for NP hardness&amp;quot;. The second is that &amp;quot;intriguing structure is not necessary for NP hardness&amp;quot;. They don't actually point to a place where the proof is wrong. But they do appear to give an obstacle to the general proof method. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;'''How to map satisfiable formulas to trivial problems with exactly the same solution space structure.''' Very easy problems can also have complicated solution distributions. The following arose from discussions with Jeremiah Blocki. Here is &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;a very &lt;/del&gt;easy way to (non-uniformly) map any infinite collection of satisfiable formulas into &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt;, the set of formulas satisfied by the all-zeroes assignment. The map completely preserves the number of variables, the number of satisfying assignments, the distances between assignments, the clusters--as far as I can tell it preserves '''every property''' that has been studied about random k-SAT. Observe &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt; is trivially decidable in polynomial time. This should be a barrier to any proof of P vs NP that attempts to argue that the solution space of an NP-hard problem is &amp;quot;too complex&amp;quot; for P. (It even blocks any proof of AC0 not equal NP with this pattern!) This is the objection which is most germane to the current proposed proof: it opposes the claim that &amp;quot;anything in P can't have a solution space with complicated structure&amp;quot;. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;::&lt;/ins&gt;'''How to map satisfiable formulas to trivial problems with exactly the same solution space structure.''' Very easy problems can also have complicated solution distributions. The following arose from discussions with Jeremiah Blocki. Here is &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;an &lt;/ins&gt;easy way to (non-uniformly&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;!&lt;/ins&gt;) map any infinite collection of satisfiable formulas into &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt;, the set of formulas satisfied by the all-zeroes assignment. The map completely preserves the number of variables, the number of satisfying assignments, the distances between assignments, the clusters--as far as I can tell it preserves '''every property''' that has been studied about random k-SAT. Observe &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt; is trivially decidable in polynomial time. This should be a barrier to any proof of P vs NP that attempts to argue that the solution space of an NP-hard problem is &amp;quot;too complex&amp;quot; for P. (It even blocks any proof of AC0 not equal NP with this pattern!) This is the objection which is most germane to the current proposed proof: it opposes the claim that &amp;quot;anything in P can't have a solution space with complicated structure&amp;quot;. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;The map is simple: for a satisfiable formula F on n variables, let &amp;lt;math&amp;gt;(A_1,\ldots,A_n)&amp;lt;/math&amp;gt; be a satisfying assignment for it, and define F' to be the formula obtained by the following procedure: &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;'&lt;/del&gt;''flip the signs of all literals involving &amp;lt;math&amp;gt;x_i&amp;lt;/math&amp;gt; in F iff &amp;lt;math&amp;gt;A_i = 1&amp;lt;/math&amp;gt;&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;'&lt;/del&gt;''. Now the space of satisfying assignments to F is transformed by the map &amp;lt;math&amp;gt;\phi_A(x_1,\ldots,x_n) = (x_1 \oplus A_1,\ldots,x_n \oplus A_n)&amp;lt;/math&amp;gt;. Observe this map completely preserves all distances between assignments, clusters, etc., and that the all-zeroes assignment satisfies F'. So for any infinite collection &amp;lt;math&amp;gt;{C}&amp;lt;/math&amp;gt; of satisfiable formulas with &amp;quot;hard solution structure&amp;quot; there is also an infinite collection of formulas &amp;lt;math&amp;gt;{C'}&amp;lt;/math&amp;gt; with analogous solution structure, where &amp;lt;math&amp;gt;C'&amp;lt;/math&amp;gt; is a subset of &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt;. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;::&lt;/ins&gt;The map is simple: for a satisfiable formula F on n variables, let &amp;lt;math&amp;gt;(A_1,\ldots,A_n)&amp;lt;/math&amp;gt; be a satisfying assignment for it, and define F' to be the formula obtained by the following procedure: ''flip the signs of all literals involving &amp;lt;math&amp;gt;x_i&amp;lt;/math&amp;gt; in F iff &amp;lt;math&amp;gt;A_i = 1&amp;lt;/math&amp;gt;''. Now the space of satisfying assignments to F is transformed by the map &amp;lt;math&amp;gt;\phi_A(x_1,\ldots,x_n) = (x_1 \oplus A_1,\ldots,x_n \oplus A_n)&amp;lt;/math&amp;gt;. Observe this map completely preserves all distances between assignments, clusters, etc., and that the all-zeroes assignment satisfies F'. So for any infinite collection &amp;lt;math&amp;gt;{C}&amp;lt;/math&amp;gt; of satisfiable formulas with &amp;quot;hard solution structure&amp;quot; there is also an infinite collection of formulas &amp;lt;math&amp;gt;{C'}&amp;lt;/math&amp;gt; with analogous solution structure, where &amp;lt;math&amp;gt;C'&amp;lt;/math&amp;gt; is a subset of &amp;lt;math&amp;gt;SAT0&amp;lt;/math&amp;gt;. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;Furthermore, the map easily extends to the generalization of k-SAT which consists of pairs &amp;lt;math&amp;gt;(F,A)&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; is a k-SAT formula, &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt; is a partial assignment to some variables, and we wish to know if it is possible to complete the partial assignment into a full satisfying assignment. By flipping literals appropriately in &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt;, there is always an instance &amp;lt;math&amp;gt;(G,A)&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt; can be satisfied by setting the rest of the variables to zero, and yet &amp;lt;math&amp;gt;G|_{A}&amp;lt;/math&amp;gt; has an isomorphic solution space to &amp;lt;math&amp;gt;F|_{A}&amp;lt;/math&amp;gt;. So for every instance of &amp;quot;k-SAT-Partial-Assignment&amp;quot; we can find an instance of &amp;quot;SAT0-Partial-Assignment&amp;quot; (which is again a trivial problem) with the same solution space structure. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;::&lt;/ins&gt;Furthermore, the map easily extends to the generalization of k-SAT which consists of pairs &amp;lt;math&amp;gt;(F,A)&amp;lt;/math&amp;gt;, where &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt; is a k-SAT formula, &amp;lt;math&amp;gt;A&amp;lt;/math&amp;gt; is a partial assignment to some variables, and we wish to know if it is possible to complete the partial assignment into a full satisfying assignment. By flipping literals appropriately in &amp;lt;math&amp;gt;F&amp;lt;/math&amp;gt;, there is always an instance &amp;lt;math&amp;gt;(G,A)&amp;lt;/math&amp;gt; where &amp;lt;math&amp;gt;G&amp;lt;/math&amp;gt; can be satisfied by setting the rest of the variables to zero, and yet &amp;lt;math&amp;gt;G|_{A}&amp;lt;/math&amp;gt; has an isomorphic solution space to &amp;lt;math&amp;gt;F|_{A}&amp;lt;/math&amp;gt;. So for every instance of &amp;quot;k-SAT-Partial-Assignment&amp;quot; we can find an instance of &amp;quot;SAT0-Partial-Assignment&amp;quot; (which is again a trivial problem) with the same solution space structure. &amp;nbsp;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;Intuitively, the &lt;/del&gt;above &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;just &lt;/del&gt;looks like a cheat. Of course it is! We are exploiting the fact that, for arbitrary problems in NP (and therefore P as well), ''the solution space of that problem is not well-defined''. (Hubie Chen raised a similar point in email correspondence.) A solution space for an instance can only be defined with respect to some polynomial time verifier for the problem: this space is the set of witnesses that make the verifier accept on the instance. ''If you change the verifier, you change the solution space.''&amp;nbsp; The usual solution space for SAT &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;is just &lt;/del&gt;the &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;one that arises from a &lt;/del&gt;verifier that checks a candidate assignment. If &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;this verifier &lt;/del&gt;is used on SAT0, we get a complex solution space. If we use a sane verifier for SAT0 instead (the verifier that checks all-zeroes) &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;then &lt;/del&gt;the solution space becomes trivial. However, if P=NP then there's also a verifier such that ''SAT has a trivial solution space'', namely the verifier which ignores its witness and just runs a polynomial time algorithm for SAT. The above argument only arises because &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;we tried to force &lt;/del&gt;the notion of solution space to be verifier-independent over P and NP problems (&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;a step &lt;/del&gt;which &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;is done in &lt;/del&gt;the P vs NP paper).&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;::The &lt;/ins&gt;above looks like a cheat. Of course it is! We are exploiting the fact that, for arbitrary problems in NP (and therefore P as well), ''the solution space of that problem is not &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;uniquely &lt;/ins&gt;well-defined &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;in general&lt;/ins&gt;''. (Hubie Chen raised a similar point in email correspondence.) A solution space for an instance can only be defined with respect to some polynomial time verifier for the problem: this space is the set of witnesses that make the verifier accept on the instance. ''If you change the verifier, you change the solution space.''&amp;nbsp; The usual solution space for SAT &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;comes from &lt;/ins&gt;the verifier &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;(call it &amp;lt;math&amp;gt;V&amp;lt;/math&amp;gt;) &lt;/ins&gt;that checks a candidate assignment. If &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;&amp;lt;math&amp;gt;V&amp;lt;/math&amp;gt; &lt;/ins&gt;is used on SAT0, we get a complex solution space. If we use a sane verifier for SAT0 instead (the verifier that checks all-zeroes)&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;, &lt;/ins&gt;the solution space becomes trivial. However, if P=NP&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;, &lt;/ins&gt;then there's also a verifier such that ''SAT has a trivial solution space'', namely the verifier which ignores its witness and just runs a polynomial time algorithm for SAT. The above argument only arises because the notion of solution space &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;was forced &lt;/ins&gt;to be verifier-independent over P and NP problems (which &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;looks critical to &lt;/ins&gt;the P vs NP paper).&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;'''How &lt;/del&gt;to &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;map satisfiable &lt;/del&gt;formulas &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;to trivial solution spaces.''' A hard distribution of solutions is not ''necessary'' for NP&lt;/del&gt;-&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;hardness&lt;/del&gt;, &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;either. A weird distribution is ''not'' what makes a problem hard, it's &lt;/del&gt;the &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;representation of that &lt;/del&gt;solution space (&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;e.g.&lt;/del&gt;, &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;a 3-CNF formula&lt;/del&gt;, &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;a 2&lt;/del&gt;-&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;CNF formula, etc.). The &amp;quot;hard&amp;quot; case of 3&lt;/del&gt;-&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;SAT is &lt;/del&gt;the &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;case where there is ''at most &lt;/del&gt;one&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;'' satisfying assignment, since there is a randomized reduction from 3&lt;/del&gt;-SAT &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;to 3-SAT with at most one satisfying assignment &lt;/del&gt;(&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;[http://en.wikipedia.org/wiki/Valiant%E2%80%93Vazirani_theorem Valiant-Vazirani]). This reduction increases the number of clauses and the number of variables, but that doesn't really matter. &lt;/del&gt;The &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;point is that you can always reduce 3&lt;/del&gt;-SAT &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;with a &lt;/del&gt;&amp;quot;&lt;del class=&quot;diffchange diffchange-inline&quot;&gt;complex&amp;quot; solution space to one with an &amp;quot;easy&amp;quot; solution space, so how can a proof separating P from NP rely on the former? (Note that, if plausible circuit lower bounds hold up, then Valiant-Vazirani can be derandomized to run in deterministic polynomial time&lt;/del&gt;.) &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;To summarize, there is essentially no correlation between the &amp;quot;hard structure&amp;quot; of the solution space for instances of some problem, and the NP-hardness of that problem.&lt;/del&gt;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;::Note there are also ways &lt;/ins&gt;to &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;construct infinitely many 2-CNF &lt;/ins&gt;formulas &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;and XOR&lt;/ins&gt;-&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;SAT formulas with &amp;quot;complex&amp;quot; distributions&lt;/ins&gt;, &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;in that &lt;/ins&gt;the solution space (&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;with respect to the verifier &amp;lt;math&amp;gt;V&amp;lt;/math&amp;gt; above!) has many clusters&lt;/ins&gt;, &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;large distances between clusters&lt;/ins&gt;, &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;frozen variables&lt;/ins&gt;--the &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;kind of properties &lt;/ins&gt;one &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;finds with random k&lt;/ins&gt;-SAT&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;. &lt;/ins&gt;(&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;See also &amp;quot;&lt;/ins&gt;The &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;XOR&lt;/ins&gt;-SAT &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;Objection&lt;/ins&gt;&amp;quot; &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;above&lt;/ins&gt;.)&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;Of course, this point on its own does not&amp;nbsp; invalidate Deolalikar's approach.&amp;nbsp;  In principle it's possible that some NP-complete problems have complex solutions spaces and some do not.&amp;nbsp; &amp;nbsp; To prove just one NP-complete problem has a complex solution space would be enough ''if'' it was also proved that all P problems have simple solution spaces.&amp;nbsp;  For this to work, it would clearly be necessary that reductions between NP problems need not preserve complexity of solution space, but the examples given above indicate this is the case.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;::'''Satisfiable formulas with 'complex' solution spaces can be efficiently mapped to satisfiable formulas with 'simple' solution spaces.''' A hard distribution of solutions is not ''necessary'' for NP-hardness, either. A weird distribution is ''not'' what makes a problem hard, it's the representation of that solution space (e.g., a 3-CNF formula, a 2-CNF formula, etc.). The &amp;quot;hard&amp;quot; case of 3-SAT is the case where there is ''at most one'' satisfying assignment, since there is a randomized reduction from 3-SAT to 3-SAT with at most one satisfying assignment ([http://en.wikipedia.org/wiki/Valiant%E2%80%93Vazirani_theorem Valiant-Vazirani]). This reduction increases the number of clauses and the number of variables, but that doesn't really matter. The point is that you can always reduce 3-SAT with a &amp;quot;complex&amp;quot; solution space to one with an &amp;quot;easy&amp;quot; solution space, so how can a proof separating P from NP rely on the former? (Note that, if plausible circuit lower bounds hold up, then Valiant-Vazirani can be derandomized to run in deterministic polynomial time.) To summarize, there is essentially no correlation between the &amp;quot;hard structure&amp;quot; of the solution space for instances of some problem, and the NP-hardness of that problem.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;#160;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot;&gt;&amp;nbsp;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;&lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;::&lt;/ins&gt;Of course, this point on its own does not&amp;nbsp; invalidate Deolalikar's approach.&amp;nbsp;  In principle it's possible that some NP-complete problems have complex solutions spaces and some do not.&amp;nbsp; &amp;nbsp; To prove just one NP-complete problem has a complex solution space would be enough ''if'' it was also proved that all P problems have simple solution spaces.&amp;nbsp;  For this to work, it would clearly be necessary that reductions between NP problems need not preserve complexity of solution space, but the examples given above indicate this is the case.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Uniformity issues ===&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;=== Uniformity issues ===&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 233:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 235:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;==== Algebrization ====&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;==== Algebrization ====&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;See Aaronson and &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;Widgerson&lt;/del&gt;, &amp;quot;[http://www.scottaaronson.com/papers/alg.pdf Algebrization: A New Barrier in Complexity Theory]&amp;quot; ''ACM Transactions on Computation Theory'' (2009).&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;See Aaronson and &lt;ins class=&quot;diffchange diffchange-inline&quot;&gt;Wigderson&lt;/ins&gt;, &amp;quot;[http://www.scottaaronson.com/papers/alg.pdf Algebrization: A New Barrier in Complexity Theory]&amp;quot; ''ACM Transactions on Computation Theory'' (2009).&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;:The paper is all about the local properties of a specific NP-complete problem (k-SAT), and for that reason, I don't think relativization is relevant. Personally, I'm more interested in why the argument makes essential use of uniformity (which is apparently why it's supposed to avoid Razborov-Rudich). ([http://scottaaronson.com/blog/?p=456#comment-44649 Scott Aaronson])&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;:The paper is all about the local properties of a specific NP-complete problem (k-SAT), and for that reason, I don't think relativization is relevant. Personally, I'm more interested in why the argument makes essential use of uniformity (which is apparently why it's supposed to avoid Razborov-Rudich). ([http://scottaaronson.com/blog/?p=456#comment-44649 Scott Aaronson])&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 247:&lt;/td&gt;
+&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 249:&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;Followup by [http://rjlipton.wordpress.com/2010/08/10/update-on-deolalikars-proof-that-p%e2%89%a0np#comment-4988 Ryan Williams]:&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;Followup by [http://rjlipton.wordpress.com/2010/08/10/update-on-deolalikars-proof-that-p%e2%89%a0np#comment-4988 Ryan Williams]:&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;lt;blockquote&amp;gt;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;lt;blockquote&amp;gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;-&lt;/td&gt;&lt;td&gt;&lt;div&gt;It is a great idea to try to formally define this barrier and develop its properties. I think the &#8220;not necessary&#8221; part is pretty well-understood, thanks to Valiant-Vazirani. But the &#8220;not sufficient&#8221; part, the part relevant to the current paper under discussion, still needs some more rigor behind it. As I related to Lenka Zdeborova, it is easy to construct, for every n, a 2-CNF formula on n variables which has many &#8220;clusters&#8221; of solutions, where each cluster has large hamming distance from each other, and within the cluster there are a lot of satisfying assignments. But one would like to say something stronger, e.g. &#8220;for any 3-CNF formula with solution space S, that space S can be very closely simulated by the solution space S&#8217; for some CSP instance &lt;del class=&quot;diffchange diffchange-inline&quot;&gt;variables &lt;/del&gt;that is polytime solvable&#8221;.&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;+&lt;/td&gt;&lt;td&gt;&lt;div&gt;It is a great idea to try to formally define this barrier and develop its properties. I think the &#8220;not necessary&#8221; part is pretty well-understood, thanks to Valiant-Vazirani. But the &#8220;not sufficient&#8221; part, the part relevant to the current paper under discussion, still needs some more rigor behind it. As I related to Lenka Zdeborova, it is easy to construct, for every n, a 2-CNF formula on n variables which has many &#8220;clusters&#8221; of solutions, where each cluster has large hamming distance from each other, and within the cluster there are a lot of satisfying assignments. But one would like to say something stronger, e.g. &#8220;for any 3-CNF formula with solution space S, that space S can be very closely simulated by the solution space S&#8217; for some CSP instance that is polytime solvable&#8221;.&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;lt;/blockquote&amp;gt;&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;div&gt;&amp;lt;/blockquote&amp;gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt; &lt;/td&gt;&lt;td&gt;&lt;/td&gt;&lt;/tr&gt;
+&lt;!-- diff generator: internal 2010-08-17 20:56:13 --&gt;
+&lt;/table&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://aeporreca.org/2010/08/15/dlt-2010-where-tetration-has-lease/" start="Sun Aug 15 2010 10:02:38 GMT+0000" title="Developments in Language Theory 2010: Where tetration has lease">Tomorrow I&amp;#8217;m leaving for Canada, to attend the 14th International Conference on Developments in Language Theory, held at University of Western Ontario, London. You can find the programme here. Our DLT 2010 paper, authored by yours truly, Alberto Leporati and Claudio Zandron, is titled On a powerful class of non-universal P systems with active membranes. [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=aeporreca.org&amp;blog=13664840&amp;post=596&amp;subd=aeporreca&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://aeporreca.org/2010/08/09/proof-that-p-isnt-np/" start="Mon Aug 09 2010 10:29:53 GMT+0000" title="A relatively serious proof that P &#8800; NP?">Note: I&amp;#8217;ve stopped updating this post, now that I&amp;#8217;ve collected quite a lot of comments. For the latest news, I suggest checking my tweets and, in particular, the Polymath wiki page (edited by more people than just yours truly). Maybe I&amp;#8217;ll have another post in the next few days, if something shocking happens. The news [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=aeporreca.org&amp;blog=13664840&amp;post=544&amp;subd=aeporreca&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://aeporreca.org/2010/08/04/popper-and-turing/" start="Wed Aug 04 2010 17:55:05 GMT+0000" title="Karl Popper and Turing&#8217;s thesis">Those of you who have been following me on Twitter may have noticed that, lately, I seem to be tormented by scientifico-philosophical questions related to Turing machines and Turing&amp;#8217;s thesis. Indeed I am, because I&amp;#8217;ve chosen this as the subject of an exam paper, a decision that I&amp;#8217;m beginning to regret: the matter is so [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=aeporreca.org&amp;blog=13664840&amp;post=503&amp;subd=aeporreca&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://aeporreca.org/2010/07/23/universal-turing-machine/" start="Fri Jul 23 2010 15:15:40 GMT+0000" title="The universal Turing machine">What did Alan Turing have in mind when he conceived his universal computing machine? One could speculate that his train of thoughts was like this: I can simulate any of my machines (there&amp;#8217;s experimental evidence, and of course I defined them to work like people doing maths on a piece of paper). I&amp;#8217;ve already formulated [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=aeporreca.org&amp;blog=13664840&amp;post=482&amp;subd=aeporreca&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://aeporreca.org/2010/07/21/hilberts-troubles/" start="Wed Jul 21 2010 08:54:39 GMT+0000" title="Hilbert&#8217;s troubles">While digging in the early computability literature, I found a precious and long forgotten account of the history of Hilbert&amp;#8217;s problems. I&amp;#8217;m proud to present my discovery to the scientific community. *** david (daboss@uni-g&#246;ttingen.de) has joined #logic &amp;#60;david&amp;#62; plz, solve integer eqns and prove completness of teh arithmetics &amp;#60;kurt&amp;#62; just 1 sec, brb * kurt [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=aeporreca.org&amp;blog=13664840&amp;post=455&amp;subd=aeporreca&amp;ref=&amp;feed=1&quot; /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://aeporreca.org/2010/07/12/length-of-proofs-2/" start="Mon Jul 12 2010 15:24:06 GMT+0000" title="On the length of proofs (episode&#160;II)">Last week I wrote a post about arithmetical theorems having very long proofs, and linked to an article on the SEP for the details. Today, me and my colleague Luca Manzoni realised that there is a much simpler proof; it is essentially the same proof described by Scott Aaronson for the uncomputability of the busy [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=aeporreca.org&amp;blog=13664840&amp;post=348&amp;subd=aeporreca&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://aeporreca.org/2010/07/11/primitive-recursion/" start="Sun Jul 11 2010 00:19:11 GMT+0000" title="How I learned to stop worrying about Turing completeness and love primitive recursion">The primitive recursive functions are a class of computable arithmetic functions with a reasonably natural definition. They do not exhaust all computable functions (e.g., Ackermann&amp;#8217;s function grows faster than any PR function). Here is a list of reasons to like them anyway. The PR functions are exactly the functions that can be computed by using [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=aeporreca.org&amp;blog=13664840&amp;post=271&amp;subd=aeporreca&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://aeporreca.org/2010/07/05/solving-pp/" start="Mon Jul 05 2010 20:53:13 GMT+0000" title="Solving PP-complete problems using P systems">Today we submitted the final version of our paper P systems with elementary active membranes: Beyond NP and coNP, which has been accepted for the Eleventh International Conference on Membrane Computing, taking place on 24&#8211;27 August. The venue of CMC11 is Friedrich-Schiller-Universit&#228;t in Jena, Germany. This university has some slightly notable alumni, such as Rudolf [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=aeporreca.org&amp;blog=13664840&amp;post=228&amp;subd=aeporreca&amp;ref=&amp;feed=1&quot; /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://aeporreca.org/2010/07/04/length-of-proofs/" start="Sun Jul 04 2010 10:16:41 GMT+0000" title="On the length of proofs">One of the most amazing facts about (un)computability is the existence of functions f : &#8469; &#8594; &#8469; that grow faster than any recursive function, that is, for all computable g : &#8469; &#8594; &#8469; we have . The most common function of this kind is the busy beaver function, described by Tibor Rad&#243; in [...]&lt;img alt=&quot;&quot; border=&quot;0&quot; src=&quot;http://stats.wordpress.com/b.gif?host=aeporreca.org&amp;blog=13664840&amp;post=168&amp;subd=aeporreca&amp;ref=&amp;feed=1&quot; /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://aeporreca.org/2010/07/03/halting-cartoon/" start="Sat Jul 03 2010 19:02:20 GMT+0000" title="A cartoon about the halting problem">Bob is a programmer and software engineer, developing automated software testing utilities for his company. However, his manager Eve wants to fire him and, in order to get an excuse, she assigns him a very hard task: developing a tool to check the termination of programs. Eve, for some reason, seems to know that he&amp;#8217;s [...]&lt;img alt="" border="0" src="http://stats.wordpress.com/b.gif?host=aeporreca.org&amp;blog=13664840&amp;post=154&amp;subd=aeporreca&amp;ref=&amp;feed=1" /&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/ScienceInTheOpen/~3/3qJE6PcpXE4/" start="Tue Aug 10 2010 15:08:06 GMT+0000" title="Interview with me by Michael Nielsen">Michael Nielsen asked me to answer some questions about practical approaches to Open Science. You can see my answers up on his blog.</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/ScienceInTheOpen/~3/TA1BpuvSeBU/" start="Tue Aug 10 2010 13:19:33 GMT+0000" title="P &#8800; NP and the future of peer review">The online maths community has lit up with excitement as a document, claiming to prove one of the major outstanding theorems in maths has been circulated. In response an online peer review process has swung into action that is very similar to the kind of post-publication peer review that many of us have advocated. Is this a one of, a special case? Or does it point the way towards successfully using the web to find a way of doing peer review effectively and efficiently?</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/ScienceInTheOpen/~3/3QOHpukwVRM/" start="Thu Aug 05 2010 20:15:02 GMT+0000" title="The triumph of document layout and the demise of Google Wave">So Google has abandoned Wave. Not really that surprising but obviously dissappointing to those of us who were excited about its potential. Here I argue that part of the problem is that most of us are still restricted in our thinking to static documents on the web. Wave was always about a next generation kind of document that was active and dynamic and that might have contributed to some of the confusion around what it was good for. The advent of the iPad and other tools for generating beautiful and dynamic content on the web may take us beyond this.</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/ScienceInTheOpen/~3/-PtOpyCfYnk/" start="Sun Jul 25 2010 17:24:16 GMT+0000" title="The Nature of Science Blog Networks">I've been watching the reflection on the Science Blogs diaspora and the wider conversation on what next for the Science Blogosphere with some interest because I remain both hopeful and not very confident that someone somewhere is really going crack the problem of effectively using the social web for advancing science. I don't really have anything to add to Bora's masterful summary of the larger picture but I wanted to pick out something that was interesting to me and that I haven't seen anyone else mention.</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://feedproxy.google.com/~r/ScienceInTheOpen/~3/yiULbNugDoc/" start="Thu Jul 22 2010 11:14:59 GMT+0000" title="Driving UK Research &#8211; Is copyright a help or a hindrance?">The following is my contribution to a collection prepared by the British Library and released today at the Wellcome Trust, called &#8220;Driving UK Research. Is copyright a help or a hindrance?&#8221; which is being released under a CC-BY-NC license. The British Library kindly allowed authors to retain copyright on their contributions so I am here releasing the text into the public domain via a CCZero waiver.</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://aeporreca.org/2010/08/09/proof-that-p-isnt-np/#comment-23" start="Thu Aug 12 2010 13:54:05 GMT+0000" title="By: Deolalikar&#8217;s manuscript &#171; Constraints">[...] 20100810: see Deolalikar&#8217;s updated paper, the clearinghouse wiki, Antonio E. Porreca&#8217;s meta-summary, and Richard Lipton&#8217;s update. Possibly related posts: (automatically generated)No [...]</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://aeporreca.org/2010/08/09/proof-that-p-isnt-np/#comment-22" start="Tue Aug 10 2010 06:14:31 GMT+0000" title="By: P&#8800;NP. at webCONSUL">[...] Antonio E. Porreca [...]</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://aeporreca.org/2010/08/09/proof-that-p-isnt-np/#comment-19" start="Mon Aug 09 2010 16:39:12 GMT+0000" title="By: How to get everyone talking about your research!">[...] Porreca has the best write-up on reactions to this [...]</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://aeporreca.org/2010/08/09/proof-that-p-isnt-np/#comment-18" start="Mon Aug 09 2010 12:59:28 GMT+0000" title="By: Tweets that mention A relatively serious proof that P &#8800; NP? &#171; Antonio E. Porreca -- Topsy.com">[...] This post was mentioned on Twitter by Tony Finch, Antonio E. Porreca. Antonio E. Porreca said: Obligatory P &#8800; NP post, with a summary of the comments by TCS people. (Please, help me keeping it up-to-date.) http://wp.me/pVkQE-8M [...]</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/97587-a-tale-of-a-serious-attempt-at-p&#8800;np" start="Sun Aug 15 2010 15:26:00 GMT+0000" title="A Tale of A Serious Attempt At P&#8800;NP">&lt;p&gt;This is a discussion of the recent claimed proof that P&amp;ne;NP by Vinay Deolalikar.&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/97527-how-to-attend-an-academic-conference" start="Fri Aug 13 2010 19:03:00 GMT+0000" title="How to Attend an Academic Conference">&lt;p&gt;&amp;nbsp;Tips on attending academic conferences for young CS researchers&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/97467-research-in-the-wild-making-research-work-in-industry" start="Thu Aug 12 2010 22:46:00 GMT+0000" title="Research in the Wild: Making Research Work in Industry">&lt;p&gt;What is the best way to organize researchers into a company?&amp;nbsp; An independent research lab?&amp;nbsp; Mixing researchers in with product teams? Or not hiring researchers at all?&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/97340-better-game-playing-using-parallel-algorithms" start="Wed Aug 11 2010 15:17:00 GMT+0000" title="Better Game Playing Using Parallel Algorithms">&lt;p&gt;Innovations in playing the game &amp;quot;Go&amp;quot; using Monte Carlo Tree Search and parallel algorithms.&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/97257-on-software-updates-and-version-changes" start="Mon Aug 09 2010 15:11:00 GMT+0000" title="On Software Updates and Version Changes">&lt;p&gt;Since the beginning of software industry, developers have produced updates for their products on&amp;nbsp;a regular basis. How do these updates affect the software itself and the end user?&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/97215-remembering-internet-dogs" start="Sun Aug 08 2010 21:04:00 GMT+0000" title="Remembering Internet Dogs">&lt;p&gt;An &lt;a href="http://en.wikipedia.org/wiki/On_the_Internet,_nobody_knows_you%27re_a_dog"&gt;iconic cartoon by Peter Steiner&lt;/a&gt;, which appeared in &lt;a href="http://www.newyorker.com/"&gt;&lt;font color="#0000ff"&gt;The New Yorker&lt;/font&gt;&lt;/a&gt; in 1993, captured the nature of the nascent Internet.&amp;nbsp;It shows a dog seated at a computer, remarking to a second dog on the floor that, &amp;ldquo;On the Internet, nobody knows you are a dog.&amp;rdquo;&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/96960-evaluating-the-success-of-mentoring-programs" start="Tue Aug 03 2010 13:22:00 GMT+0000" title="Evaluating the Success of Mentoring Programs">&lt;p&gt;Can HCI evaluation methods help to assess the effectiveness of mentoring programs for women?&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/96949-cs-education-gets-congressional-attention" start="Tue Aug 03 2010 16:43:00 GMT+0000" title="CS Education Gets Congressional Attention">&lt;p&gt;Two Congressman introduced two pieces of legislation -- The Computer Science Education Act and the Computer Science Education Week Resolution -- to offer reforms for computer science education and bring attention to the field and its issues.&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/96699-what-do-scientists-and-engineers-need-to-know-about-computer-science" start="Tue Jul 27 2010 16:18:00 GMT+0000" title="What do Scientists and Engineers Need to Know About Computer Science?">&lt;p&gt;As computational science and engineering becomes more common, it becomes important to ask what should all scientists and engineers know about computer science to be effective.&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/96690-computer-science-left-out-of-national-academies-report" start="Tue Jul 27 2010 13:40:00 GMT+0000" title="Computer Science Left out of National Academies Report">&lt;p&gt;&amp;nbsp;The new draft framework from the National Research Council on &amp;quot;science, engineering, and technology&amp;quot; makes no mention of computer science.&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/96476-why-is-great-design-so-hard" start="Wed Jul 21 2010 13:50:00 GMT+0000" title="Why is Great Design so Hard?">&lt;p&gt;Why is good design so hard to accomplish for organizations? People are coming to the realization that design really matters, but it's not that they don't want to follow suit, it seems that they are not able to.&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/96321-in-vivo-in-vitro-in-silico" start="Sat Jul 17 2010 13:40:00 GMT+0000" title="In Vivo, In Vitro, In Silico">&lt;p&gt;These &amp;ldquo;grand challenges&amp;rdquo; of systems biology encompass almost every aspect of modern computing, from numerical and symbolic methods through data management and analytics to extraordinarily high-performance computing platforms.&lt;/p&gt;
+&lt;p&gt;&amp;nbsp;&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/96103-a-connected-life" start="Mon Jul 12 2010 13:48:00 GMT+0000" title="A Connected Life">&lt;p&gt;In the Internet era, with cell phones, PDAs, and other technologies, &amp;nbsp;humans are more and more connected to each other. All day you can see people walking and at the same time chatting via cell phone. What happens to people that lives connected? What if they simply unplug from the network?&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/96002-web-applications-crave-memory" start="Fri Jul 09 2010 19:42:00 GMT+0000" title="Web Applications Crave Memory">&lt;p&gt;Big web  applications are hungry for RAM.&amp;nbsp; They want lots of low power RAM.&amp;nbsp; And they want little  else.&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/95854-interface-design-for-new-mothers" start="Tue Jul 06 2010 14:53:00 GMT+0000" title="Interface Design for New Mothers">&lt;p&gt;Why is there so little research into interface design for new mothers?&amp;nbsp;Interface designers&amp;mdash;hear my cry! Here are some pointers for designing interfaces for mothers and babies.&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/95645-2010-edition-of-campus-party-colombia" start="Thu Jul 01 2010 13:53:00 GMT+0000" title="2010 Edition of Campus Party Colombia">&lt;p&gt;A&amp;nbsp;quick look at this special event in Colombia at which people breathe, eat, and sweat technology.&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/95099-hope-vs-fear-privacy-challenges-in-online-health-communities" start="Thu Jun 24 2010 17:47:00 GMT+0000" title="Hope vs. Fear: Privacy Challenges in Online Health Communities">&lt;p&gt;&amp;nbsp;The difficulty in creating privacy controls in online health communities represents the struggle between hope and fear.&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/94517-valuing-high-school-computing-teachers" start="Thu Jun 17 2010 13:53:00 GMT+0000" title="Valuing High School Computing Teachers">&lt;p&gt;Thoughts on why high school computing teachers are so important and how we can help them.&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/94422-the-future-of-electronic-design-automation-according-to-ibms-bernie-meyerson" start="Thu Jun 17 2010 03:41:00 GMT+0000" title="The Future of Electronic Design Automation, According to IBM&#8217;s Bernie Meyerson">&lt;p&gt;To paraphrase Mr. Meyerson: A truly talented engineer is someone who, when his hand gets slammed in a doorway and is badly hurt, inserts his other hand in the doorway and slams the door shut on it, so he can get a second set of data.&amp;nbsp;&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/94419-cfp-intelligent-transportation-systems" start="Wed Jun 16 2010 14:11:00 GMT+0000" title="CFP: Intelligent Transportation Systems">&lt;p&gt;One of the goals of this years Computers Freedom and Privacy conference is to ask the question, &amp;quot;Can we be 'smart' &lt;em&gt;and&lt;/em&gt; private?&amp;quot;&amp;nbsp; Wednesday's panel on &lt;a href="http://www.cfp2010.org/wiki/index.php/Can_we_be_smart_and_private:_Intelligent_Transportation_Systems"&gt;Intellegent Transportation Systems&lt;/a&gt; (ITS) asks, as we make driving safer, more efficient and more comfortable, how can we, at the same time protect the privacy of drivers?&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/94316-cfp-activism-and-social-networking-advocating-for-privacy" start="Wed Jun 16 2010 02:03:00 GMT+0000" title="CFP: Activism and Social Networking- Advocating for Privacy">&lt;p&gt;The Computers Freedom and Privacy Conference (CFP) continued Tuesday afternoon with a panel called &lt;a href="http://www.cfp2010.org/wiki/index.php/Activism_and_Social_Networking:_Advocating_for_Privacy"&gt;Activism and Social Networking:&amp;nbsp;Advocating for Privacy&lt;/a&gt;.&amp;nbsp; Panelists each described recent campaigns utilizing both online and in-person strategies.&amp;nbsp;&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/94309-cfp-opening-plenary-privacy-and-free-speech" start="Tue Jun 15 2010 20:12:00 GMT+0000" title="CFP: Opening Plenary Privacy and Free Speech">&lt;p&gt;The opening plenary of this year's Computers Freedom and Privacy Conference (CFP) had a decidedly business orientation.&amp;nbsp; Titled &amp;quot;&lt;a href="http://www.cfp2010.org/wiki/index.php/Privacy_and_Free_Speech:_It%27s_Good_for_Business" target="_blank"&gt;Privacy and Free Speech: It's Good for Business&lt;/a&gt;,&amp;quot; the panel was moderated by &lt;a href="http://www.cfp2010.org/wiki/index.php/Speakers#Nicole_A._Ozer" title="Speakers"&gt;Nicole A. Ozer&lt;/a&gt;, Technology and Civil Liberties Policy Director, at the ACLU of Northern California. &amp;nbsp;&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/94307-are-you-invisible" start="Tue Jun 15 2010 18:50:00 GMT+0000" title="Are You Invisible?">&lt;p&gt;If you sit in your office and just focus on doing your work, you&amp;rsquo;re invisible, Patty Azzarello says. Here&amp;rsquo;s her career advice on how to be recognized at your work and be promoted.&amp;nbsp;&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/93771-robots-from-what-if-to-now-that" start="Thu Jun 10 2010 20:51:00 GMT+0000" title="Robots: From &quot;What If&quot; to &quot;Now That&quot;">&lt;p&gt;Our panel at CFP will explore how robotics implicates civil liberties today, and in the near term.  We will discuss issues from the ethical ramifications of robots in warfare and law enforcement, to the promise and perils of autonomous driving, to new issues in privacy and Fourth Amendment law.&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/93398-laptops-in-the-classroom" start="Mon Jun 07 2010 18:52:00 GMT+0000" title="Laptops in the Classroom">&lt;p&gt;Thoughts about using laptops in higher education classes, relating to a recent journal article about what students actually do with their laptops in class while pretending to study.&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/93333-research-summers" start="Sun Jun 06 2010 13:27:00 GMT+0000" title="Research Summers">&lt;p&gt;Research challenges us to pose new questions, choose exploratory avenues carefully, and be frugal and wise stewards of the financial trust invested in us.&amp;nbsp;&lt;span&gt;&lt;font color="#000000"&gt; &lt;/font&gt;&lt;/span&gt;&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/93023-virtual-education-one-more-step-to-close-the-gap" start="Wed Jun 02 2010 21:02:00 GMT+0000" title="Virtual education: One More Step  to Close the Gap">&lt;p&gt;Is virtual education a feasible option to deliver rich educational contents to country side zones where no traditional schools are available?&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/92929-online-information-brokers-the-challenges-of-privacy-in-a-networked-world" start="Tue Jun 01 2010 19:55:00 GMT+0000" title="Online Information Brokers &amp;  the Challenges of Privacy in a Networked World">&lt;p&gt;At Privacy Rights Clearinghouse, we receive phone calls and emails about online information brokers almost daily. The individuals who contact us are often those who have the most at stake&amp;ndash;they are the victims of domestic violence, law enforcement officers fearing for the safety of their families, individuals whose livelihood depends on being in the public sphere, and those dealing with identity theft.&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/91829-computers-freedom-and-privacy-in-a-networked-society-june-15-18-in-san-jose-and-cyberspace" start="Fri May 21 2010 20:33:00 GMT+0000" title="Computers, Freedom, and Privacy in a Networked Society: June 15-18, in San Jose and Cyberspace">&lt;p&gt;With Facebook privacy on the cover of &lt;em&gt;Time &lt;/em&gt;magazine and Google being investigated in Europe and the US after admitting that they collected information from home wifi networks, it&amp;rsquo;s a perfect lead-in to the 20th annual ACM conference on Computers, Freedom, and Privacy.&lt;/p&gt;</event><event icon="http://upload.wikimedia.org/wikipedia/commons/thumb/4/43/Feed-icon.svg/16px-Feed-icon.svg.png" link="http://cacm.acm.org/blogs/blog-cacm/91717-how-many-computers-will-you-own" start="Thu May 20 2010 23:34:00 GMT+0000" title="How Many Computers Will You Own?">&lt;p&gt;How many computing devices will you have in ten years?  Two?  Six?  Just  one?&lt;/p&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/nIkqDNJxNNM/" start="Sun Aug 15 2010 02:59:52 GMT+0000" title="Re: P &#8800; NP and the future of peer review">Absolutely agree with the need for credit. This is a fundamental of getting anyone to do anything, and is at the root of the many cultural problems in modern research. What I find interesting is that credit can seem trivial, people work hard to get high kudos on Stack Overflow or to get Facebook/Friendfeed "likes" or retweets. As these become important to people they can start to become "real" (Stack Overflow scores have appeared on some CVs for instance). To some extent I see these high profile examples of community review as starting to provide some legitimacy for those kind of contributions precisely because they have a high profile.&lt;br /&gt;&lt;br /&gt;In terms of the "technical quality of review" question. I've certainly heard (but can't lay my hands on any examples at the moment) CNS staff and editors say things like "the most rigorous peer review", "always send to reviewers at least twice for detailed comments", "often request extra experiments". Again this is partly confusion over what peer review does and what its for but the brands of CNS are bound up with the idea that quality and stringency of review is higher here.&lt;br /&gt;&lt;br /&gt;But at some level I'd also invoke the "extraordinary claims require extraordinary evidence" rule. The examples I gave all made extraordinary claims. If peer review is going to do anything useful it must surely subject such claims to heavy duty scrutiny? I'm sure for every bad example exposed in a high profile journal there are many lower profile examples that never get picked up of course.&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=nIkqDNJxNNM:fPE_BGSQhlw:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=nIkqDNJxNNM:fPE_BGSQhlw:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=nIkqDNJxNNM:fPE_BGSQhlw:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/nIkqDNJxNNM" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/7U-yA3k1le0/" start="Sat Aug 14 2010 11:23:34 GMT+0000" title="Re: P &#8800; NP and the future of peer review">I think it's wrong to think of Science/Nature as aiming for a higher technical standard (or, at least, achieving it).  They claim to distinguish themselves by two qualities, broad general interest and speed of publication.  Speed is pretty much antithetical to highest possible technical review. One of the problems with the current situation, as you point out, is that the community has accepted "published in Science/Nature" as equivalent to "higher technical quality than the next paper", which is not (necessarily) the case. &lt;br /&gt;&lt;br /&gt;I do understand that what you're talking about is substituting journal-driven peer review for community-driven peer review.  My point is that peer review is real work, that substitutes for work you could be doing on your own papers, and it's hard enough to get good peer review when you apply a good deal of skill and effort to obtaining it.  &lt;br /&gt;&lt;br /&gt;You express a hope that the kind of community-based commenting and review that happens in mathematics will be eventually seen as valuable.  This would be great.  It would help if such community service were seen as valuable by search committees and promotion committees.  Could we create a website where "the best public peer review" gets aggregated and credited?  Then you could put it on your CV and get some benefit out of it.&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=7U-yA3k1le0:St1X4Ixuu2Q:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=7U-yA3k1le0:St1X4Ixuu2Q:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=7U-yA3k1le0:St1X4Ixuu2Q:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/7U-yA3k1le0" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/J5ThUuFOc6c/" start="Sat Aug 14 2010 08:32:49 GMT+0000" title="Re: P &#8800; NP and the future of peer review">I guess the point for me about the GWAS paper is that the criticisms in the 23andMe post as well as comments by dgmacarthur (I think it was) and others revealed fundamental technical issues that should have been exposed by technical review at any reputable journal, let alone Science where one presumes the standard is somehow higher than average (I can't speak to this myself as I've never reviewed for them nor had a paper get to review). It my be my reading of the post but it sounded as though there were fundamental technical checks that should be done for any GWAS paper that simply weren't.&lt;br /&gt;&lt;br /&gt;In terms of the editor's problem, and I am an editor at PLoS ONE and for a new BMC journal soon to be launched, I couldn't agree more. The question is whether its worth the bother, is it more effective to simply put it out for anyone interested to review, and then allow the paper to be iteratively improved in response to those comments until it reaches some sort of level of agreement that it gets some sort of quality mark. I would argue that those papers that don't ever get a review should be released simply as data or blog posts or whatever. There will be some lost gems in there but that is a discovery problem - probably under the current system they wouldn't be published anyway.&lt;br /&gt;&lt;br /&gt;Don't imagine that I'm saying peer review isn't useful. But the question, is which peers, and when, and how do you motivate them to contribute positively to the process of improving this piece of communication?&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=J5ThUuFOc6c:KoZR2p9rBR0:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=J5ThUuFOc6c:KoZR2p9rBR0:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=J5ThUuFOc6c:KoZR2p9rBR0:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/J5ThUuFOc6c" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/M4oUJVw51rQ/" start="Fri Aug 13 2010 16:26:03 GMT+0000" title="Re: P &#8800; NP and the future of peer review">I completely agree about the problems.  (And yes, Bayh-Dole was enormously counterproductive).  I hope you're right that things are shifting.  I don't think what happened with the GWAS paper is as significant as you imply, though &#8212;&#160;essentially this was a debunking of a high-profile paper, which would have happened anyway, the Web just made it faster.  &lt;br /&gt;&lt;br /&gt;I used to be a journal editor and in that role I spent a lot of time trying to identify reviewers who met the following characteristics: not a direct competitor, but knowledgeable in the field.  Willing to spend a significant amount of time giving a paper a serious evaluation as a service to the scientific community.  Not so busy that it will take months to get the report.  Has good judgment.  These are not easy criteria to meet, and it isn't hugely surprising if the system doesn't work perfectly. On the other hand, I worked at it, and I got quite a lot of useful reviews that visibly improved the papers I dealt with. I know this is anecdotal evidence but I think it may be wrong to dismiss peer review as not useful. It is a hard thing to do well, though.&lt;div class=&quot;feedflare&quot;&gt;
+&lt;a href=&quot;http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=M4oUJVw51rQ:4PHDpwyHnGQ:yIl2AUoC8zA&quot;&gt;&lt;img src=&quot;http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA&quot; border=&quot;0&quot; /&gt;&lt;/a&gt; &lt;a href=&quot;http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=M4oUJVw51rQ:4PHDpwyHnGQ:GFqAKdhVS04&quot;&gt;&lt;img src=&quot;http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=M4oUJVw51rQ:4PHDpwyHnGQ:GFqAKdhVS04&quot; border=&quot;0&quot; /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src=&quot;http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/M4oUJVw51rQ&quot; height=&quot;1&quot; width=&quot;1&quot; /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/2oOJAtxbNL4/" start="Fri Aug 13 2010 10:23:22 GMT+0000" title="Re: P &#8800; NP and the future of peer review">Absolutely! That's partly what I was trying to argue &lt;a href="http://cameronneylon.net/blog/the-triumph-of-document-layout-and-the-demise-of-google-wave/" rel="nofollow"&gt;here&lt;/a&gt; tho I'm not sure I made myself terribly clear. But I think as a community we need to take some responsibility for not demanding more as well. It's part of the avoidance of responsibility that I feel derives from the fact that as researchers we make the decision about where to (try to) send our papers but we don't have to deal with the cost, so we don't squeal and demand good value for money.&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=2oOJAtxbNL4:igAbyp38Oxk:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=2oOJAtxbNL4:igAbyp38Oxk:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=2oOJAtxbNL4:igAbyp38Oxk:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/2oOJAtxbNL4" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/jfsm5ENJj3E/" start="Fri Aug 13 2010 10:21:12 GMT+0000" title="Re: P &#8800; NP and the future of peer review">Becky, you won't get any argument from me on that. We know that this is hard and we certainly know that making a step change from one system to another won't happen. Equally tho, some of these pressures are going to shift pretty quickly. The patenting thing is certainly shifting in some quarters as people realise just how few of those patents are making any money at all, and how many of them are just blocks to innovation and opportunities for patent trolls. &lt;br /&gt;&lt;br /&gt;Even on the issue of grant review I think things are shifting. Bottom line, organisations that judges individual people on the basis of the journal's they publish in (as opposed to the performance of their actual papers) need to take a long hard look at themselves because doing so is counterproductive and indeed highly non-scientific. You can't actually predict anything about the quality of a paper (within reasonable error bounds) from the journal it appears in. &lt;br /&gt;&lt;br /&gt;The broader question, could this work in biomedical sciences? Well I would say we have some examples, the Science GWAS paper I mention being one. Combine that with the current focus on "broader impact" and a re-calibration of what publication means and I think we will see a gradual change. It will take time - and it may only ever be applied to high profile work - but I think things are shifting away from where we were a few years back and that's what I wanted to capture.&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=jfsm5ENJj3E:6xCWzMnvIUI:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=jfsm5ENJj3E:6xCWzMnvIUI:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=jfsm5ENJj3E:6xCWzMnvIUI:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/jfsm5ENJj3E" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/tiLg9hhPmMw/" start="Fri Aug 13 2010 09:15:16 GMT+0000" title="Re: P &#8800; NP and the future of peer review">I agree that watching this happen has been wonderful.  But I have no expectation that this method of reviewing will be practical in other fields, specifically biology.  There aren't very many mathematicians in the world who can work at this level.  None of them have to worry about their contribution to the proof being overlooked.  Mathematics is (at least in principle) provably true or false, not a matter of interpretation.  There is little room for cherished beliefs, and no ability to argue that the evidence must be wrong because it contradicts your view, or that it really proves your own hypothesis.  There is no pressure from the university you work for to patent first before you share your results, and no chance of someone with a larger lab jumping on your early findings and overtaking you &#8212; having a large group doesn't actually help, in math.  And mathematicians don't have to worry about the NIH reviewers concluding that their work must be unimportant because it's not published in the right journals.  I am not arguing that the current state of peer review is good, just that there are reason to think that a transition to net review will not be easy.&lt;div class=&quot;feedflare&quot;&gt;
+&lt;a href=&quot;http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=tiLg9hhPmMw:4xrK-xLXjyA:yIl2AUoC8zA&quot;&gt;&lt;img src=&quot;http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA&quot; border=&quot;0&quot; /&gt;&lt;/a&gt; &lt;a href=&quot;http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=tiLg9hhPmMw:4xrK-xLXjyA:GFqAKdhVS04&quot;&gt;&lt;img src=&quot;http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=tiLg9hhPmMw:4xrK-xLXjyA:GFqAKdhVS04&quot; border=&quot;0&quot; /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src=&quot;http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/tiLg9hhPmMw&quot; height=&quot;1&quot; width=&quot;1&quot; /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/D9O3XN2USqs/" start="Fri Aug 13 2010 04:19:48 GMT+0000" title="Re: P &#8800; NP and the future of peer review">Moving away from paper is indeed very difficult for scientists but I blame&lt;br /&gt;the publishers. They make it harder for us by under investing in their&lt;br /&gt;website infrastructure, not only in the tools available for commenting and&lt;br /&gt;annotating but also with the lack of data available on their sites. Instead&lt;br /&gt;of enriching the manuscripts we send them with the knowledge that is already&lt;br /&gt;available on the web, they embed all the knowledge within a beautifully&lt;br /&gt;formatted pdf that is too enticing not to print.&lt;br /&gt;I'm hoping that tablets will change the way we work!&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=D9O3XN2USqs:Wse93V7qkp0:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=D9O3XN2USqs:Wse93V7qkp0:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=D9O3XN2USqs:Wse93V7qkp0:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/D9O3XN2USqs" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/D70hkd75jo8/" start="Thu Aug 12 2010 11:27:22 GMT+0000" title="Re: P &#8800; NP and the future of peer review">Fritz, thanks for your thoughtful comment. I think you're right on that there are two important characteristics here. One was the Deolalikar passed the initial "smell test" of credibility that meant serious people would take a look, and secondly as you say that Lipton was a key moderator. I've written before (tho not explicitly) about how we need to build social infrastructure that supports people like Lipton to mediate and support the application of expert attention. These people are "market makers" of a sort who provide the credibility that there is sufficient "capital" of attention to make a contribution worthwhile. These are very special people but they are often not held in high regard by people outside of their immediate circle.&lt;br /&gt;&lt;br /&gt;In my mind I can imagine a sort of technical solution that provides a place where these people can mediate interactions at lots of different levels of specialisation and with different levels of reputation. We could probably build such a thing (indeed StackOverflow is arguably precisely such a thing for programmers) but it won't work unless there is significant uptake and validation of the work of the mediators as an important contribution to the research community. So absolutely the key to this is understanding the sociology of this as a route to making it happen effectively more often.&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=D70hkd75jo8:ZQypnat-UeY:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=D70hkd75jo8:ZQypnat-UeY:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=D70hkd75jo8:ZQypnat-UeY:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/D70hkd75jo8" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/upkPf6bfPv4/" start="Thu Aug 12 2010 11:14:39 GMT+0000" title="Re: P &#8800; NP and the future of peer review">I agree there are technical issues and usability issues with current commenting systems but even with good systems you see limited comments. There simply aren't that many of us interacting with papers online. If you look at number of bookmarks in Citeulike/Mendeley or Delicious the numbers are pretty low. Most people land on the html, download the pdf, print and then annotate by hand. Nothing wrong with that, its a good workflow, but until we see more of the interaction happening in online spaces then the technical problems are relatively small.&lt;br /&gt;&lt;br /&gt;Now someone did do a pubmed-Disqus mashup that looked really interesting, but I can't find the link at the moment. Looked great, worked well, but are we using it? Even me?&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=upkPf6bfPv4:nDYSiPTwg9I:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=upkPf6bfPv4:nDYSiPTwg9I:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=upkPf6bfPv4:nDYSiPTwg9I:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/upkPf6bfPv4" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/28wPzt2nTl4/" start="Thu Aug 12 2010 04:41:36 GMT+0000" title="Re: P &#8800; NP and the future of peer review">Thank you for this post.  I am as fascinated as you are by the process of peer review that has unfolded.  It is the sort of phenomenon to restore my faith in human nature.  I have been feeling warm and fuzzy for days because of it.&lt;br /&gt;&lt;br /&gt;You have recognized that something very important is going on.  I hope that others realize it as well.  However, for anyone who wants to see this kind of thing happening more often, indeed to actively encourage it to happen more often, it is important to understand *why* it is happening.  Yes, part of the reason is the extreme importance of the topic.  Yes, part of the reason is that the Internet exists.  But these are far from sufficient reasons.  One only needs to consider what happened when the first million-dollar Millenium Prize problem was solved in 2003.  That was a parallel situation, with a purported but incomplete proof distributed over the Internet only.  In that case, however, the verification of the proof was fractured, prolonged, and bitter.  Mathematicians don't always cooperate this well.  If you don't know that story, you may read about it here: &lt;a href=&quot;http://www.newyorker.com/archive/2006/08/28/060828fa_fact2&quot; rel=&quot;nofollow&quot;&gt;http://www.newyorker.com/archive/2006/08/28/060...&lt;/a&gt;&lt;br /&gt;&lt;br /&gt;Let me add my hunch as to what happened differently in this case.  I have seen collaborative phenomena such as the current one unfold in a few widely different contexts, as well as seen it die in infancy in many, many others.  In my observation, the point of critical importance to whether teamwork happens or fails to happen is the quality of the moderator.  I conjecture that it was no accident that the P=NP discussion crystallized around Dick Lipton's blog as opposed to somewhere else, and furthermore that absent Dick Lipton, it wouldn't have happened at all.  Experts would have made scattered observations in various blogs and e-mails, collectively producing less in three weeks than the working group produced in three days.&lt;br /&gt;&lt;br /&gt;Most such discussions are smothered in the cradle because the people whose contributions are most valuable choose not to contribute.  The discussion isn't killed internally, it dies from the best people staying away in droves.  Their reasons for absence include fear of not being listened to, fear of being drowned out by idiots, expectation that the discussion will have no result, and fear of not being credited for their contributions if a positive result is produced.  Yet anyone may rapidly verify from his blog that Lipton is generous with his own ideas, respectful of others' ideas, fair-minded, competent to judge worthy from unworthy contributions, and eager to give credit where credit is due.&lt;br /&gt;&lt;br /&gt;Like you, I would love to see the whole world work like the peer review of Deolalikar&#8217;s proof.  To this end I believe we must understand it as a social phenomenon held together by the glue of an extraordinary individual.  Lipton is exceptional not by virtue of being the most brilliant mathematical mind applied to the problem, but by having the rare social wisdom necessary to bring other great minds together.&lt;br /&gt;&lt;br /&gt;I hope some sociologist picks up the ball for understanding such cooperative phenomenon better, because a society that comprehends it properly will be great above all societies.&lt;div class=&quot;feedflare&quot;&gt;
+&lt;a href=&quot;http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=28wPzt2nTl4:o6tsKKXiNjU:yIl2AUoC8zA&quot;&gt;&lt;img src=&quot;http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA&quot; border=&quot;0&quot; /&gt;&lt;/a&gt; &lt;a href=&quot;http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=28wPzt2nTl4:o6tsKKXiNjU:GFqAKdhVS04&quot;&gt;&lt;img src=&quot;http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=28wPzt2nTl4:o6tsKKXiNjU:GFqAKdhVS04&quot; border=&quot;0&quot; /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src=&quot;http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/28wPzt2nTl4&quot; height=&quot;1&quot; width=&quot;1&quot; /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/VILwk0MGz3k/" start="Wed Aug 11 2010 09:51:47 GMT+0000" title="Re: P &#8800; NP and the future of peer review">I think that one reason the commenting and "likes" etc post-review process hasn't taken off is that the infrastructure for making those comments is poor. As a researcher reviewing online, you would want a system that gives you credit for the work you are doing, also somewhere you can go to see all the comments and reviews you have made on different articles across different journals and blogs. You would want to be in control of those reviews. I have made a number of comments both on PLoS and Biomed Central but there is no way of me accessing, reviewing or deleting those comments I made and have never received any comments back from authors. If a system like DISQUS was set-up across different journals/publishers, you would have a way of being in control of the work you have done and be able to build a reputation in the research community by making reviews/comments and getting liked or disliked for it.&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=VILwk0MGz3k:g_0EHUsswmY:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=VILwk0MGz3k:g_0EHUsswmY:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=VILwk0MGz3k:g_0EHUsswmY:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/VILwk0MGz3k" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/YVaV9Pu92jQ/" start="Mon Aug 09 2010 18:32:20 GMT+0000" title="Re: The triumph of document layout and the demise of Google Wave">It is fairly straightforward to configure a server on which MathJax is hosted so that it can be used from pages hosted in another domain.  I understand not everyone would be willing or able to do this on a server, but the installation at &lt;a href="http://mathjax.org" rel="nofollow"&gt;mathjax.org&lt;/a&gt; has been configured in this way precisely to support people using Blogger.  See &lt;a href="https://sourceforge.net/projects/mathjax/forums/forum/948701/topic/3747270" rel="nofollow"&gt;https://sourceforge.net/projects/mathjax/forums...&lt;/a&gt; for details.&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=YVaV9Pu92jQ:8j8yPsa8EBM:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=YVaV9Pu92jQ:8j8yPsa8EBM:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=YVaV9Pu92jQ:8j8yPsa8EBM:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/YVaV9Pu92jQ" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/rZ6sXkf5dyY/" start="Sat Aug 07 2010 13:47:36 GMT+0000" title="Re: The triumph of document layout and the demise of Google Wave">I love jsMath and MathJax, but there's a peculiar problem I've run into with using them. Unless I'm terribly mistaken, both require a few files to be placed on the same domain as a page using the toolkits. This makes it hard to use them with, say, Blogger or WordPress (running on a shared server, anyway). Since blogs form an important channel for discussion, I would have hoped that this problem would be tractable, but so far, I've not run into a good solution. That said, I wholeheartedly agree that such toolkits are a good way forward, as they allow for LaTeX to be used as an intermediate, author-facing language, and yet retain the benefits of standards-compliant web technologies.&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=rZ6sXkf5dyY:--syJNq9VSQ:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=rZ6sXkf5dyY:--syJNq9VSQ:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=rZ6sXkf5dyY:--syJNq9VSQ:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/rZ6sXkf5dyY" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/QJqMbQQmxR4/" start="Fri Aug 06 2010 11:38:56 GMT+0000" title="Re: The triumph of document layout and the demise of Google Wave">I've just read Brian Kelly's post on formats for repositories, in which he talks about the EPUB format (see &lt;a href=&quot;http://ukwebfocus.wordpress.com/2010/08/04/epub-format-for-papers-in-repositories/&quot; rel=&quot;nofollow&quot;&gt;http://ukwebfocus.wordpress.com/2010/08/04/epub...&lt;/a&gt; for the post). He says:&lt;br /&gt;&lt;br /&gt;'EPub is described in Wikipedia as &#8220;a free and open e-book standard by the International Digital Publishing Forum (IDPF)&#8220;. The article goes on to add that &#8220;EPUB is designed for reflowable content, meaning that the text display can be optimized for the particular display device used by the reader of the EPUB-formatted book. The format is meant to function as a single format that publishers and conversion houses can use in-house, as well as for distribution and sale.&#8220;&lt;br /&gt;&lt;br /&gt;'In terms of the open standards used EPub consists of three specifications:&lt;br /&gt;&lt;br /&gt;'Open Publication Structure (OPS) 2.0, contains the formatting of its content.&lt;br /&gt;Open Packaging Format (OPF) 2.0, describes the structure of the .epub file in XML.&lt;br /&gt;OEBPS Container Format (OCF) 1.0, collects all files as a ZIP archive.&lt;br /&gt;&lt;br /&gt;'The articles states that &#8220;EPUB internally uses XHTML or DTBook (an XML standard provided by the DAISY Consortium) to represent the text and structure of the content document and a subset of CSS to provide layout and formatting. XML is used to create the document manifest, table of contents, and EPUB metadata. Finally, the files are bundled in a zip file as a packaging format.&#8220;'&lt;br /&gt;&lt;br /&gt;It sounds much more &quot;of the web&quot; than PDF, and might be very interesting!&lt;div class=&quot;feedflare&quot;&gt;
+&lt;a href=&quot;http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=QJqMbQQmxR4:bIabYFNgmL4:yIl2AUoC8zA&quot;&gt;&lt;img src=&quot;http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA&quot; border=&quot;0&quot; /&gt;&lt;/a&gt; &lt;a href=&quot;http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=QJqMbQQmxR4:bIabYFNgmL4:GFqAKdhVS04&quot;&gt;&lt;img src=&quot;http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=QJqMbQQmxR4:bIabYFNgmL4:GFqAKdhVS04&quot; border=&quot;0&quot; /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src=&quot;http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/QJqMbQQmxR4&quot; height=&quot;1&quot; width=&quot;1&quot; /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/2-mtFufeGy0/" start="Fri Aug 06 2010 11:07:55 GMT+0000" title="Re: The triumph of document layout and the demise of Google Wave">Cameron, I'm not quite sure I "get" the theme of this post, but I'll confine myself for a minute to Utopia Document. What bothers me here is the very strongly different nature of the tool. Compare this paper, for example: Ruthensteiner, B., &amp; Hess, M. (2008). Embedding 3D models of biological specimens in PDF publications. Microscopy research and technique, 71(11), 778-86. doi: 10.1002/jemt.20618. This doesn't require a different viewer, but let's you view the models in PDF. It seems to work quite well for me.&lt;br /&gt;&lt;br /&gt;I'm keen in principle on the idea of scientific articles migrating from PDF as default to HTML as default. In theory, with a little added RDF etc, we could do wonderful things, embedding supplementary data. However, in practice the result is pretty much rubbish even for quite ordinary articles. It seems to be very hard to get figures and tables to look right and be readable in context. And worse, there doesn't seem to be a standardised way of saving a packaged web page so that it can be left for years and then read even when the originating site has gone. Safari's .webarchive format is able to do that, but is not readable by anything else as far as I can see. the standard "save html" format seems to be a .html file and an associated directory of other stuff; hardly handy, even if in practice several browsers seem to be able to read it!&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=2-mtFufeGy0:hS1niWcqDvc:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=2-mtFufeGy0:hS1niWcqDvc:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=2-mtFufeGy0:hS1niWcqDvc:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/2-mtFufeGy0" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/D-OS2oOe6A8/" start="Fri Aug 06 2010 10:06:08 GMT+0000" title="Re: The triumph of document layout and the demise of Google Wave">As someone who has spent a bit of time trying to write LaTeX maths plugins for various webapps, I feel a lot of frustration about this.  The tools to do this extremely well have been around for a fairly long time, but most of them are directed towards converting complete LaTeX documents into standalone webpages and it is not easy to adapt them to plug into other systems, especially if you are trying to maintain compatibility with basic webhosting accounts.  As a result, almost all LaTeX plugins known to me are hacks and they usually have fairly serious bugs because people only test them on a few simple equations rather than writing out long mathematical expositions.&lt;br /&gt;&lt;br /&gt;The few systems that do handle mathematics very well are usually built with the idea of mathematical formulas being first class objects from the outset, rather than trying to build maths support into an existing system built by people who have no idea about the needs of mathematical scientists.  One of the best I know of is &lt;a href="http://cnx.org/" rel="nofollow"&gt;http://cnx.org/&lt;/a&gt; which will let authors submit LaTeX and convert it into very good HTML+MathML automatically.  Basically, I think we should build more systems like this, i.e. wiki and blog engines with ground level maths support, rather than trying to hack existing systems.&lt;br /&gt;&lt;br /&gt;On the rendering front, I am hopeful that adoption of MathJax &lt;a href="http://www.mathjax.org/" rel="nofollow"&gt;http://www.mathjax.org/&lt;/a&gt; will greatly improve the situation.  This is not an issue that we can sit back and wait for the browser developers to deal with, since they have clearly demonstrated that MathML support is a marginal concern even for those browsers that proudly tout their "standards compliant" status.  The situation is rather more like that of javascript, where people are resigned to browser inconsistencies, but handle it via libraries like jquery rather than waiting for the browser developers to do something about it.&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=D-OS2oOe6A8:get1TPLML1Q:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=D-OS2oOe6A8:get1TPLML1Q:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=D-OS2oOe6A8:get1TPLML1Q:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/D-OS2oOe6A8" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/oULYQZEyPI8/" start="Fri Aug 06 2010 06:45:35 GMT+0000" title="Re: The triumph of document layout and the demise of Google Wave">Indeed, maths is a serious problem on the web, and something that has definitely made people very keen on TeX. Again though my perception is that alternatives have not got traction because they are not seen as being as "pretty" as TeX. Getting TeX/MathML support into Wave was something that @axiomsofchoice and others put quite a bit of effort into and there were some nice examples but they were proof of concept not ready for the mainstream.&lt;br /&gt;&lt;br /&gt;My feeling is that what we will see is systems that enable TeX based authoring of maths which then get converted to MathML (which is a semantics and layout tool really and not much good for authoring) and displayed nicely but flexibly. The first bit exists but I'm not sure that the rendering is there yet. Ultimately it will probably be possible to just directly write the maths down and have it recognised and converted to MathML&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=oULYQZEyPI8:c22jOxMZgA8:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=oULYQZEyPI8:c22jOxMZgA8:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=oULYQZEyPI8:c22jOxMZgA8:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/oULYQZEyPI8" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/hnKeoGzUnyk/" start="Thu Aug 05 2010 19:58:43 GMT+0000" title="Re: The triumph of document layout and the demise of Google Wave">I see a few things holding back web-based alternatives to the paper model, actually. One of the most frustrating to me has to be the complete lack of widespread math support as extensible and as mature as LaTeX. I feel like having LaTeX-grade math support in a web-based platform would do a lot for adoption, as it would broaden the range of fields that could benefit rather drastically.&lt;br /&gt;&lt;br /&gt;Sure, there's some LaTeX support mixed here and there in blog engines, Wikis and similar, but from what I've seen, they tend to suffer a lack of extensibility and rely on bitmap images that don't rescale properly with the text, much less fit in typographically. MathML was supposed to deliver us from this, but between half-assed browser support and an overly verbose syntax that is unsuitable for "by-hand" use, it seems to have fizzled by comparison to LaTeX-based solutions.&lt;br /&gt;&lt;br /&gt;A distraction from Wave, to be sure, but I think that the document model will persist in the sciences at least as long as LaTeX has a monopoly on the typesetting of mathematics, frustrating implementation of Wave-like solutions.&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=hnKeoGzUnyk:necRpccpZe8:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=hnKeoGzUnyk:necRpccpZe8:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=hnKeoGzUnyk:necRpccpZe8:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/hnKeoGzUnyk" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/WZZGBGh9UkQ/" start="Mon Jul 26 2010 05:58:55 GMT+0000" title="Re: The Nature of Science Blog Networks">Absolutely! Coffee and in person are good as well!&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=WZZGBGh9UkQ:6VkYicqYOQA:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=WZZGBGh9UkQ:6VkYicqYOQA:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=WZZGBGh9UkQ:6VkYicqYOQA:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/WZZGBGh9UkQ" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/kvdZ_1Tz02M/" start="Mon Jul 26 2010 05:49:06 GMT+0000" title="Re: The Nature of Science Blog Networks">Great post. Insightful as usual. Instead of leaving a comment, can I comment in person at SciFoo over coffee? I know you like everything to be online and accessible, but I like coffee and meeting in person. Ha!&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=kvdZ_1Tz02M:YXjcM1ehDlM:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=kvdZ_1Tz02M:YXjcM1ehDlM:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=kvdZ_1Tz02M:YXjcM1ehDlM:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/kvdZ_1Tz02M" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/QGi-eonI1nw/" start="Fri Jul 09 2010 10:55:48 GMT+0000" title="Re: Capturing and connecting research objects: A pitch for @sciencehackday">The problem isn't mainly how to share data. The basic problem is that there's few to no incentives for sharing your data in the first place, and many strong incentives against.&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=QGi-eonI1nw:mrvA4xIo0AU:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=QGi-eonI1nw:mrvA4xIo0AU:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=QGi-eonI1nw:mrvA4xIo0AU:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/QGi-eonI1nw" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/ImrQ_4cz9A0/" start="Thu Jun 24 2010 14:27:56 GMT+0000" title="Re: Metrics of use: How to align researcher incentives with outcomes">Danny, you won't get any argument from me on that. I think evangelization has a role because it helps to raise the profile of the issues but on its own it is not enough. Granting agencies have to start actually caring about optimising output in the general sense - and my feeling is that that will require as you say looking at a much wider range of outputs.&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=ImrQ_4cz9A0:ei6Y8wO6hmk:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=ImrQ_4cz9A0:ei6Y8wO6hmk:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=ImrQ_4cz9A0:ei6Y8wO6hmk:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/ImrQ_4cz9A0" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/7KtpCqNeYDc/" start="Thu Jun 24 2010 12:59:17 GMT+0000" title="Re: Metrics of use: How to align researcher incentives with outcomes">Granting agencies have to reward more data generators. It seems all the glory goes to the people who use the data to publish (who in many cases don't credit the data generators properly).&lt;br /&gt;&lt;br /&gt;In my field, proteomics, sharing data is not only more work for the researcher but it also gives advantage to a competitor who might get it out before you and will get the next grant in the field instead of you.&lt;br /&gt;&lt;br /&gt;I don't think evangelization will work. Granting agencies should start looking into something esle that is not publications per se.&lt;br /&gt;&lt;br /&gt;For example, nobody cares about making programs maintainable or fast because you get the same kind of publication with a crappy program than with a good one, as long as both have the same function.&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=7KtpCqNeYDc:ymAkd3KHOmU:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=7KtpCqNeYDc:ymAkd3KHOmU:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=7KtpCqNeYDc:ymAkd3KHOmU:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/7KtpCqNeYDc" height="1" width="1" /&gt;</event><event icon="http://thewvsr.com/wp-content/themes/customtheme/images/comment-icon.gif" link="http://feedproxy.google.com/~r/CommentsForScienceInTheOpen/~3/-Z3t2wTFP1E/" start="Mon Jun 21 2010 05:52:11 GMT+0000" title="Re: Metrics of use: How to align researcher incentives with outcomes">Thought you might be interested in this short article from evolutionary biology, along very similar lines to what you've written.&lt;br /&gt;&lt;br /&gt;&lt;a href="http://www.nature.com/nature/journal/v441/n7093/full/441583a.html" rel="nofollow"&gt;http://www.nature.com/nature/journal/v441/n7093...&lt;/a&gt;&lt;div class="feedflare"&gt;
+&lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=-Z3t2wTFP1E:W5yGHaK7deY:yIl2AUoC8zA"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?d=yIl2AUoC8zA" border="0" /&gt;&lt;/a&gt; &lt;a href="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?a=-Z3t2wTFP1E:W5yGHaK7deY:GFqAKdhVS04"&gt;&lt;img src="http://feeds.feedburner.com/~ff/CommentsForScienceInTheOpen?i=-Z3t2wTFP1E:W5yGHaK7deY:GFqAKdhVS04" border="0" /&gt;&lt;/a&gt;
+&lt;/div&gt;&lt;img src="http://feeds.feedburner.com/~r/CommentsForScienceInTheOpen/~4/-Z3t2wTFP1E" height="1" width="1" /&gt;</event><event icon="http://twitter.com/favicon.ico" link="http://twitter.com/EvoMRI/statuses/21432131018" start="Tue Aug 17 2010 20:51:40 GMT+0000" title="Liked &quot;I&apos;m hacking a visualization of the #PvsNP event. Initial mock up here:...&quot; http://ff.im/pnvTi" /><event icon="http://twitter.com/favicon.ico" link="http://twitter.com/axiomsofchoice/statuses/21428493901" start="Tue Aug 17 2010 19:49:28 GMT+0000" title="I&apos;m hacking a visualization of the #PvsNP event. Initial mock up here:... http://ff.im/pnvTi" /><event icon="http://twitter.com/favicon.ico" link="http://twitter.com/p6/statuses/21325513289" start="Mon Aug 16 2010 15:58:13 GMT+0000" title="LOLWAT: http://is.gd/eki8P #PvsNP #WTF." /><event icon="http://twitter.com/favicon.ico" link="http://twitter.com/exhibitiontiz/statuses/21241954360" start="Sun Aug 15 2010 16:13:10 GMT+0000" title="#PvsNP running into trouble? - New Scientist http://t.co/78XxQ85" /><event icon="http://twitter.com/favicon.ico" link="http://twitter.com/rcvnglst/statuses/21231066765" start="Sun Aug 15 2010 13:35:59 GMT+0000" title="Proof for P!=NP? http://bit.ly/aI8yMf #compsci #math #in #pvsnp" /><event icon="http://twitter.com/favicon.ico" link="http://twitter.com/M_Turkistany/statuses/21189271271" start="Sun Aug 15 2010 00:29:01 GMT+0000" title="We must give up the tradition of equating efficient computation with polynomial running time #pnp #pvsnp" /><event icon="http://twitter.com/favicon.ico" link="http://twitter.com/M_Turkistany/statuses/21189117873" start="Sun Aug 15 2010 00:26:17 GMT+0000" title="To resolve P vs. NP Problem, we need to redefine concepts (discover alternatives) like tractable problem and efficient algorithm #pnp #pvsnp" /><event icon="http://twitter.com/favicon.ico" link="http://twitter.com/Vallenain/statuses/21129826237" start="Sat Aug 14 2010 06:19:36 GMT+0000" title="RT @4ugeek: La solution &#224; 1 million de $ de la plus grande &#233;nigme informatique contrevers&#233;e http://0z.fr/k4Tqp #science #math #PvsNP" /><event icon="http://twitter.com/favicon.ico" link="http://twitter.com/bigdougschmidt/statuses/21126179450" start="Sat Aug 14 2010 05:10:48 GMT+0000" title="If you could recognize the right solution to a problem if it was given to you does it mean the solution could be found automatically? #PvsNP" /><event icon="http://twitter.com/favicon.ico" link="http://twitter.com/4ugeek/statuses/21099539564" start="Fri Aug 13 2010 22:11:32 GMT+0000" title="La solution &#224; 1 million de $ de la plus grande &#233;nigme informatique contrevers&#233;e http://0z.fr/k4Tqp #science #math #PvsNP" /><event icon="http://twitter.com/favicon.ico" link="http://twitter.com/GISDJL/statuses/21085158454" start="Fri Aug 13 2010 18:17:13 GMT+0000" title="A speculative effort at writing about something I don&apos;t really understand: http://bit.ly/9UAyAP #P!=NP #PvsNP #geography" /><event icon="http://twitter.com/favicon.ico" link="http://twitter.com/ImaginaryTime/statuses/21048517347" start="Fri Aug 13 2010 08:43:15 GMT+0000" title="P=NP explained using #Minesweeper as an analogy. http://bit.ly/aTnsj3 #mathematics #PvsNP" /><event icon="http://twitter.com/favicon.ico" link="http://twitter.com/nishkarshs/statuses/21046297186" start="Fri Aug 13 2010 07:53:52 GMT+0000" title="RT: @cowmaaa: P=NP Solved? Indian scientist thinks so. http://alturl.com/zvzzg #pvsnp #vinaydeolalikar" /><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/35810/space-time-transform-in-solving-n-vs-np" start="Mon Aug 16 2010 23:45:31 GMT+0000" title="Space/Time Transform in solving N vs NP">&lt;p&gt;Since the solution space to that question implies available time, and the presumed physical capacity to arrange answer spaces ...&lt;/p&gt;
+
+&lt;p&gt;Why isn't the solution to this question the subject of a transform, much like a LaPlacian or other transform, which de-dimensionalizes the question.&lt;/p&gt;
+
+&lt;p&gt;I.E., removes time and/or space for its solution and solution path.?&lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/35664/why-relativization-cant-solve-np-p" start="Sun Aug 15 2010 18:11:29 GMT+0000" title="Why relativization can&apos;t solve NP !=P?">&lt;p&gt;If this problem is really stupid, please close it. But I really wanna get some answer for it. And I learnt computational complexity by reading books only.&lt;/p&gt;
+
+&lt;p&gt;When I learnt to the topic of relativization and oracle machines, I read the following theorem:&lt;/p&gt;
+
+&lt;blockquote&gt;
+  &lt;p&gt;There exist oracles A, B such that &lt;code&gt;$P^A = NP^A$&lt;/code&gt; and &lt;code&gt;$P^B \neq NP^B$&lt;/code&gt;.&lt;/p&gt;
+&lt;/blockquote&gt;
+
+&lt;p&gt;And then the book said because of this, we can't solve the problem of NP = P by using relativization. But I think what it implies is that &lt;code&gt;$NP \neq P$&lt;/code&gt;. The reasoning is like this:&lt;/p&gt;
+
+&lt;p&gt;First of all, it is quite easy to see that:&lt;/p&gt;
+
+&lt;p&gt;&lt;code&gt;$$A = B \Leftrightarrow \forall \text{oracle O, }A^O = B^O$$&lt;/code&gt;&lt;/p&gt;
+
+&lt;p&gt;Though I think it is obvious, I still give a proof to it:&lt;/p&gt;
+
+&lt;p&gt;&lt;a href="http://www.voofie.com/content/141/a-simple-proof-of-np-p/#Prove" rel="nofollow"&gt;A simple proof of NP != P ?&lt;/a&gt;&lt;/p&gt;
+
+&lt;p&gt;And the negation of it is:&lt;/p&gt;
+
+&lt;p&gt;&lt;code&gt;$$A \neq B \Leftrightarrow \exists \text{oracle O such that } A^O \neq B^O$$&lt;/code&gt;&lt;/p&gt;
+
+&lt;p&gt;Therefore since there is an oracle B such that:&lt;/p&gt;
+
+&lt;p&gt;&lt;code&gt;$$ NP^B \neq P^B$$&lt;/code&gt;&lt;/p&gt;
+
+&lt;p&gt;we can conclude that &lt;code&gt;$ NP \neq P $&lt;/code&gt;&lt;/p&gt;
+
+&lt;p&gt;What's the problem with the above reasoning?&lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/35645/is-there-anyway-this-statement-can-be-summed-up-in-a-mathematical-formula" start="Sun Aug 15 2010 11:02:07 GMT+0000" title="Is there anyway this statement can be summed up in a mathematical formula">&lt;p&gt;Imagine you have created (nanobot) fabricator. The nanobot's core function is to collect the data of every subatomic particles position, movement and all other necessary information about them. Becoming omniscient, they have the fundamental knowledge to predicting every single event from that point onward. If the deterministic view is correct.&lt;/p&gt;
+
+&lt;p&gt;I meet up with your omniscient (nanobot) fabricator, I ask them a simple question . (Though they may already know the question.) The question is what hand will I raise ? The conditions are this, you must tell me before I make the decision, also what ever hand you choice I will choice the opposite. &lt;/p&gt;
+
+&lt;p&gt;Sorry for the dull question, this will be my first and last question here.&lt;/p&gt;
+
+&lt;p&gt;The above post is suppose to be taken hypothetically as it's impossible in the real world theoretically or not. Most of you maybe aware this is similar to Kurt Godel's incompleteness theorem and maybe I'm wrong in thinking this does apply to the above post exactly I've just always wanted to have the above statement in a mathematical form if that's even possible. &lt;/p&gt;
+
+&lt;p&gt;Thanks you very much.&lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/35151/what-impact-would-pnp-have-on-the-characterization-of-bqp" start="Wed Aug 11 2010 19:31:38 GMT+0000" title="What impact would P!=NP  have on the characterization of BQP?">&lt;p&gt;Many complexity theorists assume that $P\ne NP.$ If this is proved, how would it impact quantum computing and quantum algorithms? Would the proof immediately disallow quantum algorithms from ever solving NP-Complete problems in Quantum Polynomial time?&lt;/p&gt;
+
+&lt;p&gt;&lt;a href="http://en.wikipedia.org/wiki/QMA" rel="nofollow"&gt;According to Wikipedia&lt;/a&gt;, quantum complexity classes BQP and QMA are the bounded-error quantum analogues of P and NP. Is it likely that a proof that $P\ne NP$ can be adapted to the quantum setting to show that $BQP \ne QMA?$&lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/35159/quantum-computation-implications-of-p-vs-np" start="Tue Aug 10 2010 18:49:20 GMT+0000" title="Quantum computation implications of (P vs NP)">&lt;blockquote&gt;
+  &lt;p&gt;&lt;strong&gt;Possible Duplicate:&lt;/strong&gt;&lt;br /&gt;
+  &lt;a href="http://mathoverflow.net/questions/35151/what-impact-would-pnp-have-on-the-characterization-of-bqp" rel="nofollow"&gt;What impact would P!=NP  have on the characterization of BQP?&lt;/a&gt;  &lt;/p&gt;
+&lt;/blockquote&gt;
+
+&lt;p&gt;Before I begin, I had a similar post closed for mentioning the recently released (to be verified) proof that P!=NP. This question is about the implications of P!=NP, not about the proof internals or specifics.&lt;/p&gt;
+
+&lt;p&gt;Does P!=NP imply that NP-Complete problems cannot be solved in Quantum Polynomial time?&lt;/p&gt;
+
+&lt;p&gt;According to Wikipedia, quantum complexity classes BQP and QMA which are the bounded-error quantum analogues of P and NP. If P!=NP was a know fact, does that imply that BQP!=QMA?&lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/2218/characterize-pnp" start="Mon Aug 09 2010 09:19:18 GMT+0000" title="Characterize P^NP">&lt;p&gt;What can you say about the complexity class P^NP, i.e. decision problems solvable by a polytime TM with an oracle for SAT?&lt;/p&gt;
+
+&lt;p&gt;Obviously P^NP is in PH somewhere between NP union coNP, and Sigma2 intersect Pi2. What else is known about that complexity class? &lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/34978/p-vs-np-solved" start="Mon Aug 09 2010 08:12:53 GMT+0000" title="P vs NP solved?">&lt;p&gt;This would be the story of the year: P vs NP solved by an Principial Researcher with HP:
+&lt;a href="http://gregbaker.ca/blog/2010/08/07/p-n-np/" rel="nofollow"&gt;http://gregbaker.ca/blog/2010/08/07/p-n-np/&lt;/a&gt;&lt;br /&gt;
+&lt;a href="http://rjlipton.wordpress.com/2010/08/08/a-proof-that-p-is-not-equal-to-np/" rel="nofollow"&gt;http://rjlipton.wordpress.com/2010/08/08/a-proof-that-p-is-not-equal-to-np/&lt;/a&gt;&lt;br /&gt;
+&lt;a href="http://math-frolic.blogspot.com/2010/08/p-vs-np.html" rel="nofollow"&gt;http://math-frolic.blogspot.com/2010/08/p-vs-np.html&lt;/a&gt;&lt;/p&gt;
+
+&lt;p&gt;Original paper is here:
+&lt;a href="http://www.scribd.com/doc/35539144/pnp12pt" rel="nofollow"&gt;http://www.scribd.com/doc/35539144/pnp12pt&lt;/a&gt;&lt;/p&gt;
+
+&lt;p&gt;Would anybody have the chance of giving the paper a cursory look if this is a hoax - or if this could be serious?&lt;/p&gt;
+
+&lt;p&gt;Thank you!&lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/27867/non-existence-of-algorithm-converting-np-algorithm-to-p-algorithm" start="Wed Jun 16 2010 17:40:23 GMT+0000" title="Non-existence of algorithm converting NP algorithm to P algorithm?">&lt;p&gt;[Edit: in the light of Nate Eldredge's answer below I rephrase the question]&lt;/p&gt;
+
+&lt;p&gt;P=NP is equivalent to the existence of a map of the following form:&lt;/p&gt;
+
+&lt;ul&gt;
+&lt;li&gt;&lt;p&gt;Input: a polynomial-time non-deterministic Turing machine which accepts some language (call the language L) [Edit: we are not to assume these NDTMs come with any certificate proving they run in polynomial time -- Ryan requested this clarification, below]&lt;/p&gt;&lt;/li&gt;
+&lt;li&gt;&lt;p&gt;Output: a polynomial-time deterministic Turing machine which accepts the language L&lt;/p&gt;&lt;/li&gt;
+&lt;/ul&gt;
+
+&lt;p&gt;Is it known that if such a map exists then it cannot be computable?&lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/28056/given-a-polynomial-time-algorithm-can-we-compute-an-explicit-polynomial-time-bou" start="Mon Jun 14 2010 03:18:19 GMT+0000" title="Given a polynomial-time algorithm, can we compute an explicit polynomial time bound just from the program?">&lt;p&gt;&lt;b&gt;Question.&lt;/b&gt; Given a Turing-machine program $e$, which
+is guaranteed to run in polynomial time, can we computably
+find such a polynomial?&lt;/p&gt;
+
+&lt;p&gt;In other words, is there a
+computable function $e\mapsto p_e$, such that whenever $e$
+is a Turing-machine program that runs in polynomial time,
+then $p_e$ is such a polynomial time bound? That is, $p_e$
+is a polynomial over the integers in one variable and
+program $e$ on every input $n$ runs in time at most
+$p_e(|n|)$, where $|n|$ is the length of the input $n$.&lt;/p&gt;
+
+&lt;p&gt;(Note that I impose no requirement on $p_e$ when $e$ is not
+a polynomial-time program, and I am not asking whether the
+function $e\mapsto p_e$ is polynomial-time computable, but
+rather, just whether it is computable at all.)&lt;/p&gt;
+
+&lt;p&gt;In the field of complexity theory, it is common to treat
+polynomial-time algorithms as coming equipped with an
+explicit polynomial clock, that counts steps during the
+computation and forces a halt when expired. This convention
+allows for certain conveniences in the theory. In the field
+of computability theory, however, one does not usually
+assume that a polynomial-time algorithm comes equipped with
+such a counter. My question is whether we can computably
+produce such a counter just from the Turing machine
+program.&lt;/p&gt;
+
+&lt;p&gt;&lt;em&gt;I expect a negative answer.&lt;/em&gt; I think there is no such
+computable function $e\mapsto p_e$, and the question is
+really about how we are to prove this. But I don't know...&lt;/p&gt;
+
+&lt;p&gt;Of course, given a program $e$, we can get finitely many
+sample points for a lower bound on the polynomial, but this
+doesn't seem helpful. Furthermore, it seems that the lesson
+of Rice's Theorem is
+that we cannot expect to compute nontrivial information by
+actually looking at the program itself, and I take this as
+evidence against an affirmative answer. At the same time,
+Rice's theorem does not directly apply, since the
+polynomial $p_e$ is not dependent on the set or function
+that $e$ computes, but rather on the way that it computes
+it. So I'm not sure.&lt;/p&gt;
+
+&lt;p&gt;Finally, let me mention that this question is related to and
+inspired by &lt;a href="http://mathoverflow.net/questions/27867/non-existence-of-algorithm-converting-np-algorithm-to-p-algorithm" rel="nofollow"&gt;this recent interesting MO question about the
+impossibility of converting NP algorithms to P
+algorithms&lt;/a&gt;.
+Several of the proposed answers there hinged critically on
+whether the polynomial-time counter was part of the input
+or not. In particular, an affirmative answer to the present
+question leads to a solution of that question by those
+answers. My expectation, however, is for a negative answer
+here and an answer there ruling out a computable
+transformation.&lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/27723/poly-time-algorithm-to-choose-elements-of-sets" start="Thu Jun 10 2010 20:29:07 GMT+0000" title="poly-time algorithm to choose elements of sets">&lt;p&gt;Let $A_1,A_2,\ldots,A_k$ be finite sets. Furthermore, for each $i\in{1,2,\ldots,k}$, let $B_i$ be a set whose elements are subsets of $A_i$. &lt;/p&gt;
+
+&lt;p&gt;Is there any polynomial-time algorithm that decides whether there exists a choice of precisely one element $C_i$ of each $B_i$ such that for all $x\in (C_1\cup C_2\cup\ldots\cup C_k)$ the following property is satisfied: &lt;/p&gt;
+
+&lt;p&gt;If $x\in A_i$ then $x\in C_i$ for each $i\in{1,2,\ldots,k}$?&lt;/p&gt;
+
+&lt;p&gt;Any pointer to a paper etc. would be greatly appreciated. Thanks.&lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/4953/super-linear-time-complexity-lower-bounds-for-any-natural-problem-in-np" start="Thu Jun 10 2010 15:44:25 GMT+0000" title="Super-linear time complexity lower bounds for any natural problem in NP?">&lt;p&gt;Do we know any problem in NP which has a super-linear time complexity lower bound? Ideally, we would like to show that 3SAT has super-polynomial lower bounds, but I guess we're far away from that. I'd just like to know any examples of super-linear lower bounds.&lt;/p&gt;
+
+&lt;p&gt;I know that the time hierarchy theorem gives us problems which can be solved in O(n^3) but not in O(n^2), etc. Thus I put the word "natural" in the question.&lt;/p&gt;
+
+&lt;p&gt;I ask for problems in NP, because otherwise someone would give examples of EXP-complete problems. &lt;/p&gt;
+
+&lt;p&gt;I know there are time-space tradeoffs for some problems in NP. I don't know if any of them imply a super-linear time complexity lower bound though.&lt;/p&gt;
+
+&lt;p&gt;(To address a question below about machine models, consider either multitape Turing machines or the RAM model.)&lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/19956/is-this-a-well-known-np-complete-problem" start="Thu Jun 10 2010 08:16:49 GMT+0000" title="Is this a well known NP-complete problem?">&lt;p&gt;I came across this problem recently and I wanted to know whether it was a well known NP-complete problem. I checked the library but could not find anything that matched exactly.&lt;/p&gt;
+
+&lt;p&gt;Given a directed weighted graph G. Find the minimal weight path of length 'n' in the graph.
+Which is without setting any specific start node or end node.&lt;/p&gt;
+
+&lt;p&gt;This is not my field, so you will pardon me if the solution is trivial.&lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/6418/best-case-running-time-to-solve-an-np-complete-problem" start="Sat May 15 2010 18:39:19 GMT+0000" title="Best-case Running-time to solve an NP-Complete problem">&lt;p&gt;What is the fastest algorithm that exists to solve a particular NP-Complete problem? For example, a naive implementation of &lt;a href="http://en.wikipedia.org/wiki/Travelling%5Fsalesman%5Fproblem" rel="nofollow"&gt;travelling salesman&lt;/a&gt; is $O(n!)$, but with dynamic programming it can be done in $O(n^2 2^n)$. Is there any "easier" NP-Complete problem that has a better running time?&lt;/p&gt;
+
+&lt;p&gt;Note that I'm curious about exact solutions, not approximations.&lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/23504/minimal-backtracking-proof-tree" start="Fri May 07 2010 23:32:26 GMT+0000" title="Minimal Backtracking Proof Tree">&lt;p&gt;When trying to prove that a particular instance of a problem like graph coloring or SAT is unsatisfiable, generally one explores the search tree using an algorithm like DPLL and the proof of unsatisfiability consists of the exhaustive exploration of the search tree - i.e. try an partial assignment of values to variables, do propagation and if the domain of a variable becomes empty, you know the assignment does not satisfy the instance so you backtrack. By propagation I mean, one looks at every constraint and then for every variable in the constraint, one prunes the values in the domain of the variable which are not part of any model satisfying that particular constraint.&lt;/p&gt;
+
+&lt;p&gt;My question is that how hard in terms of computational complexity is to find a search tree proving infeasibility of minimal size, i.e. if the size of the search tree is $S$, then how hard is it to find the minimal such search tree given that the size of the input is taken as $S$ (Note - that can be much larger than the size of the original problem, exponential in the worst case). &lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/10726/bpp-being-equal-to-p-under-oracle" start="Tue Jan 05 2010 03:14:15 GMT+0000" title="BPP being equal to #P under Oracle">&lt;p&gt;Luca Trevisan &lt;a href="http://lucatrevisan.wordpress.com/2008/02/16/approximate-counting/" rel="nofollow"&gt;here&lt;/a&gt; gives a randomized polynomial-time approximation algorithm for #3-coloring given an NP oracle.&lt;/p&gt;
+
+&lt;p&gt;In a similar vein, I was wondering if there were any results on $BPP^{NP}\stackrel{?}{=}$ #P - i.e. outputting a correct count for a #P problem under the presence of an NP oracle with high probability. 
+The ideal result of course would tell whether they were equal or not but since we don't know whether P=#P or P=BPP, we can't prove the above false. So I'm also interested in any results that provide evidence either way or prove the above is true (which I'm guessing it is unlikely to be).&lt;/p&gt;
+
+&lt;p&gt;If there are no such results, then is $BPP^{NP}$ generally believed to be equal to #P? &lt;/p&gt;
+
+&lt;p&gt;**Edit: ** As per Mariano's suggestion, &lt;a href="http://qwiki.stanford.edu/wiki/Complexity%5FZoo%3AB#bpp" rel="nofollow"&gt;Here&lt;/a&gt;'s the Complexity Zoo's excellent description of the complexity class BPP. And &lt;a href="http://qwiki.stanford.edu/wiki/Complexity%5FZoo%3ASymbols#sharpp" rel="nofollow"&gt;here&lt;/a&gt; is the description of the complexity class #P.&lt;/p&gt;
+
+&lt;p&gt;Thanks&lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/9512/satisfiability-of-general-boolean-formulas-with-at-most-two-occurrences-per-varia" start="Tue Dec 22 2009 04:17:07 GMT+0000" title="Satisfiability of general Boolean formulas with at most two occurrences per variable">&lt;p&gt;&lt;strong&gt;(If you know basics in theoretical computer science, you may skip immediately to the dark box below. I thought I would try to explain my question very carefully, to maximize the number of people that understand it.)&lt;/strong&gt;&lt;/p&gt;
+
+&lt;p&gt;We say that a &lt;em&gt;Boolean formula&lt;/em&gt; is a propositional formula over some 0-1 variables $x_1,\ldots,x_n$ involving AND and OR connectives, where the atoms are &lt;em&gt;literals&lt;/em&gt; which are instances of either $x_i$ or $\neg x_i$ for some $i$. That is, a Boolean formula is a "monotone" formula over the $2n$ atoms $x_1,\ldots, x_n, \neg x_1,\ldots, \neg x_n$. For example, $$((\neg x_1 ~OR~ x_2) ~AND~ (\neg x_2 ~OR~ x_1)) ~OR~ x_3$$ is a Boolean formula expressing that either $x_1$ and $x_2$ take on the same value, or $x_3$ must be $1$. We say that a 0-1 assignment to the variables of a formula is &lt;em&gt;satisfying&lt;/em&gt; if the formula evaluates to $1$ on the assignment.&lt;/p&gt;
+
+&lt;p&gt;The Cook-Levin theorem says that the general &lt;em&gt;satisfiability of Boolean formulas&lt;/em&gt; problem is $NP$-complete: given an arbitrary formula, it is $NP$-hard to find a satisfying assignment for it. In fact, even satisfying Boolean formulas where each variable $x_i$ appears at most three times in the formula is $NP$-complete. (Here is a reduction from the general case to this case: Suppose a variable $x$ appears $k &gt; 3$ times. Replace each of its occurrences with fresh new variables $x^1, x^2, \ldots, x^k$, and constrain these $k$ variables to all take on the same value, by ANDing the formula with $$(\neg x^1 ~OR~ x^2) ~AND~ (\neg x^2 ~OR~ x^3) ~AND~ \cdots ~AND~ (\neg x^{k-1} ~OR~ x^k) ~AND~ (\neg x^k ~OR~ x^1).$$ The total number of occurrences of each variable $x^j$ is now three.) On the other hand, if each variable appears only once in the formula, then the satisfiability algorithm is very easy: since we have a monotone formula in $x_1,\ldots, x_n, \neg x_1,\ldots, \neg x_n$, we set $x_i$ to $0$ if $\neg x_i$ appears, otherwise we set $x_i$ to $1$. If this assignment does not get the formula to output $1$, then no assignment will.&lt;/p&gt;
+
+&lt;blockquote&gt;
+  &lt;p&gt;My question is, suppose every variable appears at most &lt;em&gt;twice&lt;/em&gt; in a general Boolean formula. Is the satisfiability problem for this class of formulas $NP$-complete, or solvable in polynomial time? &lt;/p&gt;
+&lt;/blockquote&gt;
+
+&lt;p&gt;&lt;strong&gt;EDIT: To clarify further, here is an example instance of the problem:
+$$((x_1 ~AND~ x_3) ~OR~ (x_2 ~AND~ x_4 ~AND~ x_5)) ~AND~ (\neg x_1 ~OR~ \neg x_4) ~AND~ (\neg x_2 ~OR~ (\neg x_3 ~AND~ \neg x_5))$$&lt;/strong&gt;&lt;/p&gt;
+
+&lt;p&gt;Note that when we restrict the class of formulas further to conjunctive normal form (i.e. a depth-2 circuit, an AND of ORs of literals) then this "at most twice" problem is known to be solvable in polynomial time. In fact, applying the "resolution rule" repeatedly will work. But it is not clear (at least, not to me) how to extend resolution for the class of general formulas to get a polytime algorithm. Note when we reduce a formula to conjunctive normal form in the usual way, this reduction introduces variables with &lt;em&gt;three&lt;/em&gt; occurrences. So it seems plausible that perhaps one might be able to encode an $NP$-complete problem in the additional structure provided by a formula, even one with only two occurrences per variable.&lt;/p&gt;
+
+&lt;p&gt;My guess is that the problem is polynomial time solvable. I'm very surprised that I could not find a reference to this problem in the literature. Perhaps I'm just not looking in the right places.&lt;/p&gt;
+
+&lt;p&gt;&lt;strong&gt;UPDATE: Please think about the problem before looking below. The answer is surprisingly simple.&lt;/strong&gt;&lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/9221/what-techniques-exist-to-show-that-a-problem-is-not-np-complete" start="Fri Dec 18 2009 09:30:55 GMT+0000" title="What techniques exist to show that a problem is not NP-complete?">&lt;p&gt;The standard way to show that a problem &lt;strong&gt;is&lt;/strong&gt; NP-complete is to show that another problem known to be NP-complete reduces to it.  That much is clear.  Given a problem in NP, what's known about how to show that it is &lt;strong&gt;not&lt;/strong&gt; NP-complete?  &lt;/p&gt;
+
+&lt;p&gt;(My real question is likely to be inappropriate for this site for one or more reasons; I'm curious about what a proof that factoring isn't NP-complete would look like.)&lt;/p&gt;</event><event icon="http://www.mediawiki.org/favicon.ico" link="http://mathoverflow.net/questions/2045/how-can-one-characterize-npsat" start="Fri Oct 23 2009 23:17:48 GMT+0000" title="How can one characterize NP^SAT?">&lt;p&gt;Can you help me understand the class of problems solvable by a nondetermimistic Turing machine with an oracle for SAT running in polynomial time?&lt;/p&gt;</event><event icon="http://stackexchange.com/img/favicon.png" link="http://mathoverflow.net/questions/35567/zero-knowledge-proof-of-positivity" start="Tue Aug 17 2010 02:13:34 GMT+0000" title="Zero-knowledge proof of positivity">&lt;p&gt;If I have committed to a number x by revealing g^x mod p, can I prove that 0 &amp;lt; x mod (p-1) &amp;lt; (p-1)/2, i.e. that x is positive, without leaking any more information about x?&lt;/p&gt;</event><event icon="http://stackexchange.com/img/favicon.png" link="http://mathoverflow.net/questions/35810/space-time-transform-in-solving-n-vs-np" start="Mon Aug 16 2010 23:45:31 GMT+0000" title="Space/Time Transform in solving N vs NP">&lt;p&gt;Since the solution space to that question implies available time, and the presumed physical capacity to arrange answer spaces ...&lt;/p&gt;
+
+&lt;p&gt;Why isn't the solution to this question the subject of a transform, much like a LaPlacian or other transform, which de-dimensionalizes the question.&lt;/p&gt;
+
+&lt;p&gt;I.E., removes time and/or space for its solution and solution path.?&lt;/p&gt;</event><event icon="http://stackexchange.com/img/favicon.png" link="http://mathoverflow.net/questions/35590/is-k-xorsat-p-complete" start="Mon Aug 16 2010 07:23:12 GMT+0000" title="Is k-XORSAT #P-complete?">&lt;p&gt;k-XORSAT is the problem of deciding whether a Boolean formula $$\bigwedge_{i \in I} \oplus_{j=1}^k l_{s_{ij}}$$ is satisfiable.  Here $\oplus$ denotes the binary &lt;a href=&quot;http://en.wikipedia.org/wiki/Xor&quot; rel=&quot;nofollow&quot;&gt;XOR&lt;/a&gt; operation, $I$ is some index set, and each clause has $k$ distinct literals $l_{s_{ij}}$ each of which is either a variable $x_{s_{ij}}$ or its negation.&lt;/p&gt;
+
+&lt;p&gt;Equivalently, $k$-XORSAT requires deciding whether a set of linear equations, each of the form $\sum_{j=1}^k x_{s_{ij}}\equiv 1\; (\mod 2)$, has a solution over $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$.&lt;/p&gt;
+
+&lt;p&gt;Every decision problem has an associated counting problem, which (informally speaking) requires establishing the number of distinct solutions.  Such counting problems form the complexity class &lt;a href=&quot;http://qwiki.stanford.edu/wiki/Complexity_Zoo%3ASymbols#sharpp&quot; rel=&quot;nofollow&quot;&gt;#P&lt;/a&gt;.  The &quot;hardest&quot; problems in #P are #P-complete, as any problem in #P can be reduced to a #P-complete problem.&lt;/p&gt;
+
+&lt;p&gt;The counting problem associated with any NP-complete decision problem is #P-complete.  However, many &quot;easy&quot; decision problems (some even solvable in linear time) also lead to #P-complete counting problems.  For instance, Leslie Valiant's original 1979 paper &lt;a href=&quot;http://dx.doi.org/10.1016/0304-3975%2879%2990044-6&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;The Complexity of Computing the Permanent&lt;/em&gt;&lt;/a&gt; shows that computing the permanent of a 0-1 matrix is #P-complete.  As a second example, the list of #P-complete problems in the companion paper &lt;a href=&quot;http://dx.doi.org/10.1137/0208032&quot; rel=&quot;nofollow&quot;&gt;&lt;em&gt;The Complexity of Enumeration and Reliability Problems&lt;/em&gt;&lt;/a&gt; includes #MONOTONE 2-SAT; this problem requires counting the number of solutions to Boolean formulas in conjunctive normal form, where each clause has two variables and no negated variables are allowed.  (MONOTONE 2-SAT is of course rather trivial as a decision problem.)&lt;/p&gt;
+
+&lt;p&gt;Andrea Montanari has written about the partition function of $k$-XORSAT in some &lt;a href=&quot;http://www.stanford.edu/~montanar/TEACHING/Stat316/handouts/lecture-4.pdf&quot; rel=&quot;nofollow&quot;&gt;lecture notes&lt;/a&gt;, and his book with Marc M&#233;zard apparently discusses this (unfortunately I do not have a copy available to hand, and the relevant Chapter 17 is not included in Montanari's online draft).&lt;/p&gt;
+
+&lt;p&gt;These considerations lead to the following question:&lt;/p&gt;
+
+&lt;blockquote&gt;
+  &lt;p&gt;Is $k$-XORSAT #P-complete?&lt;/p&gt;
+&lt;/blockquote&gt;
+
+&lt;p&gt;Note that the formula in Montanari's notes does not obviously appear to answer this question.  Just because there is a nice closed form solution, doesn't mean we can evaluate it efficiently: consider the &lt;a href=&quot;http://en.wikipedia.org/wiki/Tutte_polynomial&quot; rel=&quot;nofollow&quot;&gt;Tutte polynomial&lt;/a&gt;.&lt;/p&gt;