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# Earley parser

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The Earley parser is a type of chart parser mainly used for parsing in computational linguistics, named after its inventor, Jay Earley. The algorithm uses dynamic programming.

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Earley parsers are appealing because they can parse all context-free languages. The Earley parser executes in cubic time (O(n3), where n is the length of the parsed string) in the general case, quadratic time (O(n2)) for unambiguous grammars, and linear time for almost all LR(k) grammars. It performs particularly well when the rules are written left-recursively.

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## The algorithm

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In the following descriptions, α, β, and γ represent any string of terminals/nonterminals (including the empty string), X and Y represent single nonterminals, and a represents a terminal symbol.

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Earley's algorithm is a top-down dynamic programming algorithm. In the following, we use Earley's dot notation: given a production X → αβ, the notation X → α • β represents a condition in which α has already been parsed and β is expected.

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For every input position (which represents a position between tokens), the parser generates an ordered state set. Each state is a tuple (X → α • β, i), consisting of

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• the production currently being matched (X → α β)
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• our current position in that production (represented by the dot)
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• the position i in the input at which the matching of this production began: the origin position
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(Earley's original algorithm included a look-ahead in the state; later research showed this to have little practical effect on the parsing efficiency, and it has subsequently been dropped from most implementations.)

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The state set at input position k is called S(k). The parser is seeded with S(0) consisting of only the top-level rule. The parser then iteratively operates in three stages: prediction, scanning, and completion.

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• Prediction: For every state in S(k) of the form (X → α • Y β, j) (where j is the origin position as above), add (Y → • γ, k) to S(k) for every production in the grammar with Y on the left-hand side (Y → γ).
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• Scanning: If a is the next symbol in the input stream, for every state in S(k) of the form (X → α • a β, j), add (X → α a • β, j) to S(k+1).
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• Completion: For every state in S(k) of the form (X → γ •, j), find states in S(j) of the form (Y → α • X β, i) and add (Y → α X • β, i) to S(k).
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These steps are repeated until no more states can be added to the set. The set is generally implemented as a queue of states to process (though a given state must appear in one place only), and performing the corresponding operation depending on what kind of state it is.

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## Example

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Consider the following simple grammar for arithmetic expressions:

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```+ P → S      # the start rule
+ S → S + M
+    | M
+ M → M * T
+    | T
+ T → number
+```
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With the input:

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```+ 2 + 3 * 4
+```
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This is the sequence of state sets:

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```+ (state no.) Production          (Origin) # Comment
+ ---------------------------------
+ == S(0): • 2 + 3 * 4 ==
+ (1)  P → • S         (0)    # start rule
+ (2)  S → • S + M     (0)    # predict from (1)
+ (3)  S → • M         (0)    # predict from (1)
+ (4)  M → • M * T     (0)    # predict from (3)
+ (5)  M → • T         (0)    # predict from (3)
+ (6)  T → • number    (0)    # predict from (5)
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+ == S(1): 2 • + 3 * 4 ==
+ (1)  T → number •    (0)    # scan from S(0)(6)
+ (2)  M → T •         (0)    # complete from S(0)(5)
+ (3)  M → M • * T     (0)    # complete from S(0)(4)
+ (4)  S → M •         (0)    # complete from S(0)(3)
+ (5)  S → S • + M     (0)    # complete from S(0)(2)
+ (6)  P → S •         (0)    # complete from S(0)(1)
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+ == S(2): 2 + • 3 * 4 ==
+ (1)  S → S + • M     (0)    # scan from S(1)(5)
+ (2)  M → • M * T     (2)    # predict from (1)
+ (3)  M → • T         (2)    # predict from (1)
+ (4)  T → • number    (2)    # predict from (3)
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+ == S(3): 2 + 3 • * 4 ==
+ (1)  T → number •    (2)    # scan from S(2)(4)
+ (2)  M → T •         (2)    # complete from S(2)(3)
+ (3)  M → M • * T     (2)    # complete from S(2)(2)
+ (4)  S → S + M •     (0)    # complete from S(2)(1)
+ (5)  S → S • + M     (0)    # complete from S(0)(2)
+ (6)  P → S •         (0)    # complete from S(0)(1)
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+ == S(4): 2 + 3 * • 4 ==
+ (1)  M → M * • T     (2)    # scan from S(3)(3)
+ (2)  T → • number    (4)    # predict from (1)
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+ == S(5): 2 + 3 * 4 • ==
+ (1)  T → number •    (4)    # scan from S(4)(2)
+ (2)  M → M * T •     (2)    # complete from S(4)(1)
+ (3)  M → M • * T     (2)    # complete from S(2)(2)
+ (4)  S → S + M •     (0)    # complete from S(2)(1)
+ (5)  S → S • + M     (0)    # complete from S(0)(2)
+ (6)  P → S •         (0)    # complete from S(0)(1)
+```
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The state (P → S •, 0) represents a completed parse. This state also appears in S(3) and S(1), which are complete sentences.

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## 

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##### Actions
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##### Languages
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