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Readings in Natural Language Processing An Efficient Context-Free Parsing Algorithm

Readings in Natural Language Processing An Efficient Context-Free Parsing Algorithm. J. Earley (1969). 이필용 (2003. 10. 15). Introduction. Context-free grammar: Used extensively for describing the syntax of programming language and natural languages

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Readings in Natural Language Processing An Efficient Context-Free Parsing Algorithm

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  1. Readings in Natural Language ProcessingAn Efficient Context-Free Parsing Algorithm J. Earley (1969) 이필용 (2003. 10. 15)

  2. Introduction • Context-free grammar: Used extensively for describing the syntax of programming language and natural languages • The algorithm described here seems to be the most efficient of the general algorithms, and also it can handle a larger class of grammars in linear time than most of the restricted algorithms

  3. Terminology • Example grammar of simple arithmetic expression, grammar AE (Root : E, Terminal symbols : {a, +, *}, Nonterminals : {E, T, P}) E -> T E -> E+T T -> P T -> T * P P -> a

  4. Terminology • Derivation : E => E + T => T + T => T + P => T * P + P E => E + T => E + P => T + P => T * P + P • Represented : E / \ E + T | | T P / \ T * P

  5. Informal Explanation • Ex) P -> a. 0 T -> P. 0 T -> T.*P 0 E -> T. 0 E -> E.+T 0 • Each state in the set represents • A production (currently scanning a portion of the input string) which is derived from its right side • A point in that production which shows how much of the production’s right side we have recognized so far • A pointer back to the position in the input string at which we began to look for that instance of the production • A k-symbol string which is a syntactically allowed successor to that instance of the production, with a dot in it, followed by an integer and a string

  6. Informal Explanation • input string = a+a*a, k = 1S0φ -> .Eㅓ ㅓ 0 S3 P -> a. ㅓ+* 2(X1=a) E -> .E+T ㅓ+ 0 (X4=*) T -> P. ㅓ+* 2 E -> .T ㅓ+ 0 E -> E+T. ㅓ+ 0 T -> .T+P ㅓ+* 0 T -> T.*P ㅓ+* 2 T -> .P ㅓ+* 0 P -> .a ㅓ+* 0 S4 T -> T*.P ㅓ+* 2 (X5=a) P -> .a ㅓ+* 4S1 P -> a. ㅓ+* 0(X2=+) T -> P. ㅓ+* 0 S5 P -> a. ㅓ+* 4 E -> T. ㅓ+ 0 (X6=ㅓ) T -> T*P. ㅓ+* 2 T -> T.*P ㅓ+* 0 E -> E+T. ㅓ+ 0φ -> E.ㅓ ㅓ 0 T -> T.*P ㅓ+* 2 E -> E.+T ㅓ+ 0 φ -> E.ㅓ ㅓ 0 E -> E.+T ㅓ+ 0S2 E -> E+.T ㅓ+ 0(X3=a) T -> .T*P ㅓ+* 2 S6φ -> Eㅓ. ㅓ 0 T -> .P ㅓ+* 2 T -> .a ㅓ+* 2

  7. The Recognizer • Implementation • For each nonterminal, we keep a linked list of its alternatives, for use in prediction • The states in a state set are kept in a linked list so they can be processed in order • As each state set Si is constructed, we put entries into a vector of size i. The fth entry in this vector(0<=f<=i) is a pointer to a list of all states in Si with pointer f, i.e. states of the form (p, j, f, a) ∈ Si for some p, j, a. The vector and lists can be discarded after Si is constructed • For the use of the completer, we also keep, for each state set Si and nonterminal N, a list of all states (p, j, f, a) ∈ Si such that Cp(j+1)=N

  8. The Recognizer • Implementation • If the grammar contains null productions (A -> λ), we cannot implement the completer in a straightforward way. When performing the completer on a null state (A -> · a i) we want to add to Si each state in Si with A to the right of the dot

  9. Time and Space Bounds • The general case – n^3 recognizer in general- number of state in any state set Si is proportional to i (~i)- Total time for processing the states in Si plus the scanner and predictor operations is ~i- Completer takes ~i^2 steps in Si- Summing from i = 0, …, n + 1 gives ~n^3 steps • Unambiguous grammars

  10. Time and Space Bounds • Linear time • Space

  11. Empirical Results • Tested against top-down and bottom-up parsers • Superior to the backtracking algorithmsG1 G2 G3 G4 root: root: root: root: S->Ab S->aB S->ab|aSb S->AB A->a|Ab B->aB|b A->a|Ab B->bc|bB|Bd Gram- Sen- mar tence TD STD BU SBU Ours G1 ab^n (n^2+7n+2)/2 (n^2+7n+2)/2 9n+5 9n+5 4n+7 G2 a^nb 3n+2 2n+2 11*2^n+7 4n+4 4n+4 G3 a^nb^n 5n – 1 5n-1 11*2^(n-1)-5 6n 6n+4 G4 ab^ncd ~2^(n+6) ~2^(n+2) ~2^(n+6) (n^3+21n^2+46n+15)/3 18n+8

  12. The practical use of the algorithm • The FormMake recognizer into a parserTime bounds for the parser are the same as those of the recognizer, while the space bound goes up to n^3 in general in order to store the parser tree • The Use- Probably be most useful in natural language processing systems where the power of context-free grammars is used- Do in time n any grammar that a time n parser can do at all

  13. Conclusion • Matches or surpass the best previous results for times n^3(Younger), n^2(Kasami), and n(Knuth) with one single algorithm • Knuth’s algorithm works only on LR(k) grammars and Kasami’s only on unambiguous ones, but ours works on them all and maybe others

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