Everything You Always Wanted to Know About Parsing

Transcription

1 Everything You Always Wanted to Know About Parsing Part IV : Parsing University of Padua, Italy ESSLLI, August 2013

2 Introduction First published in 1968 by Jay in his PhD dissertation, Carnegie Mellon University Inspired by LR parsing algorithm of Donald Knuth Very popular in computational linguistics, where it was rediscovered in the early seventies by Martin Kay, and called top-down chart parsing Works with general form CFGs Strictly related to top-down parsing

3 Introduction We derive the algorithm through the definition of a PDA called automaton, which specifies the parsing strategy underlying the original algorithm the application of our tabulation algorithm

4 Dotted Items Let P be a set of CFG productions; the set of dotted items for P is I P = {[A α 1 α 2 ] (A α 1 α 2 ) P} Informally, we use dotted item [A α 1 α 2 ] to indicate that in the left-to-right parsing of the input string, production A α 1 α 2 was predicted at some point prefix α 1 in the right-hand side of the production has been successfully matched against the input string

5 We assume a CFG G = (N, Σ, P, S) with a single production S σ rewriting the start symbol The push-down automaton is constructed from G as follows M E = def (Q E, Σ E, q in, q fin, E ) The stack alphabet Q E is the set of dotted items I P The input alphabet Σ E is the terminal alphabet of G The initial stack symbol q in is [S σ] The final stack symbol q fin is [S σ ]

6 The transition set E contains the following transitions : predict [A α Bβ] ε [A α Bβ] [B γ] for each A αbβ and B γ scan [A α bβ] b [A αb β] for each A αbβ complete [A α Bβ] [B γ ] ε [A αb β] for each A αbβ and B γ

7 The implemented strategy is top-down : the commitment to a production occurs as early as possible, before any of its right-hand side symbols are processed Observe the analogy with the run of an imperative program : predict transition is similar to function call complete transition is similar to control return from a function

8 Computational Complexity A production A α has exactly Aα dotted items; therefore Q E = I P = G A complete transition [A α Bβ] [B γ ] ε [A αb β] can be realized in O( G P ) different ways When the length of G s productions is bounded by some constant (does not depend on G) we have P = O( G ); therefore M E = O( E ) = O( G 2 )

9 M E Computation Example : Σ = {a, +, } N = {S, E} P contains the productions : S E E E E, E E + E, E a

10 M E Computation Example (cont d) : input string w = a + a a S E E + E S E E E a E E E + E a a a a a

11 M E Computation Example (cont d) : Σ E = {a, +, } Q E = I P = {[S E], [S E ], [E E E],...} E contains the transitions : [S E] ε [S E] [E E E] [E E E] [E E E] [S E] [E E E ] ε [S E ].

12 M E Computation Example (cont d) : input string w = a + a a [E a] [S E] [E E + E] ε [S E] [E E + E] ε [S E] 0 0 0

13 M E Computation Example (cont d) : input string w = a + a a [E a ] [E E + E] a [S E] [E E +E] ε [S E] [E E + E] + [S E] 1 1 2

14 M E Computation Example (cont d) : input string w = a + a a [E E E] [E E + E] ε [S E] [E a] [E E E] [E E + E] ε [S E] [E a ] [E E E] [E E + E] a [S E] 2 2 3

15 M E Computation Example (cont d) : input string w = a + a a [E E E] [E E + E] ε [S E] [E E E] [E E + E] [S E] [E a] [E E E] [E E + E] ε [S E] 3 4 4

16 M E Computation Example (cont d) : input string w = a + a a [E a ] [E E E] [E E + E] a [S E] [E E E ] [E E + E] ε [S E] [E E + E ] ε [S E] 5 5 5

17 M E Computation Example (cont d) : input string w = a + a a ε [S E ] 5

18 Tabulation All the transitions of M E are of the types we have seen before, as we will discuss later We can therefore apply our tabulation algorithm to M E ; the result is the algorithm

19 Remarks is strictly nondeterministic Some of its computations may never terminate in case G is left-recursive, because of the predict transition But this needs not concern us : the only purpose of the is to specify a parsing strategy nondeterminism as well as nontermination can be resolved using the tabulation algorithm

20 Items Items in algorithm have the form ([A α Bα ], j, [B β β ], i) meaning that we have detected (at least) one computation segment such that w is processed from j to i [B β β ] is pushed on top of [A α Bα ] in the stack [A α Bα ] cannot be rewritten at intermediate steps

21 Items Proposition : Assume item ([A α Bβ], j, [B γ δ], i) is constructed, and dotted item [A α Bβ ] is at the top of the stack at position j for some computation Then the algorithm also constructs item ([A α Bβ ], j, [B γ δ], i)

22 Items Sketch : The computation segment below depends on production B γδ but not on A, α and β [B γδ] [B γ δ] [A α Bβ] [A α Bβ] [A α Bβ] σ σ σ j j i

23 Items This means that we can drop the first component of an item, and use simplified items of the form (j, [B γ δ], i) The above property is specific to the stack symbols used by M E and does not hold in general

24 Deduction Rules We start with an overview of the tabular algorithm using deduction rules We will later derive the pseudocode for the complete algorithm

25 Deduction Rules M E starts with [S σ] in the stack. This corresponds to the init step of the tabular algorithm (0, [S σ], 0) [S σ] 0

26 Deduction Rules A scan transition [A α bβ] b [A αb β] can be seen as a special case of a switch transition, and is implemented as (j, [A α bβ], i 1) (j, [A αb β], i) { b = ai [A αb β] [A α bβ] j i 1 b i

27 Deduction Rules A predict transition [A α Bβ] ε [A α Bβ] [B γ] can be seen as a push transition, and is implemented as (j, [A α Bβ], i) (i, [B γ], i) [A α Bβ] [B γ] j i

28 Deduction Rules A complete transition [A α Bβ] [B γ ] ε [A αb β] can be seen as a reduce transition, and is implemented as (k, [A α Bβ], j) (j, [B γ ], i) (k, [A αb β], i) [A αb β] [A α Bβ] [B γ ] k j i

29 Pseudocode Algorithm 4 (from Tabulation of ) 1: T {(0, [S σ], 0)} 2: D {(0, [S σ], 0)} 3: for i 0,..., n do edges with right end at i 4: for each (j, [A α a i α ], i 1) T do scan step 5: T T {(j, [A α a i α ], i)} 6: D D {(j, [A α a i α ], i)} 7: end for (cont d)

30 Pseudocode 8: while D do 9: pop (j, [A α X α ], i) from D 10: if X = B then 11: for each (B β) P do predict step 12: if (i, [B β], i) T then 13: T T {(i, [B β], i)} 14: D D {(i, [B β], i)} 15: end if 16: end for 17: end if (cont d)

31 Pseudocode 18: if X α = ε then 19: for each (k, [B β Aβ ], j) T do complete 1 20: if (k, [B βa β, i) T then 21: T T {(k, [B βa β, i)} 22: D D {(k, [B βa β, i)} 23: end if 24: end for 25: end if (cont d)

32 Pseudocode 26: if X = B then 27: for each (i, [B β ], i) T do complete 2 28: if (j, [A αb α, i) T then 29: T T {(j, [A αb α, i)} 30: D D {(j, [A αb α, i)} 31: end if 32: end for 33: end if 34: end while 35: end for 36: accept iff (0, [S σ ], n) T

33 Example (cont d) : Σ = {a, +, } N = {S, E} P contains the productions : S E E E E, E E + E, E a w = a + a a

34 Example (cont d) : 0. [S E] 1. [E E E] 2. [E E + E] 3. [E a] 4. [E a ] 5. [S E ] 6. [E E E] 7. [E E +E] 8. [E E + E] 15. [E E + E ] 16. [S E ] 17. [E E +E] 18. [E E E] 9. [E E E] 10. [E E + E] 11. [E a] 12. [E a ] 13. [E E E] 14. [E E +E] a + a a

35 Example (cont d) : 20. [E E E] 27. [E E E ] 28. [E E E] 29. [E E +E] 30. [E E + E ] 31. [E E E ] 32. [S E ] 33. [E E +E] 34. [E E E] a + a a [E E E] 21. [E a ] 22. [E E E] 23. [E E +E] 24. [E a ] 25. [E E E] 26. [E E +E]

36 Example : Σ = {d} N = {S, A, B, C, D} P contains the productions : S A B A ε, B C D C, C ε, D d L(G) = {d} w = d

37 Example (cont d) : 0. [S AB] 1. [A ] 2. [S A B] 3. [B CDC] 4. [C ] 5. [B C DC] 6. [D d] 7. [D d ] 8. [B CD C] 10. [B CDC ] 11. [S AB ] 9. [C ] 0 1

38 Correctness Invariant : (j, [A α Bβ], i) is added to T if and only if S a 1 a j Aγ, for some γ; and α a j+1 a i S A γ a 1 a j α Bβ a j+1 a i Soundness and completeness directly follows from the above invariant

39 Computational Complexity We immediately derive computational results for s algorithm from the general computational complexity of the tabulation algorithm : Q has been dropped M E = O( G 2 ) time complexity is O( G 2 w 3 ) space complexity is O( G w 2 )

40 Graham-Harrison-Ruzzo Algorithm First published in 1978 by Walter Ruzzo in his PhD dissertation, University of California, Berkeley Later refined by Susan Graham, Michael Harrison and Walter Ruzzo in [Graham et al., 1980] The algorithm improves the time complexity of algorithm from O( G 2 w 3 ) to O( G w 3 )

41 Graham-Harrison-Ruzzo Algorithm The algorithm can be expressed by means of a new PDA, obtained by unfolding some of the transitions of the Use stack symbols Q GHR = Q E {[ A] A N} {[A ] A N}

42 Graham-Harrison-Ruzzo Algorithm Use the following set of transitions [A α bβ] b [A αb β] [B β ] ε [B ] [A α Bβ] [B ] ε [A αb β] [A α Bβ] ε [A α Bβ] [ B] [ B] ε [B β]

43 S.L. Graham, M.A. Harrison, and W.L. Ruzzo An improved context-free recognizer. ACM Transactions on Programming Languages and Systems, 2: