Context- Free Parsing with CKY. October 16, 2014

Transcription

1 Context- Free Parsing with CKY October 16, 2014

2 Lecture Plan Parsing as Logical DeducBon Defining the CFG recognibon problem BoHom up vs. top down Quick review of Chomsky normal form The CKY algorithm CKY as dynamic programming Complexity analysis

3 Algorithms for CFGs Given a CFG G and a string s: RecogniBon: Is s L(G)? Equivalently, find some derivabon that proves s is in G's language. Parsing: What are (all of) G's derivabons of s? What is the correct derivabon of s under G? The same core algorithms actually provide solubons to all of these!

4 Important DisBncBon DeterminisBc grammars give much faster recognibon and parsing algorithms! LR languages are parseable in linear Bme; you may have learned about these in a course on programming languages or compilers. For us, the best known algorithms are O( N n ), but these are not pracbcal. Today you will learn an O( N 3 n 3 ) solubon.

5 Parsing Strategies

6 Parsing as Search S top- down bohom- up? x1 x2... xn Trees break into pieces (parbal trees), which can be used to define a search space.

7 Top- Down Parsing (Recursive Descent) (S) s = Book that flight

8 Top- Down Parsing (Recursive Descent) (S) (S (NP) (VP)) (S Aux (NP) (VP)) (S (VP)) x = Book that flight

9 Top- Down Parsing (Recursive Descent) (S) (S (NP) (VP)) (S Aux (NP) (VP)) (S (VP)) (S (NP Pronoun) (VP)) (S (NP ProperNoun) (VP)) (S (NP Det Nominal) (VP)) x = Book that flight

10 Top- Down Parsing (Recursive Descent) (S) (S (NP) (VP)) (S Aux (NP) (VP)) (S (VP)) (S (NP Pronoun) (VP)) (S (NP ProperNoun) (VP)) (S (NP Det Nominal) (VP)) (S Aux (NP Pronoun) (VP)) (S Aux (NP ProperNoun) (VP)) (S Aux (NP Det Nominal) (VP)) x = Book that flight

11 Top- Down Parsing (Recursive Descent) (S) (S (NP) (VP)) (S Aux (NP) (VP)) (S (VP)) (S (NP Pronoun) (VP)) (S (NP ProperNoun) (VP)) (S (NP Det Nominal) (VP)) (S Aux (NP Pronoun) (VP)) (S Aux (NP ProperNoun) (VP)) (S Aux (NP Det Nominal) (VP)) (S (VP (VP) (PP))) (S (VP Verb)) (S (VP Verb (NP))) (S (VP Verb (NP) (PP))) (S (VP Verb (PP))) x = Book that flight

12 Top- Down Parsing (Recursive Descent) Never wastes Bme exploring ungrammabcal trees! Inefficiency: most search states (parbal trees) could never lead to a derivabon of our sentence. One other problem...

13 Lef Recursion! (S) (S (VP)) (S (VP (VP) (PP))) (S (VP (VP (VP) (PP)) (PP))) (S (VP (VP (VP (VP) (PP)) (PP)) (PP))) (S (VP (VP (VP (VP (VP) (PP)) (PP)) (PP)) (PP))) (S (VP (VP (VP (VP (VP (VP) (PP)) (PP)) (PP)) (PP)) (PP))) x = Book that flight

14 Top- Down RecogniBon Don't need to store the tree! Can collapse states that funcbon the same way. Store unexpanded nonterminals (in sequence) only, along with the number of words covered so far. Start state: (S, 0) Scan: If you're at (xj+1 β, j), you can get to (β, j + 1). Predict: If you're at (Zβ, j) and Z γ, you can get to (γβ, j). Final ( success ) state: (ε, n) This should remind you of genera-ng from a CFG.

15 BoHom- Up Parsing book that flight

16 BoHom- Up Parsing (Verb book) (Det that) (Noun flight) (Noun book) (Det that) (Noun flight) book that flight

17 BoHom- Up Parsing (Nominal (Noun book)) (Det that) (Nominal (Noun flight)) (Verb book) (Det that) (Noun flight) (Noun book) (Det that) (Noun flight) book that flight

18 BoHom- Up Parsing (Verb book) (Det that) (Nominal (Noun flight)) (Nominal (Noun book)) (Det that) (Nominal (Noun flight)) (Verb book) (Det that) (Noun flight) (Noun book) (Det that) (Noun flight) book that flight

19 BoHom- Up Parsing Nominal (Noun book)) (NP (Det that) (Nominal (Noun flight))) (Verb book) (Det that) (Nominal (Noun flight)) (Nominal (Noun book)) (Det that) (Nominal (Noun flight)) (Verb book) (Det that) (Noun flight) (Noun book) (Det that) (Noun flight) book that flight

20 BoHom- Up Parsing Never generates trees that are inconsistent with the sentence. Generates parbal trees that have no hope of geong to S.

21 Shif- Reduce (A BoHom- Up Method) Don't need to store the tree! Can collapse states that funcbon the same way. Store only the nodes that are parentless. Start state: (ε, 0) Shi/: If you're at (α, j), you can get to (αxj+1, j + 1). Reduce: If you're at (αγ, j) and Z γ, you can get to (αz, j). Final ( success ) state: (S, n)

22 ImplemenBng Recognizers as Search Agenda = { state0 } while(agenda not empty) s = pop a state from Agenda if s is a success- state return s // valid parse tree else if s is not a failure- state: generate new states from s push new states onto Agenda return nil // no parse!

23 Parsing as Search Basically, you need a strategy for deciding what to pursue next (how to priori-ze the agenda). Examples: depth- first, breadth- first, bohom- up, top- down, lef- to- right, right- to- lef,... The problems: Memory! Ambiguity leads to lots of bad paths In a depth- first model, this results in lots of wasted effort in backtracking. One solubon: greedy parsing. Only works if you have a really smart agenda!

24 Backtracking

25 Ambiguity Redux A sentence may have many parses. Even if a sentence has only one parse, finding it may be difficult, because there are many misleading paths you could follow. BoHom- up: fragments that can never have a home in any S Top- down: fragments that never get you to x What to do when there are many parses... how to choose? Return them all?

26 So Far... Two ways to think of parsing as search Problems: Lef- recursion (in top- down parsers) Efficiency (backtracking, memory) Ambiguity

27 CKY Kasami, 1965 Younger, 1967 Cocke and Schwartz, 1970

28 Chomsky Normal Form G = (Σ, N, S, R) Σ : Vocabulary of terminal symbols N: set of nonterminal symbols S N: special start symbol R: ProducBon rules of the form X α where X N (a nonterminal symbol) and α N 2 Σ

29 CKY Schema Data structure: chart. In each cell, store the possible phrase types. Visit each cell once. Start with narrow spans, progress to longer ones. book the flight through Chicago

30 CKY Example boo k the flight through Chicago

31 CKY Schema book Noun Verb VP Nominal S S, VP, X2 S, VP Det NP NP the Nominal Noun Nominal flight Prep PP through ProperN NP Chicago

32 CKY Algorithm (RecogniBon) input: x, G (in CNF) output: true iff x is in L(G) initialize all cells of C to for i = 1... n C[i-1, i] = {A A xi} for l = 2... n for i = 0... n - l k = i + l for j = i k 1 for each X YZ s.t. Y C[i, j] & Z C[j, k] C[i, k] = C[i, k] {X} return true if S C[0, n]

33 Visualizing CKY X 1 5 Y 1 5 Z n

34 Visualizing CKY How do we fill in C(0,2)? n

35 Visualizing CKY How do we fill in C(0,2)? Put together C(0,1) and C(1,2) n

36 Visualizing CKY How do we fill in C(0,3)? n

37 Visualizing CKY How do we fill in C(0,3)? One way n

38 Visualizing CKY How do we fill in C(0,3)? One way Another way n

39 Visualizing CKY How do we fill in C(0,n)? n

40 Visualizing CKY How do we fill in C(0,n)? n - 1 ways! n

41 Ordering in CKY n

42 CKY Algorithm for CF RecogniBon input: x, G (in CNF) output: true iff x is in L(G) initialize all cells of C to for i = 1... n C[i-1, i] = {A A xi} for l = 2... n for i = 0... n - l k = i + l for j = i k 1 for each X YZ s.t. Y C[i, j] & Z C[j, k] C[i, k] = C[i, k] {X} return true if S C[0, n]

43 CKY Algorithm input: x, G (in CNF) output: true iff x is in L(G) initialize all cells of C to for i = 1... n for each A s.t. A xi C[A, i-1, i] = 1 for l = 2... n for i = 0... n - l k = i + l for j = i k 1 for each X YZ C[X, i, k] = C[X, i, k] (C[Y, i, j] C[Z, j, k]) return C[S, 0, n]

44 CKY OperaBons given... for every add to - A x i A C[i - 1, i] Y C[i, j] Z C[j, k] X Y Z X C[i, k]

45 Visualizing CKY Operations Y Z O(n) B x i B O(n 3 ) X Y Z k X j j i i-1 i k i size of N nonterminal start end length of sentence length of sentence

46 CKY Algorithm AsymptoBcs Space: O(n 2 ) RunBme: O(n 3 ) Both also depend on grammar size

47 Parsing as Logical DeducBon

48 (Weighted) Logic Programming Parsing algorithms can be expressed as (weighted) logic programs I 1 I 2 I k I Shieber, Schabes, & Pereira (1995). Principles and ImplementaBon of DeducBve Parsing. J. Logic Programming

49 (Weighted) Logic Programming Parsing algorithms can be expressed as (weighted) logic programs I 1 I 2 I k I I 1 : w 1 I 2 : w 2 I k : w k I : w N k i=1 w i w, Shieber, Schabes, & Pereira (1995). Principles and ImplementaBon of DeducBve Parsing. J. Logic Programming

50 Understanding Logic Programs Antecedents I 1 I 2 I k I Consequent Side condibons

51 Understanding Logic Programs Items I Axioms A I Inference rules Goals G I I 1 I 2 I k I

52 Item form [N,N 1, N 1 ] CKY Algorithm

53 CKY Algorithm Item form [N,N 1, N 1 ] Goals [S, 1, x + 1]

54 CKY Algorithm Item form [N,N 1, N 1 ] Goals [S, 1, x + 1] Axioms [X, i, i + 1] (X! x i ) 2 N

55 CKY Algorithm Item form [N,N 1, N 1 ] Goals [S, 1, x + 1] Axioms [X, i, i + 1] (X! x i ) 2 N Inference rules [X, i, k] [Z, i, j] [Y,k,j] (Z! XY) 2 R

56 Top- Down Parsing Item form [(N [ ), N] Goals [", x ] Axioms [S, 0] Inference rules [x j+1,j] [,j + 1] [Z,j] [,j] (Z! ) 2 R Scan Predict

57 CKY Algorithm: Discussion Top- down or bohom- up? Parsing as deducbon: visual notabon Horn clauses, used to prove x L(G).

58 Dynamic Programming Examples: WFSA best path, Viterbi, Earley, CKY Solve sub- problems, store results in a table. Reuse work you ve done already (runbme) Pack sub- problems efficiently (memory) For parsing, many sub- parses can be used repeatedly in many larger sub- parses! Don t build the same sub- parse more than once.

59 DP ExecuBon Model Clever ordering to ensure items are built exactly once Viterbi: lef to right CKY: narrow to wide MemoizaBon Agenda DP logic execubon model

60 ImplemenBng Recognizers as Search Agenda = { state0 } while(agenda not empty) s = pop a state from Agenda if s is a success-state return s // valid parse tree else if s is not a failure-state: generate new states from s push new states onto Agenda return nil // no parse!

61 CKY as Search Agenda = [{ [A, i 1, i] A x i } while(agenda not empty) [N, i, j] = pop from Agenda if [N, i, j] = [S, 0, n] return true else: for each visited [Y, j, k] and X N Y push [X, i, k] onto Agenda for each visited [Z, k, i] and X Z N push [X, k, i] onto Agenda return false

62 What Else? So far, this only does recogni-on. Can we recover parse trees? (Need to store some more informabon.) All trees Best tree