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Chapter 12 Lexicalized and Probabilistic Parsing

Chapter 12 Lexicalized and Probabilistic Parsing. Guoqiang Shan University of Arizona November 30, 2006. Outline. Probabilistic Context-Free Grammars Probabilistic CYK Parsing PCFG Problems. Probabilistic Context-Free Grammars. Intuition Behind

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Chapter 12 Lexicalized and Probabilistic Parsing

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  1. Chapter 12Lexicalized and Probabilistic Parsing Guoqiang Shan University of Arizona November 30, 2006

  2. Outline • Probabilistic Context-Free Grammars • Probabilistic CYK Parsing • PCFG Problems

  3. Probabilistic Context-Free Grammars • Intuition Behind • To find “correct” parse for the ambiguous sentences • i.e. can you book TWA flights? • i.e. the flights include a book • Definition of Context-Free Grammar • 4-tuple G = (N, Σ, P, S) • N: a finite set of non-terminal symbols • Σ: a finite set of terminal symbols, where N ΛΣ = Φ • P: A β , where A is in N, and β is in (N V Σ)* • S: start symbol in N • Definition of Probabilistic Context-Free Grammar • 5-tuple G = (N, Σ, P, S, D) • D: A function P  [0,1] to assign a probability to each rule in P • Rules are written as A β[p], where p = D(A β) • i.e. A  a B [0.6], B  C D [0.3]

  4. PCFG Example Det  that .5 Det  the .8 Det  a .15 Noun  book .1 Noun  flights .5 Noun  meal .4 Verb  book .3 Verb  include .3 Verb  want .4 Aux  can .4 Aux  does .3 Aux  do .3 ProperN  TWA .4 ProperN  Denver .6 Pronoun  you .4 Pronoun  I .6 S  NP VP .8 S  Aux NP VP .15 S  VP .05 NP  Det Nom .2 NP  ProperN .35 NP  Noun .05 NP  ProNoun .4 Nom  Noun .75 Nom  Noun Nom .2 Nom  ProperN Nom .05 VP  Verb .55 VP  Verb NP .4 VP  Verb NP NP .05

  5. Probability of A Sentence in PCFG • Probability of any parse tree T of S • P(T,S) = Π D(r(n)) • T is the parse tree and S is the sentence to be parsed • n is a sub tree of T and r(n) is a rule to expand n • Probability of A parse tree • P(T,S) = P(T) * P(S|T) • A parse tree T uniquely corresponds a sentence S, so P(S|T) = 1 • P(T) = P(T,S) • Probability of a sentence • P(S) = Σ P(T), where T is in τ(S), the set of all the parse trees of S • In particular, for an unambiguous sentence, P(S) = P(T)

  6. Example • P(Tl) = 0.15*0.40*0.05* 0.05*0.35*0.75* 0.40*0.40*0.30* 0.40*0.50= 3.78*10-7 • P(Tr) = 0.15*0.40*0.40* 0.05*0.05*0.75* 0.40*0.40*0.30* 0.40*0.50= 4.32*10-7

  7. Probabilistic CYK Parsing of PCFG • Bottom-Up approach • Dynamic Programming: fill the tables of partial solutions to the sub-problems until they contain all the solutions to the entire problem • Input • CNF: ε-free, each production in form A β or A BC • n words, w1, w2, …, wn • Data Structure • Π[i, j, A]: the maximum probability for a constituent with non-terminal A spanning j words from wi • β[i, j, A] = {k, B, C}, where A  BC, and B spans k words from wi (for rebuilding the parse tree) • Output • The maximum probability parse will be Π[1,n,1] • The root of the parse tree is S, and spans entire string

  8. Π [i,0,A] {k, B, C} CYK Algorithm • Base case • Consider the input strings of length one • By the rules A  wi • Recursive case • For strings of words of length>1, A → wij • There exists some rules A  BC and k • 0<k<j • B → wik (known) • C → w(i+k)(j-k) (known) • Compute the probability of wij by multiplying the two probabilities • If there are more than one A  BC, pick the one that maximize the probability of wij My implementation is in lectura under directory /home/shan/538share/pcyk.c

  9. PCFG Example – Revisit to rewrite Det  that .5 Det  the .8 Det  a .15 Noun  book .1 Noun  flights .5 Noun  meal .4 Verb  book .3 Verb  include .3 Verb  want .4 Aux  can .4 Aux  does .3 Aux  do .3 ProperN  TWA .4 ProperN  Denver .6 Pronoun  you .4 Pronoun  I .6 S  NP VP .8 S  Aux NP VP .15 S  VP .05 NP  Det Nom .2 NP  ProperN .35 NP  Noun .05 NP  ProNoun .4 Nom  Noun .75 Nom  Noun Nom .2 Nom  ProperN Nom .05 VP  Verb .55 VP  Verb NP .4 VP  Verb NP NP .05

  10. Example (CYK Parsing) - Rewrite as CNF S  NP VP .8 (S  Aux NP VP .15) S  Aux NV .15 NV  NP VP 1.0 (S  VP .05) S  book .00825 S  include .00825 S  want .011 S  Verb NP .02 S  Verb DNP .0025 NP  Det Nom .2 (NP  ProperN .35) NP  TWA .14 NP  Denver .21 (NP  Nom .05) NP  book .00375 NP  flights .01875 NP  meal .015 NP  Noun Nom .01 NP  ProperN Nom .0025 (NP  ProNoun .4) NP  you .16 NP  I .24 (Nom  Noun .75) Nom  book .075 Nom  flights .375 Nom  meal .3 Nom  Noun Nom .2 Nom  ProperN Nom .05 (VP  Verb .55) VP  book .165 VP  include .165 VP  want .22 VP  Verb NP .4 (VP  Verb NP NP .05) VP  Verb DNP .05 DNP  NP NP 1.0

  11. Example (CYK Parsing) – Πmatrix

  12. Example (CYK Parsing) – Πmatrix

  13. Example (CYK Parsing) – Πmatrix

  14. Example (CYK Parsing) – Πmatrix

  15. Example (CYK Parsing) – Πmatrix

  16. Example (CYK Parsing) – βmatrix

  17. PCFG Problems • Independence Assumption • Assumption: the expansion of one nonterminal is independent of the expansion of others. • However, examination shows that how a node expands is dependent on the location of the node • 91% of the subjects are pronouns. • She’s able to take her baby to work with her. (91%) • Uh, my wife worked until we had a family. (9%) • But only 34% of the objects are pronouns. • Some laws absolutely prohibit it. (34%) • All the people signed confessions. (66%)

  18. PCFG Problems • Lack of sensitivity of words • Lexical information in a PCFG can only be represented via the probability of pre-terminal nodes (such as Verb, Noun, Det) • However, lexical information and dependencies turns out to be important in modeling syntactic probabilities. • Example: Moscow sent more than 100,000 soldiers into Afghanistan. • In PCFG, into Afghanistan may attach NP (more than 100,000 soldiers) or VP (sent) • Statistics shows that NP attachment is 67% or 52% • Thus, PCFG will produce an incorrect result. • Why? the word “Send” subcategorizes for a destination, which can be expressed with the preposition “into”. • In fact, when the verb is “send”, “into” always attaches to it

  19. However, PCFG assigns them the same probability, since the structures are using exactly the same rules. PCFG Problems • Coordination ambiguity • Look at the following case • Example: dogs in houses and cats • Semantically, dogs is a better conjunct for cats than houses • Thus, the parse [dogs in [NP houses and cats]] intuitively sounds unnatural, and should be dispreferred.

  20. References • NLTK Tutorial: Probabilistic Parsing: http://nltk.sourceforge.net/tutorial/pcfg/index.html • Stanford Probabilistic Parsing Group: http://nlp.stanford.edu/projects/stat-parsing.shtml • General CYK algorithm http://en.wikipedia.org/wiki/CYK_algorithm • General CYK algorithm web compute http://www2.informatik.hu-berlin.de/~pohl/cyk.php?action=example • Probabilistic CYK parsing http://www.ifi.unizh.ch/cl/gschneid/ParserVorl/ParserVorl7.pdf http://catarina.ai.uiuc.edu/ling306/slides/lecture23.pdf

  21. Questions? Thank You!

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