1. Issues in Computational Linguistics: Ambiguity Management Dick Crouch and Tracy King

2. The next two days Today –Mostly computational –Generalizing charts for packing ambiguity Tomorrow –Mostly linguistic –An introduction to glue semantics

3. Several Types of Ambiguity Phonetic: –“I scream” or “ice cream” Tokenization: –“I like Jan.” --- |Jan|. Or |Jan.|. (abbrev January) Morphological: –“walks” --- noun or verb –“untieable knot” --- un(tieable) or (untie)able Lexical: –“bank” --- river bank or financial institution Syntactic: –“The turkeys are ready to eat” --- fattened or hungry Semantic: –“Two boys ate fifteen pizzas” --- 15 each or 15 total Pragmatic: –“Sue won. Ed gave her a good luck charm” --- cause or result

4. Some more ambiguities Women like chocolate more than men Visiting relatives can be boring –Visiting relatives is boring –Visiting relatives are boring Pets must be carried on escalator Clothes must be worn in public Warning: keep all body parts out of window

5. PP Attachment A classic example of syntactic ambiguity PP adjuncts can attach to VPs and NPs Strings of PPs in the VP are ambiguous –I see the girl with the telescope. I see [the girl with the telescope]. I see [the girl] [with the telescope]. Ambiguities proliferate exponentially –I see the girl with the telescope in the park I see [the girl with (the telescope in the park)] I see [the (girl with the telescope) in the park] I see the girl [with the (telescope in the park)] I see the girl [with the telescope] [in the park] I see [the girl with the telescope] [in the park] –The syntax has no way to determine the attachment, even if humans can.

6. Coverage entails Ambiguity I fell in the park. + I know the girl in the park. I see the girl in the park.

7. Ambiguity: bug or feature? Bug in computer programming languages Feature in natural language –People good at resolving ambiguity in context –Ambiguity consequently often unperceived »“Cable breakage damages the printer” »“Readjust paper holding clip” Even though thousand-fold ambiguities are common –Ambiguity promotes conciseness Computers can’t resolve ambiguity like humans

8. Ambiguity can be explosive If alternatives multiply within or across components… TokenizeMorphologySyntax Semantics Discourse

9. Computational consequences of ambiguity Serious problem for computational systems –Broad coverage, hand written grammars frequently produce thousands of analyses, sometimes millions –Machine learned grammars easily produce hundreds of thousands of analyses if allowed to parse to completion Three approaches to ambiguity management: –Pruning: block unlikely analysis paths early –Procrastination: do not expand analysis paths that will lead to ambiguity explosion until something else requires them »Also known as underspecification –Packing: compact representation and computation of all possible analyses

10. The Problem with Pruning: premature disambiguation The conventional approach: Use heuristics to prune as soon as possible TokenizeMorphologySyntaxSemanticsDiscourse X X X Statistics X Strong constraints may reject the so-far-best (= only) option Fast computation, wrong result X

11. The problem with procrastination passing the buck Chunk parsing as an example: –Collect noun groups, verb groups, PP groups –Leave it to later processing to figure out the correct way of putting these together –Not all combinations are grammatically acceptable Later processing must either –Call parser to check grammatical constraints –Have its own model of grammatical constraints –In the best case, solve a set of constraints the partial parser includes with its output

12. The Problem with Packing There may just be too many analyses to pack efficiently A major problem for relatively unconstrained machine induced grammars –Grammars overgenerate massively –Statistics used to prune out unlikely sub-analyses Less of a problem for carefully hand-coded broad coverage grammars

13. Outline of rest of talk Packing –Charts for context free grammars –Generalizing free choice packing Statistical pruning –PCFGs –Viterbi and inside-outside algorithms –Lexicalized PCFGs Packing and Pruning –Maximum entropy disambiguation and avoiding premature choice

14. Packing Explosion of ambiguity results from a small number of sub-analyses combining in different ways to produce a large number of total analyses (e.g. PP attachment) Compute and represent each sub-analysis just once Compute a factored representation of how these sub- analyses combine Charts for context free grammars are an example of (free choice) packing –Compute and represent 2 N analyses in time and space N 3

15. Charts P:3NP:2V:1NP:4P:5NP:6PP:7 PP:8 NP:9 PP:10 NP:11 NP:12 or VP:13 VP:14 or VP:15 or or sawthe girlinthe parkwith the telescope

16. Context Free Charts: Free choice structures 5 analyses in 15 components –Each contains ~11 constituents –Can count analyses without unpacking chart Exploits context-freeness to factor combinations: –V:1 combines with NP:12 regardless of internal structure –Chart packs all ambiguities of NP:12 together with no sacrifice of correctness –VP:15 can freely chose any analysis of NP:12 But what happens if things are not context free? –Can we preserve free choice? P:3NP:2V:1NP:4P:5NP:6 PP:7 PP:8 NP:9 PP:10 NP:11 NP:12 or VP:13 VP:14 or VP:15 or or

17. Generalizing Free Choice Packing The sheep saw the fish. How many sheep? How many fish? The sheep-sg saw the fish-sg. The sheep-pl saw the fish-sg. The sheep-sg saw the fish-pl. The sheep-pl saw the fish-pl. Options multiplied out In principle, a verb might require agreement of Subject and Object: Have to check it out. The sheep saw the fish sg pl Options packed But English doesn’t do that: Any combination of choices is OK

18. Dependent choices Das Mädchen-nom sah die Katze-nom Das Mädchen-nom sah die Katze-acc Das Mädchen-acc sah die Katze-nom Das Mädchen-acc sah die Katze-acc bad The girl saw the cat The cat saw the girl bad Das Mädchen sah die Katze nom acc The girl saw the cat Again, packing avoids duplication … but it’s wrong It doesn’t encode all dependencies, choices are not free.

19. Solution: Label dependent choices Das Mädchen-nom sah die Katze-nom Das Mädchen-nom sah die Katze-acc Das Mädchen-acc sah die Katze-nom Das Mädchen-acc sah die Katze-acc bad The girl saw the cat The cat saw the girl bad Label each choice with distinct Boolean variables p, q, etc. Record acceptable combinations as a Boolean expression  Each analysis corresponds to a satisfying truth-value assignment (a line from  ’s truth table that assigns it “true”) Das Mädchen sah die Katze p:nom  p:acc q:nom  q:acc (p  q)  (  p  q)  = =

20. Simplifying Truth Tables Das Mädchen sah die Katze p:nom  p:acc q:nom  q:acc (p  q)  (  p  q)  = = p q  1 1 0 1 0 1 0 1 1 0 0 0 (q =  p) Das Mädchen sah die Katze p:nom  p:acc  p :nom p :acc (p   p)  = = p  1 1 0 1 Freely choose any line from the truth table

21. (Almost) Free choice for PP attachment VP V NP PP PNP PP PNP PP PNP A1 A2 A3 A4 B1 B2 B3 C1 C2 Let PPs freely attach to all NPs and VPs and then rule out crossing attachments A1 xor A2 xor A3 xor A4  1 B1 xor B2 xor B3  1 C1 xor C2  1 Independent Choices NoGoods (crossing attachments) (A2 & B1), (A2 & C1) (A3 & B1), (A3 & B2) (B2 & C1) xor : exclusive or

22. Eliminating NoGoods / Simplifying Truth Tables Starting from: –A set of N independent m-way choices between boolean variables –And a set of nogoods constraining combinations of these variables This defines: –A truth table for a nogood function on the boolean variables –Where disallowed combinations of variables correspond to lines in the truth table assigning false to the noogood function We can: –Produce a corresponding free choice truth table –Where no combinations of variables are disallowed

23. Example of NoGood Elimination VP NP PP PNP PP P NP B1 B2 C1 C2 B1 xor B2  1 C1 xor C2  1  (B1 & C1) B1 B2 C1 C2  (B1&C1) 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 0 1 1 Simplifies to VP NP PP PNP PP P NP b1 b2 c1 c2 v b1 b1 xor b2  1 c1 xor c2  b2 b1 b2 c1 c2  nogood 1 0 0 0 1 0 1 1 0 1 0 1 0 1 1

24. The Free Choice Gamble Eliminating nogoods increases number of boolean choice variables. –The more interactions between choices, the greater the additional number of variables Worst case, where everything interacts: –As many choice variables as there are readings –Packing blows up, and becomes exponential Best case, no interactions –N completely independent choices represent 2 N readings Language interactions mostly limited & local –Tends towards the best case –Free choice packing pays off for linguistic analysis

25. Advantages of Free Choice Packing Avoids procrastination –Nogoods are constraints that parser sends to other components to check when using packed structures –Eliminating nogoods: other components don’t do parser’s work Independence between choices: Allows processing relying on independence assumptions –Counting number of readings »Apparently trivial but of crucial importance, since statistical modelling requires the ability to count –Hence, statistical processing A general mechanism extending beyond parsing

26. Parsing in the XLE Extending free choice packing C-structure parsing produces a context free chart –Already a free choice packed structure F-structure constraints applied to c-str chart –Rules out certain c-structures through unification failures (i.e. contradictions in a logic of equality) –Need to impose additional no-goods on chart How do we get the f-structure logic of equality to –Operate within the free choice structure of the chart –Impose additional no-goods on the structure? That is, we know how to pack context free grammars, but how do we pack a logic of equality?

27. Let ,  be formulas from the logic you want to pack 1. New choice:    iff p     p   (p a new Boolean variable) Proof:(If)If  is true, let p be true, in which case p     p   is true. (Only if)If p is true, then  is true, in which case    is true. How to Pack a Logic 2. Ambiguity-enabled inference (by trivial logic): If      is a rule of inference, then (C 1   )  (C 2   )  (C 1  C 2   ) is also valid E.g. Substition of equals for equals: x=y     x/y is a rule of inference Therefore: (C 1  x=y )  (C 2   )  (C 1  C 2   x/y ) Goal: Ambiguity-enable components to defer choosing, unpacking 3. Nogoods: if    and C 1   then nogood(C 1 ).

28. Packing Equalities in F-structure NP  SUBJ =  NP  NUM=sg NPV NP  SUBJ=  Adj NP  =  S VP V visitingrelatives visitingrelatives  NUM=pl is  SUBJ NUM=sg Adj boring A1A2 1   SUBJ NUM=sg A1   SUBJ NUM=sg A2   SUBJ NUM=pl 1 & A1  sg=sg 1 & A2  sg=pl nogood(A2)

29. Ambiguity Enabled Language Processing Extend free choice packing to other components in language processing, e.g –Structure transfer for machine translation –Generation –Semantic interpretation –Reasoning (?) A single, uniform mechanism for managing ambiguity within and across module boundaries Allows deferment of disambiguation choices across module boundaries Efficient, provided choices are relatively independent

30. Pruning: Statistical Parsing Probabilistic Context Free Grammars (PCFGs) Viterbi parsing of PCFGs Inside-Outside training of PCFGs Lexicalized PCFGs Disambiguation of packed structures in XLE –Maximum entropy models

31. PCFGs Context free grammar with probabilities assigned to rules –Probabilities of all rules rewriting a non-terminal N sum to 1 Probability of a tree –Product of probabilities of all the rules used to derive the tree Probability of a sentence –Sum probabilities of all trees assigned to sentence by grammar Independence assumption: All rules are statistically independent –Choice of rule used to analyze a subconstituent does not influence choice rule used to embed it in a larger constituent –Empirically implausible assumption (hence lexicalized PCFGs)

32. Viterbi Parsing of PCFGs How do you find the most probable parse of a sentence without inspecting all of its parses? Observation: –most probable parse of a constituent is the most probable combination of »a rule » the most probable subconstituents matching the rule Viterbi parsing (dynamic programming) –For each word span i-j and non-terminal N, record only the most probable parse of i-j as N

33. Example of Viterbi Parsing (charts with probabilities) N  Adj N 0.1 N  relatives 0.1 … VP  V NP 1 Adj  old 0.1 NP  N 0.9 Adj  visiting 0.01 … NP  VP 0.1 V  visiting 0.1 … visitingoldrelatives V 0.1 Adj 0.01 1-1 NP 0.09 N 0.1 3-3 Adj 0.1 2-2 NP 0.0009 N 0.001 2-3 NP 9x10 -6 NP  VP NP 9x10 -7 NP  N N 1x10 -6 VP 9x10 -5 1-3

34. Training: Expectation Maximization Start with set of rules & initial estimate of rule probabilities –e.g. rule counts from annotated corpus, or random assignment –For PCFGs, results extremely sensitive to initial estimates Adjust rule probabilities to maximize probability of sentences in (unannotated) corpus –Parse corpus with existing rules and calculate sentence probabilities –Reassign rule probabilities based on rule counts weighted by the probabilities of the sentences in which those rules occurred –Iterate until probabilities settle/converge Requires efficient way of calculating rule probabilities weighted by sentence probability –Inside-Outside algorithm

35. Inside probability: IN N (i,j) = P(w i,j | N  i RHS j ) Probability that a string of words expands N Outside probability: OUT N (i,l) = P(w i,j-1, N  j RHS k,,w k+1,l ) Probability of having an expansion of N between surrounding sequences of words Inside and Outside Probabilities 1 jkl N IN OUT

36. Calculating Inside & Outside Probabilities Compute inside probabilities bottom up –Sum inside probabilities of all chart edges dominated by N-rule Compute outside probabilities top down –From outside probabilities of dominating constituents in chart Can show that weighted rule count C(N  AB) = OUT N (k,l). P(N  AB). IN A (k,m). IN B (m+1,n) 1 P(w1..n)  klm

37. Lexicalized PCFGs Probabilities of rule conditioned by lexical heads of its daughters/RHS Conceptually, each non-terminal N becomes a complex category –NP(man)  Det(the) N(man) –done more efficiently in practice Major problems with data sparseness –Unlikely to see NP(zygote)  Det(the) N(zygote) in training –Various smoothing techniques to assign reasonable non- zero probabilities to examples unseen in training

38. Packing & Pruning in XLE XLE produces (too) many candidates –All valid (with respect to grammar and OT marks) –Not all equally likely –Some applications require a single best guess Grammar writer can’t specify correct choices –Many implicit properties of words and structures with unclear significance Appeal to probability model to choose best parse –Assume: previous experience is a good guide for future decisions –Collect corpus of training sentences, build probability model that optimizes for previous good results –Apply model to choose best analysis of new sentences

39. Issues What kind of probability model? What kind of training data? Efficiency of training, efficiency of disambiguation? Benefit vs. random choice of parse

40. Probability model Conventional models: stochastic branching process –Probabilistic Context-Free grammars Sequence of decisions, each independent of previous decisions, each choice having a certain probability –PCFG: Choose from alternative expansions of a given category Probability of an analysis = product of choice probabilities Efficient algorithms –Training: inside/outside –Disambiguation: Viterbi Abney 1997 and others: Not appropriate for LFG, HPSG… –Choices are not independent: Information from different CFG branches interacts through f-structure –Probability models are biased (don’t make right choices on training set)

41. Exponential models are appropriate (aka Maximum Entropy or Log-linear models) Assign probabilities to representations, not to choices in a derivation No independence assumption Arithmetic combined with human insight –Human: »Define properties of representations that may be relevant »Based on any computable configuration of features, trees –Arithmetic: »Train to figure out the weight of each property

44. What does this mean?? Define yes-no / 1-0 features, f, that seem important Training determines weights on these features, λ, to reflect their actual importance Select parse x: count occurrences of features (0,1) and multiply by corresponding weights, λ.f(x) Convert weighted feature counts to probabilities e λ.f(x)  e λ.f(X) Un-normalized probability Normalizing factor

45. Entropy and Exponentials Sparse and interdependent training data –Not enough parse/feature combinations to uniquely determine feature weights consistent with training data –Features are not independent Entropy: reciprocal of average number of bits required to communicate a result, under an optimal coding scheme –Optimal coding: two most probable results encoded in 1 bit, next 4 in 2 bits, etc… –Dependencies between features: if f1 makes f2 more likely, can reduce size of encoding of f2 when code for f1 is present Maximize entropy of training data to smooth dependent features weights Entropy defined as (hence the exponential) H(p) = -  p(x). log p(x)

47. And what does that mean?? Variant of an Expectation-Maximization (EM) algorithm (conjugate-gradient minimization) used to compute weight distribution with maximum entropy Inside-outside algorithm used to calculate all inside and outside probabilities as determined by feature weights

48. Efficiency Properties counts –Associated with Boolean tree of XLE contexts (a1, b2) –Shared among many parses Training –Inside/outside algorithm of PCFG, but applied to Boolean tree, not parse tree –Fast algorithm for choosing best properties –Can train on sentences with relatively low-ambiguity –5 hours to train over WSJ (given file of parses) Disambiguation –Viterbi algorithm applied to Boolean tree –5% of parse time to disambiguate –30% gain in F-score

49. Conclusions Free choice packing –Extending charts beyond parsing context free grammars Efficient if limited interaction between alternatives –Generally true for natural language processing –Though sometimes hits bad cases Uniform ambiguity management architecture across all language processing modules –Not forced to make premature disambiguation choices at module boundaries Statistical disambiguation of packed representations