Formal Semantics Slides by Julia Hockenmaier, Laura McGarrity, Bill McCartney, Chris Manning, and Dan Klein

Formal Semantics It comes in two flavors: Lexical Semantics: The meaning of words Compositional semantics: How the meaning of individual units combine to form the meaning of larger units

What is meaning Meaning ≠ Dictionary entries Dictionaries define words using words. Circularity!

Reference Referent: the thing/idea in the world that a word refers to Reference: the relationship between a word and its referent

Reference Barack president Obama The president is the commander-in-chief. = Barack Obama is the commander-in-chief.

Reference Barack president Obama I want to be the president. ≠ I want to be Barack Obama.

Reference Tooth fairy? Phoenix? Winner of the 2016 presidential election?

What is meaning? Meaning ≠ Dictionary entries Meaning ≠ Reference

Sense Sense: The mental representation of a word or phrase, independent of its referent.

Sense ≠ Mental Image A word may have different mental images for different people. E.g., “mother” A word may conjure a typical mental image (a prototype), but can signify atypical examples as well.

Sense v. Reference A word/phrase may have sense, but no reference: King of the world The camel in CIS 8538 The greatest integer The A word may have reference, but no sense: Proper names: Dan McCloy, Kristi Krein (who are they?!)

Sense v. Reference A word may have the same referent, but more than one sense: The morning star / the evening star (Venus) A word may have one sense, but multiple referents: Dog, bird

Some semantic relations between words Hyponymy: subclass Poodle < dog Crimson < red Red < color Dance < move Hypernymy: superclass Synonymy: Couch/sofa Manatee / sea cow Antonymy: Dead/alive Married/single

Lexical Decomposition Word sense can be represented with semantic features:

Compositional Semantics

Compositional Semantics The study of how meanings of small units combine to form the meaning of larger units The dog chased the cat ≠ The cat chased the dog. ie, the whole does not equal the sum of the parts. The dog chased the cat = The cat was chased by the dog ie, syntax matters to determining meaning.

Principle of Compositionality The meaning of a sentence is determined by the meaning of its words in conjunction with the way they are syntactically combined.

Exceptions to Compositionality Anomaly: when phrases are well-formed syntactically, but not semantically Colorless green ideas sleep furiously. (Chomsky) That bachelor is pregnant.

Exceptions to Compositionality Metaphor: the use of an expression to refer to something that it does not literally denote in order to suggest a similarity Time is money. The walls have ears.

Exceptions to Compositionality Idioms: Phrases with fixed meanings not composed of literal meanings of the words Kick the bucket = die (*The bucket was kicked by John.) When pigs fly = ‘it will never happen’ (*She suspected pigs might fly tomorrow.) Bite off more than you can chew = ‘to take on too much’ (*He chewed just as much as he bit off.)

Idioms in other languages

Logical Foundations for Compositional Semantics We need a language for expressing the meaning of words, phrases, and sentences Many possible choices; we will focus on First-order predicate logic (FOPL) with types Lambda calculus

Truth-conditional Semantics Linguistic expressions “Bob sings.” Logical translations sings(Bob) but could be p_5789023(a_257890) Denotation: [[bob]] = some specific person (in some context) [[sings(bob)]] = true, in situations where Bob is singing; false, otherwise Types on translations: bob: e(ntity) sings(bob): t(rue or false, a boolean type)

Truth-conditional Semantics Some more complicated logical descriptions of language: “All girls like a video game.” x:e . y:e . girl(x)  [video-game(y)  likes(x,y)] “Alice is a former teacher.” (former(teacher))(Alice) “Alice saw the cat before Bob did.” x:e, y:e, z:e, t1:e, t2:e . cat(x)  see(y)  see(z)  agent(y, Alice)  patient(y, x)  agent(z, Bob)  patient(z, x)  time(y, t1)  time(z, t2)  <(t1, t2)

FOPL Syntax Summary A set of types T = {t1, … } A set of constants C = {c1, …}, each associated with a type from T A set of relations R = {r1, …}, where each ri is a subset of Cn for some n. A set of variables X = {x1, …} , , , , , , ., :

Truth-conditional semantics Proper names: Refer directly to some entity in the world Bob: bob Sentences: Are either t or f Bob sings: sings(bob) So what about verbs and VPs? sings must combine with bob to produce sings(bob) The λ-calculus is a notation for functions whose arguments are not yet filled. sings: λx.sings(x) This is a predicate, a function that returns a truth value. In this case, it takes a single entity as an argument, so we can write its type as e  t Adjectives?

Lambda calculus FOPL + λ (new quantifier) will be our lambda calculus Intuitively, λ is just a way of creating a function E.g., girl() is a relation symbol; but λx . girl(x) is a function that takes one argument. New inference rule: function application (λx . L1(x)) (L2) → L1(L2) E.g., (λx . x2) (3) → 32 E.g., (λx . sings(x)) (Bob) → sings(Bob) Lambda calculus lets us describe the meaning of words individually. Function application (and a few other rules) then lets us combine those meanings to come up with the meaning of larger phrases or sentences.

Compositional Semantics with the λ-calculus So now we have meanings for the words How do we know how to combine the words? Associate a combination rule with each grammar rule: S : β(α)  NP : α VP : β (function application) VP : λx. α(x) ∧ β(x)  VP : α and : ∅ VP : β (intersection) Example:

Composition: Some more examples Transitive verbs: likes : λx.λy.likes(y,x) Two-places predicates, type e(et) VP “likes Amy” : λy.likes(y,Amy) is just a one-place predicate Quantifiers: What does “everyone” mean? Everyone : λf.x.f(x) Some problems: Have to change our NP/VP rule Won’t work for “Amy likes everyone” What about “Everyone likes someone”? Gets tricky quickly!

Composition: Some more examples Indefinites The wrong way: “Bob ate a waffle” : ate(bob,waffle) “Amy ate a waffle” : ate(amy,waffle) Better translation: ∃x.waffle(x) ^ ate(bob, x) What does the translation of “a” have to be? What about “the”? What about “every”?

Denotation What do we do with the logical form? It has fewer (no?) ambiguities Can check the truth-value against a database More usefully: can add new facts, expressed in language, to an existing relational database Question-answering: can check whether a statement in a corpus entails a question-answer pair: “Bob sings and dances”  Q:“Who sings?” has answer A:“Bob” Can chain together facts for story comprehension

Grounding What does the translation likes : λx. λy. likes(y,x) have to do with actual liking? Nothing! (unless the denotation model says it does) Grounding: relating linguistic symbols to perceptual referents Sometimes a connection to a database entry is enough Other times, you might insist on connecting “blue” to the appropriate portion of the visual EM spectrum Or connect “likes” to an emotional sensation Alternative to grounding: meaning postulates You could insist, e.g., that likes(y,x) => knows(y,x)

More representation issues Tense and events In general, you don’t get far with verbs as predicates Better to have event variables e “Alice danced” : danced(Alice) vs. “Alice danced” : ∃e.dance(e)^agent(e, Alice)^(time(e)<now) Event variables let you talk about non-trivial tense/aspect structures: “Alice had been dancing when Bob sneezed”

More representation issues Propositional attitudes (modal logic) “Bob thinks that I am a gummi bear” thinks(bob, gummi(me))? thinks(bob, “He is a gummi bear”)? Usually, the solution involves intensions (^p) which are, roughly, the set of possible worlds in which predicate p is true. thinks(bob, ^gummi(me)) Computationally challenging Each agent has to model every other agent’s mental state This comes up all the time in language – E.g., if you want to talk about what your bill claims that you bought, vs. what you think you bought, vs. what you actually bought.

More representation issues Multiple quantifiers: “In this country, a woman gives birth every 15 minutes. Our job is to find her, and stop her.” -- Groucho Marx Deciding between readings “Bob bought a pumpkin every Halloween.” “Bob put a warning in every window.”

More representation issues Other tricky stuff Adverbs Non-intersective adjectives Generalized quantifiers Generics “Cats like naps.” “The players scored a goal.” Pronouns and anaphora “If you have a dime, put it in the meter.” … etc., etc.

Mapping Sentences to Logical Forms

CCG Parsing Combinatory Categorial Grammar Lexicalized PCFG Categories encode argument sequences A/B means a category that can combine with a B to the right to form an A A \ B means a category that can combine with a B to the left to form an A A syntactic parallel to the lambda calculus

Learning to map sentences to logical form Zettlemoyer and Collins (IJCAI 05, EMNLP 07)

Some Training Examples

CCG Lexicon

Parsing Rules (Combinators) Application Right: X : f(a)  X/Y : f Y : a Left: X : f(a)  Y : a X\Y : f Additional rules: Composition Type-raising

CCG Parsing Example

Parsing a Question

Lexical Generation Input Training Example Sentence: Texas borders Kansas. Logical form: borders(Texas, Kansas)

GENLEX Input: a training example (Si, Li) Computation: Create all substrings of consecutive words in Si Create categories from Li Create lexical entries that are the cross products of these two sets Output: Lexicon Λ

GENLEX Cross Product Input Training Example Sentence: Texas borders Kansas. Logical form: borders(Texas, Kansas) Output Lexicon Output Substrings Texas borders Kansas Texas borders borders Kansas Texas borders Kansas X (cross product) Output Categories NP : texas NP : kansas (S\NP)/NP : λx.λy.borders(y,x)

(S\NP)/NP : λx.λy.borders(y,x) GENLEX Output Lexicon Words Category Texas NP : texas NP : kansas (S\NP)/NP : λx.λy.borders(y,x) borders Borders … Texas borders Kansas

Weighted CCG Given a log-linear model with a CCG lexicon Λ, a feature vector f, and weights w: The best parse is: y* = argmax w ∙ f(x,y) where we consider all possible parses y for the sentence x given the lexicon Λ. y

Parameter Estimation for Weighted CCG Parsing Inputs: Training set {(Si,Li) | i = 1, …, n} Initial lexicon Λ, initial weights w, num. iter. T Computation: For t=1 … T, i = 1 … n: Step 1: Check correctness If y* = argmax w ∙ f(Si,y) is Li, skip to next i Step 2: Lexical generation Set λ = Λ ∪ GENLEX(Si,Li) Let y’ = argmax w ∙ f(Si,y) Define λi to be the lexical entries in y’ Set Λ = Λ ∪ λi Step 3: Update Parameters Let y’’ = argmax w ∙ f(Si,y) If y’’ ≠ Li Set w = w + f(Si, y’) – f(Si,y’’) Output: Lexicon Λ and parameters w y s.t. L(y) = Li y

Example Learned Lexical Entries

Challenge Revisited

Disharmonic Application

Missing Content Words

Missing content-free words

A complete parse

Geo880 Test Set Precision Recall F1 Zettlemoyer & Collins 2007 95.49 83.20 88.93 Zettlemoyer & Collins 2005 96.25 79.29 86.95 Wong & Mooney 2007 93.72 80.00 86.31

Summing Up Hypothesis: Principle of Compositionality Semantics of NL sentences and phrases can be composed from the semantics of their subparts Rules can be derived which map syntactic analysis to semantic representation (Rule-to-Rule Hypothesis) Lambda notation provides a way to extend FOPC to this end But coming up with rule2rule mappings is hard Idioms, metaphors and other non-compositional aspects of language makes things tricky (e.g. fake gun)