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

2. 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

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

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

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

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

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

8. What is meaning?
Meaning ≠ Dictionary entries
Meaning ≠ Reference

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

10. 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.

12. 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?!)

13. 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

14. 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

15. Lexical Decomposition
Word sense can be represented with
semantic features:

16. Compositional Semantics

17. 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.

18. 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.

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

20. 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.

21. 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.)

22. Idioms in other languages

23. 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

24. Truth-conditional Semantics
Linguistic expressions
“Bob sings.”
Logical translations
sings(Bob)
but could be p_ (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)

25. 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)

26. 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, …}
, , , , , , ., :

27. 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?

28. 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.

29. 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:

30. 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!

31. 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”?

32. 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

33. 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)

34. 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”

35. 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.

36. 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.”

37. 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.

38. Mapping Sentences to Logical Forms

39. 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

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

41. Some Training Examples

42. CCG Lexicon

43. 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

44. CCG Parsing Example

45. Parsing a Question

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

47. 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

48. 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)

49. (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

50. 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

51. 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

52. Example Learned Lexical Entries

53. Challenge Revisited

54. Disharmonic Application

55. Missing Content Words

56. Missing content-free words

57. A complete parse

58. 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

59. 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)