Modularizing Semantics Greg Kobele May 17, 2018 Universitt Leipzig - - PowerPoint PPT Presentation

Modularizing Semantics

Greg Kobele May 17, 2018

Universität Leipzig

Overview Modularity Semantic Parts This Talk

1

What it is

understand a complex phenomenon by

• factoring it into simple(r) parts
• analysing these parts

entire phenomenon = combination of parts

2

Linguistics

Complex Phenomenon ability to use language Simple(r) Parts

• Phonetics
• Phonology
• Morphology
• Syntax
• Semantics
• Pragmatics

3

Semantics

Complex Phenomenon Had a politician pushed the issue, he would have been arrested Simple(r) Parts?

• argument structure
• scope
• intentionality
• dynamics
• context sensitivity
• tense

4

What are semantic parts?

A basic idea

• semantic objects are λ-terms
• a part specifjes how to ’update’ a λ-term
• so that it refmects the structure of the part

5

Modules

An intentionality module a function INT which turns

• a non-intentional meaning
• into an intentional one

INT(no(student)(λx.can(laugh(x)))) = λw.no(student(w))(λx.∃w′.w R w′ ∧ (laugh(w′)(x w′)))

6

The Ideal

• 1. identify a domain
• 2. specify non-predictable meanings
• most things are predictable!
• 3. specify how to generate predictable meanings
• nly works if compositional:
• meanings of whole
• determined by meanings of parts

7

Homomorphisms

Strings replace a symbol

• with a string

Trees replace a symbol with k daughters

• with a tree with k empty leaves

λ-Terms replace a symbol of one type

• with a term of similar type

8

Term Homomorphisms

we want to view simpler expressions every(kitten) : (e → t) → t as ’abbreviating’ more complex ones every(kitten) : (e → t) → t in a compositional way: every(kitten) = every(kitten) (e → t) → t = (e → t) → t

9

Type homomorphism

specify what each atomic type ’abbreviates’: h(c) = α extend this homomorphically to complex types: c = h(c) α → β = α → β Single type semantics both e and t abbreviate q (with Dq = D(et)t) e → e → t = e → e → t = q → q → q

10

Lambda homomorphisms

specify what each constant ’abbreviates’: h(k) = M extend this homomorphically to complex terms: k = h(k) x = x λx.M = λx.M M N = M N You keep the structure of the term and just replace constants with their defjnitions

11

Type compatibility

To make sense, we require that lambda homomorphisms and type homomorphisms come in pairs: This means that

• you replace a symbol c : α
• with a term c : α
• of ’similar’ type

12

I will show how this works

• analyse context-sensitivity including dynamic

binding

• decompose this into two independent modules
• CON implementing context-sensitivity
• DYN implementing (static) dynamicity
• discuss options
• implementing instead dynamic dynamicity (à la DMG)
• extensions to RST/SDRT

13

Dynamics Discourse Modularizing Dynamics Example Discourse Structure

14

The dynamics of pronoun reference

Sentences can set up discourse referents for other sentences model this as scope:

• 1. S(. . . T(. . . ) . . . )
• T can access discourse refs introduced by S
• 2. S(. . . ) ∧ T(. . . )
• T cannot access discourse refs introduced by S

the primitive notion here is sentence scope

• motivated by context sensitivity
• formally independent thereof

15

Interpreting sentences in discourse

Basic observation context grows throughout the discourse Formal implementation sentences in a discourse scope over each other [[S. T.]] = [[S]] ◦ [[T]] = λx.[[S]]([[T]](x))

16

Dynamicization

On types e = e t = t → t The intuition sentences scope over the rest of the discourse

17

Inherent Dynamicity

and is internally and externally dynamic some is internally and externally dynamic if… then is internally dynamic DETs are internally dynamic

18

Break dynamicization into two steps

• 1. internal dynamicity
• 2. external dynamicity

19

Inherent Internal dynamicity

Determiners and Conservativity strong int(detS) := λP, Q.det P (P → Q)) weak int(detW) := λP, Q.det P (P ∧ Q) Implication and Classical Equivalence impl(if…then) := λφ, ψ.¬(φ ∧ ¬ψ) Accounts for Internal Dynamicity!

20

Inherent external dynamicity

and is externally dynamic ext(and) := λΦ, Ψ, ψ.Φ(Ψ ψ) = B some is externally dynamic ext(some) := λP, Q, ψ.some(λx.P x ⊤)(λx.Q x ψ)

21

Lambda homomorphisms

dyn ext ◦ impl ◦ int int int detW P Q det P P Q int detS P Q det P P Q int k k impl impl if…then impl k k ext ext and ext some P Q some x P x x Q x ext k Dyn k

22

Lambda homomorphisms

dyn ext ◦ impl ◦ int int int(detW) = λP, Q.det P (P ∧ Q) int(detS) = λP, Q.det P (P → Q) int(k) = k impl impl if…then impl k k ext ext and ext some P Q some x P x x Q x ext k Dyn k

22

Lambda homomorphisms

dyn ext ◦ impl ◦ int int int(detW) = λP, Q.det P (P ∧ Q) int(detS) = λP, Q.det P (P → Q) int(k) = k impl impl(if…then) = λφ, ψ.¬(φ ∧ ¬ψ) impl(k) = k ext ext and ext some P Q some x P x x Q x ext k Dyn k

22

Lambda homomorphisms

dyn ext ◦ impl ◦ int int int(detW) = λP, Q.det P (P ∧ Q) int(detS) = λP, Q.det P (P → Q) int(k) = k impl impl(if…then) = λφ, ψ.¬(φ ∧ ¬ψ) impl(k) = k ext ext(and) = B ext(some) = λP, Q, φ.some(λx.P(x)(⊤))(λx.Q(x)(φ)) ext(k) = Dyn(k)

22

Dynamic Lifting

Intrinsically static expressions are predictable Dynα : α → α Dyne(a) := a Dynt(φ) := λψ.φ ∧ ψ Dynαβ(f) := Dynβ ◦ f ◦ Staα = λA.Dynβ(f (Staα A)) Sta Stae A A Stat Sta F Sta F Dyn a Sta F Dyn a

23

Dynamic Lifting

Intrinsically static expressions are predictable Dyn Dyne a a Dynt Dyn f Dyn f Sta A Dyn f Sta A Staα : α → α Stae(A) := A Stat(Φ) := Φ ⊤ Staαβ(F) := Staβ ◦ F ◦ Dynα = λa.Staβ(F (Dynα a))

23

Examples

• Dyntt(not) := λΦ, ψ.not (Φ ⊤) ∧ ψ
• Dyneet(praise) := λx, y, ψ.praise x y ∧ ψ
• Dyn(et)(et)t(every) :=

λP, Q, ψ.every(λx.P x ⊤)(λx.Q x ⊤) ∧ ψ

24

An example

Start with: a : (et)(et)t boy : et jump : et laugh : et and : ttt he : e Let φ = and(a(boy)(jump))(laugh(he)) : t Then dyn(φ) : t → t Where dyn(φ) ≡ λψ.a(boy)(λx.boy(x) ∧ laugh(he) ∧ ψ)

25

Discourse Relations

• 1. John had a lovely evening
• 2. He had a great meal
• 3. He ate salmon
• 4. He devoured cheese
• 5. He won a dancing competition

Elaborating vs Narrating

• 2 and 5 elaborate on 1
• 2 and 5 are a narrative
• 3 and 4 elaborate on 2
• 3 and 4 are a narrative

26

Discorse Structure

ELAB had lovely evening NARR ELAB had great meal NARR ate salmon devoured cheese won dance-

• ing comp

27

Right Frontier Constraint

Pronouns can only refer to certain referents introduced in the previous discourse tree

• 1. start at rightmost leaf
• 2. walk anywhere except down a left branch of NARR

relation

28

Example

• 1. John had a lovely

evening

• 2. He had a great

meal

• 3. He ate salmon
• 4. He devoured

cheese

• 5. He won a dancing

competition

• 6. # It was nice and

pink ELAB had lovely evening NARR ELAB had great meal NARR ate salmon devoured cheese won dance-

• ing comp

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Interpreting Relations

Subordinating [[ELAB]] = λΦ, Ψ, φ.Φ(Ψ φ) Coordinating [[NARR]] = λΦ, Ψ, φ.(Φ ⊤) ∧ (Ψ φ) Right Frontier Constraint Pronouns can only refer to certain referents introduced in the previous discourse tree

• 1. start at rightmost leaf
• 2. walk anywhere except down a left branch of NARR

relation

30

Interpreting Relations

Subordinating [[ELAB]] = λΦ, Ψ, φ.Φ(Ψ φ) Coordinating [[NARR]] = λΦ, Ψ, φ.(Φ ⊤) ∧ (Ψ φ) Right Frontier Constraint Pronouns can only refer to certain referents introduced in the previous discourse tree

• 1. start at rightmost leaf
• 2. walk anywhere except down a left branch of NARR

relation

30

Summary

• can study logic of discourse
• independently of context-sensitivity

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Context Sensitivity Overview Pronouns as variables Pronouns as identity functions Identity functions vs variables Pronouns as defjnites Synthesis Modularizing Context Sensitivity Example Rethinking contexts

32

Modularity

we now want to study context-sensitivity

• independently of discourse dynamics

33

Multiplicity of theories

What is [[he]]? variable x7 id func λx.x defjnite the(λx.boy(x) ∧ near(x)(kim))

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De Groote (2006)

All theories agree in main respects: The fjrst thought that a completely semantically naive person might have is that a pronoun just picks out in a discourse context some contextu- ally salient individual. (Jacobson 2015) just disagree in

• 1. what contexts are
• 2. how you pick out individuals

35

Next

• why this is the case:
• pronouns as variables
• pronouns as identity functions
• pronouns as defjnites
• a synthesis

36

PRONOUNS denote variables

• a pronoun, he, denotes a variable, x7
• infjnitely many variables (x1 x2 x3

) and so pronouns are infjnitely ambiguous alternatively: push ambiguity from interface into lexicon

• he1 he2 he3
• pronoun resolution is choosing the right

disambiguation

• syntax says next to nothing about how to

disambiguate pronouns

37

PRONOUNS denote variables

• a pronoun, he, denotes a variable, x7
• infjnitely many variables (x1, x2, x3, . . .) and so

pronouns are infjnitely ambiguous alternatively: push ambiguity from interface into lexicon

• he1 he2 he3
• pronoun resolution is choosing the right

disambiguation

• syntax says next to nothing about how to

disambiguate pronouns

37

PRONOUNS denote variables

• a pronoun, he, denotes a variable, x7
• infjnitely many variables (x1, x2, x3, . . .) and so

pronouns are infjnitely ambiguous alternatively: push ambiguity from interface into lexicon

• he1 he2 he3
• pronoun resolution is choosing the right

disambiguation

• syntax says next to nothing about how to

disambiguate pronouns

37

PRONOUNS denote variables

• a pronoun, he, denotes a variable, x7
• infjnitely many variables (x1, x2, x3, . . .) and so

pronouns are infjnitely ambiguous alternatively: push ambiguity from interface into lexicon

• he1, he2, he3, . . .
• pronoun resolution is choosing the right

disambiguation

• syntax says next to nothing about how to

disambiguate pronouns

37

PRONOUNS denote variables

• a pronoun, he, denotes a variable, x7
• infjnitely many variables (x1, x2, x3, . . .) and so

pronouns are infjnitely ambiguous alternatively: push ambiguity from interface into lexicon

• he1, he2, he3, . . .
• pronoun resolution is choosing the right

disambiguation

• syntax says next to nothing about how to

disambiguate pronouns

37

PRONOUNS denote variables

• a pronoun, he, denotes a variable, x7
• infjnitely many variables (x1, x2, x3, . . .) and so

pronouns are infjnitely ambiguous alternatively: push ambiguity from interface into lexicon

• he1, he2, he3, . . .
• pronoun resolution is choosing the right

disambiguation

• syntax says next to nothing about how to

disambiguate pronouns

37

Pronouns denote VARIABLES

• what does it mean to ’denote’ a variable???
• look at the semantics of the λ calculus

[[Γ ⊢ xα]] = λg.g(x) [[Γ ⊢ (Mα→βNα)β]] = λg.[[Γ ⊢ Mα→β]] g ([[Γ ⊢ Nα]] g) [[Γ ⊢ (λxα.Mβ)α→β]] = λg, y.[[Γ ⊢ Mβ]] g[x := y]

38

Pronouns denote VARIABLES

• what does it mean to ’denote’ a variable???
• look at the semantics of the λ calculus

[[Γ ⊢ xα]] = λg.g(x) [[Γ ⊢ (Mα→βNα)β]] = λg.[[Γ ⊢ Mα→β]] g ([[Γ ⊢ Nα]] g) [[Γ ⊢ (λxα.Mβ)α→β]] = λg, y.[[Γ ⊢ Mβ]] g[x := y]

38

Pronouns as variables

We write:

• [[xi]]g = gi

This really means:

• [[xi]] = λg.gi

And so:

• xi : γ → e

Compositionality? Having pronouns denote variables is as directly compositional as anything else

39

PRONOUNS denote identity functions

• a pronoun, he, denotes the identity function λx.x
• just one identity function, so no ambiguity
• pronoun resolution is resolving syntactic ambiguity
• expressions keep track of contained pronouns
• contexts are broken up and passed to the correct

daughter

• syntax completely determines reference
• John said that Bill likes his mother

is 3 ways syntactically ambiguous

40

Pronouns denote IDENTITY FUNCTIONS

• we look at a resource sensitive λ calculus

[[xα ⊢ xα]] = λp.p [[Γ, ∆ ⊢ (Mα→βNα)β]] = λp.let (p1, p2) := splitAt |Γ| p in [[Γ ⊢ Mα→β]] p1 ([[∆ ⊢ Nα]] p2) [[Γ ⊢ (λxα.Mβ)α→β]] = λp, y.[[Γ, xα ⊢ Mβ]] (p ++ y)

41

Pronouns denote IDENTITY FUNCTIONS

• we look at a resource sensitive λ calculus

[[xα ⊢ xα]] = λp.p [[Γ, ∆ ⊢ (Mα→βNα)β]] = λp.let (p1, p2) := splitAt |Γ| p in [[Γ ⊢ Mα→β]] p1 ([[∆ ⊢ Nα]] p2) [[Γ ⊢ (λxα.Mβ)α→β]] = λp, y.[[Γ, xα ⊢ Mβ]] (p ++ y)

41

Pronouns denote IDENTITY FUNCTIONS

• we look at a resource sensitive λ calculus

[[xα ⊢ xα]] = λp.p [[Γ, ∆ ⊢ (Mα→βNα)β]] = λp.let (p1, p2) := splitAt |Γ| p in [[Γ ⊢ Mα→β]] p1 ([[∆ ⊢ Nα]] p2) [[Γ ⊢ (λxα.Mβ)α→β]] = λp, y.[[Γ, xα ⊢ Mβ]] (p ++ y)

41

Contexts

• Both require sentences to be interpreted wrt a

sequence of individuals

• represents the context in which the sentence was

uttered

• assignment functions are very poor representations
• f context!
• which assignment represents this context?

variables γ = EN id γ := ∞

n=0 En

• pronouns are defjned
• nly on contexts of size

1

42

Resolving reference in context

• Pronouns are resolved to an individual in a context
• very poor strategies for resolving reference

variables [[he7]](c1, c2, . . . ) = c7 ’pick the seventh thing in the context’ id [[he]](c) = c ’pick the only thing in the context’

43

Pronouns as paraphrases

• pronouns mean defjnite descriptions
• either because they syntactically are dds
• or because they mean the same thing as a dd
• dds pick out an individual in a ’minimal situation’
• situations are everywhere!!!

Elbourne [[it]] = λf, s.ιx.f(λs.x)(s) not type raised: [[it]] = λs.ιx.x ∈ s

44

A Synthesis

• a pronoun denotes a function of type γ → e
• i.e. pronoun resolution algorithms
• γ is the type of a context

This is:

• just what we’ve been doing all along,
• except without a priori commitments to the

metaphysics of contexts

45

γ is the type of a context

a context can be whatever you want it to be

• assignments
• a list of discourse referents

(Vermeulen)

• situations
• fjle cards

(Heim)

• a relational database of common assumptions

(Lebedeva)

• a pair of

(Asher & Pogodalla)

• available discourse references
• modal base

46

Interacting with contexts

We will revise this later:

• sel : γ → e

chooses a salient individual from the context

• upd : γ → e → γ

updates the context by introducing a discourse referent Notation gx means upd g x

47

Contextifjcation

The intuition Everything is interpreted with respect to a context On types e = γ → e t = γ → t

48

Inherent Context Sensitivity

pronouns are inherently context sensitive he := λg.sel g They must access the context! determiners are inherently context sensitive det := λP, Q, g.det (λx.P Kx gx) (λx.Q Kx gx) They update the context

49

Context homomorphism

con con(he) = sel con(det) = λP, Q, g.det(λx.P(Kx)(gx))(λx.Q(Kx)(gx)) con(k) = Con(Kk)

50

Context Lifting

Context insensitive expressions are predictable Conα : (γ → α) → α Cone(a) := a Cont(φ) := φ Conαβ(f) := Conβ ◦ Sf ◦ Nocα := λA.Conβ(λg.f g (Nocα A g)) Noc Noce A A Noct Noc F Noc F Con g a Noc F Con a g

51

Context Lifting

Context insensitive expressions are predictable Con Cone a a Cont Con f Con f Noc A Con g f g Noc A g Nocα : α → (γ → α) Noce(A) := A Noct(Φ) := Φ Nocαβ(F) := C(Nocβ ◦ F ◦ Conα ◦ K) := λg, a.Nocβ(F (Conα (Ka))) g

51

Examples

• Contt(Knot) := λΦ, g.not (Φ g)
• Coneet(Kpraise) := λA, B, g.praise (A g) (B g)
• Con(et)(et)t(Kevery) :=

λP, Q, g.every(λx.P Kx g)(λx.Q Kx g)

52

An example

Start with: every : (et)(et)t ’s : (eet)ee boy : et mother : eet kiss : eet he : e Then φ = every(boy)(λx.kiss(’s(mother)(he))(x)) : t And con(φ) : γ → t Where con(φ) ≡ λg.every(boy)(λx.kiss(’s(mother)(sel(gx))))

53

Contexts as theories

γ := t Lebedeva a context is a logical theory (representing the commitments of the discourse participants) sel : (e → t) → γ → e sel(P)(g) : {e : E|g P e}

• select an individual the context can prove has

property P upd : γ → t → γ upd g φ := φ ∧ g

54

Theories and world knowledge

I bought a car, but the rear-view-mirror was broken World knowledge cars have rear-view-mirrors [[the]]([[r-v-m]])([[car]](u) ∧ g) g ∪ wk ∃y.r-v-m(y) ∧ has(u)(y)

55

Paycheck pronouns

Every boy loves his mother, but every girl hates her. World knowledge humans have mothers Frame problem We know too much stuff Every boy loves his mother,…

• ‘mother’ world knowledge becomes salient

(Wilks ’73; Hobbes ’78)

• a semantic version of the full NP analysis

56

Summary

• can study logic of context-sensitivity
• independently of dynamics

57

Putting it all together Examples

58

Laughing jumping boys

Start with: a : (et)(et)t boy : et jump : et laugh : et and : ttt he : e Let φ = and(a(boy)(jump))(laugh(he)) : t Then con(dyn(φ)) : (g → t) → g → t Where con(dyn(φ)) ≡ λψ, g.a(boy)(λx.boy x ∧ laugh (sel gx) ∧ ψ gx)

59

Conditional Donkeys

60

Relative Donkeys

61

Summary

62

Dynamic Semantics

Dynamic semantics is the combination of two domains:

• context-sensitivity
• discourse semantics

On types e = γ → e t = (γ → t) → γ → t ≡ γ → (γ → t) → t The intuition

• individuals are interpreted wrt a context
• discourses are functions from contexts to

propositions

• a sentence takes a context, and a discourse, and

interprets that discourse in an updated context

63

Conclusion

• Modular approach allows us to
• understand logic and structure of phenomena an

sich

• and view ‘the world’ as the interaction/interleaving
• f simpler parts
• with proof that lifting preserves semantics
• Granny’s theory of pronouns very workable
• modular
• delimits role of semantics and role of ‘inference’
• extensible
• intuitive

64

Dynamics

65

Dynamic Montague Grammar

On types e = γ → e t = γ → γ → t The intuition

• Individuals are interpreted in a context
• Sentences have context-change potential

66

Dynamic Lifting

Dynα : (γ → α) → α Dyne(a) := a Dynt(φ) := λg, h.h = h ∧ φ g Dynαβ(f) := λA.Dynβ(λg.f g (Staα A g)) Sta Stae A A Stat g h g h Sta F g a Sta F Dyn a g

67

Dynamic Lifting

Dyn Dyne a a Dynt g h h h g Dyn f A Dyn g f g Sta A g Staα : α → γ → α Stae(A) := A Stat(Φ) := λg.∃h.Φ g h Staαβ(F) := λg, a.Staβ(F (Dynα a)) g

67

Examples

• Dyntt(id) := λΦ, g, h.g = h ∧ ∃k.Φ g k
• Dyntt(not) := λΦ, g, h.g = h ∧ not (∃k.Φ g k)
• Dynttt(or) := λΦ, Ψ, g, h.g = h ∧ ∃k.Φ g k ∧ Ψ g k

68

Exceptions

• dyn(and) := λΦ, Ψ, g, h.∃k.Φ g k ∧ Ψ k h
• dyn(∃) := λP, g, h.∃(λx.P Kx gx h)

69

Differences

• Truly dynamic
• Reifjcation of contexts (cf. Quine on existing)
• no (obvious) decomposition of dynamics and

context-sensitivity possible

70