LF-Interpretation, Compositionally Greg Kobele University of - - PowerPoint PPT Presentation

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LF-Interpretation, Compositionally Greg Kobele University of - - PowerPoint PPT Presentation

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LF-Interpretation, Compositionally Greg Kobele University of Chicago Dec 02 Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 1 / 45 Compositionality 1 Cooper Storage Laws 2 Formal Consequences 3 Interpreting Tucking-in 4

slide-1
SLIDE 1

LF-Interpretation, Compositionally

Greg Kobele

University of Chicago

Dec 02

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 1 / 45

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SLIDE 2

1

Compositionality

2

Cooper Storage Laws

3

Formal Consequences

4

Interpreting Tucking-in

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 2 / 45

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SLIDE 3

Plan

Give a compositional semantics for minimalism

Main claim

LF-interpretation can be viewed directly compositionally

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 3 / 45

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SLIDE 4

Semantics in Generative Grammar

Binary branching nodes

  • α

β

  • g

= [[α]]g ⊕ [[β]]g Unary branching nodes

  • α
  • g

= [[α]]g Binding

  • i

α

  • g

= λ λx.[[α]]g[i:=x] Traces [[ti]]g = g(i)

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 4 / 45

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SLIDE 5

Interpreting λ terms in type structures

Application [[(M N)]]g = [[M]]g ([[N]]g) Abstraction [[λi.M]]g = λ λx.[[M]]g[i:=x] Variables [[i]]g = g(i) Binary branching nodes

  • α

β

  • g

= [[α]]g ⊕ [[β]]g Unary branching nodes

  • α
  • g

= [[α]]g Binding

  • i

α

  • g

= λ λx.[[α]]g[i:=x] Traces [[ti]]g = g(i)

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 5 / 45

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SLIDE 6

Parts and their meanings

Most expressions don’t have any meaning

   

V V praise D D every N boy

   

g

= [[praise]]g ⊕

  • D

D every N boy

  • g

= [[praise]]g ⊕ ([[every]]g ⊕ [[boy]]g) [[every]]g ⊕ [[boy]]g : (et)t [[praise]]g : eet these cannot be combined! FA αβ → α → β BA α → αβ → β PM αt → αt → αt

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 6 / 45

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SLIDE 7

Revisiting meaningless parts

merge praise merge every boy ⇓ V V praise D D every N boy What is the contribution of praise every boy to expressions it is part of? a quantifier part every(boy)(λx. . . . and a property part praise(x) Let’s write instead: [every(boy)]x ⊢ praise(x)

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 7 / 45

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SLIDE 8

Notation and Operations

[every(boy)]x ⊢ praise(x) The general case, with multiple stored quantifiers: [Q1]x1, . . . , [Qi]xi ⊢ M

The entire point is to ignore what is stored

M

↑

⊢ M Γ ⊢ M ∆ ⊢ N

<*>

Γ, ∆ ⊢ M N

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 8 / 45

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SLIDE 9

Working with Storage

seem

↑

⊢ seem Pass

↑

⊢ Pass . . . [every(boy)]x ⊢ praise(x)

<*>

[every(boy)]x ⊢ Pass(praise(x))

<*>

[every(boy)]x ⊢ seem(Pass(praise(x)))

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 9 / 45

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SLIDE 10

Building praise every boy

praise

↑

⊢ praise every

↑

⊢ every boy

↑

⊢ boy <*> ⊢ every boy <*> type mismatch!

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 10 / 45

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SLIDE 11

Building praise every boy

praise

↑

⊢ praise every

↑

⊢ every boy

↑

⊢ boy <*> ⊢ every boy

We want to ’insert a trace’

⊢ M

  • [M]x ⊢ x

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 10 / 45

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SLIDE 12

Building praise every boy

praise

↑

⊢ praise every

↑

⊢ every boy

↑

⊢ boy <*> ⊢ every boy

  • [every boy]x ⊢ x

We want to ’insert a trace’

⊢ M

  • [M]x ⊢ x

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 10 / 45

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SLIDE 13

Building praise every boy

praise

↑

⊢ praise every

↑

⊢ every boy

↑

⊢ boy <*> ⊢ every boy

  • [every boy]x ⊢ x

<*>

[every boy]x ⊢ praise x

We want to ’insert a trace’

⊢ M

  • [M]x ⊢ x

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 10 / 45

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SLIDE 14

Taking things out of storage

seem

↑

⊢ seem Pass

↑

⊢ Pass . . . [every(boy)]x ⊢ praise(x)

<*>

[every(boy)]x ⊢ Pass(praise(x))

<*>

[every(boy)]x ⊢ seem(Pass(praise(x)))

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 11 / 45

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SLIDE 15

Taking things out of storage

seem

↑

⊢ seem Pass

↑

⊢ Pass . . . [every(boy)]x ⊢ praise(x)

<*>

[every(boy)]x ⊢ Pass(praise(x))

<*>

[every(boy)]x ⊢ seem(Pass(praise(x)))

retrieval

Γ, [Mi]xi, ∆ ⊢ N

·i

⊕

Γ, ∆ ⊢ Mi ⊕ (λxi.N)

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 11 / 45

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SLIDE 16

Taking things out of storage

seem

↑

⊢ seem Pass

↑

⊢ Pass . . . [every(boy)]x ⊢ praise(x)

<*>

[every(boy)]x ⊢ Pass(praise(x))

<*>

[every(boy)]x ⊢ seem(Pass(praise(x)))

·1

FA

⊢ every(boy)(λx.seem(Pass(praise(x))))

retrieval

Γ, [Mi]xi, ∆ ⊢ N

·i

⊕

Γ, ∆ ⊢ Mi ⊕ (λxi.N)

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 11 / 45

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SLIDE 17

Manipulating Stores

pure

M

↑

⊢ M

apply

Γ ⊢ M ∆ ⊢ N

<*>

Γ, ∆ ⊢ M N

retrieve

Γ, [Mi]xi, ∆ ⊢ N

·i

⊕

Γ, ∆ ⊢ Mi ⊕ (λxi.N)

store

⊢ M

  • [M]x ⊢ x

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 12 / 45

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SLIDE 18

Understanding stores

[M1]x1, . . . , [Mi]xi ⊢ N ⇒ λk.k M1 . . . Mi (λx1, . . . , xi.N)

Example

[every boy]x ⊢ praise x ⇒ λk.k (every boy) (λx.praise x)

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 13 / 45

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SLIDE 19

Some examples

pure

M

↑

⊢ M ⇓ M

↑

λk.k M M↑ ≡ λk.k M

storage

⊢ M

  • [M]x ⊢ x

⇓ λk.k M

  • λk.k M (λx.x)

m ≡ λk.m (λM.k M (λx.x))

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 14 / 45

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SLIDE 20

More notation

idiom brackets

write ( |f a1 . . . ai| ) for f ↑ <*> a1 <*> . . . <*> ai

application

Forward f ⊲ a := f a Backward a ⊳ f := f a

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 15 / 45

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SLIDE 21

Minimalist semantics

[[merge]] → λm, n.( |m ⊕ n| ) [[merge]] → λm, n.( |m ⊕ n| ) [[move]] → λm.m [[move]] → λm.mk

⊕

[[ℓ]] = I(ℓ)↑ for ⊕ ∈ {⊲, ⊳}

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 16 / 45

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SLIDE 22

Unpacking the notation

Recall that

λm, n.( |m ⊲ n| ) means λm, n.(⊲)↑ <*> m <*> n ⊲ ↑ ⊢ ⊲ (m) Γ ⊢ M <*> Γ ⊢ M ⊲ (n) ∆ ⊢ N <*> Γ, ∆ ⊢ M ⊲ N

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 17 / 45

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SLIDE 23

Every boy laughs

[[move]] [[merge]] [[will]] [[merge]] [[laugh]] [[merge]] [[every]] [[boy]]

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

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SLIDE 24

Every boy laughs

[[move]] [[merge]] I(will)↑ [[merge]] I(laugh)↑ [[merge]] I(every)↑ I(boy)↑

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

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SLIDE 25

Every boy laughs

[[move]] [[merge]] ⊢ will [[merge]] ⊢ laugh [[merge]] ⊢ every ⊢ boy

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

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SLIDE 26

Every boy laughs

[[move]] [[merge]] ⊢ will [[merge]] ⊢ laugh λm, n.( |m ⊲ n| ) ⊢ every ⊢ boy

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

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SLIDE 27

Every boy laughs

[[move]] [[merge]] ⊢ will [[merge]] ⊢ laugh ⊢ every boy

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

slide-28
SLIDE 28

Every boy laughs

[[move]] [[merge]] ⊢ will λm, n.( |m ⊲ n| ) ⊢ laugh ⊢ every boy

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

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SLIDE 29

Every boy laughs

[[move]] [[merge]] ⊢ will [every boy]x ⊢ laugh x

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

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SLIDE 30

Every boy laughs

[[move]] λm, n.( |m ⊲ n| ) ⊢ will [every boy]x ⊢ laugh x

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

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SLIDE 31

Every boy laughs

[[move]] [every boy]x ⊢ will (laugh x)

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

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SLIDE 32

Every boy laughs

λm.m1

⊲

[every boy]x ⊢ will (laugh x)

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

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SLIDE 33

Every boy laughs

⊢ every boy (λx.will (laugh x))

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

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SLIDE 34

1

Compositionality

2

Cooper Storage Laws

3

Formal Consequences

4

Interpreting Tucking-in

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 19 / 45

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SLIDE 35

Plan

Present algebraic laws of cooper storage Introduce delimited continuations

Main claim

LF-interpretation à la H&K is based on delimited continuations

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 20 / 45

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SLIDE 36

Applicative Functor Laws

identity id↑ <*> u = u composition ((◦↑ <*> u) <*> v) <*> w = u <*> (v <*> w) homomorphism f ↑ <*> x↑ = (f x)↑ interchange u <*> x↑ = (λP.Px)↑ <*> u

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 21 / 45

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SLIDE 37

Deriving homomorphism

homomorphism

f ↑ <*> x↑ = (f x)↑ f

↑

⊢ f x

↑

⊢ x

<*>

⊢ f x = f x

↑

⊢ f x

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 22 / 45

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SLIDE 38

Laws Particular to Cooper Storage

1 That which is stored, can be retrieved

E[kM]k

N = N↑ <*> M <*> E[kid↑]k ⊲

2 Storing and retrieving vacuity is vacuous

u <*> kid↑k

⊲ = u

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 23 / 45

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SLIDE 39

Deriving Law 1

Retrieval from storage

E[kM]k

N = N↑ ⊕ M ⊕ E[kid↑]k ⊲

⊢ M

k

[M]xk ⊢ xk . . . Γ, [M]xk, ∆ ⊢ O

·k

N

Γ, ∆ ⊢ N M (λxk.O) = N

↑

⊢ N ⊢ M

<*>

⊢ N M id

↑

⊢ id

k

[id]xk ⊢ xk . . . Γ, [id]xk, ∆ ⊢ O

·k

⊲

Γ, ∆ ⊢ id ⊲ (λxk.O) ........................... Γ, ∆ ⊢ λxk.O

<*>

Γ, ∆ ⊢ N M (λxk.O)

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 24 / 45

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SLIDE 40

shift0 and reset

x (M N) λx.M kM Mk β (λx.M) N M[x := N] η λx.(M x) M shift0 E[kM]k M (λx.E[x]k) reset V k V

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 25 / 45

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SLIDE 41

shift0 and reset

x (M N) λx.M kM Mk β (λx.M) N M[x := N] η λx.(M x) M shift0 E[kM]k M (λx.E[x]k) reset V k V E[kM]k M (λx.E[x]k) E[kM]k

⊲ = M <*> E[kid↑]k ⊲

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 25 / 45

slide-42
SLIDE 42

Minimalist semantics (via delimited continuations)

[[merge]] → λm, n.( |m ⊕ n| ) [[merge]] → λm, n.( |m ⊕ n| ) [[move]] → λm.m [[move]] → λm.mk

⊕

[[ℓ]] = I(ℓ)↑ for ⊕ ∈ {⊲, ⊳}

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 26 / 45

slide-43
SLIDE 43

Minimalist semantics (via delimited continuations)

[[merge]] → λm, n.m ⊕ n [[merge]] → λm, n.( |m ⊕ n| ) [[move]] → λm.m [[move]] → λm.mk

⊕

[[ℓ]] = I(ℓ) for ⊕ ∈ {⊲, ⊳}

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 26 / 45

slide-44
SLIDE 44

Minimalist semantics (via delimited continuations)

[[merge]] → λm, n.m ⊕ n [[merge]] → λm, n.( |m ⊕ n| ) [[move]] → λm.m [[move]] → λm.mk

⊕

[[ℓ]] = I(ℓ) for ⊕ ∈ {⊲, ⊳}

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 26 / 45

slide-45
SLIDE 45

Minimalist semantics (via delimited continuations)

[[merge]] → λm, n.m ⊕ n [[merge]] → λm, n.m ⊕ k(λo.n ⊕ o) [[move]] → λm.m [[move]] → λm.mk [[ℓ]] = I(ℓ) for ⊕ ∈ {⊲, ⊳}

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 26 / 45

slide-46
SLIDE 46

1

Compositionality

2

Cooper Storage Laws

3

Formal Consequences

4

Interpreting Tucking-in

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 27 / 45

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SLIDE 47

Plan

give examples of the benefits of viewing LF-interpretation compositionally

Main claim

LF-interpretation not only can be viewed directly compositionally, it should be.

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 28 / 45

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SLIDE 48

Generative Capacity

The set of meanings assigned to sentences of a grammar is a linear higher order IO language and thus the set of structures which yield a given λ-term can be found in polynomial time

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 29 / 45

slide-49
SLIDE 49

Decompositionality

any part of a derivation has a normal semantic value

                                    =      λx.

x

                             

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 30 / 45

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SLIDE 50

Top-down parsing

λm. m

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-51
SLIDE 51

Top-down parsing

move λm. mk

⊲

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-52
SLIDE 52

Top-down parsing

move merge λt, v. ⊲↑ <*> t <*> vk

⊲

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-53
SLIDE 53

Top-down parsing

move merge λt, v. ⊲↑ <*> t <*> vk

⊲

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-54
SLIDE 54

Top-down parsing

move merge λt, v. t <*> vk

⊲

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-55
SLIDE 55

Top-down parsing

move merge merge λt, V , s. t <*> (⊲↑ <*> V <*> ks)k

⊲

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-56
SLIDE 56

Top-down parsing

move merge merge λt, V , s. t <*> (⊲↑ <*> V <*> ks)k

⊲

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-57
SLIDE 57

Top-down parsing

move merge merge λt, V , s. t <*> (V <*> ks)k

⊲

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-58
SLIDE 58

Top-down parsing

move merge merge λt, V , s. t <*> (V <*> ks)k

⊲

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-59
SLIDE 59

Top-down parsing

move merge merge λt, V , s. s <*> t <*> (V <*> kid↑)k

⊲

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-60
SLIDE 60

Top-down parsing

move merge merge λt, V , s. s <*> t <*> (V <*> kid↑)k

⊲

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-61
SLIDE 61

Top-down parsing

move merge merge λt, V , s. s <*> (◦↑ <*> t <*> V ) <*> kid↑k

⊲

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-62
SLIDE 62

Top-down parsing

move merge merge λt, V , s. s <*> (◦↑ <*> t <*> V ) <*> kid↑k

⊲

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-63
SLIDE 63

Top-down parsing

move merge merge λt, V , s. s <*> (◦↑ <*> t <*> V )

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-64
SLIDE 64

Top-down parsing

move merge merge merge λt, V , d, n. (⊲↑ <*> d <*> n) <*> (◦↑ <*> t <*> V )

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-65
SLIDE 65

Top-down parsing

move merge merge merge λt, V , d, n. (⊲↑ <*> d <*> n) <*> (◦↑ <*> t <*> V )

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-66
SLIDE 66

Top-down parsing

move merge merge merge λt, V , d, n. (d <*> n) <*> (◦↑ <*> t <*> V )

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-67
SLIDE 67

Top-down parsing

move merge merge merge every λt, V , n. (every ↑ <*> n) <*> (◦↑ <*> t <*> V )

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-68
SLIDE 68

Top-down parsing

move merge merge merge every boy λt, V . (every ↑ <*> boy ↑) <*> (◦↑ <*> t <*> V )

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-69
SLIDE 69

Top-down parsing

move merge merge merge every boy λt, V . (every ↑ <*> boy ↑) <*> (◦↑ <*> t <*> V )

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-70
SLIDE 70

Top-down parsing

move merge merge merge every boy λt, V . (every boy)↑ <*> (◦↑ <*> t <*> V )

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-71
SLIDE 71

Top-down parsing

move merge will merge merge every boy λV . (every boy)↑ <*> (◦↑ <*> will↑ <*> V )

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-72
SLIDE 72

Top-down parsing

move merge will merge merge every boy λV . (every boy)↑ <*> (◦↑ <*> will↑ <*> V )

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-73
SLIDE 73

Top-down parsing

move merge will merge merge every boy λV . (every boy)↑ <*> ((λf .will ◦ f )↑ <*> V )

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-74
SLIDE 74

Top-down parsing

move merge will merge merge every boy λV . (every boy)↑ <*> ((λf .will ◦ f )↑ <*> V )

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-75
SLIDE 75

Top-down parsing

move merge will merge merge every boy λV . (λP.every boy(λx.will (P x)))↑ <*> V

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-76
SLIDE 76

Top-down parsing

move merge will merge laugh merge every boy (λP.every boy(λx.will (P x)))↑ <*> laugh↑

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-77
SLIDE 77

Top-down parsing

move merge will merge laugh merge every boy (λP.every boy(λx.will (P x)))↑ <*> laugh↑

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-78
SLIDE 78

Top-down parsing

move merge will merge laugh merge every boy (every boy(λx.will (laugh x)))↑

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 31 / 45

slide-79
SLIDE 79

Ellipsis

move merge

Tpast

merge

vpass

merge praise Mary move merge

Tpast

eNP→VP Harry [[ move ]]( [[ merge ]]([[Tpst]])([[ e ]]([[Harry]]))) λm.(Pass ◦ praise)↑ <*> m

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 32 / 45

slide-80
SLIDE 80

1

Compositionality

2

Cooper Storage Laws

3

Formal Consequences

4

Interpreting Tucking-in

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 33 / 45

slide-81
SLIDE 81

Plan

give compositional semantics for parasitic scope

  • ibid. for tucking-in movement

Main claim

Tucking-in type movement involves complex quantifier formation

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 34 / 45

slide-82
SLIDE 82

Parasitic Scope

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 35 / 45

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SLIDE 83

Before all movement

[same]Q, [everyone]x ⊢ serve x (the (Q waiter)) The types of the stored elements are as follows: same (Aet)et

A = (et)et

everyone (et)t

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 36 / 45

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SLIDE 84

Before all movement

[same]Q, [everyone]x ⊢ serve x (the (Q waiter)) The types of the stored elements are as follows: same (Aet)et

A = (et)et

everyone (et)t

What we want:

⊢ everyone (same (λQ, x.serve x (the (Q waiter))))

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 36 / 45

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SLIDE 85

Before all movement

[same]Q, [everyone]x ⊢ serve x (the (Q waiter)) The types of the stored elements are as follows: same (Aet)et

A = (et)et

everyone (et)t

What we want:

⊢ (everyone ◦ same) (λQ, x.serve x (the (Q waiter)))

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 36 / 45

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SLIDE 86

Parasitic Storage (Sketch)

[same]Q, [everyone]x ⊢ serve x (the (Q waiter)) [everyone ◦ same]Q;x ⊢ serve x (the (Q waiter)) ⊢ (everyone ◦ same) (λQ, x.serve x (the (Q waiter)))

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 37 / 45

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SLIDE 87

Parasitic Storage

[same]Q, [everyone]x ⊢ serve x (the (Q waiter)) [Π everyone same]z ⊢ let (x, Q) be z in serve x (the (Q waiter)) Γ, [M : (αβ)γ]x, [N : (δαη)αβ]y, ∆ ⊢ O : ζ

fuse

Γ, [Π M N : ((δ ⊗ α)η)γ]z, ∆ ⊢ let (x, y) be z in O : ζ where Π M N := M ◦ N ◦ flip ◦ curry = λR.M (N (λy, x.R (x, y)))

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 38 / 45

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SLIDE 88

Tucking-in Movement

C who C x C C T tx T T V bought what C who C x C what C y C C T tx T T V boughtty

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 39 / 45

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SLIDE 89

Before all movement

[who]x, [what]y ⊢ bought y x

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 40 / 45

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SLIDE 90

Before all movement

[who]x, [what]y ⊢ bought y x

What we want:

⊢ who (λx.what (λy.bought y x))

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 40 / 45

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SLIDE 91

Before all movement

[who]x, [what]y ⊢ bought y x

What we want:

⊢ (who ◦ (λR, x.what (λy.R y x))) bought

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 40 / 45

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SLIDE 92

Before all movement

[who]x, [what]y ⊢ bought y x

What we want:

⊢ (who ◦ (L what)) bought L := B(CBC)B = λD, R, x.D(λy.Ryx)

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 40 / 45

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SLIDE 93

About L

GQs as arity reducers (Keenan, 2016)

A DP denotes a function of type ∀n.(en+1t)ent

For D : (et)t

Ln D : (en+1t)ent = λR, x1, . . . , xn.D (λy.R y x1 . . . xn)

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 41 / 45

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SLIDE 94

Tucked Storage (Sketch)

[who]x, [what]y ⊢ bought y x [who ◦ (L what)]x;y ⊢ bought y x ⊢ (who ◦ (L what)) (λx, y.bought y x)

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 42 / 45

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SLIDE 95

Tucked Storage

[who]x, [what]y ⊢ bought y x [Π who (L what)]z ⊢ let (x, y) be z in bought y x Γ, [M : (et)t]x, [N : (et)t]y, ∆ ⊢ O : ζ

tuck

Γ, [Π M (L N) : ((e ⊗ e)t)t]z, ∆ ⊢ let (x, y) be z in O : ζ where Π M N := M ◦ N ◦ flip ◦ curry = λR.M (N (λy, x.R (x, y)))

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 43 / 45

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SLIDE 96

Parasitic vs Tucked Storage

[everyone]x, [same]Q ⊢ serve x (the (Q waiter)) [Π everyone same]z ⊢ let (x, Q) be z in serve x (the (Q waiter)) [who]x, [what]y ⊢ bought y x [Π who (L what)]z ⊢ let (x, y) be z in bought y x

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 44 / 45

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SLIDE 97

Parasitic vs Tucked Storage

[M]x, [N]y ⊢ O [Π M N]z ⊢ let (x, y) be z in O [M]x, [N]y ⊢ O [Π M (L N)]z ⊢ let (x, y) be z in O

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 44 / 45

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SLIDE 98

Parasitic vs Tucked Storage

[M]x, [N]y ⊢ O [Π M (Ln N)]z ⊢ let (x, y) be z in O

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 44 / 45

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SLIDE 99

Conclusion

LF-interpretation is in fact directly compositional This gives us access to a wealth of results and tools

generative capacity generation delimited continuations

gives new questions to explore

semantic bootstrapping?! conditioning parsing on meaning?! connex to cogsci via probabilistic programming?!

and addresses old ones

LF? Ellipsis? ACD? . . .

Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 45 / 45