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

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

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

Semantics in Generative Grammar Binary branching nodes � g � � � • = [[ α ]] g ⊕ [[ β ]] g α β Unary branching nodes � g � � � • = [[ α ]] g α Binding � g � � � • λ x . [[ α ]] g [ i := x ] = λ α i Traces [[ t i ]] g = g ( i ) Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 4 / 45

Interpreting λ terms in type structures Application Binary branching nodes [[( M N )]] g = [[ M ]] g ([[ N ]] g ) � g � � � • = [[ α ]] g ⊕ [[ β ]] g α β Unary branching nodes � g � � � • = [[ α ]] g α Abstraction Binding [[ λ i . M ]] g = λ � g λ x . [[ M ]] g [ i := x ] � � � • λ x . [[ α ]] g [ i := x ] = λ α i Variables Traces [[ i ]] g = g ( i ) [[ t i ]] g = g ( i ) Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 5 / 45

Parts and their meanings Most expressions don’t have any meaning g     � g � � � V D = [[ praise ]] g ⊕ V D D N     praise D N every boy every boy = [[ praise ]] g ⊕ ([[ every ]] g ⊕ [[ boy ]] g ) [[ every ]] g ⊕ [[ boy ]] g : ( et ) t [[ praise ]] g : eet these cannot be combined! FA αβ → α → β PM α t → α t → α t BA α → αβ → β Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 6 / 45

Revisiting meaningless parts merge What is the contribution of praise every merge praise boy to expressions it is part of? every boy a quantifier part every ( boy )( λ x . . . . ⇓ and a property part praise ( x ) V V D Let’s write instead: praise D N [ every ( boy )] x ⊢ praise ( x ) every boy Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 7 / 45

Notation and Operations [ every(boy) ] x ⊢ praise ( x ) The general case, with multiple stored quantifiers: [ Q 1 ] x 1 , . . . , [ Q i ] x i ⊢ M The entire point is to ignore what is stored Γ ⊢ M ∆ ⊢ N M ↑ <*> ⊢ M Γ , ∆ ⊢ M N Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 8 / 45

Working with Storage . . . Pass ↑ ⊢ Pass [ every ( boy )] x ⊢ praise ( x ) seem ↑ <*> ⊢ seem [ every ( boy )] x ⊢ Pass ( praise ( x )) <*> [ every ( boy )] x ⊢ seem ( Pass ( praise ( x ))) Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 9 / 45

Building praise every boy boy every ↑ ↑ praise ⊢ every ⊢ boy <*> ↑ ⊢ praise ⊢ every boy <*> type mismatch! Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 10 / 45

Building praise every boy boy every ↑ ↑ praise ⊢ every ⊢ boy <*> ↑ ⊢ praise ⊢ every boy We want to ’insert a trace’ ⊢ M � [ M ] x ⊢ x Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 10 / 45

Building praise every boy every boy ↑ ↑ ⊢ every ⊢ boy <*> praise ⊢ every boy ↑ � ⊢ praise [ every boy ] x ⊢ x We want to ’insert a trace’ ⊢ M � [ M ] x ⊢ x Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 10 / 45

Building praise every boy boy every ↑ ↑ ⊢ every ⊢ boy <*> ⊢ every boy praise ↑ � ⊢ praise [ 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

Taking things out of storage . . . Pass ↑ ⊢ Pass [ every ( boy )] x ⊢ praise ( x ) seem ↑ <*> ⊢ seem [ every ( boy )] x ⊢ Pass ( praise ( x )) <*> [ every ( boy )] x ⊢ seem ( Pass ( praise ( x ))) Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 11 / 45

Taking things out of storage . . . Pass ↑ ⊢ Pass [ every ( boy )] x ⊢ praise ( x ) seem ↑ <*> ⊢ seem [ every ( boy )] x ⊢ Pass ( praise ( x )) <*> [ every ( boy )] x ⊢ seem ( Pass ( praise ( x ))) retrieval Γ , [ M i ] x i , ∆ ⊢ N �·� i ⊕ Γ , ∆ ⊢ M i ⊕ ( λ x i . N ) Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 11 / 45

Taking things out of storage . . . Pass ↑ ⊢ Pass [ every ( boy )] x ⊢ praise ( x ) seem ↑ <*> ⊢ seem [ every ( boy )] x ⊢ Pass ( praise ( x )) <*> [ every ( boy )] x ⊢ seem ( Pass ( praise ( x ))) �·� 1 FA ⊢ every ( boy )( λ x . seem ( Pass ( praise ( x )))) retrieval Γ , [ M i ] x i , ∆ ⊢ N �·� i ⊕ Γ , ∆ ⊢ M i ⊕ ( λ x i . N ) Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 11 / 45

Manipulating Stores pure apply M Γ ⊢ M ∆ ⊢ N ↑ <*> ⊢ M Γ , ∆ ⊢ M N retrieve Γ , [ M i ] x i , ∆ ⊢ N �·� i ⊕ Γ , ∆ ⊢ M i ⊕ ( λ x i . N ) store ⊢ M � [ M ] x ⊢ x Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 12 / 45

Understanding stores [ M 1 ] x 1 , . . . , [ M i ] x i ⊢ N ⇒ λ k . k M 1 . . . M i ( λ x 1 , . . . , x i . N ) Example [ every boy ] x ⊢ praise x ⇒ λ k . k ( every boy ) ( λ x . praise x ) Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 13 / 45

Some examples storage pure ⊢ M M � ↑ [ M ] x ⊢ x ⊢ M ⇓ ⇓ λ k . k M M ↑ � λ k . k M λ k . k M ( λ x . x ) M ↑ ≡ λ k . k M � m ≡ λ k . m ( λ M . k M ( λ x . x )) Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 14 / 45

More notation idiom brackets write ( | f a 1 . . . a i | ) for f ↑ <*> a 1 <*> . . . <*> a i application Forward f ⊲ a := f a Backward a ⊳ f := f a Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 15 / 45

Minimalist semantics [[ merge ]] �→ λ m , n . ( | m ⊕ n | ) [[ merge ]] �→ λ m , n . ( | m ⊕ � n | ) [[ move ]] �→ λ m . m [[ move ]] �→ λ m . � m � k ⊕ [[ ℓ ]] = I ( ℓ ) ↑ for ⊕ ∈ { ⊲ , ⊳ } Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 16 / 45

Unpacking the notation Recall that λ m , n . ( | m ⊲ n | ) means λ m , n . ( ⊲ ) ↑ <*> m <*> n ( m ) ⊲ ↑ ( n ) ⊢ ⊲ Γ ⊢ M <*> Γ ⊢ M ⊲ ∆ ⊢ N <*> Γ , ∆ ⊢ M ⊲ N Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 17 / 45

Every boy laughs [[ move ]] [[ merge ]] [[ will ]] [[ merge ]] [[ laugh ]] [[ merge ]] [[ every ]] [[ boy ]] Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

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

Every boy laughs [[ move ]] [[ merge ]] [[ merge ]] ⊢ will ⊢ laugh [[ merge ]] ⊢ every ⊢ boy Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

Every boy laughs [[ move ]] [[ merge ]] [[ merge ]] ⊢ will ⊢ laugh λ m , n . ( | m ⊲ n | ) ⊢ every ⊢ boy Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

Every boy laughs [[ move ]] [[ merge ]] [[ merge ]] ⊢ will ⊢ laugh ⊢ every boy Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

Every boy laughs [[ move ]] [[ merge ]] λ m , n . ( | m ⊲ � n | ) ⊢ will ⊢ laugh ⊢ every boy Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

Every boy laughs [[ move ]] [[ merge ]] [ every boy ] x ⊢ laugh x ⊢ will Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

Every boy laughs [[ move ]] λ m , n . ( | m ⊲ n | ) [ every boy ] x ⊢ laugh x ⊢ will Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

Every boy laughs [[ move ]] [ every boy ] x ⊢ will ( laugh x ) Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

Every boy laughs λ m . � m � 1 ⊲ [ every boy ] x ⊢ will ( laugh x ) Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

Every boy laughs ⊢ every boy ( λ x . will ( laugh x )) Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 18 / 45

Compositionality 1 Cooper Storage Laws 2 Formal Consequences 3 Interpreting Tucking-in 4 Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 19 / 45

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

Applicative Functor Laws identity id ↑ <*> u = u composition (( ◦ ↑ <*> u ) <*> v ) <*> w = u <*> ( v <*> w ) homomorphism f ↑ <*> x ↑ = ( f x ) ↑ interchange u <*> x ↑ = ( λ P . Px ) ↑ <*> u Greg Kobele (UofC) LF-Interpretation, Compositionally Dec 02 21 / 45