October 2004csa4050: Semantics II1 CSA4050: Advanced Topics in NLP Semantics II The Lambda Calculus Semantic Representation Encoding in Prolog

October 2004csa4050: Semantics II2 The Lambda Calculus The λ-calculus allows us to write down the definition of a function without inventing a special name for it. We use the notation λx.ϕ where x is a variable marking the argument and ϕ is an expression defining the value of the function at that argument, e.g. λx.x+1. We allow the whole expression to stand in the place of a function symbol. So (λx.x+1)(3) is a well-formed term that denotes that function applied to the argument 3.

October 2004csa4050: Semantics II3 β-Reduction The rule of β-reduction says that an expression of the form λx.ϕ(a) can be reduced to ϕ{x=a}, i.e. the expression ϕ with all occurrences of x replaced with a. In this case (λx.x+1)(3) = 3+1. In the semantics we shall be developing, many intermediate LFs will have the form of propositions with certain parts missing. These can be modelled as functions over propositions expressed with λ-expressions.

October 2004csa4050: Semantics II4 λ-expressions as Partial Propositions to walk: λx.walk(x) John: john; Fido: fido λx.walk(x)(john) = walk(john) to kick: λx.λy.kick(x,y). λx.λy.kick(x,y)(john) = λy.kick(john,y) λy.kick(john,y)(fido) = kick(john,fido) λ-calculus can be used to model “semantic operations”

October 2004csa4050: Semantics II5 Rule to Rule Hypothesis: The Sentence Rule Syntactic Rule: S  NP VP Semantic Rule: [S] = [VP]([NP]) i.e. the LF of S is obtained by "applying" the LF of VP to the LF of NP. For this to be possible [VP] must be a function, and [NP] the argument to the function.

October 2004csa4050: Semantics II6 S write(bertrand,principia) NP bertand VP y.write(y,principia) V x. y.write(y,x) NP principia bertrand writes principia Parse Tree with Logical Forms

October 2004csa4050: Semantics II7 Summary Leaves of the tree are words. Words (or lexical entries) are associated with “semantic forms” by the dictionary (or lexicon) Grammar determines how to combine words and phrases syntactically. Associated semantic rules determine how to combine respective semantic forms.

October 2004csa4050: Semantics II8 Encoding the Semantic System 1.Encode logical forms. 2.Associate an encoded λ expression with each constituent. 3.Encode process of β-reduction

October 2004csa4050: Semantics II9 Encode Logical Forms LFProlog  x ϕ all(X, ϕ’ )  x ϕ exist(X, ϕ’ ) &, v,  &, v, => λx. ϕ X^ ϕ’ λx. λy. ϕ X^Y^ ϕ’

October 2004csa4050: Semantics II10 Associate an encoded λ expression with each constituent Reserve an argument position in a DCG rule to hold the logical form encoding. For example, ignoring the particular constraints governing the use of the rule, we might have s(S) --> np(NP), vp(VP). i.e. sentence with LF S can be formed by concatenating a noun phrase with LF NP and a verb phrase with LF VP.

October 2004csa4050: Semantics II11 Encode Process of β-reduction This is done by means of the predicate reduce(Fn,Arg,Result), which is defined by means of a unit clause as follows: reduce(X^F,X,F). NB. This predicate only performs a single, outermost reduction. It does not reduce to a canonical form.

October 2004csa4050: Semantics II12 A Very Simple DCG that computes Semantics % grammar s(S) --> np(NP), vp(VP), {reduce(VP,NP,S)}. vp(VP) --> v(V), np(NP), {reduce(V,NP,VP}. vp(VP) --> v(VP). % lexicon v(X^walk(X)) --> [walks]. v(X^Y^hit(X,Y)) --> [hits]. np(suzie) --> [suzie]. np(fido) --> [fido].

October 2004csa4050: Semantics II13 Demo ?- s(LF,[suzie,walks], [ ]). LF = walk(suzie). ?- s(LF,[suzie,kicks,fido], [ ]). LF = kick(suzie,fido).

October 2004csa4050: Semantics II14 Execution Trace Call: (7) s(_G471, [suzie, walks], []) Call: (8) np(_L183, [suzie, walks], _L184) Exit: (8) np(suzie, [suzie, walks], [walks]) Call: (8) vp(_L185, [walks], _L186) Call: (9) v(_L224, [walks], _L225) Exit: (9) v(_G529^walk(_G529), [walks], []) Call: (9) np(_L226, [], _L227) Fail: (9) np(_L226, [], _L227) Redo: (9) v(_L224, [walks], _L225) Redo: (8) vp(_L185, [walks], _L186) Call: (9) v(_L185, [walks], _L186) Exit: (9) v(_G529^walk(_G529), [walks], []) Exit: (8) vp(_G529^walk(_G529), [walks], []) Call: (8) reduce(_G529^walk(_G529), suzie, _G471) Exit: (8) reduce(suzie^walk(suzie), suzie, walk(suzie)) Call: (8) []=[] Exit: (8) []=[] Exit: (7) s(walk(suzie), [suzie, walks], [])