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Natural Language Processing

Natural Language Processing. Semantics II The Lambda Calculus Semantic Representation Encoding in Prolog. The Lambda Calculus. The λ -calculus allows us to write down the definition of a function without inventing a special name for it.

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Natural Language Processing

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  1. Natural Language Processing Semantics II The Lambda Calculus Semantic Representation Encoding in Prolog Semantics II

  2. 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. Semantics II

  3. 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. Semantics II

  4. λ-expressions asPartial 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” Semantics II

  5. 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. Semantics II

  6. S write(bertrand,principia) VP y.write(y,principia) NP bertrand bertrand V x.y.write(y,x) NP principia writes principia Parse Tree with Logical Forms Semantics II

  7. 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. Semantics II

  8. Encoding the Semantic System • Devise an encoding for logical forms. • Associate an encoded λ expression with each constituent. • Encode process of β-reduction This can all be done with Prolog! Semantics II

  9. Encode Logical Forms Semantics II

  10. 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 haves(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. Semantics II

  11. 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. Semantics II

  12. A Very Simple DCGthat 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]. Semantics II

  13. Demo ?- s(LF,[suzie,walks], [ ]). LF = walk(suzie). ?- s(LF,[suzie,kicks,fido], [ ]). LF = kick(suzie,fido). Semantics II

  14. 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], []) Semantics II

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