1. Natural Language Processing
Semantics II
The Lambda Calculus
Semantic Representation
Encoding in Prolog
November 2006
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.
November 2006
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.
November 2006
Semantics II

4. 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”
November 2006
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.
November 2006
Semantics II

6. Parse Tree with Logical Forms
write(bertrand,principia) NP
bertrand VP
y.write(y,principia) V
x.y.write(y,x)
principia
writes
November 2006
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.
November 2006
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!
November 2006
Semantics II

9. Encode Logical Forms LF Prolog x ϕ all(X,ϕ’) x ϕ exist(X,ϕ’) &, v, 
λx. λy. ϕ
X^Y^ϕ’
November 2006
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 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.
November 2006
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.
November 2006
Semantics II

12. 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].
November 2006
Semantics II

13. Demo ?- s(LF,[suzie,walks], [ ]). LF = walk(suzie).
?- s(LF,[suzie,kicks,fido], [ ]).
LF = kick(suzie,fido).
November 2006
Semantics II

14. Execution Trace Call: (8) np(_L183, [suzie, walks], _L184)
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: (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], [])
November 2006
Semantics II