Notes on computational linguistics.pdf - UCLA Department of ...

Stabler - Lx 185/209 2003
compute sums on the natural numbers, as we will see.
From these axioms, prolog can refute ?- human(montague). This cannot be refuted using the proof rule
shown above, though, since no axiom has human(montague) asitshead.Theessenceofprologiswhatitdoes
here: it unifies the goal ?- human(montague) with the head of the axiom, human(X) :- mathematician(X).
We need to see exactly how this works.
Two expressions unify with each other just in case there is a substitution of terms for variables that makes
them identical. To unify human(montague) and human(X) we substitute the term montague for the variable X.
We will represent this substitution by the expression {X ↦ montague}. Letting θ ={X ↦ montague}, and
writing the substitution in “postfix” notation – after the expression it applies to, we have
human(X)θ =human(montague)θ =human(montague).
Notice that the substitution θ has no effect on the term human(montague) since this term has no occurrences
of the variable X.
We can replace more than one variable at once. For example, we can replace X by s(Y) and replace Y by Z.
Letting θ ={X ↦ s(Y),Y ↦ Z}, wehave:
sum(X,Y,Y)θ =sum(s(Y),Z,Z).
Notice that the Y in the first term has not been replaced by Z. This is because all the elements of the substitution
are always applied simultaneously, not one after another.
After a little practice, it is not hard to get the knack of finding substitutions that make two expressions
identical, if there is one. These substitutions are called (most general) unifiers, and the step of finding and
applying them is called (term) unification. 2
To describe unification we need two preliminaries. In the first place, we need to be able to recognize
subexpressions. Consider the formula:
whats_it(s(Y,r(Z,g(Var))),func(g(Argument),X),W).
The subexpression beginning with whats_it is the whole expression.
The subexpression beginning with s is s(Y, r(Z, g(Var))).
The subexpression beginning with Argument is Argument.
No subexpression begins with a parenthesis or comma.
The second preliminary we need is called composition of substitutions – we need to be able to build up
substitutions by, in effect, applying one to another. Remember that substitutions are specified by expressions
of the form
{V1 ↦ t1,...,Vn ↦ tn}
where the Vi are distinct variables and the ti are terms (ti = Vi) which are substituted for those variables.
Definition 1 The composition of substitutions η, θ is defined as follows: Suppose
The composition of η and θ, ηθ, is
η ={X1 ↦ t1,...,Xn ↦ tn}
θ ={Y1 ↦ s1,...,Ym ↦ sm}.
ηθ = {X1 ↦ (t1θ),...,Xn ↦ (tnθ), Y1 ↦ s1,...,Ym ↦ sm}−
({Yi ↦ si | Yi ∈{X1,...,Xn}} ∪ {Xi ↦ tiθ | Xi = tiθ}).
2So-called “unification grammars” involve a related, but slightly more elaborate notion of unification (Pollard and Sag, 1987; Shieber,
1992; Pollard and Sag, 1994).
Earlier predicate logic theorem proving methods like the one in Davis and Putnam (1960) were significantly improved by the discovery
in Robinson (1965) that term unification provided the needed matching method for exploiting the insight from the doctoral thesis of
Herbrand (1930) that proofs can be sought in a syntactic domain defined by the language of the theory.
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