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5.10.1 Theorem Proving in FOL with Resolution Principle
Suppose, we have to prove a theorem Th from a set of axioms. We denote it
by
Let
{ A1, A2, ...., An}_ Th
A1 = Biscuit (coconut-crunchy)
A2 = Child (mary) ∧ Takes (mary, coconut-crunchy)
A3 = ∀ X (Child(X) ∧ ∃ Y (Takes (X,Y) ∧ Biscuit (Y))) →
Loves (john, X)
and Th = Loves (john, mary) = A4 (say).
Now, to prove the above theorem, we would use Robinson’s inference
rule. First of all, let us express A1 through A4 in CNF. Expressions A1 and A4
are already in CNF. Expression A2 can be converted into CNF by breaking it
into two clauses:
Child (mary) and
Takes (mary, coconut-crunchy).
Further, the CNF of expression A3 is
¬Child (X) ∨ ¬Takes (X,Y) ∨ ¬Biscuit (Y) ∨ Loves (john, X)
It can now be easily shown that the negation of the theorem (goal) if
resolved with the CNF form of expressions A1 through A3, the resulting
expression would be a null clause for a valid theorem. To illustrate this, we
will now form pairs of clauses, one of which contains a positive predicate,
while the other contains the same predicate in negated form. Thus by
Robinson’s rule, both the negated and positive predicates will drop out and
the value of the variables used for unification should be substituted in the
resulting expression. The principle of resolution is illustrated below (fig. 5.4)
to prove the goal that Loves (john, mary).
5.11 Different Types of Resolution
The principle of resolution can be extended to different forms. But an overextension
may cause fatal errors. This section illustrates the diversified use of
the resolution principle with the necessary precautions to avoid the scope of
mistakes by the beginners.