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e proved for linear resolution, it can be proved for other resolutions (like
unit resolution) as well. To prove the soundness of the resolution theorem,
we use the following three steps:
Step 1: After the proof procedure terminates, back substitute the constants by
variables in the tree.
Step 2: Now instantiate these clauses with all possible constants. We thus get
the Herbrand base corresponding to the clauses that participated in the proof
procedure.
Step 3: The resolution theorem of propositional logic is now applied to that
subset of the Herbrand base. Note that the propositional resolution theorem,
employed here, is sound.
Since the elements of Herbrand base also include the clauses that participated
in the resolution proof of predicate logic, the proof procedure of the resolution
theorem in predicate logic is also sound [2].
5.14 Conclusions
The chapter presented the syntax and semantics of propositional and predicate
logics and demonstrated their applications in logical theorem proving. Many
AI problems, which can be represented as theorem proving problems, thus can
be handled with the concept outlined in the chapter. The resolution theorem,
being the fundamental theorem under the proposed framework of knowledge,
its soundness and completeness have been discussed in detail. The semidecidablity
of FOL has also been covered briefly. The shortcomings of double
resolution, as a common mistake, have also been pointed out. This will help
the students to properly identify the use of the resolution theorem.
Exercises
1. Prove that for the atomic propositions p, q, r and s
a) p, ¬ q ⇒ r ≡ p ⇒ q, r and
b) p, q ⇒ ¬ r, s ≡ p, q, r ⇒ s
Could you remember the use of the above tautologies in Wang’s algorithm? If
yes, in which steps did you use them?
2. Verify the following theorems by Wang’s algorithm.
a) p \/ q, p → r, q → r ⇒ r