This note is based on lecture notes for the Caltech course Math 6c, prepared with A. Kechris and M. Shulman.
We would like to have a mechanical procedure (algorithm) for checking whether a given set of formulas logically implies another, that is, given , whether
is a tautology (i.e., it is true under all truth-value assignments.)
This happens iff
So it suffices to have an algorithm to check the (un)satisfiability of a single propositional formula. The method of truth tables gives one such algorithm. We will now develop another method which is often (with various improvements) more efficient in practice.
It will be also an example of a formal calculus. By that we mean a set of rules for generating a sequence of strings in a language. Formal calculi usually start with a certain string or strings as given, and then allow the application of one or more “rules of production” to generate other strings.
A formula is in conjunctive normal form iff it has the form
where each has the form
and each is either a propositional variable, or its negation. So is in conjunctive normal form iff it is a conjunction of disjunctions of variables and negated variables. The common terminology is to call a propositional variable or its negation a literal.
Suppose is a propositional statement which we want to test for satisfiability. First we note (without proof) that although there is no known efficient algorithm for finding in cnf (conjunctive normal form) equivalent to , it is not hard to show that there is an efficient algorithm for finding in cnf such that:
is satisfiable iff is satisfiable.
(But, in general, has more variables than .)
So from now on we will only consider s in cnf, and the Resolution Method applies to such formulas only. Say
with literals. Since order and repetition in each conjunct (1):
Such a set of literals is called a clause. It corresponds to the formula (1). So the formula above can be simply written as a set of clauses (again since the order of the conjunctions is irrelevant):
Satisfiability of means then simultaneous satisfiability of all of its clauses , i.e., finding a valuation which makes true for each , i.e., which for each makes some true.
From now on we will deal only with a set of clauses . It is possible to consider also infinite sets , but we will not do that here.
Satisfying means (again) that there is a valuation which satisfies all , i.e. if , then for all there is so that it makes true.
Notice that if the set of clauses is associated as above to (in cnf) and to , then
By convention we also have the empty clause , which contains no literals. The empty clause is (by definition) unsatisfiable, since for a clause to be satisfied by a valuation, there has to be some literal in the clause which it makes true, but this is impossible for the empty clause, which has no literals. For a literal , let denote its “conjugate”, i.e. if and if
Definition 1 Suppose now are three clauses. We say that is a resolvent of if there is a such that , and
We allow here the case , i.e. .