Definition:Decision Procedure/Refutation Procedure

Definition

Given a decision procedure for satisfiability, one can craft a decision procedure for tautologies in the following way:

Suppose one wanted to decide if a propositional formula $\mathbf A$ is a tautology.

Then apply the given procedure to decide if its negation $\neg \mathbf A$ is satisfiable.

Now:

If $\neg \mathbf A$ is not satisfiable, then by Tautology iff Negation is Unsatisfiable, $\mathbf A$ is a tautology.

If $\neg \mathbf A$ is satisfiable, then by Satisfiable iff Negation is Falsifiable, $\mathbf A$ is falsifiable, so cannot be a tautology.

Hence we have crafted a decision procedure for tautologies.

Such a procedure is called a refutation procedure, because it proceeds by refuting, i.e. proving unsatisfiability of the negation of the formula at hand.

Sources

• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): $\S 2.5.1$