# Definition:Decision Procedure

## Definition

Let $S$ be a set of propositional formulas.

A **decision procedure for $S$** is an algorithm which, given a propositional formula $\mathbf A$, always terminates, returning the answer:

**Yes**if $\mathbf A \in S$;**No**if $\mathbf A \notin S$.

### Decision Procedure for Satisfiability

Let $U$ be the set of satisfiable propositional formulas.

Then a decision procedure for $U$ is called a **decision procedure for satisfiability**.

### Decision Procedure for Tautologies

Let $U$ be the set of propositional formulas that are tautologies.

Then a decision procedure for $U$ is called a **decision procedure for tautologies**.

### Refutation Procedure

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): $\S 2.5.1$: Definition $2.40$