Resolutiomenetelmän

Resolutiomenetelmä, often translated as the resolution method, is a fundamental proof procedure in automated theorem proving and mathematical logic. It is a refutation complete proof technique, meaning that if a statement is logically valid, the resolution method can prove it. It operates on clauses, which are disjunctions of literals. A literal is a propositional variable or its negation.

The core of the resolution method lies in its inference rule, the resolution rule. Given two clauses, say A or P and B or not P, where A and B are disjunctions of literals not containing P or not P respectively, the resolution rule produces a new clause A or B. This new clause, called the resolvent, is a logical consequence of the original two clauses. The process aims to derive a contradiction, specifically the empty clause, which represents falsehood.

To use resolution for proving a statement S, one first converts S into a set of clauses in Conjunctive Normal Form (CNF). Then, the negation of S, also converted into CNF, is added to this set. The resolution method is then applied iteratively to pairs of clauses in the combined set. If the empty clause can be derived, it signifies a contradiction, meaning that the initial set of clauses (statement S and its negation) is unsatisfiable. This, in turn, proves that the original statement S is valid. The resolution method is particularly significant for its completeness and its role in developing automated reasoning systems.