Redundancy and Deletion

Abstract

The problems of showing the existence of proofs and finding proofs of given theorems mark the borderline between mathematical logic and computer science. So far we merely proved results about the existence of refutations for unsatisfiable sets of clauses under various types of refinements; we have not spoken about how to (really) obtain refutations. In our formalism the situation can be described in the following way: Let Ψ be an arbitrary complete refinement and C be an unsatisfiable set of clauses. By the completeness of Ψ there exists a refutation Γ ∈ Ψ(∁); finding such a Γ (within reasonable computing time) is the main problem of automated deduction. At this point we face the problem of search which is of central importance to all fields of Artificial Intelligence. With regard to a resolution refinement, search is an algorithmic method for producing the elements of Ψ(∁) until a refutation is found (in principle we can try to find all Ψ-refutations of ∁, but such a procedure usually is nonterminating). The computational cost of proof search is, in practice, the main obstruction to automated theorem proving. For this reason, several techniques have been invented to reduce search. We list three of them: