The Concept of Demodulation in Theorem Proving

Abstract

In many fields of mathematics the richness of the underlying axiom set leads to the establishment of a number of very general equalities. For example, it is easy to prove that in groups (x ^-1)^-1= x and that in rings -x•-y = x •y. In the presence of such an equality, each new inference made during a proof search by a theorem-proving program may immediately yield a set of very closely related inferences. If, for example, b •a = c is inferred in the presence of (x^-1)^-1= x, substitution immediately yields obviously related inferences such as (b^-1)^-1 • a = c. Retention of many members of each such set of inferences has seriously impeded the effectiveness of automatic theorem proving. Similar to the gain made by discarding instances of inferences already present is that made by discarding instances of repeated application of a given equality. The latter is achieved by use of demodulation. Its definition, evidence of its value, and a related rule of inference are given. In addition a number of concepts are defined the implementation of which reduces both the number and sensitivity to choice of parameters governing the theorem-proving procedures.