Hyperparamodulation: A refinement of paramodulation

Abstract

A refinement of paramodulation, called hyperparamodulation, is the focus of attention in this paper. Clauses obtained by the use of this inference rule are, in effect, the result of a sequence of paramodulations into one common nucleus. Among the interesting properties of hyperparamodulation are: first, clauses are chosen from among the input and designated as nuclei or "into" clauses for paramodulation; second, terms in the nucleus are starred to restrict the domain of generalized equality substitution; third, total control is thus iteratively established over all possible targets for paramodulation during the entire run of the theorem-proving program; and fourth, application of demodulation is suspended until the hyperparamodulant is completed. In contrast to these four properties which are reminiscent of the spirit of hyper-resolution, the following differences exist: first, the nucleus and the starred terms therein, which are analogous to negative literals, are determined by the user rather than by syntax; second, nuclei are not restricted to being mixed clauses; and third, while hyper-resolution requires inferred clauses to be positive, no corresponding requirement exists for clauses inferred by hyperparamodulation.

To illustrate the value of this refinement of paramodulation, we have chosen certain conjectures which arose during the study of Robbins algebra. A Robbins algebra is a set on which the functions o and n are defined such that o is both commutative and associative and such that for all x and y the following identity

$$n(o(n(o(x,y)),n(o(x,n(y))))) = x$$

holds. One may think of o as union and n as complement. The main interest in such algebras arises from the following open question: If S is a Robbins algebra, is S necessarily a Boolean algebra? The study of this open question entailed heavy use of an automated theorem-proving program to examine various conjectures. Certain computer proofs therein were obtained only after recourse to hyperparamodulation. (These lemmas were actually obtained prior to the work reported on here by Winker and Wos using a non-standard theorem-proving approach developed by Winker.)

This work was funded by the Applied Mathematical Sciences Program of the Office of Energy Research, U.S. Department of Energy and supported in part by the National Science Foundation under grants MCS-7903870 and MCS-7913252.