Automated theorem-proving for the theories of partial and total ordering

Abstract

To make further progress, resolution principle programs need to make better inferences and to make them faster. Previous papers of the authors have presented a fairly general approach for taking some advantage of the structure of special theories, for example, the theories of equality, partial ordering and sets, and described experiments with a program base on this approach. The object of the approach is to replace some or all of the axioms of the given theory by (refutation) complete, valid, efficient (in time) inference rules. In this paper, the approach is used to develop an improved procedure for ‘building in’ partial ordering and a procedure for total ordering. These results may be stated roughly as follows. 1. If the five (not all independent) partial ordering axioms for {=, ≤, <} are replaced by the irreflexivity rule ri and the transitivity rule rt (for <), by an expansion rule, and by an extension to hyper-resolution, then refutation completeness is preserved. 2. If only the connectivity axiom, {x < y ∨ y] ≤ x}, is retained from the five total ordering axioms for {=, ≤, <} and if the other four are replaced by ri, rt, and an antisymmetry rule, refutation completeness is preserved. A program using total ordering inference rules is then described. Differences between the rules as presented in the theoretical development and as implemented in the program are noted. The paper concludes with a discussion of the program's successful performance on a large collection of problems taken from published papers.