Subsumption algorithms based on search trees

Abstract

Clause subsumption is of fundamental importance for reducing the search space in theorem proving systems. Since the subsumption problem is NP-complete, the design of efficient heuristics is of significant interest. The core of all subsumption algorithms is the search for suitable substitutions for the variables in a given clause, which in all previously known algorithms is implicitly embedded in the control structure of the algorithm. In this paper we adopt a more abstract view of subsumption and introduce the concept of a subsumption search tree to separate the search control from other computational tasks, such as computing substitutions and verifying their consistency. We study key algorithmic aspects of search trees and of heuristics for constructing them. For instance, the complexity of a search-tree based algorithm depends on the height of the search tree. We show that the problem of constructing minimal-height search trees is NP-complete. We also derive improved upper bounds on the height of search trees constructed according to an analysis based on variable dependencies, as proposed by Gottlob and Leitsch; and show that the bound is essentially tight in the worst case, by establishing suitable lower bounds for arbitrary search trees. In addition to these theoretical results, we propose further algorithmic improvements based on more sophisticated data structures for computing and representing substitutions. Finally, we have implemented several variants of our proposed algorithm and report on corresponding experiments.

This work was supported in part by NSF grants CCR-9404921, CDA-9303181, CDA-9504275 and INT-9314412.