Fast Subsumption Checks Using Anti-Links

Abstract

The concept of anti-link is defined (an anti-link consists of two occurrences of the same literal in a formula), and useful equivalence-preserving operations based on anti-links are introduced. These operations eliminate a potentially large number of subsumed paths in a negation normal form formula. Those anti-links that directly indicate the presence of subsumed paths are characterized. The operations have linear time complexity in the size of that part of the formula containing the anti-link. The problem of removing all subsumed paths in an NNF formula is shown to be NP-hard, even though such formulas may be small relative to the size of their path sets. The general problem of determining whether there exists a pair of subsumed paths associated with an arbitrary anti-link is shown to be NP-complete. Additional techniques that generalize the concept of pure literals are introduced and are also shown to eliminate redundant subsumption checks. The effectiveness of these techniques is examined with respect to some benchmark examples from the literature.