Linear response time for implicate and implicant queries

Abstract

Knowledge bases can be represented as propositional Formulas. A query of such a theory typically has the form, Is a clause an implicate of the theory? Answering such queries can require exponential time. In Kautz and Selman (Proceedings of the international workshop on processing declarative knowledge (PDK), Kaiserslautern, Germany, 1991), knowledge compilation was proposed as a solution to this problem: Pay the exponential penalty once by compiling the knowledge base into a target language that would guarantee fast response to queries. The reduced implicate trie (ri-trie), introduced in Murray and Rosenthal (Proceedings of the international conference TABLEAUX 2005—analytic tableaux and related methods, Koblenz, Germany. Lecture notes in artificial intelligence, vol 3702. Springer, Berlin, pp 231–244, 2005), may be used as a target language for knowledge compilation. It has the property that a query is processed in time linear in the size of the query, regardless of the size of the compiled knowledge base. In this paper, structures dual to ri-tries, the reduced implicant tries are investigated, and the dual problem—determining the implicants of a formula—is considered. The main result is that, for a given formula, the two structures can be merged into a single reduced implicate/implicant trie that can serve dual roles, representing both implicates and implicants. Furthermore, rii-tries can be computed directly, without separately computing and then merging the dual structures.