On some tractable classes in deduction and abduction

Abstract

We address the identification of propositional theories for which entailment is tractable, so that every query about logical consequences of the theory can be answered in polynomial time. We map tractable satisfiability classes to tractable entailment classes, including hierarchies of tractable problems; and show that some initially promising conditions for tractability of entailment, proposed by Esghi (1993) and del Val (1994), surprisingly only identify a subset of renamable Horn. We then consider a potential application of tractable entailment, through a reduction due to Esghi (1993) of certain abduction problems to a sequence of entailment problems. Besides clarifying the range of applicability of Esghi's results from the semantic point of view, we show that the reduction can almost trivially fail to be in any of the basic tractable classes discussed in the first part of the paper. We leave open the question of how to more broadly identify tractable entailment classes, as our examples suggest that there is room for progress in this area.