Is intractability of nonmonotonic reasoning a real drawback?

Abstract

Several studies about computational complexity of nonmonotonic reasoning (NMR) showed that nonmonotonic inference is significantly harder than classical, monotonic inference. This contrasts with the general idea that NMR can be used to make knowledge representation and reasoning simpler, not harder. In this paper we show that, to some extent, NMR fulfills the representation goal. In particular, we prove that nonmonotonic formalisms such as circumscription and default logic allow for a much more compact and natural representation of propositional knowledge than propositional calculus. Proofs are based on a suitable definition of a compilable inference problem, and on non-uniform complexity classes. Some results about intractability of circumscription and default logic can therefore be interpreted as the price one has to pay for having such an extra-compact representation. On the other hand, intractability of inference and compactness of representation are not equivalent notions: we exhibit intractable nonmonotonic formalisms whose nonmonotonic assumptions are representable by few propositional formulae. Finally, sometimes NMR really makes reasoning simpler. We present prototypical scenarios where closed-world reasoning and well-founded semantics account for a faster, complete and unsound approximation of classical reasoning.