The complexity of path-based defeasible inheritance

Abstract

Touretzky [22] proposed a formalism for nonmonotonic multiple inheritance reasoning which is sound in the presence of ambiguities and redundant links. We show that Touretzky's inheritance notion is NP-hard, and thus, provided P ≠ NP, computationally intractable. This result holds even when one only considers unambiguous, totally acyclic inheritance networks. A direct consequence of this result is that the conditioning strategy proposed by Touretzky to allow for fast parallel inference is also intractable. We also analyze the influence of various design choices made by Touretzky. We show that all versions of so-called downward inheritance, i.e., on-path or off-path preemption and skeptical or credulous reasoning, are intractable. However, on the positive side, tractability can be achieved when using upward inheritance. Finally, we consider the problem of determining what holds in all extensions of an ambiguous inheritance network. Without any preemption—corresponding to a very cautious form of inheritance—this problem is tractable, but when allowing even the most basic forms of preemption, it becomes intractable.