The complexity of Model-Preference Default theories
Most formal theories of default inference have very poor computational properties, and are easily shown to be intractable, or worse, undecidable. We are therefore investigating limited but efficiently computable theories of default reasoning. This paper defines systems of Propositional Model Preference Defaults, which provide a true model-theoretic account of default inference with exceptions. Some of our results extend to other nonmonotonic formalisms, such as Default Logic.
The most general system of Model Preference Defaults is decidable but still intractable. Inspired by the very good (linear) complexity of propositional Horn theories, we consider systems of Horn Defaults. Surprisingly, finding a most-preferred model in even this very limited system is shown to be NP-Hard. Tractability can be achieved in two ways: by eliminating the “specificity ordering” among default rules, thus limiting the system's expressive power; and by restricting our attention to systems of Acyclic Horn Defaults. These acyclic theories can encode inheritance hierarchies of the form examined by Touretzky, but are strictly more general.
This analysis suggests several directions for future research: finding other syntactic restrictions which permit efficient computation; or more daringly, investigation of default systems whose implementations do not require checking global consistency — that is, fast “approximate” inference.
KeywordsExpressive Power Search Problem Maximal Model Default Rule Default Theory
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