A dichotomy in the complexity of propositional circumscription

The inference problem for propositional circumscription is known to be highly intractable and, in fact, harder than the inference problem for classical propositional logic. More precisely, in its full generality this problem is II₂^P-complete, which means that it has the same inherent computational complexity as the satisfiability problem for quantified Boolean formulas with two alternations (universal-existential) of quantifiers. We use Schaefer's framework of generalized satisfiability problems to study the family of all restricted cases of the inference problem for propositional circumscription. Our main result yields a complete classification of the "truly hard" (II₂^P-complete) and the "easier" cases of this problem (reducible to the inference problem for classical propositional logic). Specifically, we establish a dichotomy theorem which asserts that each such restricted case either is II₂^P-complete or is in coNP. Morever, we provide efficiently checkable criteria that tell apart the "truly hard" cases from the "easier" ones.