We study the complexity of the inference problem for propositional circumscription (the minimal inference problem) over arbitrary finite domains. The problem is of fundamental importance in nonmonotonic logics and commonsense reasoning. The complexity of the problem for the two-element domain has been completely classified [Durand, Hermann, and Nordh, Trichotomy in the complexity of minimal inference, LICS 2009]. In this paper, we classify the complexity of the problem over all conservative languages. We consider a version of the problem parameterized by a set of relations (a constraint language), from which we are allowed to build a knowledge base, and where a linear order used to compare models is a part of an input. We show that in this setting the problem is either Π_2~P-complete, coNP-complete, or in P. The classification is based on a coNP-hardness proof for a new class of languages, an analysis of languages that do not express any member of the class and a new general polynomial-time algorithm solving the minimal inference problem for a large class of languages.
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