Reasoning under minimal upper bounds in propositional logic

Reasoning from the minimal models of a theory, as fostered by circumscription, is in the area of Artificial Intelligence an important method to formalize common sense reasoning. However, as it appears, minimal models may not always be suitable to capture the intuitive semantics of a knowledge base, aiming intuitively at an exclusive interpretation of disjunctions of atoms, i.e., if possible then assign at most one of the disjuncts the value true in a model. In this paper, we consider an approach which is more lenient and also admits non-minimal models, such that inclusive interpretation of disjunction also may be possible in cases where minimal model reasoning adopts an exclusive interpretation. Nonetheless, in the spirit of minimization, the approach aims at including only positive information that is necessary. This is achieved by closing the set of admissible models of a theory under minimal upper bounds in the set of models of the theory, which we refer to as curbing. We demonstrate this method on some examples, and investigate its semantical and computational properties. We establish that curbing is an expressive reasoning method, since the main reasoning tasks are shown to be PSPACE-complete. On the other hand, we also present cases of lower complexity, and in particular cases in which the complexity is located, just as for ordinary minimal model reasoning, at the second level of the Polynomial Hierarchy, or even below.