Non-Monotonic Reasoning with Quantified Boolean Constraints

In this paper, we define and investigate the complexity of several nonmonotonic logics with quantified Boolean formulas as constraints. We give quantified constraint versions of the constraint programming formalism of Marek, Nerode, and Remmel [15] and of the natural extension of their theory to default logic. We also introduce a new formalism which adds constraints to circumscription. We show that standard complexity results for each of these formalisms generalize in the quantified constraint case. Gogic, Kautz, Papadimitriou, and Selman [8] have introduced a new method for measuring the strengths of reasoning formalisms based on succinctness of model representation. We show a natural hierarchy based on this measure exists between our versions of logic programming, circumscription, and default logic. Finally, we discuss some results about the relative succinctness of our reasoning formalisms versus any formalism for which model checking can be done somewhere in the polynomial time hierarchy.