Complexity of Nested Circumscription and Nested Abnormality Theories

The need for a circumscriptive formalism that allows for simple yet elegant modular problem representation has led Lifschitz (AIJ, 1995) to introduce nested abnormality theories (NATs) as a tool for modular knowledge representation, tailored for applying circumscription to minimize exceptional circumstances. Abstracting from this particular objective, we propose L_{CIRC}, which is an extension of generic propositional circumscription by allowing propositional combinations and nesting of circumscriptive theories. As shown, NATs are naturally embedded into this language, and are in fact of equal expressive capability. We then analyze the complexity of L_{CIRC} and NATs, and in particular the effect of nesting. The latter is found to be a source of complexity, which climbs the Polynomial Hierarchy as the nesting depth increases and reaches PSPACE-completeness in the general case. We also identify meaningful syntactic fragments of NATs which have lower complexity. In particular, we show that the generalization of Horn circumscription in the NAT framework remains CONP-complete, and that Horn NATs without fixed letters can be efficiently transformed into an equivalent Horn CNF, which implies polynomial solvability of principal reasoning tasks. Finally, we also study extensions of NATs and briefly address the complexity in the first-order case. Our results give insight into the ``cost'' of using L_{CIRC} (resp. NATs) as a host language for expressing other formalisms such as action theories, narratives, or spatial theories.Comment: A preliminary abstract of this paper appeared in Proc. Seventeenth International Joint Conference on Artificial Intelligence (IJCAI-01), pages 169--174. Morgan Kaufmann, 200

Value minimization in circumscription

Minimization in circumscription has focussed on minimizing the extent of a set of predicates (with or without priorities among them), or of a formula. Although functions and other constants may be left varying during circumscription, no earlier formalism to the best of our knowledge minimized functions. In this paper we introduce and motivate the notion of value minimizing a function in circumscription. Intuitively, value minimizing a function consists in choosing those models where the value of the function is minimal relative to an ordering on its range. We first give the formulation of value minimization of a single function based on a syntactic transformation and then give a formulation in model-theoretic terms. We then discuss value minimization of a set of functions with and without priorities. We show how Lifschitz's Nested Abnormality Theories can be used to express value minimization, and discuss the prospect of its use for knowledge representation, particularly in formalizing reasoning about actions

Value minimization in circumscription

Minimization in circumscription has focussed on minimizing the extent of a set of predicates (with or without priorities among them), or of a formula. Although functions and other constants may be left varying during circumscription, no earlier formalism to the best of our knowledge minimized functions. In this paper we introduce and motivate the notion of value minimizing a function in circumscription. Intuitively, value minimizing a function consists in choosing those models where the value of the function is minimal relative to an ordering on its range. We show how Lifschitz's Nested Abnormality Theories can be used to express value minimization, and discuss the prospect of its use for knowledge representation with classical logic. 1 Introduction Ever since Circumscription was defined, it has been possible to vary function symbols in the minimization process, i.e. to select minimal models irrespective of how function symbols are interpreted. The best known example is perhaps Baker's..