The idea of circumscription can be explained on a simple example. We would like to represent information about the locations of blocks in a blocks world, using the "default":

Circumscription and the closed world assumption with its variants are well-known nonmonotonic techniques for reasoning with incomplete knowledge. Their complexity in the propositional case has been studied in detail for fragments of propositional logic. One open problem is whether the deduction problem for arbitrary propositional theories under the extended closed world assumption or under circumscription is $\Pi^P_2$-complete, i.e., complete for a class of the second level of the polynomial hierarchy. We answer this question by proving these problems $\Pi^P_2$-complete, and we show how this result applies to other variants of closed world reasoning.

this paper we carry out a detailed analysis of the computational complexity of model checking for propositional circumscriptive formulae. Following Schaefer's approach [6], we classify propositional formulae according to the logical relations which are used to represent them. Using the results reported in that paper we prove polynomial as well as co-NP-hardness results, thus providing a detailed picture of the tractability threshold of the problem. In particular we answer negatively to Kolaitis and Papadimitriou's question, showing that the model checking problem is co-NP-complete for a subclass of the propositional formulae. The structure of the paper is as follows: In Section 2 we recall the basic notions on circumscription and formally define our problem, and in Section 3 we present the results of the complexity analysis. 2 Preliminaries

In recent years, a great deal of attention has been devoted to logics of "commonsense" reasoning. Among the candidates proposed, circumscription has been perceived as an elegant mathematical technique for modeling nonmonotonic reasoning, but di cult to apply in practice. The major reason for this is the 2nd-order nature of circumscription axioms and the di culty in nding proper substitutions of predicate expressions for predicate variables. One solution to this problem is to compile, where possible, 2nd-order formulas into equivalent 1st-order formulas. Although some progress has been made using this approach, the results are not as strong as one might desire and they are isolated in nature. In this article, we provide a general method which can be used in an algorithmic manner to reduce circumscription axioms to 1st-order formulas. The algorithm takes as input an arbitrary 2nd-order formula and either returns as output an equivalent 1st-order formula, or terminates with failure. The class of 2nd-order formulas, and analogously the class of circumscriptive theories which can be reduced, provably subsumes those covered by existing results. We demonstrate the generality of the algorithm using circumscriptive theories with mixed quanti ers (some involving Skolemization), variable constants, non-separated formulas, and formulas with n-ary predicate variables. In addition, we analyze the strength of the algorithm and compare it with existing approaches providing formal subsumption results.

. We prove that -- unless the polynomial hierarchy collapses at the second level -- the size of a purely propositional representation of the circumscription CIRC(T ) of a propositional formula T grows faster than any polynomial as the size of T increases. We then analyze the significance of this result in the related field of closed-world reasoning. Appeared on the Proceedings of the 12th Symposium on Theoretical Aspects of Computer Science (STACS'95) March 2-4, 1995, Munchen, Germany Lecture Notes in Computer Science, 900, pages 205--216, Springer-Verlag 1 Introduction Reasoning with selected (or intended) models of a logical formula is a common reasoning technique used in Databases, Logic Programming, Knowledge Representation and Artificial Intelligence (AI). One of the most popular criteria for selecting intended models is minimality wrt the set of true atoms. The idea behind minimality is to assume that a fact is false whenever possible. Such a criterion allows one to represent o...

We investigate the space e#ciency of a Propositional Knowledge Representation (PKR) formalism. Intuitively, the space e#ciency of a formalism F in representing a certain piece of knowledge #, is the size of the shortest formula of F that represents #. In this paper we assume that knowledge is either a set of propositional interpretations (models) or a set of propositional formulae (theorems). We provide a formal way of talking about the relative ability of PKR formalisms to compactly represent a set of models or a set of theorems. We introduce two new compactness measures, the corresponding classes, and show that the relative space e#ciency of a PKR formalism in representing models/theorems is directly related to such classes. In particular, we consider formalisms for nonmonotonic reasoning, such as circumscription and default logic, as well as belief revision operators and the stable model semantics for logic programs with negation. One interesting result is that formalisms ...

We provide a general method which can be used in an algorithmic manner to reduce certain classes of 2nd-order circumscription axioms to logically equivalent 1st-order formulas. The algorithm takes as input an arbitrary 2nd-order formula and either returns as output an equivalent 1st-order formula, or terminates with failure. In addition to demonstrating the algorithm by applyingittovarious circumscriptive theories, we analyze its strength and provide formal subsumption results based on comparison with existing approaches. 1

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Various semantics for logic programs with negation are described in terms of a dualized program together with additional axioms, some of which are second order formulas. The semantics of Clark, Fitting, and Kunen are characterized in this framework, and a nite rst-order presentation of Kunen's semantics is described. A new axiom to represent \common sense " reasoning is proposed for logic programs. It is shown that the well-founded semantics and stable models are de nable with this axiom. The roles of domain augmentation and domain closure are examined. A \domain foundation " axiom is proposed to replace the domain closure axiom. 1

OF THE DISSERTATION The Complexity of Querying Indefinite Information: Defined Relations, Recursion and Linear Order by Ronald van der Meyden, Ph.D. Dissertation Director: L.T. McCarty This dissertation studies the computational complexity of answering queries in logical databases containing indefinite information arising from two sources: facts stated in terms of defined relations, and incomplete information about linearly ordered domains. First, we consider databases consisting of (1) a DATALOG program and (2) a description of the world in terms of the predicates defined by the program as well as the basic predicates. The query processing problem in such databases is related to issues in database theory, including view updates and DATALOG optimization, and also to the Artificial Intelligence problems of reasoning in circumscribed theories and sceptical abductive reasoning. If the program is non-recursive, the meaning of the database can be represented by Clark's Predicate Completion,...