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18
Circumscription
, 1996
"... 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": ..."
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Cited by 324 (13 self)
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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":
Propositional Circumscription and Extended Closed World Reasoning are $\Pi^P_2$complete
 Theoretical Computer Science
, 1993
"... Circumscription and the closed world assumption with its variants are wellknown 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 prob ..."
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Cited by 99 (22 self)
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Circumscription and the closed world assumption with its variants are wellknown 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.
The Complexity of Model Checking for Circumscriptive Formulae
 Information Processing Letters
, 1992
"... 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 repo ..."
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Cited by 52 (5 self)
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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 coNPhardness 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 coNPcomplete 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
Computing circumscription revisited: A reduction algorithm
 J. Automated Reasoning
, 1997
"... 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 ..."
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Cited by 44 (14 self)
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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 2ndorder 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, 2ndorder formulas into equivalent 1storder 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 1storder formulas. The algorithm takes as input an arbitrary 2ndorder formula and either returns as output an equivalent 1storder formula, or terminates with failure. The class of 2ndorder 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, nonseparated formulas, and formulas with nary predicate variables. In addition, we analyze the strength of the algorithm and compare it with existing approaches providing formal subsumption results.
On Compact Representations of Propositional Circumscription
 Theoretical Computer Science
, 1997
"... . 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 res ..."
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Cited by 33 (12 self)
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. 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 closedworld reasoning. Appeared on the Proceedings of the 12th Symposium on Theoretical Aspects of Computer Science (STACS'95) March 24, 1995, Munchen, Germany Lecture Notes in Computer Science, 900, pages 205216, SpringerVerlag 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...
Space Efficiency of Propositional Knowledge Representation Formalisms
 IN PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON THE PRINCIPLES OF KNOWLEDGE REPRESENTATION AND REASONING (KR'96
, 2000
"... We investigate the space efficiency of a Propositional Knowledge Representation (PKR) formalism. Intuitively, the space efficiency 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 ..."
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Cited by 26 (3 self)
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We investigate the space efficiency of a Propositional Knowledge Representation (PKR) formalism. Intuitively, the space efficiency 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 efficiency 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 ...
Elimination of Predicate Quantifiers
 UWE REYLE, HANS JÜRGEN OHLBACH (EDS.): LOGIC, LANGUAGE AND REASONING  ESSAYS IN HONOUR OF DOV GABBAY
"... ..."
Commonsense axiomatizations for logic programs
 Journal of Logic Programming
, 1993
"... 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 rstorder presentation of Kunen's seman ..."
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Cited by 11 (1 self)
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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 rstorder 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 wellfounded 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
The Complexity Of Querying Indefinite Information: Defined Relations, Recursion And Linear Order
, 1992
"... 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 indefin ..."
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Cited by 7 (0 self)
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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 nonrecursive, the meaning of the database can be represented by Clark's Predicate Completion,...
Existential SecondOrder Logic over Strings
, 1998
"... Existential secondorder logic (ESO) and monadic secondorder logic (MSO) have attracted much interest in logic and computer science. ESO is a much more expressive logic over successor structures than MSO. However, little was known about the relationship between MSO and syntactic fragments of ESO ..."
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Cited by 6 (1 self)
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Existential secondorder logic (ESO) and monadic secondorder logic (MSO) have attracted much interest in logic and computer science. ESO is a much more expressive logic over successor structures than MSO. However, little was known about the relationship between MSO and syntactic fragments of ESO. We shed light on this issue by completely characterizing this relationship for the prefix classes of ESO over strings, (i.e., finite successor structures). Moreover, we determine the complexity of model checking over strings, for all ESOprefix classes. Let ESO(Q) denote the prefix class containing all sentences of the shape 9RQ' where R is a list of predicate variables, Q is a firstorder quantifier prefix from the prefix set Q, and ' is quantifierfree. We show that ESO(9 88) are the maximal standard ESOprefix classes contained in MSO, thus expressing only regular languages.