Computing Circumscription Revisited: A Reduction Algorithm

In recent years, a great deal of attention has been devoted to logics of common-sense reasoning. Among the candidates proposed, circumscription has been perceived as an elegant mathematical technique for modeling nonmonotonic reasoning, but difficult to apply in practice. The major reason for this is the second-order nature of circumscription axioms and the difficulty in finding proper substitutions of predicate expressions for predicate variables. One solution to this problem is to compile, where possible, second-order formulas into equivalent first-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 that can be used in an algorithmic manner to reduce certain circumscription axioms to first-order formulas. The algorithm takes as input an arbitrary second-order formula and either returns as output an equivalent first-order formula, or terminates with failure. The class of second-order formulas, and analogously the class of circumscriptive theories that can be reduced, provably subsumes those covered by existing results. We demonstrate the generality of the algorithm using circumscriptive theories with mixed quantifiers (some involving Skolemization), variable constants, nonseparated formulas, and formulas with n-ary predicate variables. In addition, we analyze the strength of the algorithm, compare it with existing approaches, and provide formal subsumption results.

[1]  W. Ackermann.  Untersuchungen über das Eliminationsproblem der mathematischen Logik , 1935 .

[2]  J. McCarthy.  Circumscription|a Form of Nonmonotonic Reasoning , 1979 .

[3]  Jean-Pierre Bourguignon,et al.  Mathematische Annalen , 1893 .

[5]  Matthew L. Ginsberg.  A Circumscriptive Theorem Prover , 1989, Artif. Intell..

[6]  Patrick Doherty,et al.  A characterization result for circumscribed normal logic programs. Revised version accepted for publication : Special issue of honor of H. Rasiowa, Fundamenta Informaticae. , 1995 .

[7]  Raymond Reiter,et al.  A Logic for Default Reasoning , 1987, Artif. Intell..

[8]  Wilhelm Ackermann,et al.  Solvable Cases Of The Decision Problem , 1954 .

[9]  Vladimir Lifschitz,et al.  Computing Circumscription , 1985, IJCAI.

[10]  Johan van Benthem,et al.  Semantic Parallels in Natural Language and Computation , 1989 .

[11]  Leopold Löwenheim.  Über Möglichkeiten im Relativkalkül , 1915 .

[12]  Christos H. Papadimitriou,et al.  Some computational aspects of circumscription , 1988, JACM.

[13]  J.F.A.K. van Benthem,et al.  Modal logic and classical logic , 1983 .

[14]  Teodor C. Przymusinski.  An Algorithm to Compute Circumscription , 1989, Artif. Intell..

[15]  R. Labrecque.  The Correspondence Theory , 1978 .

[16]  John McCarthy,et al.  Circumscription - A Form of Non-Monotonic Reasoning , 1980, Artif. Intell..

[17]  Patrick Doherty,et al.  General Domain Circumscription in its First-Order Reduction , 1996, FAPR.

[18]  Dov M. Gabbay,et al.  Quantifier Elimination in Second-Order Predicate Logic , 1992, KR.

[19]  Vladimir Lifschitz,et al.  Nested Abnormality Theories , 1995, Artif. Intell..

[20]  Vladimir Lifschitz,et al.  Pointwise circumscription , 1987 .

[21]  Michael Gelfond,et al.  Compiling Circumscriptive Theories into Logic Programs , 1989, NMR.

[22]  Arkady Rabinov.  A Generalization of Collapsible Cases of Circumscription , 1989, Artif. Intell..

[23]  Andrzej Szalas.  On the Correspondence between Modal and Classical Logic: An Automated Approach , 1993, J. Log. Comput..

[24]  Vladimir Lifschitz,et al.  A Simple Formalization of Actions Using Circumscription , 1995, IJCAI.