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Tableau-based characterization and theorem proving for default logic

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Abstract

In this paper, we present a new method for computing extensions and for deriving formulae from a default theory. It is based on the semantic tableaux method and works for default theories with a finite set of defaults that are formulated over a decidable subset of first-order logic. We prove that all extensions (if any) of a default theory can be produced by constructing the semantic tableau ofone formula built from the general laws and the default consequences. This result allows us to describe an algorithm that provides extensions if there are any, and to decide if there are none. Moreover, the method gives a necessary and sufficient criterion for the existence of extensions of default theories with finitely many defaults provided they are formulated on a decidable subset of FOL.

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References

  1. Besnard, P., Quiniou, R., and Quinton, P.: A theorem-prover for a decidable subset of default logic,Proc. AAAI, (1983), pp. 27–30.

  2. Beth, E. W.:The Foundations of Mathematics, North-Holland, 1959.

  3. Besnard, P. and Siegel, P.: Supposition-based logic for automated nonmonotonic reasoning,Proc. 9th Conference on Automated Deduction, Argonne, 1988.

  4. Bossu, G. and Siegel, P.: Saturation, nonmonotonic reasoning and the closed-world assumption,Artificial Intelligence 25 (1985), 13–63.

    Google Scholar 

  5. Brown, F. M.: A common sense theory of nonmonotonic reasoning,Proc. 8th Conference on Automated Deduction, Oxford, Lecture Notes in Computer Science,230, Springer-Verlag, 1986, pp. 209–228.

  6. Etherington, D. W.: Formalizing nonmonotonic reasoning systems,Artificial Intelligence 31(1), (1987), 81–132.

    Google Scholar 

  7. Fitting, M.:First-Order Logic and Automated Theorem Proving, Texts and Monographs in Computer Science, Springer-Verlag, 1990.

  8. Fitting, M., Marek, V., and Truszczynski, M.: The pure logic of necessitation,J. Logic and Computation 2(3) (1992), 349–373.

    Google Scholar 

  9. Ginsberg, M. L.: A circumscriptive theorem prover,Artificial Intelligence 39 (1989), 209–230.

    Google Scholar 

  10. Gueirreiro, R. A., Casanova, M. A., and Hermely, A. S.: Contributions to a proof theory for generic defaults,Proc. 9th European Conference on Artificial Intelligence, 1990, pp. 213–218.

  11. Junker, U. and Brewka, G.: Handling partially ordered defaults in TMS,Proc. European Conference on Symbolic and Quantitative Approaches for Uncertainty, Marseille, France, Lecture Notes in Computer Science548, Springer-Verlag, 1991, pp. 211–218.

  12. Junker, U. and Konolige, K.: Computing the extensions of autoepistemic and default logics with a truth maintenance system,Proc. AAAI-90, Boston, 1990, pp. 278–283.

  13. Kuhna, P.: Circumscription and minimal models for propositional logics,Workshop on Theorem Proving with Analytic Tableaux and Related Methods, Marseille, France, pp. 143–155, 1991, MPI-I-93-213.

  14. Lafon, E. and Schwind, C.: A theorem prover for action performance,Proc. 8th European Conference on Artificial Intelligence, 1988, pp. 541–546.

  15. Lifschitz, V.: Computing circumscription,Proc. International Joint Conference on Artificial Intelligence, 1985, pp. 121–127.

  16. Levy, F.: Computing extensions of default theories,Proc. European Conference on Symbolic and Quantitative Approaches for Uncertainty, Marseille, France, Lecture Notes in Computer Science548, Springer-Verlag, 1991, pp. 219–226.

  17. Lukaszewicz, W.: Considerations on default logic — an alternative approach,Computational Intelligence 4 (1988), 1–16.

    Google Scholar 

  18. McCarthy, J.: Circumscription a form of non-monotonic reasoning,Artificial Intelligence 13 (1980), 27–39.

    Google Scholar 

  19. Moore, R.: Autoepistemic logic, inNonstandard Logics for Automated Reasoning (eds P. Smets, A. Mamdani, D. Dubois, and H. Prade), Academic Press, pp. 105–136.

  20. Niemelä, I.: Decision procedure for autoepistemic logic,Proc. 9th Conference on Automated Deduction, Springer-Verlag, 1988, pp. 676–684.

  21. Olivetti, N.: Tableaux and sequent calculus for minimal entailment,J. Automated Reasoning 9 (1992), 99–139.

    Google Scholar 

  22. Przymusinski, T. C.: An algorithm to compute circumscription,Artificial Intelligence 38 (1989), 49–73.

    Google Scholar 

  23. Reiter, R.: A logic for default reasoning,Artificial Intelligence 13 (1980), 81–132.

    Google Scholar 

  24. Risch, V.: Une caractérisation en termes de tableaux sémantiques pour la logique des défaults au sens de Lukaszewicz, 1992,Revue d'Intelligence Artificielle 7(1) (1993).

  25. Schütte, K.: Beweistheorie, Springer-Verlag, 1960.

  26. Schwind, C.: Un démonstrateur de théorèmes pour des logiques modales et temporelles en PROLOG,5th Congress AFCET Reconnaissance des Formes et Intelligence Artificielle, Grenoble, France, 1985, pp. 897–913.

  27. Schwind, C.: A tableau based theorem prover for a decidable subset of default logic,Proc. 10th Conference on Automated Deduction, Springer-Verlag, 1990, pp. 541–546.

  28. Smullyan, R.:First-Order Logic, Springer-Verlag, 1968.

  29. Thistlewaite, P. B., McRobbie, M. D., and Meyer, R. R.:Automated Theorem Proving in Non-Classical Logics, Pitman, 1988.

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Authors and Affiliations

  1. Department of Computer Science, University of Western Ontario, Middlesex College, N6A 6B7, London, Ontario, Canada

    Vincent Risch

  2. Faculté des Sciences de Luminy, Laboratoire d'Informatique de Marseille, CNRS, 16 Avenue de Luminy Case 901, 13288, Marseille Cedex 9, France

    Camilla B. Schwind

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  1. Vincent Risch
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  2. Camilla B. Schwind
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This work was completed while the author was at CNRS, Marseille.

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Risch, V., Schwind, C.B. Tableau-based characterization and theorem proving for default logic. J Autom Reasoning 13, 223–242 (1994). https://doi.org/10.1007/BF00881957

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  • DOI: https://doi.org/10.1007/BF00881957

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