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Restricted semantics for default reasoning

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Abstract

In nonmonotonic reasoning there are the problems of inconsistency and incoherence in general, and in default reasoning there may be only one trivial extension or no extension in special. We propose the restricted semantics of four-valued logic for default reasoning to resolve the problems of inconsistency and incoherence and in the meantime retain classical extensions in the presence of consistency and coherency. The restricted semantics can maintain both the expressiveness and reasoning ability of default logic. We provide a transformation approach to compute the restricted extensions by reducing them into classical ones.

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References

  1. van den Akker, J., Tan, Y.H.: Qml: a paraconsistent default logic. Logique Anal. 143(143-144), 311–328 (1993)

    MathSciNet  MATH  Google Scholar 

  2. Arieli, O.: Paraconsistent preferential reasoning by signed quantified boolean formulae. In: Proceedings of the 16th Eureopean Conference on Artificial Intelligence, ECAI’2004, including Prestigious Applicants of Intelligent Systems, pp. 773-777. PAIS 2004, Valencia (2004)

  3. Arieli, O., Avron, A.: The value of the four values. Artif. Intell. 102(1), 97–141 (1998)

    Article  MathSciNet  Google Scholar 

  4. Arieli, O., Denecker, M.: Modeling paraconsistent reasoning by classical logic Eiter, T., Schewe, K.D. (eds.) , vol. 2284. Springer Berlin, Heidelberg. https://doi.org/10.1007/3-540-45758-5_1 (2002)

    Google Scholar 

  5. Arieli, O., Denecker, M.: Reducing preferential paraconsistent reasoning to classical entailment. J. Log. Comput. 13(4), 557–580 (2003)

    Article  MathSciNet  Google Scholar 

  6. Arruda, A.I.: Non-classical logics, model theory, and computability on the imaginary logic of n. a. vasil’év. Stud. Log. Found. Math. 89, 3–24 (1977). https://doi.org/10.1016/S0049-237X(08)70642-6

    Article  Google Scholar 

  7. Belnap, Jr, N.D.: A useful four-valued logic. In: Modern uses of multiple-valued logic, pp. 5–37. Springer (1977)

  8. Besnard, P., Schaub, T.: A simple signed system for paraconsistent reasoning. In: Proceedings of the European Workshop on Logics in Artificial Intelligence, JELIA ’96, pp. 404–416. Springer-Verlag, London (1996)

    Chapter  Google Scholar 

  9. Besnard, P., Besnard, G., Ramon, S.: A default logic patch for default logic. In: Sossai, C., Chemello, G. (eds.) Symbolic and Quantitative Approaches to Reasoning with Uncertainty, Lecture Notes in Computer Science, vol. 5590, pp. 578–589. Springer Berlin, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02906-6_50

    Google Scholar 

  10. Brewka, G.: Cumulative default logic: in defense of nonmonotonic inference rules. Artif. Intell. 50(2), 183–205 (1991)

    Article  MathSciNet  Google Scholar 

  11. Caminada, M.W., Carnielli, W.A., Dunne, P.E.: Semi-stable semantics. J. Log. Comput. 22(5), 1207–1254 (2011)

    Article  MathSciNet  Google Scholar 

  12. Costa, N.C.A.D.: On the Theory of Inconsistent Formal Systems. Notre Dame J. Form. Log. 15(4), 497–510 (1974)

    Article  MathSciNet  Google Scholar 

  13. Denecker, M., Marek, V.W., Truszczyński, M.: Uniform semantic treatment of default and autoepistemic logics. Artif. Intell. 143(1), 79–122 (2003)

    Article  MathSciNet  Google Scholar 

  14. Gottlob, G.: Complexity results for nonmonotonic logics. J. Log. Comput. 2(3), 397–425 (1992)

    Article  MathSciNet  Google Scholar 

  15. Han, Q., Lin, Z.: Paraconsistent default reasoning. In: 10th International Workshop on Non-Monotonic Reasoning (NMR 2004), pp. 197–203. Proceedings, Whistler (2004)

  16. Lin, Z., Ma, Y., Lin, Z.: A fault-tolerant default logic. In: Fisher, M., van der Hoek, W., Konev, B., Lisitsa, A. (eds.) Logics in Artificial Intelligence, Lecture Notes in Computer Science., vol. 4160, pp. 253–265. Springer, Berlin Heidelberg (2006) https://doi.org/10.1007/11853886_22

    Chapter  Google Scholar 

  17. Łukaszewicz, W.: Considerations on default logic: an alternative approach. Comput. Intell. 4(1), 1–16 (1988)

    Article  Google Scholar 

  18. Papadimitriou, C.H., Sideri, M.: Default theories that always have extensions. Artif. Intell. 69(1), 347–357 (1994)

    Article  MathSciNet  Google Scholar 

  19. Pequeno, T.H.C., Buchsbaum, A.: The logic of epistemic inconsistency. In: Proceedings of the 2nd International Conference on Principles of Knowledge Representation and Reasoning (KR’91), pp. 453–460, Cambridge (1991)

  20. Przymusinska, H., Przymusinski, T.C.: Stationary default extensions. Fundam. Inform. 21(1/2), 67–87 (1994)

    MathSciNet  MATH  Google Scholar 

  21. Reiter, R.: A logic for default reasoning. Artif. Intell. 13(1), 81–132 (1980)

    Article  MathSciNet  Google Scholar 

  22. Rosati R: Model checking for nonmonotonic logics: algorithms and complexity. In: IJCAI, pp. 76–83 (1999)

  23. Schaub, T.: On constrained default theories. In: Proceedings of the 10th European Conference on Artificial Intelligence, ECAI ’92, pp. 304–308. Wiley, New York (1992)

  24. Verheij, B.: Two approaches to dialectical argumentation: admissible sets and argumentation stages. Proc. NAIC 96, 357–368 (1996)

    Google Scholar 

  25. Yue, A., Lin, Z.: A Coherent and Paraconsistent Variant of the Default Logic. In: Logical Formalizations of Commonsense Reasoning, Papers from the 2007 AAAI Spring Symposium, Technical Report SS-07-05, Stanford, pp. 162 (2007)

  26. Yue, A., Ma, Y., Lin, Z.: Four-valued semantics for default logic. In: Lamontagne, L., Marchand, M. (eds.) Advances in Artificial Intelligence, Lecture Notes in Computer Science, vol. 4013, pp. 195–205. Springer, Berlin Heidelberg (2006). https://doi.org/10.1007/11766247_17

    Chapter  Google Scholar 

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Acknowledgments

The preliminary version of this paper received the best paper award in KSEM 2015. We thank anonymous referees for valuable comments and related references.

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Authors and Affiliations

  1. Department of Information Science, School of Mathematical Science, Peking University, Beijing, 100871, China

    Zuoquan Lin & Chen Chen

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  1. Zuoquan Lin
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  2. Chen Chen
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Correspondence to Zuoquan Lin.

Additional information

The work is supported in part by Natural Science Fund of China under numbers 61672049/61732001 and Advance Programs Fund of Ministry of Education of China.

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Lin, Z., Chen, C. Restricted semantics for default reasoning. Ann Math Artif Intell 84, 185–200 (2018). https://doi.org/10.1007/s10472-018-9592-z

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