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Paraconsistent Reasoning via Quantified Boolean Formulas, II: Circumscribing Inconsistent Theories

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Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2003)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2711))

Abstract

Through minimal-model semantics, three-valued logics provide an interesting formalism for capturing reasoning from inconsistent information. However, the resulting paraconsistent logics lack so far a uniform implementation platform. Here, we address this and specifically provide a translation of two such paraconsistent logics into the language of quantified Boolean formulas (QBFs). These formulas can then be evaluated by off-the-shelf QBF solvers. In this way, we benefit from the following advantages: First, our approach allows us to harness the performance of existing QBF solvers. Second, different paraconsistent logics can be compared with in a unified setting via the translations used. We alternatively provide a translation of these two paraconsistent logics into quantified Boolean formulas representing circumscription, the well-known system for logical minimization. All this forms a case study inasmuch as the other existing minimization-based many-valued paraconsistent logics can be dealt with in a similar fashion.

This work was partially supported by the Austrian Science Foundation under grant P15068.

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References

  1. Arieli, O., Avron, A.: Logical bilattices and inconsistent data. In: Proc. LICS, pp. 468–476 (1994)

    Google Scholar 

  2. Arieli, O., Avron, A.: Automatic diagnoses for properly stratified knowledge-bases. In: Proc. ICTAI 1996, pp. 392–399. IEEE Computer Society Press, Los Alamitos (1996)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Arieli, O., Denecker, M.: Modeling paraconsistent reasoning by classical logic. In: Eiter, T., Schewe, K.-D. (eds.) FoIKS 2002. LNCS, vol. 2284, pp. 1–14. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  5. Avron, A.: Simple consequence relations. Information and Computation 92, 105–139 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ayari, A., Basin, D.: QUBOS: Deciding quantified Boolean logic using propositional satisfiability solvers. In: Aagaard, M.D., O’Leary, J.W. (eds.) FMCAD 2002. LNCS, vol. 2517, pp. 187–201. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  7. Belnap, N.: A useful four-valued logic. In: Dunn, J., Epstein, G. (eds.) Modern Uses of Multiple-Valued Logic. Reidel, Dordrechtz (1977)

    Google Scholar 

  8. Benferhat, S., Cayrol, C., Dubois, D., Lang, J., Prade, H.: Inconsistency management and prioritized syntax-based entailment. In: Proc. IJCAI 1993, pp. 640–647 (1993)

    Google Scholar 

  9. Besnard, P., Schaub, T.: Circumscribing inconsistency. In: Proc. IJCAI 1997, pp. 150–155 (1997)

    Google Scholar 

  10. Besnard, P., Schaub, T.: Signed systems for paraconsistent reasoning. Journal of Automated Reasoning 20(1-2), 191–213 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Besnard, P., Schaub, T., Tompits, H., Woltran, S.: Paraconsistent reasoning via quantified Boolean formulas, I: Axiomatising signed systems. In: Flesca, S., Greco, S., Leone, N., Ianni, G. (eds.) JELIA 2002. LNCS (LNAI), vol. 2424, pp. 320–331. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  12. Cadoli, M., Giovanardi, A., Schaerf, M.: An algorithm to evaluate quantified Boolean formulae. In: Proc. AAAI 1998, pp. 262–267. AAAI Press, Menlo Park (1998)

    Google Scholar 

  13. Cadoli, M., Schaerf, M.: On the complexity of entailment in propositional multivalued logics. Annals of Mathematics and Artificial Intelligence 18, 29–50 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  14. Carnielli, W., Fariñas del Cerro, L., Lima Marques, M.: Contextual negations and reasoning with contradictions. In: Proc. IJCAI 1991, pp. 532–537 (1991)

    Google Scholar 

  15. Coste-Marquis, S., Marquis, P.: Complexity results for paraconsistent inference relations. In: Proc. KR 2002, pp. 61–72 (2002)

    Google Scholar 

  16. Doherty, P., Lukaszewicz, W., Szalas, A.: Computing circumscription revisited: Areduction algorithm. Journal of Automated Reasoning 18, 297–334 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  17. D’Ottaviano, I., da Costa, N.: Sur un problème de Jaśkowski. In: Comptes Rendus de l’Académie des Sciences de Paris, vol. 270, pp. 1349–1353 (1970)

    Google Scholar 

  18. Feldmann, R., Monien, B., Schamberger, S.: A distributed algorithm to evaluate quantified Boolean formulas. In: Proc. AAAI 2000, pp. 285–290. AAAI Press, Menlo Park (2000)

    Google Scholar 

  19. Frisch, A.: Inference without chaining. In: Proc. IJCAI 1987, pp. 515–519 (1987)

    Google Scholar 

  20. Giunchiglia, E., Narizzano, M., Tacchella, A.: QuBE: A system for deciding quantified Boolean formulas satisfiability. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 364–369. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  21. Konieczny, S., Marquis, P.: Three-valued logics for inconsistency handling. In: Flesca, S., Greco, S., Leone, N., Ianni, G. (eds.) JELIA 2002. LNCS (LNAI), vol. 2424, pp. 332–344. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  22. Levesque, H.: A knowledge-level account of abduction. In: Proc. IJCAI 1989, pp. 1061–1067 (1989)

    Google Scholar 

  23. Lin, F.: Reasoning in the presence of inconsistency. In: Proc. AAAI 1987, pp. 139–143 (1987)

    Google Scholar 

  24. Manor, R., Rescher, N.: On inferences from inconsistent information. Theory and Decision 1, 179–219 (1970)

    Article  MATH  Google Scholar 

  25. McCarthy, J.: Applications of circumscription to formalizing common-sense knowledge. Artificial Intelligence 28, 89–116 (1986)

    Article  MathSciNet  Google Scholar 

  26. Mundici, D.: Satisfiability in many-valued sentential logic is NP-complete. Theoretical Computer Science 52(1-2), 145–153 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  27. Priest, G.: Logic of paradox. Journal of Philosophical Logic 8, 219–241 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  28. Priest, G.: Reasoning about truth. Artificial Intelligence 39, 231–244 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  29. Sandewall, E.: A functional approach to non-monotonic logic. Computational Intelligence 1, 80–87 (1985)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. IRIT-CNRS, 118, route de Narbonne, F–31062, Toulouse Cedex, France

    Philippe Besnard

  2. Institut für Informatik, Universität Potsdam, Postfach 90 03 27, D–14439, Potsdam, Germany

    Torsten Schaub

  3. Institut für Informationssysteme 184/3, Technische Universität Wien, Favoritenstraße 9–11, A–1040, Vienna, Austria

    Hans Tompits & Stefan Woltran

Authors
  1. Philippe Besnard
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  2. Torsten Schaub
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  3. Hans Tompits
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  4. Stefan Woltran
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Aalborg University, Fredrik Bajersvej 7E, 9220, Aalborg, Denmark

    Thomas Dyhre Nielsen

  2. Department of Computer Science, Hong Kong University of Science and Technology, ClearWater Bay Road, Kowloon, Hong Kong, China,

    Nevin Lianwen Zhang

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© 2003 Springer-Verlag Berlin Heidelberg

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Besnard, P., Schaub, T., Tompits, H., Woltran, S. (2003). Paraconsistent Reasoning via Quantified Boolean Formulas, II: Circumscribing Inconsistent Theories. In: Nielsen, T.D., Zhang, N.L. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2003. Lecture Notes in Computer Science(), vol 2711. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45062-7_43

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  • DOI: https://doi.org/10.1007/978-3-540-45062-7_43

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40494-1

  • Online ISBN: 978-3-540-45062-7

  • eBook Packages: Springer Book Archive

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