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On the Complexity of Paraconsistent Inference Relations

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3300))

Abstract

Reasoning in a non-trivial way from inconsistent pieces of information is a major challenge in artificial intelligence, and its importance is reflected by the number of techniques designed so far for dealing with inconsistency (especially the few ones reported in this handbook). Many of these techniques have been investigated in depth from a logical point of view, but far less to what concerns the computational complexity aspects. The purpose of this chapter is to present in a structured way the main complexity results identified so far for paraconsistent inference based on multi-valued propositional logics.

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References

  1. Amgoud, L., Cayrol, C.: On the acceptability of arguments in preference-based argumentation framework. In: Proceedings of the 14th Conference on Uncertainty in Artificial Intelligence, Madison (WI), pp. 1–7 (1998)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Arieli, O., Avron, A.: A model-theoretic approach for recovering consistent data from inconsistent knowledge bases. Journal of Automated Reasoning 22(2), 263–309 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arieli, O., Denecker, M.: Reducing preferential paraconsistent reasoning to classical entailment. Journal of Logic and Computation 13(4), 557–580 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baral, C., Kraus, S., Minker, J.: Combining multiple knowledge bases. IEEE Transactions on Knowledge and Data Engineering 3(2), 208–220 (1991)

    Article  Google Scholar 

  6. Belnap, N.: A useful four-valued logic. In: Modern Uses of Multiple-Valued Logic, pp. 8–37. Reidel, Dordrechtz (1977)

    Google Scholar 

  7. Benferhat, S., Cayrol, C., Dubois, D., Lang, J., Prade, H.: Inconsistency management and prioritized syntax-based entailment. In: Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI 1993), Chambéry (France), pp. 640–645 (1993)

    Google Scholar 

  8. Benferhat, S., Dubois, D., Prade, H.: How to infer from inconsistent beliefs without revising. In: Proceedings of the 14th International Joint Conference on Artificial Intelligence (IJCAI 1995), Montreal (Canada), pp. 1449–1455 (1995)

    Google Scholar 

  9. Benferhat, S., Kaci, S., Le Berre, D., Williams, M.-A.: Weakening conflicting information for iterated revision and knowledge integration. Artificial Intelligence 153(1–2), 339–371 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Besnard, P., Hunter, A.: Introduction to actual and potential contradictions. In: Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol. 2, pp. 1–11. Kluwer Academic, Dordrecht (1998)

    Google Scholar 

  11. Besnard, P., Schaub, T., Tompits, H., Woltran, S.: Paraconsistent reasoning via Quantified Boolean Formulas I: Axiomatizing 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. Besnard, P., Schaub, T., Tompits, H., Woltran, S.: Paraconsistent reasoning via Quantified Boolean Formulas II: Circumscribing inconsistent theories. In: Nielsen, T.D., Zhang, N.L. (eds.) ECSQARU 2003. LNCS (LNAI), vol. 2711, pp. 528–539. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  13. Besnard, P., Schaub, T.: Circumscribing inconsistency. In: Proceedings of the 15th International Joint Conference on Artificial Intelligence (IJCAI 1997), Nagoya (Japan), pp. 150–155 (1997)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Besnard, P., Hunter, A.: Quasi-classical logic: Non-trivializable classical reasoning from inconsistent information. In: Froidevaux, C., Kohlas, J. (eds.) ECSQARU 1995. LNCS (LNAI), vol. 946, pp. 44–51. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  16. Bloch, I., Lang, J.: Towards mathematical morpho-logics. In: Proceedings of the 8th International Conference on Information Processing and Management of Uncertainty in Knowledge based Systems (IPMU 2000), Madrid (Spain), pp. 1405–1412 (2000)

    Google Scholar 

  17. Bondarenko, A., Dung, P., Kowalski, R.A., Toni, F.: An abstract, argumentationtheoretic framework for default reasoning. Artificial Intelligence 93, 63–101 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Brewka, G.: Preferred subtheories: An extended logical framework for default reasoning. In: Proceedings of the 11th International Joint Conference on Artificial Intelligence (IJCAI 1989), Detroit (MI), pp. 1043–1048 (1989)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Cayrol, C., Lagasquie-Schiex, M.-C., Schiex, T.: Nonmonotonic reasoning: From complexity to algorithms. Annals of Mathematics and Artificial Intelligence 22(3–4), 207–236 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Coste-Marquis, S., Marquis, P.: Complexity results for paraconsistent inference relations. In: Proceedings of the 8th International Conference on Knowledge Representation and Reasoning (KR 2002), Toulouse (France), pp. 61–72 (2002)

    Google Scholar 

  22. Coste-Marquis, S., Marquis, P.: On stratified belief base compilation. Annals of Mathematics and Artificial Intelligence 42(4), 399–442 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Darwiche, A., Marquis, P.: Compiling propositional weighted bases. Artificial Intelligence 157, 81–113 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. da Costa, N.C.A.: On the theory of inconsistent formal systems. Notre Dame Journal of Formal Logic 15, 497–510 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dimopolos, Y., Nebel, B., Toni, F.: Finding admissible and preferred arguments can be very hard. In: Proceedings of the 7th International Conference on Knowledge Representation and Reasoning (KR 2000), Breckenridge (CO), pp. 53–61 (2000)

    Google Scholar 

  26. Dimopoulos, Y., Nebel, B., Toni, F.: Preferred arguments are harder to compute than stable extensions. In: Proceedings of the 16th International Joint Conference on Artificial Intelligence (IJCAI 1999), Stockholm (Sweden), pp. 36–41 (1999)

    Google Scholar 

  27. D’Ottaviano, I.M.L., da Costa, N.C.A.: Sur un problème de Jaśkowski. Technical report, Comptes Rendus de l’Académie des Sciences de Paris (1970)

    Google Scholar 

  28. Dubois, D., Konieczny, S., Prade, H.: Quasi-possibilistic logic and its measures of information and conflict. Fundamenta Informaticae 57(2–4), 101–125 (2003)

    MathSciNet  MATH  Google Scholar 

  29. Dung, P.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence 77, 321–357 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Dunne, P., Bench-Capon, T.J.M.: Coherence in finite argument systems. Artificial Intelligence 141, 187–203 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Eiter, T., Gottlob, G.: On the complexity of propositional knowledge base revision, updates, and counterfactuals. Artificial Intelligence 57, 227–270 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. Eiter, T., Lukasiewicz, T.: Default reasoning from conditional knowledge bases: Complexity and tractable cases. Artificial Intelligence 124(2), 169–241 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. Elvang-Goransson, M., Hunter, A.: Argumentative logics: Reasoning from classically inconsistent information. Data and Knowledge Engineering 16, 125–145 (1995)

    Article  MATH  Google Scholar 

  34. Enderton, H.B.: A mathematical introduction to logic. Academic Press, New York (1972)

    MATH  Google Scholar 

  35. Epstein, R.L.: The Semantic Foundations of Logic. In: Propositional Logics, vol. 1. Kluwer Academic, Dordrecht (1990)

    Google Scholar 

  36. Fagin, R., Ullman, J.D., Vardi, M.Y.: On the semantics of updates in databases. In: Proceedings of the 2nd ACM Symposium on Principles of Database Systems (PODS 1983), pp. 352–355 (1983)

    Google Scholar 

  37. Frisch, A.M.: Inference without chaining. In: Proceedings of the 10th International Joint Conference on Artificial Intelligence (IJCAI 1987), Milan (Italy), pp. 515–519 (1987)

    Google Scholar 

  38. Gärdenfors, P., Makinson, D.: Relations between the logic of theory change and nonmonotonic logic. In: Fuhrmann, A., Morreau, M. (eds.) The Logic of Theory Change. LNCS, vol. 465, pp. 185–205. Springer, Heidelberg (1991)

    Chapter  Google Scholar 

  39. Gärdenfors, P.: Belief revision and nonmonotonic logic: Two sides of the same coin? In: Proceedings of the 9th European Conference on Artificial Intelligence, Stockholm (Sweden), pp. 768–773 (1990)

    Google Scholar 

  40. Garey, M.R., Johnson, D.S.: Computers and intractability: A guide to the theory of NP-completeness. Freeman, New York (1979)

    MATH  Google Scholar 

  41. Ginsberg, M.L.: Counterfactuals. Artificial Intelligence 30, 35–79 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  42. Gottlob, G.: Complexity results for nonmonotonic logics. Journal of Logic and Computation 2, 397–425 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  43. Grant, J., Subrahmanian, V.S.: Reasoning in inconsistent knowledge bases. IEEE Transactions on Knowledge and Data Engineering 7(1), 177–189 (1995)

    Article  MATH  Google Scholar 

  44. Hunter, A.: Paraconsistent logics. In: Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol. 2, pp. 11–36. Kluwer Academic, Dordrecht (1998)

    Google Scholar 

  45. Hunter, A.: Reasoning with contradictory information using quasi-classical logic. Journal of Logic and Computation 10(5), 677–703 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  46. Jaśkowski, S.: Propositional calculus for contradictory deductive systems. Studia Logica 24, 143–167 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  47. Konieczny, S., Lang, J., Marquis, P.: DA2 merging operators. Artificial Intelligence 157(1-2), 49–79 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  48. 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 

  49. Konieczny, S., Pino Pérez, R.: On the logic of merging. In: Proceedings of the 6th International Conference on Knowledge Representation and Reasoning (KR 1998), Trento (Italy), pp. 488–498 (1998)

    Google Scholar 

  50. Konieczny, S.: On the difference between merging knowledge bases and combining them. In: Proceedings of the 7th International Conference on Knowledge Representation and Reasoning (KR 2000), Breckenridge (CO), pp. 135–144 (2000)

    Google Scholar 

  51. Kraus, S., Lehmann, D., Magidor, M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence 44(1-2), 167–207 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  52. Lang, J., Marquis, P.: Resolving inconsistencies by variable forgetting. In: Proceedings of the 8th International Conference on Knowledge Representation and Reasoning (KR 2002), Toulouse (France), pp. 239–250 (2002)

    Google Scholar 

  53. Levesque, H.J.: A knowledge-level account of abduction (preliminary version). In: Proceedings of the 11th International Joint Conference on Artificial Intelligence (IJCAI 1989), Detroit (MI), pp. 1061–1067 (1989)

    Google Scholar 

  54. Lin, J.: Integration of weighted knowledge bases. Artificial Intelligence 83(2), 363–378 (1996)

    Article  MathSciNet  Google Scholar 

  55. Marquis, P., Porquet, N.: Computational aspects of quasi-classical entailment. Journal of Applied Non-Classical Logics 11(3–4), 295–312 (2001)

    MathSciNet  MATH  Google Scholar 

  56. Marquis, P., Porquet, N.: Resource-bounded paraconsistent inference. Annals of Mathematics and Artificial Intelligence 39(4), 349–384 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  57. Meyer, J.-J.C., Van der Hoek, W.: Modal logics for representing incoherent knowledge. In: Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol. 2, pp. 37–75. Kluwer Academic, Dordrecht (1998)

    Google Scholar 

  58. Nebel, B.: Syntax-based approaches to belief revision. In: Gärdenfors, P. (ed.) Belief revision. Cambridge Tracts in Theoretical Computer Science, vol. 29, pp. 52–88. Cambridge University Press, Cambridge (1992)

    Chapter  Google Scholar 

  59. Nebel, B.: Base revision operations and schemes: Semantics, representation and complexity. In: Proceedings of the 11th European Conference on Artificial Intelligence, Amsterdam (Netherlands), pp. 341–345 (1994)

    Google Scholar 

  60. Nebel, B.: How hard is it to revise a belief base? In: Dubois, D., Prade, H. (eds.) Handbook of Defeasible Reasoning and Uncertainty Management Systems, vol. 3, pp. 77–145. Kluwer Academic, Dordrecht (1998)

    Google Scholar 

  61. Papadimitriou, C.H.: Computational complexity. Addison-Wesley, Amsterdam (1994)

    MATH  Google Scholar 

  62. Pinkas, G., Loui, R.P.: Reasoning from inconsistency: A taxonomy of principles for resolving conflict. In: Proceedings of the 3rd International Conference on Knowledge Representation and Reasoning (KR 1992), Cambridge (MA), pp. 709–719 (1992)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  64. Priest, G.: Minimally inconsistent LP. Studia Logica 50, 321–331 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  65. Priest, G.: Paraconsistent Logic. In: Handbook of Philosophical Logic, vol. 6, pp. 287–393. Kluwer Academic, Dordrecht (2002)

    Chapter  Google Scholar 

  66. Reiter, R.: A theory of diagnosis from first principles. Artificial Intelligence 32, 57–95 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  67. Rescher, N., Manor, R.: On inference from inconsistent premises. Theory and Decision 1, 179–219 (1970)

    Article  MATH  Google Scholar 

  68. Revesz, P.Z.: On the semantics of arbitration. International Journal of Algebra and Computation 7(2), 133–160 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  69. Schaerf, M., Cadoli, M.: Tractable reasoning via approximation. Artificial Intelligence 74, 249–310 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  70. Stillman, J.: The complexity of propositional default logics. In: Proceedings of the 10th National Conference on Artificial Intelligence, San Jose (CA), pp. 794–799 (1992)

    Google Scholar 

  71. Winslett, M.: Updating logical databases. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge (1990)

    Google Scholar 

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Coste-Marquis, S., Marquis, P. (2005). On the Complexity of Paraconsistent Inference Relations. In: Bertossi, L., Hunter, A., Schaub, T. (eds) Inconsistency Tolerance. Lecture Notes in Computer Science, vol 3300. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30597-2_6

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  • DOI: https://doi.org/10.1007/978-3-540-30597-2_6

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