Abstract
Minimal inconsistent subsets of knowledge bases in monotonic logics play an important role when investigating the reasons for conflicts and trying to handle them. In the context of non-monotonic reasoning this notion is not as meaningful due to the possibility of resolving conflicts by adding information. In this paper we investigate inconsistency in non-monotonic logics while taking this issue into account. In particular, we show that the well-known classical duality between hitting sets of minimal inconsistent subsets and maximal consistent subsets generalizes to arbitrary logics even if we allow adding novel information to a given knowledge base. We illustrate the versatility of the main theorems by covering more sophisticated situations and demonstrate how to utilize our results to analyze inconsistency in abstract argumentation.
This work was funded by Deutsche Forschungsgemeinschaft DFG (Research Training Group 1763; project BR 1817/7-2).
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- 1.
S is upward closed if \(B \in S\), \(B \subseteq B'\) implies \(B' \in S\).
References
Amgoud, L., Ben-Naim, J.: Measuring disagreement in argumentation graphs. In: Moral, S., Pivert, O., Sánchez, D., Marín, N. (eds.) SUM 2017. LNCS (LNAI), vol. 10564, pp. 208–222. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-67582-4_15
Baroni, P., Caminada, M., Giacomin, M.: An introduction to argumentation semantics. Knowl. Eng. Rev. 26(4), 365–410 (2011)
Baumann, R., Ulbricht, M.: If nothing is accepted - repairing argumentation frameworks. In: Principles of Knowledge Representation and Reasoning: Proceedings of the Sixteenth International Conference, KR 2018, Tempe, Arizona, 30 October–2 November 2018, pp. 108–117 (2018). https://aaai.org/ocs/index.php/KR/KR18/paper/view/17979
Benferhat, S., Dubois, D., Prade, H.: A local approach to reasoning under inconsistency in stratified knowledge bases. In: Froidevaux, C., Kohlas, J. (eds.) ECSQARU 1995. LNCS, vol. 946, pp. 36–43. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-60112-0_5
Berge, C.: Hypergraphs: Combinatorics of Finite Sets, vol. 45. North-Holland, Amsterdam (1989)
Béziau, J.Y., Carnielli, W., Gabbay, D. (eds.): Handbook of Paraconsistency. College Publications, London (2007)
Brachman, R.J., Levesque, H.J.: Knowledge Representation and Reasoning. The Morgan Kaufmann Series in Artificial Intelligence. Morgan Kaufmann Publishers, San Francisco (2004)
Brewka, G., Eiter, T.: Equilibria in heterogeneous nonmonotonic multi-context systems. In: Proceedings of the Twenty-Second AAAI Conference on Artificial Intelligence, Vancouver, British Columbia, Canada, 22–26 July 2007, pp. 385–390 (2007)
Brewka, G., Eiter, T., Truszczynski, M.: Answer set programming at a glance. Commun. ACM 54(12), 92–103 (2011). https://doi.org/10.1145/2043174.2043195
Brewka, G., Thimm, M., Ulbricht, M.: Strong inconsistency in nonmonotonic reasoning. In: Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, IJCAI 2017, pp. 901–907 (2017)
Brewka, G., Thimm, M., Ulbricht, M.: Strong inconsistency. Artif. Intell. 267, 78–117 (2019)
Cholvy, L., Hunter, A.: Information fusion in logic: a brief overview. In: Gabbay, D.M., Kruse, R., Nonnengart, A., Ohlbach, H.J. (eds.) ECSQARU/FAPR -1997. LNCS, vol. 1244, pp. 86–95. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0035614
Dubois, D., Lang, J., Prade, H.: Inconsistency in possibilistic knowledge bases: to live with it or not live with it. In: Zadeh, L., Kacprzyk, J. (eds.) Fuzzy Logic for the Management of Uncertainty, pp. 335–351. Wiley, New York (1992)
Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif. Intell. 77(2), 321–358 (1995)
Eiter, T., Fink, M., Schüller, P., Weinzierl, A.: Finding explanations of inconsistency in multi-context systems. Artif. Intell. 216, 233–274 (2014). https://doi.org/10.1016/j.artint.2014.07.008
Garcia, L., Lefèvre, C., Papini, O., Stéphan, I., Würbel, É.: A semantic characterization for ASP base revision. In: Moral, S., Pivert, O., Sánchez, D., Marín, N. (eds.) SUM 2017. LNCS (LNAI), vol. 10564, pp. 334–347. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-67582-4_24
Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: ICLP/SLP, vol. 88, pp. 1070–1080 (1988)
Grant, J.: Classifications for inconsistent theories. Notre Dame J. Formal Log. 19(3), 435–444 (1978)
Hansson, S.O.: A Textbook of Belief Dynamics. Kluwer Academic Publishers, Norwell (2001)
Hunter, A., Konieczny, S.: Approaches to measuring inconsistent information. In: Bertossi, L., Hunter, A., Schaub, T. (eds.) Inconsistency Tolerance. LNCS, vol. 3300, pp. 191–236. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-30597-2_7
Hunter, A.: Measuring inconsistency in argument graphs. Technical report. arXiv:1708.02851 (2017)
Hunter, A.: Measuring inconsistency in argument graphs. In: Grant, J., Martinez, M.V. (eds.) Measuring Inconsistency in Information. College Publications (2018)
Konieczny, S., Lang, J., Marquis, P.: Reasoning under inconsistency: the forgotten connective. In: Proceedings of IJCAI 2005, pp. 484–489 (2005)
Konieczny, S., Perez, R.P.: Logic based merging. J. Philos. Log. 40, 239–270 (2011)
Lang, J., Marquis, P.: Reasoning under inconsistency: a forgetting-based approach. Artif. Intell. 174(12–13), 799–823 (2010)
Reiter, R.: A theory of diagnosis from first principles. Artif. Intell. 32(1), 57–95 (1987). https://doi.org/10.1016/0004-3702(87)90062-2
Ulbricht, M., Thimm, M., Brewka, G.: Inconsistency measures for disjunctive logic programs under answer set semantics. In: Grant, J., Martinez, M.V. (eds.) Measuring Inconsistency in Information, Studies in Logic, vol. 73. College Publications, February 2018
Ulbricht, M., Thimm, M., Brewka, G.: Measuring strong inconsistency. In: Proceedings of the 32nd AAAI Conference on Artificial Intelligence, AAAI 2018, February 2018
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Ulbricht, M. (2019). Repairing Non-monotonic Knowledge Bases. In: Calimeri, F., Leone, N., Manna, M. (eds) Logics in Artificial Intelligence. JELIA 2019. Lecture Notes in Computer Science(), vol 11468. Springer, Cham. https://doi.org/10.1007/978-3-030-19570-0_10
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