Abstract
Hunter and Konieczny explored the relationships between measures of inconsistency for a belief base and the minimal inconsistent subsets of that belief base in several of their papers. In particular, an inconsistency value termed MIV C , defined from minimal inconsistent subsets, can be considered as a Shapley Inconsistency Value. Moreover, it can be axiomatized completely in terms of five simple axioms. MinInc, one of the five axioms, states that each minimal inconsistent set has the same amount of conflict. However, it conflicts with the intuition illustrated by the lottery paradox, which states that as the size of a minimal inconsistent belief base increases, the degree of inconsistency of that belief base becomes smaller. To address this, we present two kinds of revised inconsistency measures for a belief base from its minimal inconsistent subsets. Each of these measures considers the size of each minimal inconsistent subset as well as the number of minimal inconsistent subsets of a belief base. More specifically, we first present a vectorial measure to capture the inconsistency for a belief base, which is more discriminative than MIV C . Then we present a family of weighted inconsistency measures based on the vectorial inconsistency measure, which allow us to capture the inconsistency for a belief base in terms of a single numerical value as usual. We also show that each of the two kinds of revised inconsistency measures can be considered as a particular Shapley Inconsistency Value, and can be axiomatically characterized by the corresponding revised axioms presented in this paper.
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References
Aumann R, Hart S (2002) Handbook of game theory with economic applications. Elsevier, Amsterdam, pp 2025–2054
Bagheri E, Ghorbani A (2010) The analysis and management of non-canonical requirement specifications through a belief integration game. Knowl Inf Syst 22(1): 27–64
Barrangans-Martinez B, Pazos-Arias J, Fernandez-Vilas A (2008) On the interplay between inconsistency and incompleteness in multi-perspective requirements specifications. Inf Softw Technol 50(4): 296–321
Bertossi L, Hunter A, Schaub T (2004) Introduction to inconsistency tolerance. In: Bertossi L, Hunter A, Schaub T (eds) Inconsistency tolerance, lecture notes in computer science vol. 3300. Springer, New York, pp 1–16
Cristani M, Burato E (2009) Approximate solutions of moral dilemmas in multiple agent system. Knowl Inf Syst 18(2): 157–181
Fan H, Zaiane O, Foss A, Wu J (2009) Resolution-based outlier factor: detecting the top-n most outlying data points in engineering data. Knowl Inf Syst 19(1): 31–51
Grant J, Hunter A (2006) Measuring inconsistency in knowledgebases. J Intell Inf Syst 27(2): 159–184
Grant J, Hunter A (2008) Analysing inconsistent first-order knowledge bases. Artif Intell 172(8–9): 1064–1093
Grant J (1978) Classifications for inconsistent theories. Notre Dame J Formal Log 19(3): 435–444
Hunter A, Konieczny S (2004) Approaches to measuring inconsistent information. In: Bertossi L, Hunter A, Schaub T (eds) Inconsistency tolerance, lecture notes in computer science, vol. 3300. Springer, New York, pp 189–234
Hunter A, Konieczny S (2006) Shapley inconsistency values. In: Doherty P, Mylopoulos J, Welty C (eds) Principles of knowledge representation and reasoning: proceedings of the 10th international conference(KR06). AAAI Press, Menlo Park, US, pp 249–259
Hunter A, Konieczny S (2008) Measuring inconsistency through minimal inconsistent sets. In: Brewka G, Lang J (eds) Principles of knowledge representation and reasoning: proceedings of the eleventh international conference(KR08). AAAI Press, Menlo Park, US, pp 358–366
Hunter A (2002) Measuring inconsistency in knowledge via quasi-classical models. In: Proceedings of the national conference on artificial intelligence (AAAI2002), AAAI Press, Menlo Park, US, pp 68–73
Hunter A (2004) Logical comparison of inconsistent perspectives usingscoring functions. Knowl Inf Syst 6(5): 528–543
Knight K (2002) Measuring inconsistency. J Philos Log 31(1): 77–98
Knight K (2003) Two information measures for inconsistent sets. J Log Lang Inf 12(2): 227–248
Konieczny S, Lang J, Marquis P (2003) Quantifying information and contradiction in propositional logic through epistemic actions. In: Proceedings of the 18th international joint conference on artificial intellignce (IJCAI2003), pp 106–111
Kyburg E (1961) Probability and the logic of rational belief. Wesleyan University Press, Middletown, pp 197–197
Ma Y, Qi G, Hitzler P, Lin Z (2007) Measuring inconssitency for description logics based on paraconsistent semantics. In: Mellouli K (eds) Proceedings of European conference on symbolic and quantitative approaches to reasoning about uncertainty 2007, Lecture notes in computer science, vol 4724. Springer, Berlin Heidelberg, pp 30–41
Mu K, Jin Z, Lu R, Liu W(2005) Measuring inconsistency in requirements specifications. In: Godo L (ed) Proceedings of European conference symbolic and quantitative approaches to reasoning about uncertainty 2005, Lecture notes in computer science, vol. 3571, Springer, pp 440–451
Mu K, Jin Z, Lu R, Peng Y (2007) Handling non-canonical software requirements based on annotated predicate calculus. Knowl Inf Syst 11(1): 85–104
Mu K, Jin Z, Zowghi D (2008) A priority-based negotiations approach for handling inconsistency in multiperspective software requirements. J Syst Sci Complex 21(4): 574–596
Paris J, Vencovska A (1998) Proof systems for probabilistic uncertain reasoning. J Symbolic Log 63(3): 1007–1039
Paris J (1994) The uncertain reasoner’s companion: a mathematical perspective. In Cambridge Tracts in Theoret. Comput. Sci. 39, Cambridge University Press.
Qi G, Liu W, Bell D (2005) Measuring conflict and agreement between two prioritized belief bases. In: Proceedings of the nineteenth international joint conference on artificial intelligence(IJCAI05), pp 552–557
Reiter R (1987) A theory of diagnosis from first priniciples. Artif Intell 32(1): 57–95
Resconi G, Kovalerchuk B (2009) Agents’ model of uncertainty. Knowl Inf Syst 18(2): 213–229
Shapley L (1953) A value for n-person games. In: Kuhn H, Tucker M (eds) Contributions to the theory of games, vol. II (Annals of Mathematical Studies vol. 28). Princeton University Press, Princeton, pp 307– 317
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Mu, K., Liu, W. & Jin, Z. A general framework for measuring inconsistency through minimal inconsistent sets. Knowl Inf Syst 27, 85–114 (2011). https://doi.org/10.1007/s10115-010-0295-y
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DOI: https://doi.org/10.1007/s10115-010-0295-y