Skip to main content
SpringerLink
Log in

On the Compliance of Rationality Postulates for Inconsistency Measures: A More or Less Complete Picture

  • Technical Contribution
  • Published:
KI - Künstliche Intelligenz Aims and scope Submit manuscript

Abstract

An inconsistency measure is a function mapping a knowledge base to a non-negative real number, where larger values indicate the presence of more significant inconsistencies in the knowledge base. In order to assess the quality of a particular inconsistency measure, a wide range of rationality postulates has been proposed in the literature. In this paper, we survey 15 recent approaches to inconsistency measurement and provide a comparative analysis on their compliance with 18 rationality postulates. In doing so, we fill the gaps in previous partial investigations and provide new insights into the adequacy of certain measures and the significance of certain postulates.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. Note that slightly different formalizations of this idea have been given in [16, 29, 30].

  2. http://tweetyproject.org/w/incmes/.

  3. Consider a lottery of n tickets and let \(a_{i}\) be the proposition that ticket i, \(i=1,\ldots ,n\) will win. It is known that exactly one ticket will win (\(a_{1}\vee \ldots \vee a_{n}\)) but each ticket owner assumes that his ticket will not win (\(\lnot a_{i}\), \(i=1,\ldots ,n\)). For \(n=1000\) it is reasonable for each ticket owner to believe that he will not win but for e. g., \(n=2\) it is not. Therefore larger minimal inconsistent subsets can be regarded less inconsistent than smaller ones.

  4. http://www.mthimm.de/misc/mt_ratposim_appendix.

  5. Note that proofs of [43] are for propositional probabilistic logic. As this is a generalization of propositional logic, the results apply here as well.

References

  1. Ammoura M, Raddaoui B, Salhi Y, Oukacha B (2015) On measuring inconsistency using maximal consistent sets. In: Proceedings of the 13th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU’15), Springer, pp 267–276

  2. Besnard P (2014) Revisiting postulates for inconsistency measures. In: Proceedings of the 14th European Conference on Logics in Artificial Intelligence (JELIA’14), pp 383–396

  3. Béziau JY, Carnielli W, Gabbay D (eds) (2007) Handbook of paraconsistency. College Publications, London

    MATH  Google Scholar 

  4. Chan H, Darwiche A (2005) On the revision of probabilistic beliefs using uncertain evidence. Artif Intell 163(1):67–90

    Article  MathSciNet  MATH  Google Scholar 

  5. Daniel L (2008) Haziness for common sensical inference from uncertain and inconsistent linear knowledge base. In: Proceedings of the 20th IEEE international conference on tools with artificial intelligence (ICTAI’08), pp 163–170

  6. De Bona G, Finger M (2015) Measuring inconsistency in probabilistic logic: rationality postulates and dutch book interpretation. Artif Intell 227:140–164

    Article  MathSciNet  MATH  Google Scholar 

  7. De Bona G, Finger M, Ribeiro MM, Santos YD, Wassermann R (2016) Consolidating probabilistic knowledge bases via belief contraction. In: Proceedings of the 15th international conference on principles of knowledge representation and reasoning (KR’16)

  8. Doder D, Raskovic M, Markovic Z, Ognjanovic Z (2010) Measures of inconsistency and defaults. Int J Approx Reason 51:832–845

    Article  MathSciNet  MATH  Google Scholar 

  9. Grant J, Hunter A (2006) Measuring inconsistency in knowledgebases. J Intell Inf Syst 27:159–184

    Article  Google Scholar 

  10. Grant J, Hunter A (2011) Measuring consistency gain and information loss in stepwise inconsistency resolution. In: Proceedings of the 11th European conference on symbolic and quantitative approaches to reasoning with uncertainty (ECSQARU 2011), no. 6717 in LNAI, Springer, pp 362–373

  11. Grant J, Hunter A (2013) Distance-based measures of inconsistency. In: Proceedings of the 12th Europen conference on symbolic and quantitative approaches to reasoning with uncertainty (ECSQARU’13), LNCS, vol 7958, Springer, pp 230–241

  12. Hansson SO (2001) A textbook of belief dynamics: theory change and database updating. Kluwer Academic Publishers

  13. Hunter A, Konieczny S (2004) Approaches to measuring inconsistent information. In: Inconsistency tolerance, LNCS, vol 3300, Springer, pp 189–234

  14. Hunter A, Konieczny S (2006) Shapley inconsistency values. In: Proceedings of the 10th international conference on knowledge representation (KR’06), AAAI Press, pp 249–259

  15. Hunter A, Konieczny S (2008) Measuring inconsistency through minimal inconsistent sets. In: Proceedings of the eleventh international conference on principles of knowledge representation and reasoning (KR’2008), AAAI Press, pp 358–366

  16. Hunter A, Konieczny S (2010) On the measure of conflicts: Shapley inconsistency values. Artif Intell 174(14):1007–1026

    Article  MathSciNet  MATH  Google Scholar 

  17. Hunter A, Parsons S, Wooldridge M (2014) Measuring inconsistency in multi-agent systems. Künstliche Intelligenz 28:169–178

    Article  Google Scholar 

  18. Jabbour S, Ma Y, Raddaoui B (2014) Inconsistency measurement thanks to mus decomposition. In: Proceedings of the 13th international conference on autonomous agents and multiagent systems (AAMAS 2014)

  19. Jabbour S, Ma Y, Raddaoui B, Sais L (2014) Prime implicates based inconsistency characterization. In: Proceedings of the 21st European conference on artificial intelligence (ECAI’14), pp 1037–1038

  20. Jabbour S, Ma Y, Raddaoui B, Sais L, Salhi Y (2015) On structure-based inconsistency measures and their computations via closed set packing. In: Proceedings of the 14th international conference on autonomous agents and multiagent systems (AAMAS’15)

  21. Jabbour S, Raddaoui B (2013) Measuring inconsistency through minimal proofs. In: Proceedings of the 12th European conference on symbolic and quantitative approaches to reasoning with uncertainty, ECSQARU’13, Springer, pp 290–301

  22. Kern-Isberner G, Rödder W (2003) Belief revision and information fusion in a probabilistic environment. Proceedings of the 16th international FLAIRS conference., FLAIRS’03AAAI Press, Menlo Park, California, pp 506–510

  23. Knight KM (2001) Measuring inconsistency. J Philos Logic 31:77–98

    Article  MathSciNet  MATH  Google Scholar 

  24. Knight KM (2002) A theory of inconsistency. Ph.D. thesis, University Of Manchester

  25. Konieczny S, Perez RP (2011) Logic based merging. J Philos Logic 40:239–270

    Article  MathSciNet  MATH  Google Scholar 

  26. Kourousias G, Makinson DC (2007) Parallel interpolation, splitting, and relevance in belief change. J Symb Logic 72:994–1002

    Article  MathSciNet  MATH  Google Scholar 

  27. Kyburg HE (1961) Probability and the logic of rational belief. Wesleyan University Press

  28. Lozinskii EL (1994) Information and evidence in logic systems. J Exp Theor Artif Intell 6:163–193

    Article  MATH  Google Scholar 

  29. Ma Y, Qi G, Hitzler P (2011) Computing inconsistency measure based on paraconsistent semantics. J Logic Comput 21(6):1257–1281

    Article  MathSciNet  MATH  Google Scholar 

  30. Ma Y, Qi G, Hitzler P, Lin Z (2007) Measuring inconsistency for description logics based on paraconsistent semantics. In: Proceedings of the 9th European conference on symbolic and quantitative approaches to reasoning with uncertainty, ECSQARU ’07, Springer, pp 30–41

  31. Ma Y, Qi G, Xiao G, Hitzler P, Lin Z (2009) An anytime algorithm for computing inconsistency measurement. In: Knowledge science, engineering and management, no. 5914 in LNCS, Springer, pp 29–40

  32. McAreavey K, Liu W, Miller P (2014) Computational approaches to finding and measuring inconsistency in arbitrary knowledge bases. Int J Approx Reason 55:1659–1693

    Article  MathSciNet  MATH  Google Scholar 

  33. Mu K, Liu W, Jin Z (2011) A general framework for measuring inconsistency through minimal inconsistent sets. Knowl Inf Syst 27:85–114

    Article  Google Scholar 

  34. Mu K, Liu W, Jin Z, Bell D (2011) A syntax-based approach to measuring the degree of inconsistency for belief bases. Int J Approx Reason 52(7):978–999

    Article  MathSciNet  MATH  Google Scholar 

  35. Mu K, Wang K, Wen L (2014) Approaches to measuring inconsistency for stratified knowledge bases. Int J Approx Reason 55:529–556

    Article  MathSciNet  MATH  Google Scholar 

  36. Muiño DP (2011) Measuring and repairing inconsistency in probabilistic knowledge bases. Inte J Approx Reason 52(6):828–840

    Article  MathSciNet  MATH  Google Scholar 

  37. Parikh R (1999) Beliefs, belief revision, and splitting languages. In: Moss LS, Ginzburg J, de Rijke M (eds) Logic, language and computation, vol 2, CSLI Publications, pp 266–278

  38. Potyka N, Thimm M (2014) Consolidation of probabilistic knowledge bases by inconsistency minimization. In: Proceedings of the 21st European conference on artificial intelligence (ECAI’14), pp 729–734

  39. Potyka N, Thimm M (2015) Probabilistic reasoning with inconsistent beliefs using inconsistency measures. In: Proceedings of the 24th international joint conference on artificial intelligence (IJCAI’15)

  40. Priest G (1979) Logic of paradox. J Philos Logic 8:219–241

    Article  MathSciNet  MATH  Google Scholar 

  41. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423

    Article  MathSciNet  MATH  Google Scholar 

  42. Thimm M (2009) Measuring inconsistency in probabilistic knowledge bases. In: Bilmes J, Ng A (eds) Proceedings of the twenty-fifth conference on uncertainty in artificial intelligence (UAI’09), AUAI Press, pp 530–537

  43. Thimm M (2013) Inconsistency measures for probabilistic logics. Artif Intell 197:1–24

    Article  MathSciNet  MATH  Google Scholar 

  44. Thimm M (2014) Coherence and compatibility of markov logic networks. In: Proceedings of the 21st European conference on artificial intelligence (ECAI’14)

  45. Thimm M (2014) Tweety—a comprehensive collection of java libraries for logical aspects of artificial intelligence and knowledge representation. In: Proceedings of the 14th international conference on principles of knowledge representation and reasoning (KR’14), pp 528–537

  46. Thimm M (2016) On the expressivity of inconsistency measures. Artif Intell 234:120–151

    Article  MathSciNet  MATH  Google Scholar 

  47. Thimm M (2016) Stream-based inconsistency measurement. Int J Approx Reason 68:68–87

    Article  MathSciNet  MATH  Google Scholar 

  48. Thimm M, Wallner JP (2016) Some complexity results on inconsistency measurement. In: Proceedings of the 15th international conference on principles of knowledge representation and reasoning (KR’16)

  49. Xiao G, Ma Y (2012) Inconsistency measurement based on variables in minimal unsatisfiable subsets. In: Proceedings of the 20th European conference on artificial intelligence (ECAI’12)

Download references

Author information

Authors and Affiliations

  1. Institute for Web Science and Technologies, Universität Koblenz-Landau, Universitätsstr. 1, 56070, Koblenz, Germany

    Matthias Thimm

Authors
  1. Matthias Thimm
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthias Thimm.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Thimm, M. On the Compliance of Rationality Postulates for Inconsistency Measures: A More or Less Complete Picture. Künstl Intell 31, 31–39 (2017). https://doi.org/10.1007/s13218-016-0451-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13218-016-0451-y

Keywords

Search

Navigation