Skip to main content
Springer Nature Link
Log in

Computing MUS-Based Inconsistency Measures

  • Conference paper
  • First Online:
Logics in Artificial Intelligence (JELIA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 14281))

Included in the following conference series:

  • 737 Accesses

Abstract

We detail two instantiations of a generic algorithm for the problematic and MUS-variable-based inconsistency measures, based on answer set programming and Boolean satisfiability (SAT). Empirically, the SAT-based approach allows for more efficiently computing the measures when compared to enumerating all minimal correction subsets of a knowledge base.

Work financially supported by Deutsche Forschungsgemeinschaft (grant 506604007/IK) and by Academy of Finland (grants 347588/AN and 322869, 356046/MJ).

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    Not to be confused with the notion of variable minimal unsatisfiability [5, 18].

References

  1. Agrawal, R., Srikant, R.: Fast algorithms for mining association rules in large databases. In: Proceedings of the VLDB 1994, pp. 487–499 (1994)

    Google Scholar 

  2. Alviano, M.: Model enumeration in propositional circumscription via unsatisfiable core analysis. Theory Pract. Log. Program. 17(5–6), 708–725 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alviano, M.: Query answering in propositional circumscription. In: Proceedings of the IJCAI 2018, pp. 1669–1675. ijcai.org (2018)

    Google Scholar 

  4. Bacchus, F., Davies, J., Tsimpoukelli, M., Katsirelos, G.: Relaxation search: A simple way of managing optional clauses. In: Proc. AAAI 2014. pp. 835–841. AAAI Press (2014)

    Google Scholar 

  5. Belov, A., Ivrii, A., Matsliah, A., Marques-Silva, J.: On efficient computation of variable MUSes. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 298–311. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31612-8_23

    Chapter  MATH  Google Scholar 

  6. Bendík, J.: On decomposition of maximal satisfiable subsets. In: Proceedings of the FMCAD 2021, pp. 212–221. IEEE (2021)

    Google Scholar 

  7. Bendík, J., Benes, N., Cerná, I., Barnat, J.: Tunable online MUS/MSS enumeration. In: Proceedings of the FSTTCS 2016. LIPIcs, vol. 65, pp. 50:1–50:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

    Google Scholar 

  8. Bendík, J., Černá, I.: MUST: minimal unsatisfiable subsets enumeration tool. In: TACAS 2020. LNCS, vol. 12078, pp. 135–152. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45190-5_8

    Chapter  Google Scholar 

  9. Bendík, J., Černá, I.: Replication-guided enumeration of minimal unsatisfiable subsets. In: Simonis, H. (ed.) CP 2020. LNCS, vol. 12333, pp. 37–54. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58475-7_3

    Chapter  Google Scholar 

  10. Bendík, J., Cerna, I.: Rotation based MSS/MCS enumeration. In: Proceedings of the LPAR 2020. EPiC Series in Computing, vol. 73, pp. 120–137. EasyChair (2020)

    Google Scholar 

  11. Bendík, J., Černá, I., Beneš, N.: Recursive online enumeration of all minimal unsatisfiable subsets. In: Lahiri, S.K., Wang, C. (eds.) ATVA 2018. LNCS, vol. 11138, pp. 143–159. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-01090-4_9

    Chapter  Google Scholar 

  12. Bendík, J., Meel, K.S.: Approximate counting of minimal unsatisfiable subsets. In: Lahiri, S.K., Wang, C. (eds.) CAV 2020. LNCS, vol. 12224, pp. 439–462. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-53288-8_21

    Chapter  Google Scholar 

  13. Bendík, J., Meel, K.S.: Counting maximal satisfiable subsets. In: Proceedings of the AAAI 2021, pp. 3651–3660. AAAI Press (2021)

    Google Scholar 

  14. Bendík, J., Meel, K.S.: Counting minimal unsatisfiable subsets. In: Silva, A., Leino, K.R.M. (eds.) CAV 2021. LNCS, vol. 12760, pp. 313–336. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-81688-9_15

    Chapter  Google Scholar 

  15. Besnard, P., Doutre, S., Herzig, A.: Encoding argument graphs in logic. In: Laurent, A., Strauss, O., Bouchon-Meunier, B., Yager, R.R. (eds.) IPMU 2014. CCIS, vol. 443, pp. 345–354. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08855-6_35

    Chapter  MATH  Google Scholar 

  16. Biere, A., Fazekas, K., Fleury, M., Heisinger, M.: CaDiCaL, Kissat, Paracooba, Plingeling and Treengeling entering the SAT competition 2020. In: Proceedings of the SAT Competition 2020 - Solver and Benchmark Descriptions. Department of Computer Science Report Series B, vol. B-2020-1, pp. 51–53. University of Helsinki (2020)

    Google Scholar 

  17. Brewka, G., Eiter, T., Truszczynski, M.: Answer set programming at a glance. Commun. ACM 54(12), 92–103 (2011)

    Article  Google Scholar 

  18. Chen, Z., Ding, D.: Variable minimal unsatisfiability. In: Cai, J.-Y., Cooper, S.B., Li, A. (eds.) TAMC 2006. LNCS, vol. 3959, pp. 262–273. Springer, Heidelberg (2006). https://doi.org/10.1007/11750321_25

    Chapter  Google Scholar 

  19. Clarke, E.M., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement for symbolic model checking. J. ACM 50(5), 752–794 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Clarke, E.M., Gupta, A., Strichman, O.: SAT-based counterexample-guided abstraction refinement. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 23(7), 1113–1123 (2004)

    Google Scholar 

  21. Doder, D., Raskovic, M., Markovic, Z., Ognjanovic, Z.: Measures of inconsistency and defaults. Int. J. Approx. Reason. 51(7), 832–845 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Eiter, T., Gottlob, G.: On the computational cost of disjunctive logic programming: propositional case. Ann. Math. Artif. Intell. 15, 289–323 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gebser, M., Kaminski, R., Kaufmann, B., Ostrowski, M., Schaub, T., Wanko, P.: Theory solving made easy with Clingo 5. In: Technical Communications of ICLP, pp. 2:1–2:15. OASICS, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2016)

    Google Scholar 

  25. Gebser, M., Kaminski, R., Kaufmann, B., Schaub, T.: Answer set solving in practice. In: Synthesis Lectures on Artificial Intelligence and Machine Learning, vol. 6, no. 3, pp. 1–238 (2012)

    Google Scholar 

  26. Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Proceedings of the ICLP/SLP, pp. 1070–1080. MIT Press (1988)

    Google Scholar 

  27. Grant, J.: Classifications for inconsistent theories. Notre Dame J. Formal Log. 19(3), 435–444 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  28. Grant, J., Hunter, A.: Measuring consistency gain and information loss in stepwise inconsistency resolution. In: Liu, W. (ed.) ECSQARU 2011. LNCS (LNAI), vol. 6717, pp. 362–373. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22152-1_31

    Chapter  MATH  Google Scholar 

  29. Grant, J., Martinez, M.V. (eds.): Measuring Inconsistency in Information, Studies in Logic, vol. 73. College Publications (2018)

    Google Scholar 

  30. Grégoire, É., Izza, Y., Lagniez, J.: Boosting MCSes enumeration. In: Proceedings of the IJCAI 2018, pp. 1309–1315. ijcai.org (2018)

    Google Scholar 

  31. Grégoire, É., Lagniez, J., Mazure, B.: An experimentally efficient method for (MSS, CoMSS) partitioning. In: Proceedings of AAAI 2014, pp. 2666–2673. AAAI Press (2014)

    Google Scholar 

  32. Hunter, A., Konieczny, S.: Measuring inconsistency through minimal inconsistent sets. In: Proceedings KR 2008, pp. 358–366. AAAI Press (2008)

    Google Scholar 

  33. Ignatiev, A., Morgado, A., Marques-Silva, J.: PySAT: a python toolkit for prototyping with SAT oracles. In: Beyersdorff, O., Wintersteiger, C.M. (eds.) SAT 2018. LNCS, vol. 10929, pp. 428–437. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94144-8_26

    Chapter  MATH  Google Scholar 

  34. Jabbour, S., Ma, Y., Raddaoui, B., Sais, L., Salhi, Y.: A MIS partition based framework for measuring inconsistency. In: Baral, C., Delgrande, J.P., Wolter, F. (eds.) Proceedings of the KR 2016, pp. 84–93. AAAI Press (2016)

    Google Scholar 

  35. Janota, M., Grigore, R., Marques-Silva, J.: Counterexample guided abstraction refinement algorithm for propositional circumscription. In: Janhunen, T., Niemelä, I. (eds.) JELIA 2010. LNCS (LNAI), vol. 6341, pp. 195–207. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15675-5_18

    Chapter  Google Scholar 

  36. Janota, M., Marques-Silva, J.: cmMUS: a tool for circumscription-based MUS membership testing. In: Delgrande, J.P., Faber, W. (eds.) LPNMR 2011. LNCS (LNAI), vol. 6645, pp. 266–271. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-20895-9_30

    Chapter  Google Scholar 

  37. Janota, M., Marques-Silva, J.: On deciding MUS membership with QBF. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 414–428. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-23786-7_32

    Chapter  Google Scholar 

  38. Kleine Büning, H., Kullmann, O.: Minimal unsatisfiability and autarkies. In: Handbook of Satisfiability - Second Edition, Frontiers in Artificial Intelligence and Applications, vol. 336, pp. 571–633. IOS Press (2021)

    Google Scholar 

  39. Kuhlmann, I., Gessler, A., Laszlo, V., Thimm, M.: A comparison of ASP-based and SAT-based algorithms for the contension inconsistency measure. In: Dupin de Saint-Cyr, F., Öztürk-Escoffier, M., Potyka, N. (eds.) Scalable Uncertainty Management. SUM 2022. Lecture Notes in Computer Science, vol. 13562, pp. 139–153. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-18843-5_10

  40. Kuhlmann, I., Gessler, A., Laszlo, V., Thimm, M.: Comparison of SAT-based and ASP-based algorithms for inconsistency measurement. arXiv p. 2304.14832 (2023). preprint

    Google Scholar 

  41. Kuhlmann, I., Thimm, M.: An algorithm for the contension inconsistency measure using reductions to answer set programming. In: Davis, J., Tabia, K. (eds.) SUM 2020. LNCS (LNAI), vol. 12322, pp. 289–296. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58449-8_23

    Chapter  Google Scholar 

  42. Kuhlmann, I., Thimm, M.: Algorithms for inconsistency measurement using answer set programming. In: Proceedings of the NMR 2021, pp. 159–168 (2021)

    Google Scholar 

  43. Kullmann, O.: Constraint satisfaction problems in clausal form II: minimal unsatisfiability and conflict structure. Fundam. Informaticae 109(1), 83–119 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. Liberatore, P.: Redundancy in logic I: CNF propositional formulae. Artif. Intell. 163(2), 203–232 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  45. Liffiton, M.H., Previti, A., Malik, A., Marques-Silva, J.: Fast, flexible MUS enumeration. Constraints An Int. J. 21(2), 223–250 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  46. Liffiton, M.H., Sakallah, K.A.: Algorithms for computing minimal unsatisfiable subsets of constraints. J. Autom. Reason. 40(1), 1–33 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  47. Lifschitz, V.: Answer Set Programming. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-24658-7

    Book  MATH  Google Scholar 

  48. Marques-Silva, J., Heras, F., Janota, M., Previti, A., Belov, A.: On computing minimal correction subsets. In: Proceedings of the IJCAI 2013, pp. 615–622. IJCAI/AAAI (2013)

    Google Scholar 

  49. Marques-Silva, J., Ignatiev, A., Morgado, A., Manquinho, V.M., Lynce, I.: Efficient autarkies. In: Proceedings of the ECAI 2014. Frontiers in Artificial Intelligence and Applications, vol. 263, pp. 603–608. IOS Press (2014)

    Google Scholar 

  50. Marques-Silva, J., Mencía, C.: Reasoning about inconsistent formulas. In: Proceedings of the IJCAI 2020, pp. 4899–4906. ijcai.org (2020)

    Google Scholar 

  51. Mencía, C., Ignatiev, A., Previti, A., Marques-Silva, J.: MCS extraction with sublinear oracle queries. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 342–360. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40970-2_21

    Chapter  Google Scholar 

  52. Mencía, C., Kullmann, O., Ignatiev, A., Marques-Silva, J.: On computing the union of MUSes. In: Janota, M., Lynce, I. (eds.) SAT 2019. LNCS, vol. 11628, pp. 211–221. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-24258-9_15

    Chapter  MATH  Google Scholar 

  53. Mencía, C., Previti, A., Marques-Silva, J.: Literal-based MCS extraction. In: Proceedings of the IJCAI 2015, pp. 1973–1979. AAAI Press (2015)

    Google Scholar 

  54. Nadel, A.: Boosting minimal unsatisfiable core extraction. In: Proceedings of the FMCAD 2010, pp. 221–229. IEEE (2010)

    Google Scholar 

  55. Narodytska, N., Bjørner, N.S., Marinescu, M.V., Sagiv, M.: Core-guided minimal correction set and core enumeration. In: Proceedings of the IJCAI 2018, pp. 1353–1361. ijcai.org (2018)

    Google Scholar 

  56. Niemelä, I.: Logic programs with stable model semantics as a constraint programming paradigm. Ann. Math. Artif. Intell. 25(3–4), 241–273 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  57. Niskanen, A., Kuhlmann, I., Thimm, M., Järvisalo, M.: MaxSAT-Based inconsistency measurement. In: Proceedings of the ECAI 2023. IOS Press (2023)

    Google Scholar 

  58. Previti, A., Mencía, C., Järvisalo, M., Marques-Silva, J.: Improving MCS enumeration via caching. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 184–194. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66263-3_12

    Chapter  Google Scholar 

  59. Previti, A., Mencía, C., Järvisalo, M., Marques-Silva, J.: Premise set caching for enumerating minimal correction subsets. In: Proceedings of the AAAI 2018, pp. 6633–6640. AAAI Press (2018)

    Google Scholar 

  60. Reiter, R.: A theory of diagnosis from first principles. Artif. Intell. 32(1), 57–95 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  61. Thimm, M., Wallner, J.P.: On the complexity of inconsistency measurement. Artif. Intell. 275, 411–456 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  62. Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Siekmann, J.H., Wrightson, G. (eds.) Automation of Reasoning, pp. 466–483. Springer, Heidelberg (1983). https://doi.org/10.1007/978-3-642-81955-1_28

  63. Xiao, G., Ma, Y.: Inconsistency measurement based on variables in minimal unsatisfiable subsets. In: Proceedings of the ECAI 2012, pp. 864–869. IOS Press (2012)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Hagen, Hagen, Germany

    Isabelle Kuhlmann

  2. University of Helsinki, Helsinki, Finland

    Andreas Niskanen & Matti Järvisalo

Authors
  1. Isabelle Kuhlmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreas Niskanen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matti Järvisalo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matti Järvisalo .

Editor information

Editors and Affiliations

  1. TU Dresden, Dresden, Germany

    Sarah Gaggl

  2. Artificial Intelligence Research Institute - IIIA CSIC, Barcelona, Spain

    Maria Vanina Martinez

  3. Umeå University, Umeå, Sweden

    Magdalena Ortiz

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kuhlmann, I., Niskanen, A., Järvisalo, M. (2023). Computing MUS-Based Inconsistency Measures. In: Gaggl, S., Martinez, M.V., Ortiz, M. (eds) Logics in Artificial Intelligence. JELIA 2023. Lecture Notes in Computer Science(), vol 14281. Springer, Cham. https://doi.org/10.1007/978-3-031-43619-2_50

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-43619-2_50

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-43618-5

  • Online ISBN: 978-3-031-43619-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics