Skip to main content
SpringerLink
Log in

Efficient Reasoning for Inconsistent Horn Formulae

  • Conference paper
  • First Online:
Logics in Artificial Intelligence (JELIA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10021))

Included in the following conference series:

Abstract

Horn formulae are widely used in different settings that include logic programming, answer set programming, description logics, deductive databases, and system verification, among many others. One concrete example is concept subsumption in lightweight description logics, which can be reduced to inference in propositional Horn formulae. Some problems require one to reason with inconsistent Horn formulae. This is the case when providing minimal explanations of inconsistency. This paper proposes efficient algorithms for a number of decision, function and enumeration problems related with inconsistent Horn formulae. Concretely, the paper develops efficient algorithms for finding and enumerating minimal unsatisfiable subsets (MUSes), minimal correction subsets (MCSes), but also for computing the lean kernel. The paper also shows the practical importance of some of the proposed algorithms.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Arif, M.F., Mencía, C., Ignatiev, A., Manthey, N., Peñaloza, R., Marques-Silva, J.: BEACON: an efficient SAT-based tool for debugging \({\cal EL}^{+}\) ontologies. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 521–530. Springer, Heidelberg (2016). doi:10.1007/978-3-319-40970-2_32

    Chapter  Google Scholar 

  2. Arif, M.F., Mencía, C., Marques-Silva, J.: Efficient axiom pinpointing with EL2MCS. In: Hölldobler, S., Krötzsch, M., Peñaloza, R., Rudolph, S. (eds.) KI 2015. LNCS (LNAI), vol. 9324, pp. 225–233. Springer, Heidelberg (2015). doi:10.1007/978-3-319-24489-1_17

    Chapter  Google Scholar 

  3. Arif, M.F., Mencía, C., Marques-Silva, J.: Efficient MUS enumeration of horn formulae with applications to axiom pinpointing. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 324–342. Springer, Heidelberg (2015). doi:10.1007/978-3-319-24318-4_24

    Chapter  Google Scholar 

  4. Baader, F., Brandt, S., Lutz, C.: Pushing the \({\cal EL}\) envelope. In: IJCAI, pp. 364–369 (2005)

    Google Scholar 

  5. Baader, F., Peñaloza, R., Suntisrivaraporn, B.: Pinpointing in the description logic \({\cal EL}^{+}\). In: Hertzberg, J., Beetz, M., Englert, R. (eds.) KI 2007. LNCS (LNAI), vol. 4667, pp. 52–67. Springer, Heidelberg (2007). doi:10.1007/978-3-540-74565-5_7

    Chapter  Google Scholar 

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

    Google Scholar 

  7. Bacchus, F., Katsirelos, G.: Using minimal correction sets to more efficiently compute minimal unsatisfiable sets. In: Kroening, D., Păsăreanu, C.S. (eds.) CAV 2015. LNCS, vol. 9207, pp. 70–86. Springer, Heidelberg (2015). doi:10.1007/978-3-319-21668-3_5

    Chapter  Google Scholar 

  8. Bakker, R.R., Dikker, F., Tempelman, F., Wognum, P.M.: Diagnosing and solving over-determined constraint satisfaction problems. In: IJCAI, pp. 276–281 (1993)

    Google Scholar 

  9. Bate, A., Motik, B., Grau, B.C., Simancik, F., Horrocks, I.: Extending consequence-based reasoning to SRIQ. In: KR, pp. 187–196 (2016)

    Google Scholar 

  10. Belov, A., Lynce, I., Marques-Silva, J.: Towards efficient MUS extraction. AI Commun. 25(2), 97–116 (2012)

    MathSciNet  MATH  Google Scholar 

  11. Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press, Amsterdam (2009)

    MATH  Google Scholar 

  12. Bjørner, N., Fioravanti, F., Rybalchenko, A., Senni, V. (eds.) Proceedings First Workshop on Horn Clauses for Verification and Synthesis, HCVS 2014, Vienna, Austria, 17 July 2014. EPTCS, vol. 169 (2014)

    Google Scholar 

  13. Bjørner, N., Gurfinkel, A., McMillan, K., Rybalchenko, A.: Horn clause solvers for program verification. In: Beklemishev, L.D., Blass, A., Dershowitz, N., Finkbeiner, B., Schulte, W. (eds.) Fields of Logic and Computation II: Essays Dedicated to Yuri Gurevich on the Occasion of his 75th Birthday. LNCS, vol. 9300, pp. 24–51. Springer, Heidelberg (2015). doi:10.1007/978-3-319-23534-9_2

    Chapter  Google Scholar 

  14. Buss, S.R., Krajíček, J., Takeuti, G.: Provably total functions in the bounded arithmetic theories \(R^i_3\), \(U^i_2\), and \(V^i_2\). In: Clote, P., Krajíček, J. (eds.) Arithmetic, Proof Theory, and Computational Complexity, pp. 116–161. OUP (1995)

    Google Scholar 

  15. Chinneck, J.W., Dravnieks, E.W.: Locating minimal infeasible constraint sets in linear programs. INFORMS J. Comput. 3(2), 157–168 (1991)

    Article  MATH  Google Scholar 

  16. Dowling, W.F., Gallier, J.H.: Linear-time algorithms for testing the satisfiability of propositional Horn formulae. J. Log. Program. 1(3), 267–284 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Eiter, T., Gottlob, G.: The complexity of logic-based abduction. J. ACM 42(1), 3–42 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Henschen, L.J., Wos, L.: Unit refutations and Horn sets. J. ACM 21(4), 590–605 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  19. Horn, A.: On sentences which are true of direct unions of algebras. J. Symb. Log. 16(1), 14–21 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  20. Itai, A., Makowsky, J.A.: Unification as a complexity measure for logic programming. J. Log. Program. 4(2), 105–117 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  21. Janota, M., Marques-Silva, J.: On the query complexity of selecting minimal sets for monotone predicates. Artif. Intell. 233, 73–83 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: On generating all maximal independent sets. Inf. Process. Lett. 27(3), 119–123 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kazakov, Y., Krötzsch, M., Simancik, F.: The incredible ELK - from polynomial procedures to efficient reasoning with ontologies. J. Autom. Reasoning 53(1), 1–61 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kleine Büning, H., Kullmann, O.: Minimal unsatisfiability and autarkies. In: Biere, A., et al. [11], pp. 339–401

    Google Scholar 

  25. Kullmann, O.: Investigations on autark assignments. Discrete Appl. Math. 107(1–3), 99–137 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kullmann, O.: On the use of autarkies for satisfiability decision. Electron. Notes Discrete Math. 9, 231–253 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kullmann, O., Lynce, I., Marques-Silva, J.: Categorisation of clauses in conjunctive normal forms: minimally unsatisfiable sub-clause-sets and the lean kernel. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 22–35. Springer, Heidelberg (2006). doi:10.1007/11814948_4

    Chapter  Google Scholar 

  28. Kullmann, O., Marques-Silva, J.: Computing maximal autarkies with few and simple oracle queries. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 138–155. Springer, Heidelberg (2015). doi:10.1007/978-3-319-24318-4_11

    Chapter  Google Scholar 

  29. Li, C.M., Manyà, F., Mohamedou, N.O., Planes, J.: Resolution-based lower bounds in MaxSAT. Constraints 15(4), 456–484 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Liberatore, P.: Redundancy in logic II: 2CNF and Horn propositional formulae. Artif. Intell. 172(2–3), 265–299 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Manthey, N., Peñaloza, R., Rudolph, S.: Efficient axiom pinpointing in \({\cal EL}\) using SAT technology. In: DL (2016)

    Google Scholar 

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

    Google Scholar 

  34. Marques-Silva, J., Ignatiev, A., Morgado, A., Manquinho, V.M., Lynce, I.: Efficient autarkies. In: ECAI, pp. 603–608 (2014)

    Google Scholar 

  35. 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, Heidelberg (2016). doi:10.1007/978-3-319-40970-2_21

    Chapter  Google Scholar 

  36. Mencía, C., Previti, A., Marques-Silva, J.: Literal-based MCS extraction. In: IJCAI, pp. 1973–1979 (2015)

    Google Scholar 

  37. Minoux, M.: LTUR: a simplified linear-time unit resolution algorithm for Horn formulae and computer implementation. Inf. Process. Lett. 29(1), 1–12 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  38. Monien, B., Speckenmeyer, E.: Solving satisfiability in less than \(2^{n}\) steps. Discrete Appl. Math. 10(3), 287–295 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  39. Papadimitriou, C.H.: Computational Complexity. Addison Wesley, Redwood City (1993)

    MATH  Google Scholar 

  40. Papadimitriou, C.H.: NP-completeness: a retrospective. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 2–6. Springer, Heidelberg (1997). doi:10.1007/3-540-63165-8_160

    Chapter  Google Scholar 

  41. Peñaloza, R.: Axiom-pinpointing in description logics and beyond. Ph.D. thesis, Dresden University of Technology, Germany (2009)

    Google Scholar 

  42. Peñaloza, R., Sertkaya, B.: On the complexity of axiom pinpointing in the \({\cal EL}\) family of description logics. In: KR (2010)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  44. Sebastiani, R., Vescovi, M.: Axiom pinpointing in lightweight description logics via Horn-SAT encoding and conflict analysis. In: Schmidt, R.A. (ed.) CADE 2009. LNCS (LNAI), vol. 5663, pp. 84–99. Springer, Heidelberg (2009). doi:10.1007/978-3-642-02959-2_6

    Chapter  Google Scholar 

  45. Sebastiani, R., Vescovi, M.: Axiom pinpointing in large \({\cal EL}^{+}\) ontologies via SAT and SMT techniques. Technical report DISI-15-010, DISI, University of Trento, Italy. http://disi.unitn.it/~rseba/elsat/elsat_techrep.pdf

  46. Vescovi, M.: Exploiting SAT and SMT techniques for automated reasoning and ontology manipulation in description logics. Ph.D. thesis, University of Trento (2011)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Lisbon, Lisbon, Portugal

    Joao Marques-Silva & Alexey Ignatiev

  2. University of Oviedo, Oviedo, Spain

    Carlos Mencía

  3. Free University of Bozen-Bolzano, Bolzano, Italy

    Rafael Peñaloza

  4. ISDCT SB RAS, Irkutsk, Russia

    Alexey Ignatiev

Authors
  1. Joao Marques-Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexey Ignatiev
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Carlos Mencía
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Rafael Peñaloza
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joao Marques-Silva .

Editor information

Editors and Affiliations

  1. Open University of Cyprus, Nicosia, Cyprus

    Loizos Michael

  2. University of Cyprus, Nicosia, Cyprus

    Antonis Kakas

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing AG

About this paper

Cite this paper

Marques-Silva, J., Ignatiev, A., Mencía, C., Peñaloza, R. (2016). Efficient Reasoning for Inconsistent Horn Formulae. In: Michael, L., Kakas, A. (eds) Logics in Artificial Intelligence. JELIA 2016. Lecture Notes in Computer Science(), vol 10021. Springer, Cham. https://doi.org/10.1007/978-3-319-48758-8_22

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-48758-8_22

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-48757-1

  • Online ISBN: 978-3-319-48758-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation