Skip to main content
SpringerLink
Log in

An Entailment Procedure for Kleene Answer Set Programs

  • Conference paper
  • First Online:
Multi-disciplinary Trends in Artificial Intelligence (MIWAI 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10053))

  • 637 Accesses

Abstract

Classical Answer Set Programming is a widely known knowledge representation framework based on the logic programming paradigm that has been extensively studied in the past decades. Semantic theories for classical answer sets are implicitly three-valued in nature, yet with few exceptions, computing classical answer sets is based on translations into classical logic and the use of SAT solving techniques. In this paper, we introduce a variation of Kleene three-valued logic with strong connectives, \({\text {R}_3}\), and then provide a sound and complete proof procedure for \({\text {R}_3}\) based on the use of signed tableaux. We then define a restriction on the syntax of \({\text {R}_3}\) to characterize Kleene ASPs. Strongly-supported models, which are a subset of \({\text {R}_3}\) models are then defined to characterize the semantics of Kleene ASPs. A filtering technique on tableaux for \({\text {R}_3}\) is then introduced which provides a sound and complete tableau-based proof technique for Kleene ASPs. We then show a translation and semantic correspondence between Classical ASPs and Kleene ASPs, where answer sets for normal classical ASPs are equivalent to strongly-supported models. This implies that the proof technique introduced can be used for classical normal ASPs as well as Kleene ASPs. The relation between non-normal classical and Kleene ASPs is also considered.

This work is partially supported by the Swedish Research Council (VR) Linnaeus Center CADICS, the ELLIIT network organization for Information and Communication Technology, the Swedish Foundation for Strategic Research (CUAS, SymbiCloud Project), and Vinnova NFFP6 Project 2013-01206.

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

Notes

  1. 1.

    For clarity, we restrict the results to the propositional case, but they are easily generalized to the first-order case with certain restrictions.

  2. 2.

    Strong negation, conjunction and disjunction have also been used in [26].

  3. 3.

    The implication connective, \(\rightarrow _s\,\), is in fact equivalent to that used in [35].

  4. 4.

    This signed approach to semantic tableaux for multi-valued logic was discovered independently, by [6, 18] who generalized the technique for a larger class of multi-valued logics. The approach of [6] was used as a basis for [28].

  5. 5.

    In understanding the theorems, recall that the syntax for ASP\(^{K}\) programs and classical ASP programs is identical.

  6. 6.

    Note that both \(\{p\}\) and \(\{\lnot q, p\}\) are \({\text {R}_3}\) models for the considered formula, so the fact that q is entailed by \(\varPi \) cannot be proved using only rules provided in Sect. 3.1.

References

  1. Baral, C.: Knowledge Representation, Reasoning, and Declarative Problem Solving. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  2. Bogaerts, B., Vennekens, J., Denecker, M.: Grounded fixpoints and their applications in knowledge representation. Artif. Intell. 224, 51–71 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brewka, G., Niemelä, I., Truszczynski, M.: Answer set optimization. In: Gottlob, G., Walsh, T. (eds.) Proceedings of the 18th IJCAI, pp. 867–872. Morgan Kaufmann (2003)

    Google Scholar 

  4. Chan, P.: A possible world semantics for disjunctive databases. IEEE Trans. Knowl. Data Eng. 5(2), 282–292 (1993)

    Article  Google Scholar 

  5. Denecker, M., Marek, V., Truszczynski, M.: Stable operators, well-founded fixpoints and applications in nonmonotonic reasoning. In: Minker, J. (ed.) Logic-Based Artificial Intelligence, pp. 127–144. Kluwer Academic Publishers, Dordrecht (2000)

    Chapter  Google Scholar 

  6. Doherty, P.: A constraint-based approach to proof procedures for multi-valued logics. In: Proceedings of the 1st World Conference Fundamentals of AI (WOCFAI), pp. 165–178. Springer (1991)

    Google Scholar 

  7. Doherty, P., Szałas, A.: Stability, supportedness, minimality and Kleene Answer Set Programs. In: Eiter, T., Strass, H., Truszczyński, M., Woltran, S. (eds.) Advances in Knowledge Representation, Logic Programming, and Abstract Argumentation. LNCS (LNAI), vol. 9060, pp. 125–140. Springer, Heidelberg (2015). doi:10.1007/978-3-319-14726-0_9

    Google Scholar 

  8. Eiter, T., Faber, W., Fink, M., Woltran, S.: Complexity results for answer set programming with bounded predicate arities and implications. Ann. Math. Artif. Intell. 51(2–4), 123–165 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Eiter, T., Gottlob, G.: Complexity results for disjunctive logic programming and application to nonmonotonic logics. In: Miller, D. (ed.) Proceedings of the Logic Programming, pp. 266–278 (1993)

    Google Scholar 

  10. Fages, F.: Consistency of Clark’s completion and existence of stable models. Methods Logic Comput. Sci. 1, 51–60 (1994)

    Google Scholar 

  11. Fitting, M.: A Kripke-Kleene semantics for logic programs. J. Logic Program. 2(4), 295–312 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fitting, M.: The family of stable models. J. Logic Program. 17(2–4), 197–225 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gebser, M., Kaufmann, B., Schaub, T.: Conflict-driven answer set solving: from theory to practice. Artif. Intell. 187, 52–89 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gebser, M., Schaub, T.: Tableau calculi for logic programs under answer set semantics. ACM Trans. Comput. Log. 14(2), 15:1–15:40 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gelfond, M., Kahl, Y.: Knowledge Representation, Reasoning, and the Design of Intelligent Agents - The Answer-Set Programming Approach. Cambridge University Press, Cambridge (2014)

    Book  Google Scholar 

  16. Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Kowalski, R., Bowen, K. (eds.) Proceedings of the International Logic Programming, pp. 1070–1080. MIT Press (1988)

    Google Scholar 

  17. Gelfond, M., Lifschitz, V.: Classical negation in logic programs and disjunctive databases. New Gener. Comput. 9(3/4), 365–386 (1991)

    Article  MATH  Google Scholar 

  18. Hähnle, R.: Uniform notation of tableau rules for multiple-valued logics. In: ISMVL, pp. 238–245 (1991)

    Google Scholar 

  19. Kleene, S.C.: On a notation for ordinal numbers. Symbolic Logic 3, 150–155 (1938)

    Article  MATH  Google Scholar 

  20. Kunen, K.: Negation in logic programming. J. Log. Program. 4(4), 289–308 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  21. Leone, N., Pfeifer, G., Faber, W., Eiter, T., Gottlob, G., Perri, S., Scarcello, F.: The DLV system for knowledge representation and reasoning. ACM Trans. Comput. Logic 7(3), 499–562 (2006)

    Article  MathSciNet  Google Scholar 

  22. Leone, N., Rullo, P., Scarcello, F.: Disjunctive stable models: unfounded sets, fixpoint semantics, and computation. Inf. Comput. 135(2), 69–112 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lierler, Y.: Relating constraint answer set programming languages and algorithms. Artif. Intell. 207, 1–22 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lifschitz, V.: Thirteen definitions of a stable model. In: Blass, A., Dershowitz, N., Reisig, W. (eds.) Fields of Logic and Computation. LNCS, vol. 6300, pp. 488–503. Springer, Heidelberg (2010). doi:10.1007/978-3-642-15025-8_24

    Chapter  Google Scholar 

  25. Lin, F., Zhao, Y.: ASSAT: computing answer sets of a logic program by SAT solvers. Artif. Intell. 157(1–2), 115–137 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Łukasiewicz, J.: O logice trójwartościowej (in Polish). Ruch filozoficzny 5, 170–171 (1920). English translation: On three-valued logic. In: Borkowski, L. (ed.) Selected works by Jan Łukasiewicz, pp. 87–88. North-Holland, Amsterdam (1970)

    Google Scholar 

  27. McCarthy, J.: Circumscription - a form of non-monotonic reasoning. Artif. Intell. 13(1–2), 27–39 (1980)

    Article  MATH  Google Scholar 

  28. Murray, N., Rosenthal, E.: Adapting classical inference techniques to multiple-valued logics using signed formulas. Fundam. Inform. 21(3), 237–253 (1994)

    MathSciNet  MATH  Google Scholar 

  29. Pearce, D.: Equilibrium logic. Ann. Math. AI 47(1–2), 3–41 (2006)

    MathSciNet  MATH  Google Scholar 

  30. Pearce, D., Guzmán, I.P., Valverde, A.: Computing equilibrium models using signed formulas. In: Lloyd, J., et al. (eds.) CL 2000. LNCS (LNAI), vol. 1861, pp. 688–702. Springer, Heidelberg (2000). doi:10.1007/3-540-44957-4_46

    Chapter  Google Scholar 

  31. Przymusinski, T.: Stable semantics for disjunctive programs. New Gener. Comput. 9(3/4), 401–424 (1991)

    Article  MATH  Google Scholar 

  32. Reiter, R.: A logic for default reasoning. Artif. Intell. 13(1–2), 81–132 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sakama, C., Inoue, K.: An alternative approach to the semantics of disjunctive logic programs and deductive databases. J. Autom. Reasoning 13(1), 145–172 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  34. Seipel, D., Minker, J., Ruiz, C.: A characterization of the partial stable models for disjunctive databases. In: Małuszyński, J. (ed.) Logic Programming Symposium, pp. 245–259 (1997)

    Google Scholar 

  35. Shepherdson, J.C.: A sound and complete semantics for a version of negation as failure. Theor. Comput. Sci. 65(3), 343–371 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  36. Simons, P., Niemelä, I., Soininen, T.: Extending and implementing the stable model semantics. Artif. Intell. 138(1–2), 181–234 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  37. Smullyan, R.: First-Order Logic. Dover Publications, Mineola (1968)

    Book  MATH  Google Scholar 

  38. Stamate, D.: Assumption based multi-valued semantics for extended logic programs. In: 36th IEEE International Symposium on ISMVL, p. 10. IEEE Computer Society (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer and Information Science, Linköping University, SE-581 83, Linköping, Sweden

    Patrick Doherty & Andrzej Szałas

  2. Institute of Informatics, University of Warsaw, Banacha 2, 02-097, Warsaw, Poland

    Andrzej Szałas

Authors
  1. Patrick Doherty
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrzej Szałas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrzej Szałas .

Editor information

Editors and Affiliations

  1. Mahasarakham University , Maha Sarakham, Thailand

    Chattrakul Sombattheera

  2. Harz University of Applied Sciences , Wernigerode, Germany

    Frieder Stolzenburg

  3. University of Science and Technology , Hong Kong, China

    Fangzhen Lin

  4. Macquarie University , North Ryde, New South Wales, Australia

    Abhaya Nayak

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing AG

About this paper

Cite this paper

Doherty, P., Szałas, A. (2016). An Entailment Procedure for Kleene Answer Set Programs. In: Sombattheera, C., Stolzenburg, F., Lin, F., Nayak, A. (eds) Multi-disciplinary Trends in Artificial Intelligence. MIWAI 2016. Lecture Notes in Computer Science(), vol 10053. Springer, Cham. https://doi.org/10.1007/978-3-319-49397-8_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-49397-8_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-49396-1

  • Online ISBN: 978-3-319-49397-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation