Skip to main content
SpringerLink
Log in

The Complexity of Theorem Proving in Autoepistemic Logic

  • Conference paper
Book cover Theory and Applications of Satisfiability Testing – SAT 2013 (SAT 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7962))

Abstract

Autoepistemic logic is one of the most successful formalisms for nonmonotonic reasoning. In this paper we provide a proof-theoretic analysis of sequent calculi for credulous and sceptical reasoning in propositional autoepistemic logic, introduced by Bonatti and Olivetti [5]. We show that the calculus for credulous reasoning obeys almost the same bounds on the proof size as Gentzen’s system LK. Hence proving lower bounds for credulous reasoning will be as hard as proving lower bounds for LK. This contrasts with the situation in sceptical autoepistemic reasoning where we obtain an exponential lower bound to the proof length in Bonatti and Olivetti’s calculus.

Research supported by a grant from the John Templeton Foundation.

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Beyersdorff, O., Kutz, O.: Proof complexity of non-classical logics. In: Bezhanishvili, N., Goranko, V. (eds.) ESSLLI 2010/2011. LNCS, vol. 7388, pp. 1–54. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  2. Beyersdorff, O., Meier, A., Müller, S., Thomas, M., Vollmer, H.: Proof complexity of propositional default logic. Archive for Mathematical Logic 50(7), 727–742 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beyersdorff, O., Meier, A., Thomas, M., Vollmer, H.: The complexity of reasoning for fragments of default logic. Journal of Logic and Computation 22(3), 587–604 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bonatti, P.A.: A Gentzen system for non-theorems. Technical Report CD/TR 93/52, Christian Doppler Labor für Expertensysteme (1993)

    Google Scholar 

  5. Bonatti, P.A., Olivetti, N.: Sequent calculi for propositional nonmonotonic logics. ACM Transactions on Computational Logic 3(2), 226–278 (2002)

    Article  MathSciNet  Google Scholar 

  6. Bonet, M.L., Pitassi, T., Raz, R.: On interpolation and automatization for Frege systems. SIAM Journal on Computing 29(6), 1939–1967 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cook, S.A., Nguyen, P.: Logical Foundations of Proof Complexity. Cambridge University Press (2010)

    Google Scholar 

  8. Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. The Journal of Symbolic Logic 44(1), 36–50 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  9. Creignou, N., Meier, A., Vollmer, H., Thomas, M.: The complexity of reasoning for fragments of autoepistemic logic. ACM Transactions on Computational Logic 13(2) (2012)

    Google Scholar 

  10. Egly, U., Tompits, H.: Proof-complexity results for nonmonotonic reasoning. ACM Transactions on Computational Logic 2(3), 340–387 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gentzen, G.: Untersuchungen über das logische Schließen. Mathematische Zeitschrift 39, 68–131 (1935)

    Google Scholar 

  12. Gottlob, G.: Complexity results for nonmonotonic logics. Journal of Logic and Computation 2(3), 397–425 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gottlob, G.: Translating default logic into standard autoepistemic logic. J. ACM 42(4), 711–740 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hrubeš, P.: On lengths of proofs in non-classical logics. Annals of Pure and Applied Logic 157(2-3), 194–205 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jeřábek, E.: Substitution Frege and extended Frege proof systems in non-classical logics. Annals of Pure and Applied Logic 159(1-2), 1–48 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Krajíček, J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory. Encyclopedia of Mathematics and Its Applications, vol. 60. Cambridge University Press (1995)

    Google Scholar 

  17. Krajíček, J., Pudlák, P.: Quantified propositional calculi and fragments of bounded arithmetic. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 36, 29–46 (1990)

    Article  MATH  Google Scholar 

  18. Marek, V.W., Truszczyński, M.: Nonmonotonic Logics—Context-Dependent Reasoning. Springer, Heidelberg (1993)

    Google Scholar 

  19. Moore, R.C.: Semantical considerations on modal logic. Artificial Intelligence 25, 75–94 (1985)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computing, University of Leeds, UK

    Olaf Beyersdorff

Authors
  1. Olaf Beyersdorff
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. HIIT and Department of Computer Science, University of Helsinki, Gustaf Hällströmin katu 2 b, 00014, Helsinki, Finland

    Matti Järvisalo

  2. Computer Science Department, University of California at Santa Cruz, 1156 High Street, SOE-3, 95064, Santa Cruz, CA, USA

    Allen Van Gelder

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Beyersdorff, O. (2013). The Complexity of Theorem Proving in Autoepistemic Logic. In: Järvisalo, M., Van Gelder, A. (eds) Theory and Applications of Satisfiability Testing – SAT 2013. SAT 2013. Lecture Notes in Computer Science, vol 7962. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39071-5_27

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-39071-5_27

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39070-8

  • Online ISBN: 978-3-642-39071-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation