Advertisement

Archive for Mathematical Logic

, Volume 50, Issue 7–8, pp 727–742 | Cite as

Proof complexity of propositional default logic

  • Olaf BeyersdorffEmail author
  • Arne Meier
  • Sebastian Müller
  • Michael Thomas
  • Heribert Vollmer
Article

Abstract

Default logic is one of the most popular and successful formalisms for non-monotonic reasoning. In 2002, Bonatti and Olivetti introduced several sequent calculi for credulous and skeptical reasoning in propositional default logic. In this paper we examine these calculi from a proof-complexity perspective. In particular, 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. On the other hand, we show an exponential lower bound to the proof size in Bonatti and Olivetti’s enhanced calculus for skeptical default reasoning.

Keywords

Default logic Sequent calculus Proof complexity 

Mathematics Subject Classification (2000)

03F20 03B60 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amati G., Aiello L.C., Gabbay D.M., Pirri F.: A proof theoretical approach to default reasoning I: tableaux for default logic. J. Log. Comput. 6(2), 205–231 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Antoniou G.: A tutorial on default logics. ACM Comput. Surv. 31(4), 337–359 (1999)CrossRefGoogle Scholar
  3. 3.
    Beyersdorff, O., Meier, A., Müller, S., Thomas, M., Vollmer, H.: Proof complexity of propositional default logic. In: Proceedings of 13th International Conference on Theory and Applications of Satisfiability Testing, vol. 6175 of Lecture Notes in Computer Science, pp. 30–43. Springer, Berlin, Heidelberg (2010)Google Scholar
  4. 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. 5.
    Bonatti P.A., Olivetti N.: Sequent calculi for propositional nonmonotonic logics. ACM Trans. Comput. Log. 3(2), 226–278 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bonet M.L., Buss S.R., Pitassi T.: Are there hard examples for Frege systems?. In: Clote, P., Remmel, J. (eds) Feasible Mathematics II, pp. 30–56. Birkhäuser, Basel (Boston/Stuttgart) (1995)Google Scholar
  7. 7.
    Bonet M.L., Pitassi T., Raz R.: On interpolation and automatization for Frege systems. SIAM J. Comput. 29(6), 1939–1967 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cadoli M., Schaerf M.: A survey of complexity results for nonmonotonic logics. J. Log. Program. 17(2/3&4), 127–160 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cook S.A., Reckhow R.A.: The relative efficiency of propositional proof systems. J. Symb. Log. 44(1), 36–50 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dix, J., Furbach, U., Niemelä, I.: Nonmonotonic reasoning: towards efficient calculi and implementations. In: Handbook of Automated Reasoning, pp. 1241–1354. Elsevier and MIT Press (2001)Google Scholar
  11. 11.
    Egly U., Tompits H.: Proof-complexity results for nonmonotonic reasoning. ACM Trans. Comput. Log. 2(3), 340–387 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Gabbay, D.: Theoretical foundations of non-monotonic reasoning in expert systems. In: Logics and Models of Concurrent Systems, pp. 439–457. Springer, Berlin, Heidelberg (1985)Google Scholar
  13. 13.
    Gentzen G.: Untersuchungen über das logische Schließen. Mathematische Zeitschrift 39, 68–131 (1935)Google Scholar
  14. 14.
    Gottlob G.: Complexity results for nonmonotonic logics. J. Log. Comput. 2(3), 397–425 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Hrubeš P.: On lengths of proofs in non-classical logics. Ann. Pure Appl. Log. 157(2–3), 194–205 (2009)zbMATHCrossRefGoogle Scholar
  16. 16.
    Jeřábek E.: Frege systems for extensible modal logics. Ann. Pure Appl. Log. 142, 366–379 (2006)zbMATHCrossRefGoogle Scholar
  17. 17.
    Jeřábek E.: Substitution Frege and extended Frege proof systems in non-classical logics. Ann. Pure Appl. Log. 159(1–2), 1–48 (2009)zbMATHCrossRefGoogle Scholar
  18. 18.
    Krajíček J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory, vol. 60 of Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  19. 19.
    Kraus S., Lehmann D.J., Magidor M.: Nonmonotonic reasoning, preferential models and cumulative logics. Artif. Intell. 44(1–2), 167–207 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Makinson, D.: General theory of cumulative inference. In: Proceedings of 2nd International Workshop on Non-Monotonic Reasoning, pp. 1–18. Springer, Berlin, Heidelberg (1989)Google Scholar
  21. 21.
    Marek V.W., Truszczyński M.: Nonmonotonic Logics—Context-Dependent Reasoning. Springer, Berlin, Heidelberg (1993)Google Scholar
  22. 22.
    Reiter R.: A logic for default reasoning. Artif. Intell. 13, 81–132 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Risch V., Schwind C.: Tableaux-based characterization and theorem proving for default logic. J. Autom. Reason. 13(2), 223–242 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Tiomkin, M.L.: Proving unprovability. In: Proceedings of 3rd Annual Symposium on Logic in Computer Science, pp. 22–26 (1988)Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Olaf Beyersdorff
    • 1
    Email author
  • Arne Meier
    • 1
  • Sebastian Müller
    • 2
  • Michael Thomas
    • 1
  • Heribert Vollmer
    • 1
  1. 1.Institute of Theoretical Computer ScienceLeibniz University HanoverHanoverGermany
  2. 2.Faculty of Mathematics and PhysicsCharles University PraguePragueCzech Republic

Personalised recommendations