Skip to main content
Log in

Sequent calculi for skeptical reasoning in predicate default logic and other nonmonotonic logics

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

Sequent calculi for skeptical consequence in predicate default logic, predicate stable model logic programming, and infinite autoepistemic theories are presented and proved sound and complete. While skeptical consequence is decidable in the finite propositional case of all three formalisms, the move to predicate or infinite theories increases the complexity of skeptical reasoning to being Π 11 -complete. This implies the need for sequent rules with countably many premises, and such rules are employed.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. K.R. Apt, Introduction to logic programming, Technical Report TR-97-35, University of Texas (1987).

  2. K.R. Apt, Logic programming, in: Handbook of Theoretical Computer Science, ed. J. van Leeuwen (MIT Press, Cambridge, MA, 1990) pp. 493–574.

    Google Scholar 

  3. S.N. Artemov, Explicit provability and constructive semantics, Bull. Symbolic Logic 7 (2001) 1–36.

    Google Scholar 

  4. J. Barwise, An introduction to first-order logic, in: Handbook of Mathematical Logic, ed. J. Barwise (North-Holland, Amsterdam, 1977) pp. 5–46.

    Google Scholar 

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

  6. P. Bonatti, Reasoning with infinite stable models, Artificial Intelligence (2004) 75–111.

  7. P. Bonatti and N. Olivetti, Sequent calculi for propositional nonmonotonic logics, ACM Trans. Comput. Logic 3 (2002) 226–278.

    Google Scholar 

  8. D. Cenzer and J.B. Remmel, Π 01 classes in mathematics, in: Handbook of Recursive Mathematics, Vol. 2, eds. Y.L. Ershov, S.S. Goncharov, V.W. Marek, A. Nerode and J.B. Remmel (North-Holland, Amsterdam, 1998) pp. 623–821.

    Google Scholar 

  9. M. Gelfond and V. Lifschitz, The stable semantics for logic programs, in: Proceedings of the 5th Annual Symposium on Logic Programming, eds. R.A. Kowalski and K.A. Bowen (1988) pp. 1070–1080.

  10. K. Konolige, Autoepistemic logic, in: Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 3, eds. D.M. Gabbay, C.J. Hogger and J.A. Robinson (Clarendon Press, Oxford, 1994) pp. 217–296.

    Google Scholar 

  11. V. Lifschitz, On open defaults, in: Computational Logic. Symposium Proceedings, ed. J.W. Lloyd (1990) pp. 80–95.

  12. V. Lifschitz, Foundations of logic programming, in: Principles of Knowledge Representation, ed. G. Brewka (CSLI Publications, 1996) pp. 69–127.

  13. J.W. Lloyd, Foundations of Logic Programming, 2nd edition (Springer, Berlin, 1987).

    Google Scholar 

  14. V.W. Marek, A. Nerode and J.B. Remmel, Nonmonotonic rule systems I, Ann. Math. Artificial Intelligence 1 (1990) 241–273.

    Google Scholar 

  15. V.W. Marek, A. Nerode and J.B. Remmel, The stable models of a predicate logic program, J. Logic Progr. 21 (1994) 129–154.

    Google Scholar 

  16. V.W. Marek and M. Truszczyński, Nonmonotonic Logic: Context-Dependent Reasoning (Springer, Berlin, 1993).

    Google Scholar 

  17. J. McCarthy, Circumscription – A form of nonmonotonic reasoning, Artificial Intelligence 13 (1980) 27–39.

    Google Scholar 

  18. D. McDermott and J. Doyle, Nonmonotonic logic I, Artificial Intelligence 13 (1980) 41–72.

    Google Scholar 

  19. R.S. Milnikel, Nonmonotonic logic: A monotonic approach, Ph.D. thesis, Cornell University (1999).

  20. R.S. Milnikel, The complexity of predicate default logic over a countable domain, Ann. Pure Appl. Logic 120 (2003) 151–163.

    Google Scholar 

  21. R.S. Milnikel, A sequent calculus for skeptical reasoning in predicate default logic, in: Proceedings of the 7th European Conference for Symbolic and Quantitative Approaches to Reasoning with Uncertainty, eds. T.D. Nielsen and N.L. Zhang (2003) pp. 564–575.

  22. R.S. Milnikel, A sequent caclulus for skeptical reasoning in autoepistemic logic, in: Proceedings of the 10th International Conference on Nonmonotonic Reasoning, eds. J. Delgrande and T. Schaub (2004).

  23. G. Mints, Several formal systems of logic programming, Comput. Artif. Intell. 9 (1990) 19–41.

    Google Scholar 

  24. G. Mints, A complete calculus for pure prolog, Proc. Acad. Sci. Estonian SSR 35 (1996) 367–380. (In Russian.)

    Google Scholar 

  25. R.C. Moore, Possible-world semantics for the autoepistemic logic, in: Proceedings of the Workshop on Non-Monotonic Reasoning, ed. R. Reiter (1984) pp. 344–354.

  26. R.C. Moore, Semantical considerations on non-monotonic logic, Artificial Intelligence 25 (1985) 75–94.

    Google Scholar 

  27. A. Nerode and R.A. Shore, Logic for Applications, 2nd edition (Springer, Berlin, 1997).

    Google Scholar 

  28. R. Reiter, A logic for default reasoning, Artificial Intelligence 13 (1980) 81–132.

    Google Scholar 

  29. J.S. Schlipf, Decidability and definability with circumscription, Ann. Pure Appl. Logic 35 (1987) 173–191.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Robert Saxon Milnikel.

Additional information

AMS subject classification

03B42, 68N17, 68T27

This paper grew directly out of the author’s dissertation, written under the direction of Anil Nerode.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Milnikel, R.S. Sequent calculi for skeptical reasoning in predicate default logic and other nonmonotonic logics. Ann Math Artif Intell 44, 1–34 (2005). https://doi.org/10.1007/s10472-005-1808-3

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-005-1808-3

Keywords

Navigation