Advertisement

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

  • Robert Saxon MilnikelEmail author
Article

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 1 1 -complete. This implies the need for sequent rules with countably many premises, and such rules are employed.

Keywords

default logic stable models autoepistemic logic sequent calculus 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    K.R. Apt, Introduction to logic programming, Technical Report TR-97-35, University of Texas (1987). Google Scholar
  2. [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. [3]
    S.N. Artemov, Explicit provability and constructive semantics, Bull. Symbolic Logic 7 (2001) 1–36. Google Scholar
  4. [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. [5]
    P. Bonatti, A Gentzen system for non-theorems, Technical Report CD-TR 93/52, Christian Doppler Labor für Expertensysteme (1993). Google Scholar
  6. [6]
    P. Bonatti, Reasoning with infinite stable models, Artificial Intelligence (2004) 75–111. Google Scholar
  7. [7]
    P. Bonatti and N. Olivetti, Sequent calculi for propositional nonmonotonic logics, ACM Trans. Comput. Logic 3 (2002) 226–278. Google Scholar
  8. [8]
    D. Cenzer and J.B. Remmel, Π10 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. [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. Google Scholar
  10. [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. [11]
    V. Lifschitz, On open defaults, in: Computational Logic. Symposium Proceedings, ed. J.W. Lloyd (1990) pp. 80–95. Google Scholar
  12. [12]
    V. Lifschitz, Foundations of logic programming, in: Principles of Knowledge Representation, ed. G. Brewka (CSLI Publications, 1996) pp. 69–127. Google Scholar
  13. [13]
    J.W. Lloyd, Foundations of Logic Programming, 2nd edition (Springer, Berlin, 1987). Google Scholar
  14. [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. [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. [16]
    V.W. Marek and M. Truszczyński, Nonmonotonic Logic: Context-Dependent Reasoning (Springer, Berlin, 1993). Google Scholar
  17. [17]
    J. McCarthy, Circumscription – A form of nonmonotonic reasoning, Artificial Intelligence 13 (1980) 27–39. Google Scholar
  18. [18]
    D. McDermott and J. Doyle, Nonmonotonic logic I, Artificial Intelligence 13 (1980) 41–72. Google Scholar
  19. [19]
    R.S. Milnikel, Nonmonotonic logic: A monotonic approach, Ph.D. thesis, Cornell University (1999). Google Scholar
  20. [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. [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. Google Scholar
  22. [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). Google Scholar
  23. [23]
    G. Mints, Several formal systems of logic programming, Comput. Artif. Intell. 9 (1990) 19–41. Google Scholar
  24. [24]
    G. Mints, A complete calculus for pure prolog, Proc. Acad. Sci. Estonian SSR 35 (1996) 367–380. (In Russian.) Google Scholar
  25. [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. Google Scholar
  26. [26]
    R.C. Moore, Semantical considerations on non-monotonic logic, Artificial Intelligence 25 (1985) 75–94. Google Scholar
  27. [27]
    A. Nerode and R.A. Shore, Logic for Applications, 2nd edition (Springer, Berlin, 1997). Google Scholar
  28. [28]
    R. Reiter, A logic for default reasoning, Artificial Intelligence 13 (1980) 81–132. Google Scholar
  29. [29]
    J.S. Schlipf, Decidability and definability with circumscription, Ann. Pure Appl. Logic 35 (1987) 173–191. Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of MathematicsKenyon CollegeGambierUSA

Personalised recommendations