Advertisement

A sequent calculus for skeptical Default Logic

  • P. A. Bonatti
  • N. Olivetti
Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1227)

Abstract

In this paper, we contribute to the proof-theory of Reiter's Default Logic by introducing a sequent calculus for skeptical reasoning. The main features of this calculus are simplicity and regularity, and the fact that proofs can be surprisingly concise and, in many cases, involve only a small part of the default theory.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. Amati, L. Carlucci Aiello, D. Gabbay, F. Pirri. A proof theoretical approach to default reasoning I: tableaux for default logic. Journal of Logic and Computation, 6(2):205–231, 1996.Google Scholar
  2. 2.
    F. Baader, B. Hollunder. Embedding defaults into terminological knowledge representation formalisms. Journal of Automated Reasoning, 14(1):149–180, 1995.Google Scholar
  3. 3.
    C. Bell, A. Nerode, R. Ng and V.S. Subrahmanian. Implementing deductive databases by linear programming. In Proc. of ACM-PODS, 1992.Google Scholar
  4. 4.
    C. Bell, A. Nerode, R. Ng and V.S. Subrahmanian. Implementing stable semantics by linear programming. In [33].Google Scholar
  5. 5.
    A. Bochman. On the relation between default and modal consequence relations. In Proc of KR'94, 63–74, Morgan Kaufmann, 1994.Google Scholar
  6. 6.
    P.A.Bonatti. Sequent calculi for default and autoepistemic logics. In Proc. of TABLEAUX'96, LNAI 1071, pp. 127–142, Springer-Verlag, Berlin, 1996.Google Scholar
  7. 7.
    P.A.Bonatti. Autoepistemic logic programming. Journal of Automated Reasoning, 13:35–67, 1994.Google Scholar
  8. 8.
    P.A. Bonatti. A Gentzen system for non-theorems. Technical Report CD-TR 93/52, Christian Doppler Labor für Expertensysteme, Technische Universität, Wien, September 1993.Google Scholar
  9. 9.
    G. Brewka. Cumulative default logic: in defense of nonmonotonic inference rules. Artificial Intelligence 50:183–205, 1991.Google Scholar
  10. 10.
    X. Caicedo. A formal system for the non-theorems of the propositional calculus. Notre Dame Journal of Formal Logic, 19:147–151, (1978).Google Scholar
  11. 11.
    R. Dutkiewicz. The method of axiomatic rejection for the intuitionistic propositional calculus. Studia Logica, 48:449–459, (1989).Google Scholar
  12. 12.
    U. Egly, H. Tompits. Non-elementary speed-ups in default logic. Submitted.Google Scholar
  13. 13.
    D. Gabbay et al. (eds). Handbook of Logic in Artificial Intelligence and Logic Programming, Vol.III, Clarendon Press, Oxford, 1994.Google Scholar
  14. 14.
    D. Gabbay. Theoretical foundations for non-monotonic reasoning in expert systems. In K.R. Apt (ed.) Logics and Models of Concurrent Systems. Springer-Verlag, Berlin, 1985.Google Scholar
  15. 15.
    M.L. Ginsberg. A circumscriptive theorem prover. Artificial Intelligence, 39(2):209–230, (1989).Google Scholar
  16. 16.
    Y.J. Jiang. A first step towards autoepistemic logic programming. Computers and Artificial Intelligence, 10(5):419–441, (1992).Google Scholar
  17. 17.
    K. Konolige. On the Relationship between Default and Autoepistemic Logic. Artificial Intelligence, 35:343–382, 1988. + Errata, same journal, 41:115, 1989/90.Google Scholar
  18. 18.
    K. Konolige. On the Relation Between Autoepistemic Logic and Circumscription. In Proceedings IJCAI-89, 1989.Google Scholar
  19. 19.
    S. Kraus, D. Lehmann and M. Magidor. Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44(1):167–207, (1990).Google Scholar
  20. 20.
    H.J. Levesque. All I know: a study in autoepistemic logic. Artificial Intelligence, 42:263–309, (1990).Google Scholar
  21. 21.
    J. Łukasiewicz. Aristotle's syllogistic from the standpoint of modern formal logic. Clarendon Press, Oxford, 1951.Google Scholar
  22. 22.
    W. Lukasziewicz. Non-Monotonic Reasoning. Ellis Horwood Limited, Chichester, England, 1990.Google Scholar
  23. 23.
    J. McCarthy. Circumscription: a form of non-monotonic reasoning. Artificial Intelligence, 13:27–39, (1980).Google Scholar
  24. 24.
    D. Makinson. General theory of cumulative inference. In M. Reinfrank, J. De Kleer, M.L. Ginsberg and E. Sandewall (eds.) Non-monotonic Reasoning, LNAI 346, Springer-Verlag, Berlin, 1989, 1–18.Google Scholar
  25. 25.
    W. Marek, A. Nerode, M. Trusczyński (eds). Logic Programming and Non-monotonic Reasoning: Proc. of the Third Int. Conference. LNAI 928, Springer-Verlag, Berlin, 1995.Google Scholar
  26. 26.
    W. Marek, M. Truszczyński. Nonmonotonic Logics — Context-Dependent Reasoning. Springer, 1993.Google Scholar
  27. 27.
    W. Marek, M. Trusczyński. Computing intersections of autoepistemic expansions. In [29].Google Scholar
  28. 28.
    M.A. Nait Abdallah. An extended framework for default reasoning. In Proc. of FCT'89, LNCS 380, 339–348, Springer-Verlag, 1989.Google Scholar
  29. 29.
    A. Nerode, W. Marek, V.S. Subrahmanian (eds.). Logic Programming and Non-monotonic Reasoning: Proc. of the First Int. Workshop, MIT Press, Cambridge, Massachusetts, 1991.Google Scholar
  30. 30.
    I. Niemela. Decision procedures for autoepistemic logic. Proc. CADE-88, LNCS 310, Springer-Verlag, 1988.Google Scholar
  31. 31.
    I. Niemela. Toward efficient default reasoning. Proc. IJCAI'95, 312–318, Morgan Kaufmann, 1995.Google Scholar
  32. 32.
    N. Olivetti. Tableaux and sequent calculus for minimal entailment. Journal of Automated Reasoning, 9:99–139, (1992).Google Scholar
  33. 33.
    L. M. Pereira, A. Nerode (eds.). Logic Programming and Non-monotonic Reasoning: Proc. of the Second Int. Workshop, MIT Press, Cambridge, Massachusetts, 1993.Google Scholar
  34. 34.
    R. Reiter. A logic for default reasoning. Artificial Intelligence, 13:81–132, (1980).Google Scholar
  35. 35.
    V. Risch, C.B. Schwind. Tableau-based characterization and theorem proving for default logic. Journal of Automated Reasoning, 13:223–242, 1994.Google Scholar
  36. 36.
    T. Schaub. A new methodology for query answering in default logics via structureoriented theorem proving. Journal of Automated Reasoning, 15(1):95–165, 1995.Google Scholar
  37. 37.
    D. Scott. Completeness proofs for the intuitionistic sentential calculus. Summaries of Talks Presented at the Summer Institute for Symbolic Logic (Itaha, Cornell University, July 1957), Princeton: Institute for Defense Analyses, Communications Research Division, 1957, 231–242.Google Scholar
  38. 38.
    J. Słupecki, G. Bryll, U. Wybraniec-Skardowska. Theory of rejected propositions. Studia Logica, 29:75–115, (1971).Google Scholar
  39. 39.
    M. Tiomkin. Proving unprovability. In Proc. of LICS'88, 1988.Google Scholar
  40. 40.
    A. Varzi. Complementary sentential logics. Bulletin of the Section of Logic, 19:112–116, (1990).Google Scholar
  41. 41.
    A. Varzi. Complementary logics for classical prepositional languages. Kriterion. Zeitschrift für Philosophie, 4:20–24 (1992).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • P. A. Bonatti
    • 1
  • N. Olivetti
    • 1
  1. 1.Dip. di InformaticaUniversità di TorinoTorinoItaly

Personalised recommendations