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A theory of nonmonotonic rule systems I

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Abstract

We introduce here the study of generalnonmonotonic rule systems. These deal with situations where a conclusion is drawn from a “system of beliefs”S (and seen to be inS), basedboth on some “premises” being inS and on some “restraints” not being inS. In the monotone systems of traditional logic there are no restraints, conclusions are drawn solely based on premises being inS. Nonmonotonic rule systems capture the essential syntactic, semantic, and algorithmic features of many nonmonotone systems such as default logic, negation as failure, truth maintenance, autoepistemic logic, and also important combinatorial questions from mathematics such as the marriage problem. This reveals semantics and syntax and proof procedures and algorithms for computing belief sets in many cases where none were previously available and entirely uniformly. In particular, we introduce and study deductively closed sets, extensions and weak extensions. Semantics of nonmonotonic rule systems is studied in part II of this paper and extensions to predicate classical, intuitionistic, and modal logics are left to a later paper.

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References

  1. K.R. Apt., Introduction to logic programming, Technical Report TR-87-35, University of Texas (1988).

  2. K.R. Apt, H.A. Blair and A. Walker, Towards a theory of declarative knowledge, in:Foundations of Deductive Databases and Logic Programming, ed. J. Minker (Morgan Kaufmann, Los Altos, CA, 1987).

    Google Scholar 

  3. N. Bidoit and C. Froidevaux, General logical databases and programs, default logic semantics, and stratification, J. Information and Comput., to appear.

  4. H.A. Blair, A.L. Brown and V.S. Subrahmanian, Monotone logic programming, Technical Report CS-TR-2375, University of Maryland (1989).

  5. J. de Kleer, An assumption-based TMS, Artificial Intelligence 28 (1986) 127–162.

    Article  Google Scholar 

  6. R.P. Dilworth, A decomposition theorem for partially ordered sets, Ann. Math. 51 (1950) 161–165.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Doyle, A truth maintenance system, Artificial Intelligence J. 12 (1979) 231–272.

    Article  MathSciNet  Google Scholar 

  8. M. Gelfond and V. Lifschitz, Stable semantics for logic programs, in:Proc. 5th Int. Symp. on Logic Programming, Seattle (1988).

  9. M. Gelfond and V. Lifschitz, Logic programming with classical negation, unpublished manuscript (1989).

  10. M. Gelfond and H. Przymusińska, On the relationship between circumscription and autoepistemic logic, in:Proc. ISMIS Conf. (1986).

  11. M. Gelfond and H. Przymusińska, Inheritance reasoning in autoepistemic logic, to appear in Fund. Inform.

  12. D. Gries,The Science of Programming (Springer, 1981).

  13. P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935) 26–30.

    Google Scholar 

  14. M. Hall, Distinct representatives of subsets, Bull. Amer. Math. Soc. 54 (1948) 922–926.

    Article  MathSciNet  MATH  Google Scholar 

  15. J.Y. Halpern and Y.O. Moses, Knowledge and common knowledge in a distributed environment,3rd ACM Conf. on the Principles of Distributed Computing (1984) pp. 50–61.

  16. J. Hintikka,Knowledge and Belief (Cornell University Press, 1962).

  17. W-Q. Huang and A. Nerode, Applications of pure recursion theory to recursive analysis, Acta Sinica 28 (1985).

  18. K. Konolige, On the relation between default and autoepistemic logic, Artificial Intelligence J. 35 (1988) 343–382.

    Article  MathSciNet  MATH  Google Scholar 

  19. W. Marek and A. Nerode, Decision procedure for default logic, Mathematical Sciences Institute Reports, Cornell University (1990).

  20. W. Marek and M. Truszczyński, Relating autoepistemic and default logics, in:Principles of Knowledge Representation and Reasoning (Morgan Kaufmann, San Mateo, 1989). (Full version available as Technical Report 144-89, Computer Science, University of Kentucky, Lexington, KY 40506-0027, 1989.)

    Google Scholar 

  21. W. Marek and M. Truszczyński, Stable models for logic programs and default logic, in:Proc. North American Conf. Logic Programming (MIT Press, 1989). (Full version available as Technical Report, Computer Science Department, University of Kentucky, Lexington, KY 40506-0027, 1989.)

  22. J. McCarthy, Circumscription — a form of nonmonotonic reasoning, Artificial Intelligence J. 13 (1980) 27–39.

    Article  MATH  Google Scholar 

  23. G. Metakides and A. Nerode, Effective content of field theory, Ann. Math. Logic 17 (1977) 289–320.

    Article  MathSciNet  Google Scholar 

  24. M. Minsky, A framework for representing knowledge, in:The Psychology of Computer Vision (McGraw-Hill, 1975) pp. 211–272.

  25. L. Mirsky,Transversal Theory (Academic Press, New York, 1971).

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  27. A. Nerode and J.B. Remmel, A survey of r.e. substructures, Proc. Symp. Math. 42, Amer. Math. Soc. (1985) 323–376.

  28. A. Nerode and J.B. Remmel, Complexity-theoretic algebra I: vector spaces over finite fields, in:Structures in Complexity (1987) pp. 218–241.

  29. A. Nerode and J.B. Remmel, Complexity-theoretic algebra II: Boolean algebras, Ann. Pure Appl. Logic 44 (1989) 71–99.

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Nerode and J.B. Remmel, Complexity-theoretic algebra III: bases of vector spaces, in:Feasible Mathematics (Springer, 1990).

  31. M. Reinfrank and O. Dressler, On the relation between truth maintenance and non-monotonic logics, in:Proc. Int. Joint Conf. on Artificial Intelligence (1989).

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

    Article  MathSciNet  MATH  Google Scholar 

  33. J.B. Remmel, Recursive Boolean algebras, in:Handbook of Boolean Algebras, ed. J.D. Monk (North-Holland, 1989) chap. 25, pp. 1099–1165.

  34. H. Rogers, Jr.Theory of Recursive Functions and Effective Computability (McGraw-Hill, New York, 1967).

    MATH  Google Scholar 

  35. A. Tarski,Logic, Semantics, Metamathematics (Oxford, 1956).

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Authors and Affiliations

  1. Department of Computer Science, University of Kentucky, 40506-0027, Lexington, KY, USA

    W. Marek

  2. Mathematical Sciences Institute, Cornell University, 14853, Ithaca, NY, USA

    A. Nerode

  3. Department of Mathematics, University of California at San Diego, 92903, La Jolla, CA, USA

    J. Remmel

Authors
  1. W. Marek
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  2. A. Nerode
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  3. J. Remmel
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Additional information

Work partially supported by NSF grant RII-8610671 and Kentucky EPSCoR program and ARO contract DAAL03-89-K-0124.

Work partially supported by NSF grant DMS-8902797 and ARO contract DAAG629-85-C-0018.

Work partially supported by NSF grant DMS-8702473.

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Marek, W., Nerode, A. & Remmel, J. A theory of nonmonotonic rule systems I. Ann Math Artif Intell 1, 241–273 (1990). https://doi.org/10.1007/BF01531080

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