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A Maximum Entropy Approach to Nonmonotonic Reasoning
March 1993 (vol. 15 no. 3)
pp. 220-232

An approach to nonmonotonic reasoning that combines the principle of infinitesimal probabilities with that of maximum entropy, thus extending the inferential power of the probabilistic interpretation of defaults, is proposed. A precise formalization of the consequences entailed by a conditional knowledge base is provided, the computational machinery necessary for drawing these consequences is developed, and the behavior of the maximum entropy approach is compared to related work in default reasoning. The resulting formalism offers a compromise between two extremes: the cautious approach based on the conditional interpretations of defaults and the bold approach based on minimizing abnormalities.

[1] E. W. Adams,The Logic of Conditionals. Dordrecht, Netherlands: D. Reidel, 1975.
[2] M. Aoki,Introduction to Optimization Techniques. New York: MacMillan, 1971, ch. 5.
[3] F. Bacchus, A. Grove, J. Halpern, and D. Koller, "Statistical foundations of default reasoning indifference and irrelevance," inProc. Fourth Int. Workshop Nonmonotonic Reasoning, 1992.
[4] R. Ben-Eliyahu, "NP-complete problems in optimal horn clauses satisfiability," Tech. Rep. R-158, Cognitive Syst. Lab., Univ. Calif. Los Angeles, 1990.
[5] M. Born,Natural Philosophy of Cause and Chance. Oxford, UK: Clarendon, 1949.
[6] P. Cheeseman, "A method of computing generalized Bayesian probability values for expert systems, " inProc. Int. Joint Conf. Artificial Intell. (IJCAI-83)(Karlsruhe, W. Germany), 1983, pp. 198-202.
[7] J. P. Delgrande, "An approach to default reasoning based on a first-order conditional logic: Revised report,"Artificial Intell., vol. 36, pp. 63-90, 1988.
[8] W. Dowling and J. Gallier, "Linear-time algorithms for testing the satisfiability of propositional Horn formulae,"J. Logic Programming, vol. 3, pp. 267-284, 1984.
[9] D. Etherington and R. Reiter, "On inheritance hierarchies with exceptions," inProc. Amer. Assoc. Artificial Intell. Conf.(Washington DC), 1983, pp. 104-108, 1983.
[10] D. Gabbay, "Theoretical foundations for nonmonotonic reasoning in expert systems," inLogic and Models of Concurrent Systems(K. R. Apt, Ed.). Berlin: Springer-Verlag, 1985.
[11] H. A. Geffner and J. Pearl, "A framework for reasoning with defaults," inKnowledge Representation and Defeasible Reasoning(H. Kyburg, R. Loui, and G. Carlson, Eds.). London: Kluwer, 1990, pp. 245-265.
[12] H. A. Geffner,Default Reasoning: Causal and Conditional Theories. Cambridge, MA: MIT Press, 1992.
[13] M. Goldszmidt and J. Pearl, "On the relation between rational closure and system Z." inProc. Third Int. Workshop Nonmonotonic Reasoning(South Lake Tahoe), 1990, pp. 130-140, 1990.
[14] M. Goldszmidt and J. Pearl, "On the consistency of defeasible databases,"Artificial Intell., vol. 52, pp. 121-149, 1991.
[15] M. Goldszmidt and J. Pearl, "SystemZ+: A formalism for reasoning with variable strength defaults," inProc. Amer. Assoc. Artificial Intell. Conf.(Anaheim, CA), 1991, pp. 399-404, San Mateo, CA: Morgan Kaufmann.
[16] M. Goldszmidt and J. Pearl, "Reasoning with qualitative probabilities can be tractable," inProc. Eighth Conf. Uncertainty Artificial Intell.(Stanford, CA), 1992, San Mateo, CA: Morgan Kaufmann.
[17] M. Goldszmidt and J. Pearl, "Rank-based systems: A simple approach to belief revision, belief update, and reasoning about evidence and actions," inProc. Third Int. Conf. Principles Knowledge Representation Reasoning(Boston, MA), 1992, pp. 661-671, San Mateo, CA: Morgan Kaufmann.
[18] S. Hanks and D. McDermott, "Non-monotonic logics and temporal projection,"Artificial Intell., vol. 33, pp. 379-412, 1987.
[19] D. Hunter, "Causality and maximum entropy updating,"Int. J. Approx. Reasoning, vol. 3, pp. 87-114, 1989.
[20] E. Jaynes, "Where do we stand on maximum entropy?" inThe Maximum Entropy Formalism(R. Levine and M. Tribus, Eds.). Cambridge, MA: MIT Press, 1979.
[21] S. Kraus, D. Lehmann, and M. Magidor, "Nonmonotonic reasoning, preferential models and cumulative logics,"Artificial Intell., vol. 44, pp. 167-207, 1990.
[22] D. Lehmann and M. Magidor, "What does a conditional knowledge base entail?"Artificial Intell., vol. 55, pp. 1-60, 1992.
[23] D. Lehmann, "What does a conditional knowledge base entail?" inProc. Principles Knowledge Represent. Reasoning(Toronto), 1989, pp. 212-222.
[24] V. Lifschitz, "Open problems on the border of logic and artificial intelligence," Tech. Rep., Dept. of Comput. Sci., Stanford Univ., 1989.
[25] D. Makinson, "General theory of cummulative inference, inNonmonotonic Reasoning(M. Reinfrank, J. de Kleer, M. Ginsberg, and E. Sandewall, Eds.). Berlin: Springer-Verlag, Lecture Notes on Artificial Intelligence 346, 1989.
[26] J. McCarthy, "Circumscription--A form of nonmonotonic reasoning,"Artificial Intell., vol. 13, pp. 27-39, 1980.
[27] McCarthy, J., 1986. "Applications of Circumscription to Formalizing Common Sense Knowledge."Artificial Intelligence, April 1986.
[28] R. Moore, "Semantical considerations on nonmonotonic logic,"Art. Intell., vol. 25, pp. 75-94, 1985.
[29] J. Pearl,Probabilistic Reasoning in Intelligent Systems. San Mateo, CA: Morgan Kaufmann, 1988.
[30] J. Pearl, "System Z: A natural ordering of defaults with tractable applications to default reasoning," inProceedings of Theoretical Aspects of Reasoning about Knowledge(R. Parikh, Ed.). San Mateo, CA: Morgan Kaufmann, 1990, pp. 121-135.
[31] J. Pearl, "Probabilistic semantics for nonmonotonic reasoning: A survey," inPhilosophy and AI - Essays at the Interface(R. Cummins and J. Pollock, Eds.). Cambridge, MA: MIT Press, pp. 157-187, 1991.
[32] R. Reiter and G. Criscuolo, " Some representational issues in default reasoning,"Int. J. Comput. Math., vol. 9, pp. 1-13, 1983.
[33] R. Reiter, "A logic for default reasoning.Artificial Intell., vol. 13, pp. 81-132, 1980.
[34] R. Reiter, "A theory of diagnosis from first principles,"Artif. Intell., vol. 32, pp. 57-95, 1987.
[35] K. Satoh, "A probabilistic interpretation for lazy nonmonotonic reasoning," inProc. Amer. Assoc. Artificial Intell. Conf.(Boston), 1990, pp. 659-664.
[36] Y. Shoham, "Chronological ignorance: Time, necessity, and causal theories," inProc. Amer. Assoc. Artificial Intell. Conf.(Philadelphia), 1986, pp. 389-393.
[37] Y. Shoham, "Nonmonotonic logics: Meaning and utility," inProc. Int. Conf. Artificial Intell. (IJCAI_87)(Milan, Italy,) 1987, pp. 388-393.
[38] M. Tribus,Rational Descriptions, Decisions and Designs. Elmsford, NY: Pergamon, 1969.

Index Terms:
abnormality minimisation; inference; maximum entropy; nonmonotonic reasoning; infinitesimal probabilities; probabilistic interpretation; conditional knowledge base; conditional interpretations; knowledge based systems; nonmonotonic reasoning; probabilistic logic
Citation:
M. Goldzsmidt, P. Morris, J. Pearl, "A Maximum Entropy Approach to Nonmonotonic Reasoning," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 15, no. 3, pp. 220-232, March 1993, doi:10.1109/34.204904
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