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# Asymptotic conditional probabilities: The non-unary case

## Abstract

Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences φ and θ, we consider the structures with domain {1, …, N} that satisfy θ, and compute the fraction of them in which φ is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering asymptotic conditional probabilities. As shown by Liogon'kiĭ [24], if there is a non-unary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kiĭ also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well-defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable.

## References

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[1]Bacchus, F., Grove, A. J., Halpern, J. Y., and Koller, D., From statistics to belief, Proceedings of the national conference on artificial intelligence (AAAI '92), 1992, pp. 602608.
[2]Bacchus, F., Grove, A. J., Halpern, J. Y., and Koller, D., Statistical foundations for default reasoning, Proceedings of the Thirteenth International Joint Conference on Artificial Intelligence (IJCAI '93), 1993, A longer version, entitled From statistical knowledge bases to degrees of belief will appear in Artificial Intelligence.
[3]Carnap, R., Logical foundations of probability, University of Chicago Press, Chicago, 1950.
[4]Carnap, R., The continuum of inductive methods, University of Chicago Press, Chicago, 1952.
[5]Cheeseman, P. C., A method of computing generalized Bayesian probability values for expert systems, Proceedings of the eighth international joint conference on artificial intelligence (IJCAI '83), 1983, pp. 198202.
[6]Compton, K., 0-1 laws in logic and combinatorics, Proceedings of the 1987 NATO advanced study institute on algorithms and order (Rival, I., editor), Reidel, Dordrecht, Netherlands, 1988, pp. 353383.
[7]de Laplace, P. S., Essai philosophique sur les probabilités, 1820, English translation is Philosophical Essay on Probabilities, Dover Publications, New York, 1951.
[8]Denbigh, K. G. and Denbigh, J. S., Entropy in relation to incomplete knowledge, Cambridge University Press, Cambridge, UK, 1985.
[9]Dreben, B. and Goldfarb, W. D., The decision problem: Solvable classes of quantificational formulas, Addison-Wesley, Reading, MA, 1979.
[10]Fagin, R., Probabilities on finite models, this Journal, vol. 41 (1976), no. 1, pp. 5058.
[11]Fagin, R., The number of finite relational structures, Discrete Mathematics, vol. 19 (1977). pp. 1721.
[12]Gaifman, H., Probability models and the completeness theorem, International congress of logic methodology and philosophy of science, 1960, This is the abstract of which [13] is the full paper, pp. 7778.
[13]Gaifman, H., Concerning measures in first order calculi, Israel Journal of Mathematics, vol. 2 (1964), pp. 118.
[14]Glebskiĭ, Y. V., Liogon'kiĭ, M. I.Kogan, D. I., and Talanov, V. A., Range and degree of realizability of formulas in the restricted predicate calculus, Kibernetika, vol. 2 (1969), pp. 1728.
[15]Graham, R. L., Knuth, D. E., and Patashnik, O., Concrete mathematics—a foundation for computer science, Addison-Wesley, Reading, MA, 1989.
[16]Grove, A. J., Halpern, J. Y., and Koller, D., Asymptotic conditional probabilities for first-order logic, Part II: the unary case, Research report, IBM, 1993, SIAM Journal on Computing, in press.
[17]Grove, A. J., Halpern, J. Y., and Koller, D., Asymptotic conditional probabilities for first-order logic, Proc. 24th ACM symp. on theory of computing, 1992, pp. 294305.
[18]Grove, A. J., Halpern, J. Y., and Koller, D., Random worlds and maximum entropy, Journal of Artificial Intelligence Research, vol. 2 (1994), pp. 3338.
[19]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.
[20]Jaynes, E. T., Where do we stand on maximum entropy?, The maximum entropy formalism (Levine, R. D. and Tribus, M., editors), MIT Press, 1978, pp. 15118.
[21]Keynes, J. M., A treatise on probability, Macmillan, London, 1921.
[22]Kolaitis, Ph. G. and Vardi, M. Y., 0-1 laws and decision problems for fragments of second-order logic, Information and Computation, vol. 87 (1990), pp. 302338.
[23]Lewis, H. R., Unsolvable classes of quantificational formulas, Addison-Wesley, New York, 1979.
[24]Liogon'kiĭ, M. I., On the conditional satisfiability ratio of logical formulas, Mathematical Notes of the Academy of the USSR, vol. 6 (1969), pp. 856861.
[25]Lynch, J., Almost sure theories, Annals of Mathematical Logic, vol. 18 (1980), pp. 91135.
[26]Manna, Z. and Pnueli, A., The temporal logic of reactive and concurrent systems, vol. 1, Springer-Verlag, Berlin, 1992.
[27]Paris, J. B. and Vencovska, A., On the applicability of maximum entropy to inexact reasoning, International Journal of Approximate Reasoning, vol. 3 (1989), pp. 134.
[28]Powell, R. E. and Shah, S. M., Summability theory and applications, Van Nostrand Reinhold, 1972.
[29]Trakhtenbrot, B. A., Impossibility of an algorithm for the decision problem in finite classes, Doklady Akademii Nauk SSSR, vol. 70 (1950), pp. 569572.
[30]von Kries, J., Die principien der wahrscheinlichkeitsrechnung und rational expectation, Freiburg, 1886.

# Asymptotic conditional probabilities: The non-unary case

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