Abstract
We prove some general results about the existence of 0–1 and convergence laws for L k∞,ω and L k∞,ω on classes of finite structures equipped with a sequence of arbitrary probability measures {μ n }, as well as a few results for particular classes. First, two new proofs of the characterization theorem of Kolaitis and Vardi [9] are given. Then this theorem is generalized to obtain a characterization of the existence of L ω∞,ω convergence laws on a class with arbitrary measure. We use this theorem to obtain some results about the nonexistence of L ω∞,ω convergence laws for particular classes of structures. We also disprove a conjecture of Tyszkiewicz [16] relating the existence of L ω∞,ω and MSO 0–1 laws on classes of structures with arbitrary measures.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
Chang, C. C., and Keisler, H. J., Model Theory, North-Holland, Amsterdam, 1990.
Compton, K. J., 0–1 Laws in Logic and Combinatorics, in I. Rival, ed., Proc. NATO Advanced Study Institute on Algorithms and Order, Reidel, Dordrecht (1988), pp. 353–383.
Dawar, A., Lindell, S., and Weinstein, S., Infinitary Logic and Inductive Definability over Finite Structures, University of Pennsylvania Tech. Report IRCS 92-20 (1992).
Ehrenfeucht, A., An Application of Games to the Completeness Problem for Formalized Theories, Fund. Math. 49 (1961), pp. 129–141.
Fagin, R., Probabilities on Finite Models, J. Symbolic Logic 41 (1976), pp. 50–58.
Fraïssé, R., Sur quelques classifications des systèmes de relations, Publ. Sci. Univ. Alger. Sér. A 1 (1954), pp. 35–182.
Gurevich, Y., Immerman, N., and Shelah, S., McColm's Conjecture, Proc. of the 9th IEEE Symposium on Logic in Computer Science, 1994.
Kolaitis, Ph., On Asymptotic Probabilities of Inductive Queries and their Decision Problem, in R. Parikh, ed., Logics of Programs '85, Lecture Notes in Computer Science 193 (1985), Springer-Verlag, pp. 153–166.
Kolaitis, Ph., and Vardi, M., Infinitary Logics and 0–1 Laws, Information and Computation 98 (1992), pp. 258–294.
Lynch, J. F., Probabilities of First-Order Sentences about Unary Functions, Transactions of the AMS 287 (1985), pp. 543–568.
Lynch, J. F., Infinitary Logics and Very Sparse Random Graphs, Proc. of the 8th IEEE Symposium on Logic in Computer Science, 1993.
Pillay, A., An Introduction to Stability Theory, Oxford University Press, New York, 1983.
Shelah, S., and Spencer, J., Zero-One Laws for Sparse Random Graphs, Journal of the AMS 1 (1988), pp. 97–115.
Tyszkiewicz, J., Infinitary Queries and their Asymptotic Probabilities I: Properties Definable in Transitive Closure Logic, in E. Börger, et. al., eds., Proc. Computer Science Logic '91, Springer LNCS 626.
Tyszkiewicz, J., On Asymptotic Probabilities of Logics that Capture DSPACE(log n) in the Presence of Ordering, Proc. CAAP '91.
Tyszkiewicz, J., On Asymptotic Probabilities of Monadic Second Order Properties, Proc. Computer Science Logic '92, Pisa, Italy.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
McArthur, M. (1995). Convergence and 0–1 laws for L k∞,ω under arbitrary measures. In: Pacholski, L., Tiuryn, J. (eds) Computer Science Logic. CSL 1994. Lecture Notes in Computer Science, vol 933. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0022259
Download citation
DOI: https://doi.org/10.1007/BFb0022259
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60017-6
Online ISBN: 978-3-540-49404-1
eBook Packages: Springer Book Archive