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Strong convergence in finite model theory

Published online by Cambridge University Press:  12 March 2014

Wafik Boulos Lotfallah
Wafik Boulos Lotfallah*
Affiliation:
Department of Engineering Mathematics and Physics, Faculty of Engineering, Cairo University, Cairo, Egypt, 11451, E-mail: lotfalla@alphal-eng.cairo.edu.eg
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Abstract

In [9] we introduced a new framework for asymptotic probabilities, in which a σ-additive measure is defined on the sample space of all sequences of finite models, where the universe of , is {1,2,…,n}. In this framework we investigated the strong 0-1 law for sentences, which states that each sentence either holds in eventually almost surely or fails in eventually almost surely.

In this paper we define the strong convergence law for formulas, which carries over the ideas of the strong 0-1 law to formulas with free variables, and roughly states that for each formula ϕ(x), the fraction of tuples a in , which satisfy the formula ϕ(x), almost surely has a limit as n tends to infinity.

We show that the infinitary logic with finitely many variables has the strong convergence law for formulas for the uniform measure, and further characterize the measures on random graphs for which the strong convergence law holds.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 3 , September 2002 , pp. 1083 - 1092
Copyright
Copyright © Association for Symbolic Logic 2002

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References

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