Almost everywhere equivalence of logics in finite model theory

Bulletin of Symbolic Logic 2 (4):422-443 (1996)
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Abstract

We introduce a new framework for classifying logics on finite structures and studying their expressive power. This framework is based on the concept of almost everywhere equivalence of logics, that is to say, two logics having the same expressive power on a class of asymptotic measure 1. More precisely, if L, L ′ are two logics and μ is an asymptotic measure on finite structures, then $\scr{L}\equiv _{\text{a.e.}}\scr{L}^{\prime}(\mu)$ means that there is a class C of finite structures with μ (C)=1 and such that L and L ′ define the same queries on C. We carry out a systematic investigation of $\equiv _{\text{a.e.}}$ with respect to the uniform measure and analyze the $\equiv _{\text{a.e.}}$ -equivalence classes of several logics that have been studied extensively in finite model theory. Moreover, we explore connections with descriptive complexity theory and examine the status of certain classical results of model theory in the context of this new framework

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Citations of this work

Generalized quantifiers.Dag Westerståhl - 2008 - Stanford Encyclopedia of Philosophy.
How to define a linear order on finite models.Lauri Hella, Phokion G. Kolaitis & Kerkko Luosto - 1997 - Annals of Pure and Applied Logic 87 (3):241-267.
Strong convergence in finite model theory.Wafik Boulos Lotfallah - 2002 - Journal of Symbolic Logic 67 (3):1083-1092.
Zero-one law and definability of linear order.Hannu Niemistö - 2009 - Journal of Symbolic Logic 74 (1):105-123.
Strong 0-1 laws in finite model theory.Wafik Boulos Lotfallah - 2000 - Journal of Symbolic Logic 65 (4):1686-1704.

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References found in this work

Probabilities on finite models.Ronald Fagin - 1976 - Journal of Symbolic Logic 41 (1):50-58.
Monadic generalized spectra.Ronald Fagin - 1975 - Mathematical Logic Quarterly 21 (1):89-96.
Fixed-point extensions of first-order logic.Yuri Gurevich & Saharon Shelah - 1986 - Annals of Pure and Applied Logic 32:265-280.

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