Almost everywhere equivalence of logics in finite model theory

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, L0 are two logics and ì is an asymptotic measure on finite structures, then L a.e. L0 (ì) means that there is a class C of finite structures with ì(C ) = 1 and such that L and L0 define the same queries on C. We carry out a systematic investigation of a.e. with respect to the uniform measure and analyze the 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. x