Approximating the Expressive Power of Logics in Finite Models

Abstract

We present a probability logic (essentially a first order language extended with quantifiers that count the fraction of elements in a model that satisfy a first order formula) which, on the one hand, captures uniform circuit classes such as AC⁰ and TC⁰ over arithmetic models, namely, finite structures with linear order and arithmetic relations, and, on the other hand, their semantics, with respect to our arithmetic models, can be closely approximated by giving interpretations of their formulas on finite structures where all relations (including the order) are restricted to be “modular” (i.e. to act subject to an integer modulo). In order to give a precise measure of the proximity between satisfaction of a formula in an arithmetic model and satisfaction of the same formula in the “approximate” model, we define the approximate formulas and work on a notion of approximate truth. We also indicate how to enhance the expressive power of our probability logic in order to capture polynomial time decidable queries.

There are various motivations for this work. As of today, there is not known logical description of any computational complexity class below NP which does not requires a built–in linear order. Also, it is widely recognized that many model theoretic techniques for showing definability in logics on finite structures become almost useless when order is present. Hence, if we want to obtain significant lower bound results in computational complexity via the logical description we ought to find ways of by-passing the ordering restriction. With this work we take steps towards understanding how well can we approximate, without a true order, the expressive power of logics that capture complexity classes on ordered structures.