Syntactic approximations to computational complexity classes

Abstract

We present a formal syntax of approximate formulas suited for the logic with counting quantifiers SOLP. This logic was studied by us in [1] where, among other properties, we showed: (i) In the presence of a built–in (linear) order, SOLP can describe NP–complete problems and fragments of it capture classes like P and NL; (ii) weakening the ordering relation to an almost order we can separate meaningful fragments, using a combinatorial tool adapted to these languages. The purpose of the approximate formulas presented here, is to provide a syntactic approximation to logics contained in SOLP with built-in order, that should be complementary of the semantic approximation based on almost orders, by producing approximating logics where problems are described within a small counting error. We state and prove a Bridge Theorem that links expressibility in fragments of SOLP over almostordered structures to expressibility with respect to approximate formulas for the corresponding fragments over ordered structures. A consequence of these results is that proving inexpressibility results over fragments of SOLP with built-in order could be done by proving inexpressibility over the corresponding fragments with built-in almost order, where separation proofs are allegedly easier.