On a syntactic approximation to logics that capture complexity classes

We formulate a formal syntax of approximate formulas for the logic with counting quantifiers, SOLP, studied by us in [1], where we showed the following facts: (i) In the presence of a built–in (linear) order, SOLP can describe NP–complete problems and fragments of it capture classes like P andNL; (ii) weakening the ordering relation to an almost order (in the sense of [7]) we can separate meaningful fragments, using a combinatorial tool suited for these languages. The purpose of the approximate formulas 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 introduce a concept of strong expressibility based on approximate formulas, and show that for many fragments of SOLP with built-in order, including ones that capture P andNL, expressibility and strong expressibility are equivalent. We state and prove a Bridge Theorem that links expressibility in fragments of SOLP over almost-ordered structures to strong 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 can be done by proving inexpressibility over the corresponding fragments with built-in almost order, where separation proofs are easier. Subject Classification: Logic in computer science; Descriptive Complexity.