Approximate formulae for a logic that capture classes of computational complexity
Argimiro Arratia, C. E. Ortiz · Logic Journal of IGPL · 2008
This paper presents a syntax of approximate formulae suited for the logic with counting quantifiers 𝒮𝒪ℒ𝒫. This logic was formalised by us in [1] where, among other properties, we showed the following facts: (i) In the presence of a built–in (linear) order, 𝒮𝒪ℒ𝒫 can describe NP–complete problems and some of its fragments capture the classes 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 our approximate formulae is to provide a syntactic approximation to the logic 𝒮𝒪ℒ𝒫, enhanced with a built-in order, that should be complementary of the semantic approximation based on almost orders, by means of producing logics where problems are syntactically described within a small counting error. We introduce a concept of strong expressibility based on approximate formulae, and show that for many fragments of 𝒮𝒪ℒ𝒫 with built-in order, including ones that capture P and NL, 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 formulae for the corresponding fragments over ordered structures. A consequence of these results is that proving inexpressibility 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.