Completeness results and syntactic characterizations of complexity classes over arbitrary structures

Abstract

We focus on the BBS model of computation over arbitrary structures. We provide new completeness results for geometrical problems when this structure is the set of real numbers with addition and order. We also provide several machine independent characterizations of complexity classes over arbitrary structures. We extend some results by Gradel, Gurevich and Meer in descriptive complexity, characterizing deterministic and non deterministic pol-nomial time decision problems in terms of logics over metafinite structures. We extend some results by Bellantoni and Cook, characterizing functions computable in sequential deterministic polynomial time, and by Leivant and Marion, characterizing functions computable in provide some characterizations of functions computable within the polynomial hierarchy and in polynomial alternating time. Keywords : BSS model of computation, algebraic complexity, descriptive complexity, implicate complexity, logic algebra of recursive functions.