Logics for complexity classes

A new syntactic characterization of problems complete via Turing reductions is presented. General canonical forms are developed to define such problems. One of these forms allows us to define complete problems on ordered structures, and another form to define them on unordered structures with at least one binary relation. Using the canonical forms, logics are developed for complete problems in various complexity classes. Evidence is shown that there cannot be any complete problem on structures containing only unary relations. Our approach is extended beyond complete problems. Using a similar form, a logic is developed to capture (without the additional requirement of effectiveness in the sense of Gurevich) the complexity class NPa (c) coNP which very likely contains no complete problem.