Logics for complexity classes

A new syntactic characterization of problems complete via Turing reductions is presented. General canonical forms are developed in order 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 non-Aristotelian structures. 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 Aristotelian structures for several complexity classes. Our approach is extended beyond complete problems. Using a similar form, a logic is developed to capture the complexity class $NP\cap coNP$ which very likely contains no complete problem.Comment: This article has been accepted for publication in Logic Journal of IGPL Published by Oxford University Press; 23 pages, 2 figure

Universal First-Order Logic is Superfluous for NL, P, NP and coNP

In this work we continue the syntactic study of completeness that began with the works of Immerman and Medina. In particular, we take a conjecture raised by Medina in his dissertation that says if a conjunction of a second-order and a first-order sentences defines an NP-complete problems via fops, then it must be the case that the second-order conjoint alone also defines a NP-complete problem. Although this claim looks very plausible and intuitive, currently we cannot provide a definite answer for it. However, we can solve in the affirmative a weaker claim that says that all ``consistent'' universal first-order sentences can be safely eliminated without the fear of losing completeness. Our methods are quite general and can be applied to complexity classes other than NP (in this paper: to NLSPACE, PTIME, and coNP), provided the class has a complete problem satisfying a certain combinatorial property