A Logic for PTIME and a Parameterized Halting Problem

Abstract

In [9] Yuri Gurevich addresses the question whether there is a logic captures polynomial time. He conjectures there is no such logic. He considers a logic, we denote it by L≤, allows to express precisely the polynomial time properties of structures; however, apparently, there is no algorithm that given an L≤-sentence ϕ produces a polynomial time Turing machine recognizes the class of models of ϕ. In [12] Nash, Remmel, and Vianu have raised the question whether one can prove there is no such algorithm. They give a reformulation of this question in terms of a parameterized halting problem p-ACC≤ for nondeterministic Turing machines. We analyze the precise relationship between L≤ and p-ACC≤. Moreover, we show p-ACC≤ is not fixed-parameter tractable if P ≠ NP holds for all time constructible and increasing functions. A slightly stronger complexity theoretic hypothesis implies L≤ does not capture polynomial time. Furthermore, we analyze the complexity of various variants of p-ACC≤ and address the construction problem associated with p-ACC≤.