Invariant definability and P/poly

Abstract

We look at various uniform and non-uniform complexity classes within P/poly and its variations L/poly, NL/poly, NP/poly and PSpace/poly, and look for analogues of the Ajtai-Immerman theorem which characterizes AC 0 as the non-uniformly First Order Definable classes of finite structures. We have previously observed that the Ajtai-Immerman theorem can be rephrased in terms of invariant definability: A class of finite structures is FOL invariantly definable iff it is in AC 0. Invariant definability is a notion closely related to but different from implicit definability and Δ-definability. Its exact relationship to these other notions of definability has been determined in [Mak97]. Our first results are a slight generalization of similar results due to Molzan and can be stated as follows: let C be one of L, NL, P, NP, PSpace and L be a logic which captures C on ordered structures. Then the non-uniform L-invariantly definable classes of (not necessarily ordered) finite structures are exactly the classes in C/poly. We also consider uniformity conditions imposed on invariant definability and relate them to uniformity conditions on the advice sequences. This approach is different from imposing uniformity conditions on the circuit families. The significance of our investigation is conceptual, rather than technical: We identify exactly the logical analogue of uniform and non-uniform complexity classes.