Invariant Definability and P/poly

. 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 AC0 as the non-uniformly First Order Definable classes of finite structures. We have previously observed that the AjtaiImmerman theorem can be rephrased in terms of invariant definability: A class of finite structures is FOL invariantly definable iff it is in AC0 . Invariant definability is a notion closely related to but different from implicit definability and \Delta-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..