The Kolmogorov Expression Complexity of Logics

Abstract

The purpose of the paper is to propose a completely new notion of complexity of logics in finite-model theory. It is the Kolmogorov variant of the Vardi'sexpression complexity. We define it by considering the value of the Kolmogorov complexityC(L[A]) of the infinite stringL[A] of all truth values of sentences ofLin A. The higher is this value, the more expressive is the logicLin A. If D is a class of finite models, then the value ofC(L[A]) over all A∈D is a measure of expressive power ofLin D. Unboundedness ofC(L[A])−C(L′[A]) for A∈D implies nonexistence of a recursive interpretation ofLinL′. A version of this statement with complexities modulo oracles implies the nonexistence of any interpretation ofLinL′. Thus the valuesC(L[A]) modulo oracles constitute an invariant of the expressive power of logics over finite models, depending on their real (absolute) expressive power, and not on the syntax. We investigate our notion for fragments of the infinitary logic Lω∞ω: least fixed point logic (LFP) and partial fixed point logic (PFP). We prove a precise characterization of 0–1 laws for these logics in terms of a certain boundedness condition placed onC(L[A]). We get an extension of the notion of a 0–1 law by imposing an upper bound on the value ofC(L[A]) growing not too fast with cardinality of A, which still implies inexpressibility results similar to those implied by 0–1 laws. We also discuss classes D in whichC(PFPk[A]) is very high. It appears that then PFP or its simple extension can define all the PSPACE subsets of D.