Truth definitions in finite models

Abstract

The paper discusses the notion of finite model truth definitions (or FM-truth definitions), introduced by M. Mostowski as a finite model analogue of Tarski’s classical notion of truth definition.

We compare FM-truth definitions with Vardi’s concept of the combined complexity of logics, noting an important difference: the difficulty of defining FM-truth for a logic ℒ does not depend on the syntax of ℒ, as long as it is decidable. It follows that for a natural ℒ there exist FM-truth definitions whose evaluation is much easier than the combined complexiy of ℒ would suggest.

We apply the general theory to give a complexity-theoretical characterization of the logics for which the Σ^d_m classes (prenex classes of higher order logics) define FM-truth. For any d≥ 2, m≥ 1 we construct a family {[Σ^d_m]^{≤ k}}_k∈ω of syntactically defined fragments of Σ^d_m which satisfy this characterization. We also use the [Σ^d_m]^{≤ k} classes to give a refinement of known results on the complexity classes captured by Σ^d_m.

We close with a few simple corollaries, one of which gives a sufficient condition for the existence, given a vocabulary σ, of a fixed number k such that model checking for all first order sentences over σ can be done in deterministic time n^k.