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 Σdm classes (prenex classes of higher order logics) define FM-truth. For any d≥ 2, m≥ 1 we construct a family {[Σdm]≤ k}k∈ω of syntactically defined fragments of Σdm which satisfy this characterization. We also use the [Σdm]≤ k classes to give a refinement of known results on the complexity classes captured by Σdm.
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 nk.
Citation
Leszek Aleksander Kołodziejczyk. "Truth definitions in finite models." J. Symbolic Logic 69 (1) 183 - 200, March 2004. https://doi.org/10.2178/jsl/1080938836
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