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March 2004 Truth definitions in finite models
Leszek Aleksander Kołodziejczyk
J. Symbolic Logic 69(1): 183-200 (March 2004). DOI: 10.2178/jsl/1080938836

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

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Leszek Aleksander Kołodziejczyk. "Truth definitions in finite models." J. Symbolic Logic 69 (1) 183 - 200, March 2004. https://doi.org/10.2178/jsl/1080938836

Information

Published: March 2004
First available in Project Euclid: 2 April 2004

zbMATH: 1068.03025
MathSciNet: MR2039356
Digital Object Identifier: 10.2178/jsl/1080938836

Rights: Copyright © 2004 Association for Symbolic Logic

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Vol.69 • No. 1 • March 2004
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