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Truth Definitions in Finite Models

Leszek Aleksander Kołodziejczyk
The Journal of Symbolic Logic
Vol. 69, No. 1 (Mar., 2004), pp. 183-200
Stable URL: http://www.jstor.org/stable/30041718
Page Count: 18
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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 ᵍ does not depend on the syntax of L, as long as it is decidable. It follows that for a natural ᵍ there exist FM-truth definitions whose evaluation is much easier than the combined complexiy of ᵍ would suggest. We apply the general theory to give a complexity-theoretical characterization of the logics for which the $\Sigma_{m}^{d}$ classes (prenex classes of higher order logics) define FM-truth. For any $d \geq 2.m \geq 1$ we construct a family $\{[\Sigma_{m}^{d}]^{\leg\kappa}\}\kappa\in\omega$ of syntactically defined fragments of $\Sigma_{m}^{d}$ which satisfy this characterization. We also use the $[\Sigma_{m}^{d}]\leg\kappa$ classes to give a refinement of known results on the complexity classes captured by $\Sigma_{m}^{d}$. We close with a few simple corollaries, one of which gives a sufficient condition for the existence, given a vocabulary $\sigma$, of a fixed number $\kappa$ such that model checking for all first order sentences over $\sigma$ can be done in deterministic time $n^{\kappa}$

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