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Trakhtenbrot Theorem and First-Order Axiomatic Extensions of MTL

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An Erratum to this article was published on 09 October 2015

Abstract

In 1950, B.A. Trakhtenbrot showed that the set of first-order tautologies associated to finite models is not recursively enumerable. In 1999, P. Hájek generalized this result to the first-order versions of Łukasiewicz, Gödel and Product logics, w.r.t. their standard algebras. In this paper we extend the analysis to the first-order versions of axiomatic extensions of MTL. Our main result is the following. Let \({\mathbb{K}}\) be a class of MTL-chains. Then the set of all first-order tautologies associated to the finite models over chains in \({\mathbb{K}}\), fTAUT\({_{\forall}^{\mathbb{K}}}\), is \({\Pi_{1}^{0}}\) -hard. Let TAUT\({_\mathbb{K}}\) be the set of propositional tautologies of \({\mathbb{K}}\). If TAUT\({_{\mathbb{K}}}\) is decidable, we have that fTAUT\({_{\forall}^{\mathbb{K}}}\) is in \({\Pi_{1}^{0}}\). We have similar results also if we expand the language with the Δ operator.

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Authors and Affiliations

  1. Department of Computer Science, Università degli Studi di Milano, Via Comelico 39/41, 20135, Milan, Italy

    Matteo Bianchi

  2. Department of Information Engineering and Mathematics, Università degli Studi di Siena, via Roma 56, 53100, Siena, Italy

    Franco Montagna

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  1. Matteo Bianchi
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  2. Franco Montagna
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Correspondence to Matteo Bianchi.

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Note by the first author

Sadly, Franco Montagna passed away on 18 February 2015. It is a great loss for the scientific community, and also for the people who personally knew him, including me. I am glad I had the opportunity to work with him, and I remember his kindness, openness in discussions, and his ability in approaching and solving mathematical problems. I was not one of his students, but I can say that he contributed in a significant way to improve my knowledge in the area of mathematical logic.

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Bianchi, M., Montagna, F. Trakhtenbrot Theorem and First-Order Axiomatic Extensions of MTL. Stud Logica 103, 1163–1181 (2015). https://doi.org/10.1007/s11225-015-9614-3

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