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Studia Logica

, Volume 103, Issue 6, pp 1163–1181 | Cite as

Trakhtenbrot Theorem and First-Order Axiomatic Extensions of MTL

  • Matteo Bianchi
  • Franco Montagna
Article

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.

Keywords

Trakhtenbrot theorem Many-valued logics MTL logic Residuated lattices Completeness Arithmetical complexity 

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversità degli Studi di MilanoMilanItaly
  2. 2.Department of Information Engineering and MathematicsUniversità degli Studi di SienaSienaItaly

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