A wrong argument in the proof of one of the main results in the paper is corrected.The result itself remains true.The right proof incorporates the basic ideas in the originally alleged proof, but in a more restricted construction.

Journal Article Logics Preserving Degrees of Truth from Varieties of Residuated Lattices Get access Félix Bou, Félix Bou Artificial Intelligence Research Institute (IIIA - CSIC), Bellaterra, Spain.E-mail: [email protected]; [email protected] Search for other works by this author on: Oxford Academic Google Scholar Francesc Esteva, Francesc Esteva Artificial Intelligence Research Institute (IIIA - CSIC), Bellaterra, Spain.E-mail: [email protected]; [email protected] Search for other works by this author on: Oxford Academic Google Scholar Josep Maria Font, Josep Maria Font Department of Probability, Logic and Statistics, Faculty of Mathematics, University of Barcelona, Spain.E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Àngel J. Gil, Àngel J. Gil Departament d'Economia i Empresa, Universitat Pompeu Fabra, Barcelona, Spain.E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Lluís Godo, Lluís Godo Artificial Intelligence Research Institute (IIIA - CSIC), Bellaterra, Spain.E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Antoni Torrens, Antoni Torrens Department of Probability, Logic and Statistics, Faculty of Mathematics, University of Barcelona, Spain.E-mail: [email protected]; [email protected] Search for other works by this author on: Oxford Academic Google Scholar Ventura Verdú Ventura Verdú Department of Probability, Logic and Statistics, Faculty of Mathematics, University of Barcelona, Spain.E-mail: [email protected]; [email protected] Search for other works by this author on: Oxford Academic Google Scholar Journal of Logic and Computation, Volume 19, Issue 6, December 2009, Pages 1031–1069, https://doi.org/10.1093/logcom/exp030 Published: 26 June 2009 Article history Received: 03 March 2008 Published: 26 June 2009

Given a finite set of vectors over a finite totally ordered domain, we study the problem of computing a constraint in conjunctive normal form such that the set of solutions for the produced constraint is identical to the original set. We develop an efficient polynomial-time algorithm for the general case, followed by specific polynomial-time algorithms producing Horn, dual Horn, and bijunctive formulas for sets of vectors closed under the operations of conjunction, disjunction, and median, respectively. Our results generalize the work of Dechter and Pearl on relational data, as well as the papers by Hébrard and Zanuttini. They complement the results of Hähnle et al. on multivalued logics and Jeavons et al. on the algebraic approach to constraints.

In this paper we obtain a characterization of the algebraizability of an m-dimensional Gentzen system in line with the characterization obtained for m-dimensional deductive systems and the characterization of 2-dimensional Gentzen systems. We also prove that if S(m) is the finite linear MV-algebraof m elements, then the m-dimensional Gentzen system obtained by using the sequent calculi associated with S(m) is equivalent to the m-valued Łukasiewicz logic Łm and to the equational consequence relation associated with S(m). Taking the two-element Boolean algebra we obtain the expected result concerning the relationship between the sequent calculus LK, the Classical Prepositional Calculus and the variety of Boolean algebras.