Journal of Intelligent & Fuzzy Systems 31 (2016) 1795–1806 DOI:10.3233/JIFS-152489 IOS Press 1795 Closure properties for fuzzy recursively enumerable languages and fuzzy recursive languages Antonio Diego Silva Farias a,b,∗ , Luiz Ranyer de Ara´ ujo Lopes b , Benjam´ ın Bedregal b and Regivan H.N. Santiago b a Federal Rural University of Semi-Arid – UFERSA, Universitary Campus of Pau dos Ferros, S˜ ao Geraldo, Pau dos Ferros, RN, Brazil b Department of Informatics and Applied Mathematics – DIMAp, Federal University of Rio Grande do Norte – UFRN, Universitary Campus of Lagoa Nova, Natal, RN, Brazil Abstract. There are several variations of fuzzy Turing machines in the literature, many of them require a t-norm in order to establish their accepted language. This paper generalize the concept of non-deterministic fuzzy Turing machine - NTFM, replacing the t-norm operator for several aggregation functions. We establish the languages accepted by these machines, called fuzzy recursively enumerable languages or simply LFRE and show, among other results, which classes of LFRE are closed under unions and intersections. Keywords: Turing machines, fuzzy turing machines, fuzzy languages, aggregation functions 1. Introduction The starting point for the Computer Science was to model the so called “effective procedure”; i.e. a method which can be carried out by machines. The first proposed model is due to Alan Turing, called Turing Machines (TM), was proposed in 1936 [28, 29], and since then has been the most accepted model. In the 1960’s Lofti Zadeh [38] generalized the notion of membership and non-membership value from the set {0, 1} to the continuous interval [0, 1], Giving rise to the Fuzzy Set Theory. This allowed several extensions of classical mathematical theories [7], including that of Turing machines [2, 20, 21, 37]. The first formulation of Fuzzy Turing machine was introduced by Zadeh [39], in the late of 1960s, through what he called fuzzy algorithm. Zadeh, ∗ Corresponding author. Antonio Diego Silva Farias. Tel.: +55 84 99939 9065; E-mail: antonio.diego@ufersa.edu.br. together with Lee, have defined the concept of Fuzzy Languages [19]. After these papers, Santos [30, 31] showed that fuzzy algorithms and fuzzy Turing machines are equivalent models. For a long time there were no researches in fuzzy Turing machines, however the work of Harkleroad [16] pointed some themes about it (see [5, 6, 8, 11, 12, 24–26]). In recent years, mainly motivated by the possibility of extrapolating the Church-Turing thesis, there were several other approaches to computability (for exam- ple: [9, 10, 14, 34]). Besides all this, Wiedermann [35, 36] show that it is possible to solve the halting problem. With the work of Wiedermann, Bedregal and Figueira [2] proved that the idea of a Universal Fuzzy Turing Machine is not possible. The proposed models of Turing machine found in literature require t-norm operators to calculate the degree of acceptability of a string, and thus determine the language that this machine is able to 1064-1246/16/$35.00 © 2016 – IOS Press and the authors. All rights reserved