Fuzzy Linear Automata and Some Equivalences

Abstract

ABSTRACT. In this paper, we present two new normal forms for fuzzy linear grammars. Moreover, we introduce two new classes of fuzzy machines. The first is the class of fuzzy linear automata (FLA). We show that automata of this type is equivalent to the fuzzy linear grammar, i.e., we prove that the class of fuzzy languages is recognized by fuzzy linear automata. Finally, we introduce the class of fuzzy nondeterministic 2-tape automata (FNTA), show the equivalence between them and fuzzy linear automata.

On typical hesitant fuzzy automata

“…Remark 1. Notice that by the Chomsky's hierarchy [11], as mentioned in proof of Theorem 2 in [12], there is only an enumerable set of regular languages.…”

“…The concept of fuzzy set is a generalization of the classical notion of set, introduced by Zadeh in his seminal work (Zadeh 1965). As said in Costa and Bedregal (2018), the key concept of fuzzy sets is that the elements of the universe belong to the fuzzy set with some membership degree, typically a value in [0, 1]. Definition 6 (Zadeh 1965) Let U be a non-empty universe set.…”

“…Remark 1 Notice that Σ 0 contains only the empty word, i.e., Σ 0 = {λ}. Definition 7 (Costa and Bedregal 2018) For any alphabet Σ is defined, the positive closure is Σ + = ∞ i=1 Σ i and the Kleene closure, denoted by Σ * , is…”

“…Fuzzy sets (Zadeh 1965) have been used to solve computing problems in which there is inherent uncertainty in the computing processor itself (Asveld 2005;Mordeson and Malik 2002), and this has influenced the development of research in various fuzzy computer models and fuzzy languages (Bedregal and Figueira 2008;Costa and Bedregal 2018;Farias et al 2016a, b;Ghorani 2019;Komejwar 2012;Lee and Zadeh 1969;Mordeson and Malik 2002;Wang et al 2013;Wee and Fu 1993;Wiedermann 2004;Yang and Li 2018;Yu 2001). However, fuzzy computing requires that the level of uncertainty tolerated between state changes in computer is a single value, precisely defined and without hesitation, which is a problem because even the best engineer cannot guarantee that a certain level of uncertainty tolerated is best for the functioning of the computer.…”

“…This model is widely used for modeling applications in hardware and software [5], and it is also essential for building compilers [4,6]. In recent years, several generalizations of the concept of finite automata have been presented, such as fuzzy automata [7,8,9,10,11], probabilistic automata [12,13,14], and quantum automata [15,16,17,18,19].…”