On algebraic study of fuzzy automata
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ORIGINAL ARTICLE
On algebraic study of fuzzy automata S. P. Tiwari • Vijay K. Yadav • Anupam K. Singh
Received: 12 September 2013 / Accepted: 16 January 2014 Ó Springer-Verlag Berlin Heidelberg 2014
Abstract This paper is towards the characterization of algebraic concepts such as subautomaton, retrievability and connectivity of a fuzzy automaton in terms of its layers, and to associate upper semilattices with fuzzy automata. Meanwhile, we provide a decomposition of a fuzzy automaton in terms of its layers and propose a construction of a fuzzy automaton corresponding to a given finite partially ordered set (poset). Finally, we establish an isomorphism between the poset of class of subautomata of a fuzzy automaton and an upper semilattice. Keywords Fuzzy automaton Layer Subautomaton Upper semilattice
1 Introduction Algebraic study of automata have been done by many authors in many forms (cf., eg., [1, 4, 6, 17, 18]). Among these studies, in [1], the concepts like separatedness, connectedness and retrievability of automata were introduced and studied; in [4], decompositions and several products of automata were studied; while [6] is the recent contribution in this area to determine the structure of an automaton. In [1], it has also been pointed out that the study of such concepts of automata naturally contributes toward a better S. P. Tiwari V. K. Yadav (&) A. K. Singh Department of Applied Mathematics, Indian School of Mines, Dhanbad 826004, India e-mail: [email protected]
understanding of the structure of automata and their applications; while in [4], it has been stated that such concepts have arisen from a desire to understand the behaviour of a system in an environment and played a large role in the development of the fundamentals of computer science. In view of the above, the algebraic study of fuzzy automata has been initiated by Malik [15] (cf., [16] for details), and afterthat a number of works were reported in this direction (cf., e.g., [7–12]). Also, the works done in [19–23, 25] showed that it was possible to put (fuzzy) topologies on the state-sets of fuzzy automata in natural ways and that these (fuzzy) topologies could then be used to prove some of the algebraic results of fuzzy automata studied in [16] with considerable ease. In different directions, the recent study on fuzzy automata were appeared in [2, 3, 13, 14, 26]. Chiefly inspired from [6], in this paper, we introduce a new concept of ‘layer’ of a fuzzy automaton to strengthen the algebraic study of fuzzy automata. Specifically, in Sect. 3, we provide the characterization of some algebraic concepts such as subautomaton, retrievability and connectivity of a fuzzy automaton in terms of its layers. Also, we show that the maximal layer of a cyclic fuzzy automaton and minimal layer of a directable fuzzy automaton are unique. Finally, we provide a decomposition of a fuzzy automaton in terms of its layers. In Sect. 4, we introduce and study the relationship between fuzzy automata and upper semilattices, and provide an isomorphism between
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