Homomorphisms of fuzzy recognizers

Abstract

PurposeThe purpose of this research paper is to develop an algorithm/methodology for the reduction of a fuzzy recognizer through an algebraic concept such as homomorphism.Design/methodology/approachThe approach of this research is to introduce the concepts, compatible with the purpose of this paper, and then to find a necessary and sufficient condition for the reduction of a fuzzy recognizer.FindingsA fuzzy Σ‐recognizer M with non‐empty fuzzy initial state and the behavior A is reduced if and only if it is isomorphic to MA.Research limitations/implicationsThe research proposes an algebraic method for the reduction of a fuzzy recognizer. The problem of finding dispensable fuzzy productions of a regular fuzzy grammar may be tackled by the use this algebraic method, as fuzzy recognizers and fuzzy regular grammars are equivalent.Originality/valueThe concepts and the methodology are original. The work is useful to the researchers in the field of fuzzy automata, fuzzy grammars and pattern recognition.

Computing behavior of finite fuzzy machines – Algorithm and its application to reduction and minimization

“…We implement expressions (1) and (2) and software from [21]. We apply (2) to the last three matrices and obtain: It is obvious that this method of computation is tremendous even for words having fixed small length. This motivated us to develop functions for computing T A , as explained in the next II way.II way.…”

“…It is semi-infinite matrix with n ¼ jQ j rows and with columns Tðu j vÞ, ðu j vÞ 2 ðX j YÞ Ã , computed by (2) and ordered according to the lexicographical order in ðX j YÞ Ã , see Table 1 (if X 6 ¼ ; and Y 6 ¼ ; then X Ã and Y Ã are countably infinite and ðX j YÞ Ã is lexicographically ordered).…”

“…Published papers concern theory and some algorithms: for equivalence of states and equivalence of machines [10,18,31,34]; for reduction [2,[9][10][11]18,31,32] and minimization [3,9,11,12,15,16,27,31]. None of them pretends to present software for obtaining behavior matrix, for establishing equivalence of states, for finding redundant states, for finding reduced and minimal forms of a fuzzy machine -still open problems for finite fuzzy machines.…”

“…During the period 1968-2000 few references were published, see the monographs [7] by Klir and Yuan, [17] by Mordeson and Malik, [21] by Peeva and Kyosev. Recently the interest in this subject has highly increased. Many authors contributed to algebraic approach and introduced variety of fuzzy machines on suitable algebraic structures (lattices [20,34] or lattice-ordered monoids [8][9][10][11][12]14,16], as well as formal power series [27], semirings [18], categorical approach [13], homomorphisms [2], etc.) with extension for fuzzy languages [2,8,15,16,26].…”