Minimization of states in automata theory based on finite lattice-ordered monoids

Fuzzy relation equations and reduction of fuzzy automata

“…Reduction of number of states of fuzzy automata was studied in [1,14,41,44,50,53], and the algorithms given there were also based on the idea of computing and merging indistinguishable states. They were called minimization algorithms, but the term minimization is not adequate because it does not mean the usual construction of the minimal one in the set of all fuzzy automata recognizing a given fuzzy language, but just the procedure of computing and merging indistinguishable states.…”

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

“…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].…”

“…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.…”

Diagnosability of fuzzy discrete event systems

“…Extending the range of logical values {0, 1} to the interval [0, 1] yields fuzzy automata (FA) [1,19,35], which are also called fuzzy finite-state machines [1,23]. They are similar to stochastic automata [18], except that their transition probabilities are replaced by transition correlations, which also may be interpreted as transition possibilities. Fuzzy methods based on fuzzy extensions of the theory of languages and automata exist.…”