Determinization of fuzzy automata via factorization of fuzzy states

Fuzzy finite-state automata over the algebra ([0,1],max,⊗,0,1), in which the monoid ([0,1],⊗,1) (⊗ denotes a continuous triangular norm) is not locally finite, can accept fuzzy languages of infinite range. For a given fuzzy finite-state automaton which accepts a fuzzy language of infinite range, we define the determinization of the fuzzy automaton via factorization of fuzzy states, i.e., the computation of an equivalent deterministic fuzzy automaton whether it is finite. This method of determinization is a generalization of the well-known accessible subset construction. Our main contribution is to determine that the representable-cycles property is the necessary and sufficient condition for determinization of a fuzzy finite-state automaton via a maximal factorization of fuzzy states. This property is more general than the twins property (adapted for fuzzy automata) which is the sufficient condition for weighted automata over the tropical semiring.