Fuzzy languages with infinite range accepted by fuzzy automata: Pumping Lemma and determinization procedure

The formulation of fuzzy automata allows us to select a great variety of triangular norms. Depending on the selected triangular norm, a fuzzy automaton can accept a fuzzy language (FA-language) with infinite range. These fuzzy automata are not equivalent to the so-called deterministic fuzzy automata (deterministic automata with a fuzzy subset of final states) which only accept fuzzy languages with finite range. In this paper, we study FA-languages with infinite range and a determinization procedure in order to obtain an equivalent fuzzy deterministic automaton for a given fuzzy automaton. A fuzzy deterministic automaton is a fuzzy automaton which satisfies the deterministic condition in its state transition function. The main contributions of our paper are: (1) a Pumping Lemma of FA-languages with infinite range; (2) the formulation of fuzzy deterministic automata and a Pumping Lemma of FDA-languages; (3) the necessary conditions for the determinization of fuzzy automata under continuous triangular norms which accept fuzzy languages of infinite range; and (4) a determinization algorithm for fuzzy automata, its correctness proof and performance.