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In this paper we provide further improvements of determinization methods for fuzzy finite automata. These methods perform better than all previous determinization methods for fuzzy finite automata, developed by Bělohlávek [3], Li and Pedrycz [21], Ignjatović et al. [12], and Jančić et al. [16], in the sense that they produce smaller automata, while require the same computation time. The only exception is the Brzozowski type determinization algorithm developed recently by Jančić andĆirić [17], which produces a minimal crispdeterministic fuzzy automaton, but the algorithms created here can also be used within the Brzozowski type algorithm and improve its performance.

In this paper we adapt the well-known Brzozowski determinization method to fuzzy automata. This method gives better results than all previously known methods for determinization of fuzzy automata developed by Bělohlávek [Inform Sciences 143 (2002) 205-209], Li and Pedrycz [Fuzzy Set Syst 156 (2005) 68-92], Ignjatović et al. [Inform Sciences 178 (2008) 164-180], and Jančić et al. [Inform Sciences 181 (2011) 1358-1368]. Namely, as in the case of ordinary nondeterministic automata, Brzozowski type determinization of a fuzzy automaton results in a minimal crisp-deterministic fuzzy automaton equivalent to the starting fuzzy automaton, and we show that there are cases when all previous methods result in infinite automata, while Brzozowski type determinization results in a finite one. The paper deals with fuzzy automata over complete residuated lattices, but identical results can also be obtained in a more general context, for fuzzy automata over lattice-ordered monoids, and even for weighted automata over commutative semirings.

Determinization of fuzzy finite automata is understood here as a procedure of their conversion into equivalent crisp-deterministic fuzzy automata, which can be viewed as being deterministic with possibly infinitely many states, but with fuzzy sets of terminal states. Particularly significant determinization methods are those that provide a minimal crisp-deterministic fuzzy automaton equivalent to the original fuzzy finite automaton, called canonization methods. One canonization method for fuzzy finite automata, the Brzozowski type determinization, has been developed recently by Jančić andĆirić in [10]. Here we provide another canonization method for a fuzzy finite automaton A = (A, σ, δ, τ ) over a complete residuated lattice L, based on the degrees of inclusion of the right fuzzy languages associated with states of A into the left derivatives of the fuzzy language recognized by A. The proposed procedure terminates in a finite number of steps whenever the membership values taken by δ, σ and τ generate a finite subsemiring of the semiring reduct of L. This procedure is generally faster than the Brzozowski type determinization, and if the basic operations in the residuated lattice L can be performed in constant time, it has the same computational time as all other determinization procedures provided in [8], [11], [12].

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