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Determinization of fuzzy automata by means of the degrees of language inclusion

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 and Ćirić in [10]. Here we provide another canonization method for a fuzzy finite automaton $\cal A=(A,σ, δ,τ)$ over a complete residuated lattice $\cal L$, based on the degrees of inclusion of the right fuzzy languages associated with states of $\cal A$ into the left derivatives of the fuzzy language recognized by $\cal A$. 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 $\cal L$. This procedure is generally faster than the Brzozowski type determinization, and if the basic operations in the residuated lattice $\cal L$ can be performed in constant time, it has the same computational time as all other determinization procedures provided in [8], [11], [12].