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 Jan\v{c}i\'{c} and \'{C}iri\'{c} in [10]. Here we provide another canonization method for a fuzzy finite automaton $\cal A=(A,\sigma, \delta,\tau)$ 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 $\delta $, $\sigma $ and $\tau $ 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].