arXiv subjects
Publications and source records attributed to Ivana Micić.
In this paper we introduce a new type of approximate state reductions where the behaviors of the reduced and the original automaton do not have to be identical, but they must match on all words of length less than or equal to some given natural number. We provide four methods for performing such reductions.
Simulations and bisimulations are well-established notions in crisp/fuzzy automata theory and are widely used to compare the behaviors of automata. Their main drawback is that they compare the behaviors of fuzzy automata in a crisp manner. Recently, fuzzy simulations and fuzzy bisimulations have been defined for fuzzy automata as a kind of approximate simulations and approximate bisimulations that compare the behaviors of fuzzy automata in a fuzzy manner. However, they still suffer from serious shortcomings. First, they still cannot correlate all fuzzy automata that are intuitively "more or less" (bi)similar. Second, the currently known algorithms for computing the greatest fuzzy simulation or bisimulation between two finite fuzzy automata have an exponential time complexity when the Łukasiewicz or product structure of fuzzy values is used. This work deals with these problems, providing approximations of fuzzy simulations and fuzzy bisimulations. We define such approximations via a novel notion of decreasing sequences of fuzzy relations whose infima are, under some conditions, fuzzy simulations (respectively, bisimulations). We call such a sequence a depth-bounded fuzzy simulation (respectively, bisimulation), as the $n$th element from the sequence compares the behaviors of fuzzy automata, but only for words with a length bounded by $n$. We further provide a logical characterization of the greatest depth-bounded fuzzy simulation or bisimulation between two fuzzy automata by proving that it satisfies the corresponding Hennessy-Milner property. Finally, we provide polynomial-time algorithms for computing the $n$th component of the greatest depth-bounded fuzzy simulation (respectively, bisimulation) between two finite fuzzy automata.
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].
In this paper we combine determinization and state reduction methods into two-in-one algorithms that simultaneously perform determinization and state reduction. These algorithms perform better than all previous determinization algorithms for fuzzy finite automata, developed by Belohlavek [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], 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 and Ćirić [Fuzzy Set Syst (2014), to appear], which produces a minimal crisp-deterministic fuzzy automaton, but the algorithms created here can also be used within the Brzozowski type algorithm and improve its performances.