We introduce depth-bounded fuzzy bisimulation between fuzzy Kripke models. Roughly speaking, a depth-bounded fuzzy bisimulation is a decreasing sequence of fuzzy binary relations whose infimum is a fuzzy bisimulation. We provide logical characterizations of depth-bounded fuzzy bisimulations between fuzzy Kripke models w.r.t. a fuzzy multimodal logic fK over complete residuated lattices, including fuzzy invariance of formulas of fK with a modal depth bounded by n under the nth component of a depth-bounded fuzzy bisimulation, as well as the Hennessy-Milner property of depth-bounded fuzzy bisimulations. We also provide a polynomial-time algorithm for computing the nth component of the greatest depth-bounded fuzzy bisimulation between two finite fuzzy Kripke models when the underlying complete residuated lattice is linear.

In this article, we introduce fuzzy minimax nets as a novel tool to compute the greatest fuzzy bisimulation/simulation between two finite fuzzy labeled graphs. Fuzzy labeled graphs are a universal data structure for representing fuzzy systems, such as fuzzy automata, fuzzy labeled transition systems, fuzzy Kripke models, fuzzy social networks, and fuzzy interpretations in description logic. The greatest fuzzy bisimulation between two such systems characterizes the similarity between their states, actors, or individuals. Using fuzzy minimax nets, we design the first algorithms for the mentioned computational problems in the case of using the product t-norm, as well as the first algorithms whose complexity order does not depend on the fuzzy values occurring in the inputs for those problems in the case of using the Łukasiewicz t-norm.

In this article, we define$\lambda$-approximate simulations and bisimulations for fuzzy automata over complete Heyting algebras. The value$\lambda$presents the degree of language similarity or equality between observed fuzzy automata. Algorithms for computing the greatest$\lambda$-approximate simulations and bisimulations are given. We show that$\lambda$-approximate simulations and bisimulations on a fuzzy automaton can be effectively used for factorization of fuzzy automata. We present the algorithm that splits the interval of the degrees of language similarity or equality into subintervals with the same minimal corresponding factor 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 possible 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 Jancic and Ciric in [10]. Here we provide another canonization method for a fuzzy finite automaton A = (A, σ, δ, r) over a complete residuated lattice £, 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 r generate a finite subsemiring of the semiring reduct of £. This procedure is generally faster than the Brzozowski type determinization, and if the basic operations in the residuated lattice £ can be performed in constant time, it has the same computational time as all other determinization procedures provided in [8], [11], and [12].