Bisimulations have been widely used in many areas of computer science to model equivalence between various systems, and to reduce the number of states of these systems, whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets. Here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata. We prove that a uniform fuzzy relation between fuzzy automata $\cal A$ and $\cal B$ is a forward bisimulation if and only if its kernel and co-kernel are forward bisimulation fuzzy equivalences on $\cal A$ and $\cal B$ and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalences. As a consequence we get that fuzzy automata $\cal A$ and $\cal B$ are UFB-equivalent, i.e., there is a uniform forward bisimulation between them, if and only if there is a special isomorphism between the factor fuzzy automata of $\cal A$ and $\cal B$ with respect to their greatest forward bisimulation fuzzy equivalences. This result reduces the problem of testing UFB-equivalence to the problem of testing isomorphism of fuzzy automata, which is closely related to the well-known graph isomorphism problem. We prove some similar results for backward-forward bisimulations, and we point to fundamental differences. Because of the duality with the studied concepts, backward and forward-backward bisimulations are not considered separately. Finally, we give a comprehensive overview of various concepts on deterministic, nondeterministic, fuzzy, and weighted automata, which are related to bisimulations.
Algorithms for computing the greatest simulations and bisimulations between fuzzy automata
University of Niš（尼什大学）；University of Kragujevac（克拉古耶瓦茨大学）
最近，两种类型的模拟（向前和向后模拟）和四种类型的互模拟（向前，向后，向前-向后，向后-向前互模拟）之间的模糊自动机已被引入。如果在给定的模糊自动机之间至少有一个这些类型的模拟/互模拟，则已经证明存在这种类型的最大模拟/互模拟。在本文中，对于任何上述类型的模拟/互模拟，我们提供了一个有效的算法，用于决定是否有一个模拟/互模拟这种类型之间的给定的模糊自动机，并计算最大的，每当它存在。该算法是基于在[J. J. J. Jatović，M. S. Bogdanović，On the greatest solutions to certain systems of fuzzy relation inequalities and equations，Fuzzy Sets and Systems 161（2010）3081-3113]，其归结为计算模糊关系格上的保序函数的包含在给定模糊关系中的最大后不动点。
状态最小化是自动机理论中的一个基本问题。该问题在模糊自动机的研究中也具有重要意义。然而，大多数文献只考虑模糊自动机的状态约简，而模糊自动机的状态最小化问题几乎没有涉及。因此，在本文中，我们专注于后一个问题。形式上，模糊自动机极小化问题的判定形式如下：\begin{itemize} \item Given a fuzzy automaton $\mathcal{A}$ and a natural number $k$, that is, a pair $\langle \mathcal{A}, k\rangle$, is there a $k$-state fuzzy automaton equivalent to $\mathcal{A}$? \end{itemize}我们首次证明了全序格上模糊自动机的上述问题是可判定的。为此，我们首先给出了模糊多项式方程组的概念，然后提出了一个程序来解决这些系统。然后，我们将模糊多项式方程组的可解性应用于上述极小化问题，得到了可判定性。最后，我们指出，上述问题至少是PSAPCE-完全一样难。
Construction of fuzzy automata from fuzzy regular expressions
Li和Pedrycz [Y. M. Li，W. Pedrycz，Fuzzy finite automata and fuzzy regular expressions with membership values in lattice-ordered monoid，Fuzzy Sets and Systems 156（2005）68- 92]已经证明了基本结果，这些结果提供了在格序monoid中表示具有隶属值的模糊语言的不同等价方式，并且推广了形式语言的经典理论的众所周知的结果。特别是，他们已经证明，一个模糊语言的整体格序幺半群可以表示为模糊正则表达式，当且仅当它可以识别的模糊有限自动机。然而，他们没有给出任何有效的方法来构造一个等价的模糊有限自动机从一个给定的模糊正则表达式。在本文中，我们提供了这样一个有效的方法。通过将模糊正则表达式α中的标量变换为新的扩展字母表中的字母，将模糊正则表达式α转化为普通正则表达式α R。然后，从识别语言的任意非确定性有限自动机A开始，||α_R||用正则表达式α R表示，构造了模糊有限自动机A_α和A_α^r，它们的状态数与自动机A相同或更少，它们识别模糊语言||α||由模糊正则表达式α表示。起始的非确定性有限自动机A可以从α_R中使用任何一种众所周知的将正则表达式转换为非确定性有限自动机的构造来获得，例如Glushkov-McNaughton-Y…
Soft state reduction of fuzzy automata over residuated lattices
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw（华沙大学数学、信息学和力学学院）；Faculty of Information Technology, Nguyen Tat Thanh University（阮友景大学信息技术学院）；Faculty of Information Technology, School of Engineering and Technology, Vinh University（荣市大学工程与技术学院信息技术系）
