谢正卫, 翟莹, 黄飞丹, 易忠, 邓培民. 两类模糊有限状态机积的交换性[J]. 计算机研究与发展, 2014, 51(9): 2130-2136.
 引用本文: 谢正卫, 翟莹, 黄飞丹, 易忠, 邓培民. 两类模糊有限状态机积的交换性[J]. 计算机研究与发展, 2014, 51(9): 2130-2136.
Xie Zhengwei, Zhai Ying, Huang Feidan, Yi Zhong, Deng Peimin. On Commutativity of Products of Two Types Fuzzy Finite State Machines[J]. Journal of Computer Research and Development, 2014, 51(9): 2130-2136.
 Citation: Xie Zhengwei, Zhai Ying, Huang Feidan, Yi Zhong, Deng Peimin. On Commutativity of Products of Two Types Fuzzy Finite State Machines[J]. Journal of Computer Research and Development, 2014, 51(9): 2130-2136.

## On Commutativity of Products of Two Types Fuzzy Finite State Machines

• 摘要: 利用代数工具矩阵、半群等对两类模糊有限状态机的交换性作了进一步的研究.首先给出了模糊有限状态机是可交换的几个等价刻画，即模糊有限状态机交换与其状态转移矩阵关于模糊矩阵乘法交换等价，与其输入集上字符串关于同余关系构成的乘法半群交换等价，并讨论了模糊有限状态机的直积、级联积、圈积以及和的交换性.其次提出了Mealy-型模糊有限状态机是可交换的概念,同时在新的概念下详细地研究了Mealy-型模糊有限状态机的直积、级联积、圈积以及和、商的交换性.得到了两个(Mealy-型)模糊有限状态机的完全直积、和交换的充要条件；得到两个(Mealy-型)模糊有限状态机的圈积、级联积交换的一个充分条件；证明了商Mealy-型模糊有限状态机保持原Mealy-型模糊有限状态机的交换性.最后给出判别模糊有限状态机交换性的算法.

Abstract: Automata theory is one of the basic and important theories in computer science. The use of algebraic techniques in determining the structure of automata has been significant. Afterword， Malik et al. applied algebraic techniques to study fuzzy automata or fuzzy finite state machines(ffsm). In this article, the further research on the commutativity of two types ffsm is investigated by algebraic tools such as matrices, semgroups, and so on.Some equivalent characterizations of the commutativity of ffsm are given. It is proved that ffsm are commutative if and only if their state transition matrices are commutative for fuzzy matrix multiplication or the semigroup of strings over input alphabet by congruence relations is commutative. The commutativity of direct product, cascade product,wreath product, and sum of ffsm are discussed. Meanwhile,the concept of commutativity of Mealy-type fuzzy finite state machines (Mffsm) is defined.The commutativity of several products, sum, and quotient for Mffsm are studied in detail.Furthermore,the sufficient and necessary conditions of the commutativity of direct product, sum for ffsm (Mffsm) are obtained as well as the sufficient conditions of the commutativity of wreath product, cascade product for Mffsm.It is also proved that quotient Mffsm maintains the commutativity of Mffsm. Moreover,the algorithm for commutativity of ffsm is presented.

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