Lattice Structure of Binary-Equivalent Fuzzy Logical Functions under Max-Min Logic
Lattice Structure of Binary-Equivalent Fuzzy Logical Functions under Max-Min Logic
复制标题
最大-最小逻辑下二元等价模糊逻辑函数的格结构
DOI：
--
复制
发表时间：
2005
期刊：
影响因子：
--
通讯作者：
K. Hirota
中科院分区：
文献类型：
--
作者：
S. Yoshida;K. Hirota
分析了二元等价模糊逻辑函数在Max-Min模糊逻辑下的格结构，传统的模糊触发器研究表明，在Max-Min模糊逻辑下，存在一些不同类型的模糊触发器，它们的二进制逻辑特性是相同的。这些模糊触发器的模糊特征构成了歧义偏序下的格结构(布尔格或分配格)。本文证明了在Max-Min模糊逻辑下，二元逻辑函数的一组模糊扩张构成了任意二元逻辑函数的分配格。对于在最大-最小模糊逻辑下的模糊逻辑函数的分析，只需对变量值的0、1/2、1进行检查即可。(即，模糊逻辑函数的特征是在使用Max-Min运算的三值逻辑下完全确定的。)当一个二进制逻辑函数被扩展到最大-最小逻辑时，一些函数的值可以是不含糊的值(0或1)或含糊的值(1/2)。这种差异导致了二元逻辑函数的模糊扩张的变化。在变量取值固定的情况下，模糊可拓的变化量最多为2个，并且这两个变化量在两个模糊偏序下都是线性排序的。这是两个元素的布尔格。如果模糊扩张的变量对于任意变量值是独立决定的，则模糊扩张的结构构成二元布尔格的直积，即n元布尔格。在模糊延拓的某些情况下，函数值的变化不能独立于其他函数值来决定。在这种情况下，扩张的结构构成了非布尔分配格。
Lattice structure of binary-equivalent fuzzy logical functions are analyzed under Max-Min fuzzy logic, Conventional studies of fuzzy flip-flops has shown that some different types of fuzzy flip-flops, whose binary logical characteristics are same, exist under Max-Min fuzzy logic. The fuzzy characteristics of these fuzzy flip-flops construct lattice structures (a Boolean lattice or a distributive lattice) under the partial order of ambiguity. This paper shows that a set of fuzzy extensions of a binary logical function constructs a distributive lattice for any binary logical function under Max-Min fuzzy logic. For analysis of a fuzzy logical function under Max-Min fuzzy logic, it is sufficient that the functions's value are examined for 0, 1/2, 1 of variables's values. (i.e, The characteristics of a fuzzy logical function are completely decided under the ternary logic using Max-Min operations.) When a binary logical function is extended to Max-Min logic, some of the functions's values can take a non-ambiguous value (0 or 1) or a ambiguous value (1/2). This difference derives the variations of fuzzy extensions of a binary logical funciton. If the variables' values are fixed, the variations of fuzzy extensions are at most 2. And the two variations are linearly ordered under both of the partial order of ambiguity. This is two-element Boolean lattice. If the variations of fuzzy extensions are decided independently for any variables' value, the structure of the fuzzy extensions contstruct the direct product of two-element Boolean lattice, i.e., n-element Boolean lattice. In some cases of fuzzy extensions, variations of function's value cannot be decided independently from other function's value. In this case, the structure of the extensions constructs non-Boolean distributive lattice.