ABSTRACT

A minimal solution to a fuzzy relational equation is, loosely speaking, a solution such that the values of all its elements cannot be decreased. To find some minimal solution is simple – one can just identify the greatest solution to the equation and then lower the values of all its elements as much as possible. However, finding all minimal solutions is significantly more difficult since when decreasing the values one needs to go through all combinations of the elements. The paper shows that when considering an n-element scale of truth values equipped with Łukasiewicz or Goguen operations, the set of all minimal solutions can be represented by constrained solutions which allows to skip a considerable amount of these combinations.

抽象的

笼统地说，模糊关系方程的最小解决方案是使所有元素的值都不能减小的解决方案。找到一些最小解很简单–只需找出方程的最大解，然后尽可能降低其所有元素的值即可。但是，找到所有最小解的难度要大得多，因为当减小值时，需要遍历元素的所有组合。本文表明，当考虑配备Łukasiewicz或Goguen运算的n元素真值标度时，所有最小解的集合都可以由受约束的解表示，从而可以跳过大量的这些组合。