Maximizing sets and fuzzy Markoff algorithms

A fuzzy algorithm is an ordered set of fuzzy instructions that upon execution yield an approximate solution to a given problem. Two unrelated aspects of fuzzy algorithms are considered in this paper. The first is concerned with the problem of maximization of a reward function. It is argued that the conventional notion of a maximizing value for a function is not sufficiently informative and that a more useful notion is that of a maximizing set. Essentially, a maximizing set serves to provide information not only concerning the point or points at which a function is maximized, but also about the extent to which the values of the reward function approximate to its supremum at other points in its range. The second is concerned with the formalization of the notion of a fuzzy algorithm. In this connection, the notion of a fuzzy Markoff algorithm is introduced and illustrated by an example. It is shown that the generation of strings by a fuzzy algorithm bears a resemblance to a birth-and-death process and that the execution of the algorithm terminates when no more “live” strings are left.