The Fibonacci sequence in the description of maximal discrete Archimedean t-norms

Abstract

There are many arguments for counting with more than two “truth values”; this allows to imitate human reasoning of facts which are not binary. For theoretical reasons, it is natural to use the whole real interval as the scale. However, this brings practical problems: it is difficult, and even impossible, to represent exact values. Often only a small scale of values suffices to express what we need. Therefore, finite chains are frequently used as domains of fuzzy logical operations. Their representation and manipulation are easy. In this paper, we focus on triangular norms (t-norms). The choice of a finite domain admits some operations (Gödel, Łukasiewicz), while it excludes others (all strict ones, including the product). A disadvantage of the Gödel (minimum) t-norm is that repetition of arguments does not change their meaning. This is often desirable to emphasize the statement. (“Words, words, words!”) Thus we do not consider the Gödel operations sufficient for representation of all fuzzy logical statements in human reasoning. Together with them, we discard all operations with idempotent elements other than 0, 1, thus we restrict attention to Archimedean t-norms (and t-conorms as their duals, not treated explicitly in the sequel). A disadvantage of many Archimedean t-norms is that, when applied to several arguments, the result is very often zero. Then, it gives no clue in comparing the outcomes, e.g., in giving priorities to alternatives evaluated by fuzzy rules. For instance, when the continuous Łukasiewicz t-norm is applied to 5 entries, it is nonzero only on 1/5!=1/120 of the volume of its 5-dimensional domain; in its discrete versions, nonzero results are even more rare. Thus we are interested in Archimedean t-norms with values “as large as possible”, here in the maximal Archimedean t-norms (=those which are not majorized by other Archimedean t-norms). It was shown in previous works that there is an abundance of discrete t-norms; their number grows fast with the number of elements of the underlying chain (no exponential bound seems to be known). There is also an abundance of Archimedean t-norms. In contrast to that, when we counted the number of maximal Archimedean t-norms, it grows asymptotically exponentially with a mild base. What is more interesting, these numbers follow the Fibonacci sequence. We have found a description of maximal Archimedean t-norms and explained also the processes behind their construction and the role of the Fibonacci sequence. These results link the t-norms to a construction using a kind of additive generators, although this could not be used without modifications. We do not describe all Archimedean t-norms and their numbers, but we have completed at least a significant step towards this goal, too.