Membership Functions or alpha-Cuts? Algorithmic (Constructivist) Analysis Justifies an Interval Approach

In his pioneering papers, Igor Zaslavsky started an algorithmic (constructivist) analysis of fuzzy logic. In this paper, we extend this analysis to fuzzy mathematics and fuzzy data processing. Specifically, we show that the two mathematically equivalent representations of a fuzzy number – by a membership function and by α-cuts – are not algorithmically equivalent, and only the α-cut representation enables us to efficiently process fuzzy data. 1 Need for Fuzzy Logic and Fuzzy Mathematics: Brief Reminder Need for automating expert knowledge. In many application areas, we actively use expert knowledge: skilled medical doctors know how to diagnose and cure diseases, skilled pilots know how to deal with extreme situations, etc. The big problem with expert knowledge is that there are only a few top experts, and it is not realistically possible to use them every time. For example, there are only a few top experts in heart diseases, and there are millions of patients; similarly, there are a few top pilots, and there are thousands of daily flights. Since we cannot use the top experts in all situations, it is desirable to design automated systems that would incorporate the knowledge of these experts. Need for fuzzy knowledge. One of the main challenges in incorporating expert knowledge into an automated system is that experts often cannot describe their knowledge in precise terms. Instead, they express a significant part of their knowledge by using imprecise (“fuzzy”) words from natural languages such as “small”, “slightly”, “a little bit”, etc. So, to design automated expert systems, we need to translate these words into a language that a computer can understand – i.e., translate this knowledge in precise terms. Fuzzy logic: truth values. Techniques for translating imprecise (fuzzy) knowledge into precise formulas were pioneered by Lotfi Zadeh who called them fuzzy logic; see, e.g., [4, 11]. The main idea is that in contrast to well-defined properties (like x < 1.0) which are either true or not for every real value x, a property like “x is small” is fuzzy: for very small values x, everyone would agree that x is small; for large values x, everyone agrees that these values are not small; however, for intermediate values x, some experts may consider them small, and some not. A reasonable way