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

Abstract

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.