Fast implementations of fuzzy arithmetic operations using fast Fourier transform (FFT)

Abstract

In engineering applications of fuzzy logic, the main goal is not to simulate the way the experts really think, but to come up with a good engineering solution that would (ideally) be better than the expert's control. In such applications, it makes perfect sense to restrict ourselves to simplified approximate expressions for membership functions. If we need to perform arithmetic operations with the resulting fuzzy numbers, then we can use simple and fast algorithms that are known for operations with simple membership functions. In other applications, especially the ones that are related to humanities, simulating experts is one of the main goals. In such applications, we must use membership functions that capture every nuance of the expert's opinion; these functions are therefore complicated, and fuzzy arithmetic operations with the corresponding fuzzy numbers become a computational problem. In this paper, we design a new algorithm for performing such operations. This algorithm uses Fast Fourier Transform (FFT) to reduce computation time from O( n 2) to O( nlog( n)) (where n is the number of points x at which we know the membership functions μ( x)). To compute FFT even faster, we propose to use special hardware. The results of this paper were announced in the work of Kosheleva et al. [Proc. 1996 IEEE Int. Conf. on Fuzzy Systems, Vol. 3, pp. 1958–1964].