Comments on Singh and Zeng: "Approximation theory of fuzzy systems-SISO case" [with reply]

The author comments on the paper by Singh and Zeng (see ibid., vol.2, no.2, p.162-76, 1994). He states that every bounded function f: R/spl rarr/R has an exact representation as an additive fuzzy system. If f is not constant, one fuzzy set and two rules define the system. Otherwise, a single rule suffices. This result shows that the approximation properties of one-input fuzzy systems derive solely from interpolation between output extrema. The basis for the interpolation at any point is the value of the input fuzzy sets at that point. In reply Singh and Zeng state that in the comments by Watkins, it is proven that every SISO function can be exactly represented by a fuzzy system, which implies that fuzzy approximation (i.e., to approximate functions by fuzzy systems) is unnecessary or moot. However, they state that this conclusion is invalid because his presented representation scheme does not meet the basic requirements in the applications of fuzzy systems and is impractical.