From Fuzzy Universal Approximation to Fuzzy Universal Representation: It All Depends on the Continuum Hypothesis

It is known that fuzzy systems have a universal approximation property. A natural question is: can this property be extended to a universal representation property? Somewhat surprisingly, the answer to this question depends on whether the following Continuum Hypothesis holds: every infinite subset of the real line has • either the same number of elements as the real line itself • or as many elements as natural numbers. I. WHEN CAN WE GO FROM FUZZY UNIVERSAL APPROXIMATION TO FUZZY UNIVERSAL REPRESENTATION: FORMULATION OF THE PROBLEM Need to translate expert statements into precise terms. In many practical situations: • there is a correlation between two quantities x and y, and • the only information that we have to describe this correlation are expert statements formulated in terms of imprecise (fuzzy) words from natural language, such as “small”. For example, an expert can say that: • if x is small, • then y is big, and vice versa. Fuzzy logic provides the desired translation. Fuzzy logic (see, e.g., [4], [5], [6]) is a technique that translates this knowledge into precise mathematical terms. In this technique, each fuzzy term A is described by a function A(x) assigning, • to each possible value x of the corresponding quantity, • a degree A(x) to which this value has the appropriate property (e.g., is small). Once we have rules Ai(x) ⇒ Bi(y), the degree d(x, y) to which each pair (x, y) is possible can be described as the degree to which: • either the first rule is satisfied (i.e., A1(x) and B1(y)) • or the second rule is satisfied, etc. One possible way to interpret “and” is to use product: namely, if we know: • the degrees a to which the statement A is satisfied and • the degree b to which a statement B is satisfied, then it is reasonable to estimate the degree to which the conjunction A&B is satisfied as a · b. Similarly, a possible way to interpret “or” is to use sum: namely, if we know: • the degrees a to which the statement A is satisfied and • the degree b to which a statement B is satisfied, then it is reasonable to estimate the degree to which the disjunction A ∨ B is satisfied as a + b (t be more precise, min(a+ b, 1)). Under these interpretations of ”and” and ”or”, • the degree to which the i-th rule is satisfied for a given pair (x, y) can be estimated as the product Ai(x) ·Bi(y), and • the desired degree to which one of the rules is satisfied, i.e., to which: • either the first rule is satisfies, • or the second rule is satisfied, etc.,