Publication Date



Technical Report UTEP-CS-07-03


It is well-known that t-norms are widely applicable in certain models, which describe human reasoning about uncertainty, and that for different applications, different t-norms fit better. Thus, given a practical problem, it is important to be able to find a t-norm which is the most suitable for that particular problem. To solve such optimization problems, it would be desirable to know the structure of the class of all possible t-norms. Toward this -- probably unreachable -- goal there are many interesting open problems. If the corresponding mathematical problems are expressed in terms of quantifiers and logical connectives, then we get formulas which are very similar to formulas about real numbers. A. Tarski has proved that there is a deciding algorithm -- i.e., an algorithm that, given a formula for real numbers, decides whether it is true or not -- for real numbers. So, the natural question is whether we can extend Tarski's algorithm to a class of mathematical statements about t-norms? The answer is "no": once we allow quantifiers over t-norms, no deciding algorithm exists. In this sense, in general, the analysis of the mathematical properties of t-norms is logically non-trivial.