Comparative analysis of logic operators: A perspective of statistical testing and granular computing

• We perform a series of statistical analysis on a pairwise and parametric comparison of triangular norms.
• We extend the comparison of composition operations involving different triangular norms.
• We introduce a concept of granular t-norms and discuss an idea of granular equivalence of logic operators.

Fuzzy logic operations realized by triangular norms (t-norms and t-conorms) have been intensively studied in their theoretical setting and used in various applications. These operations are inherently numeric constructs yielding numeric results. We pose a question whether two t-norms produce distinct results. The problem is formally expressed as a certain hypothesis testing where a null hypothesis concerns equality of medians of membership grades produced by the two triangular norms. In the sequel, we extend a discussion on composition operations constructed with the aid of both t-norms and t-conorms (s–t and t–s composition operators). A comprehensive suite of experiments deals with a collection of commonly encountered triangular norms including those endowed with some parameters. The null hypothesis is tested with regard to the pairs of triangular norms as well as the same triangular norm but having two values of its parameter. We introduce a concept of granular t-norms (more specifically, t-norms operating on granular arguments) and discuss an idea of granular equivalence of logic operators.