Novel analysis and design of fuzzy inference systems based on lattice theory

This work presents a fuzzy inference system (FIS) as a look-up table for function approximation by interpolation involving fuzzy interval numbers or FINs for short. It is shown that the cardinality of the set F of FINs equals N₁, that is the cardinality of the totally ordered lattice R of real numbers. Hence a FIS can implement in principle all N₂=2^N₁>N₁ real functions, moreover a FIS is endowed with a capacity for local generalization. It follows a unification of Mamdani- with Sugeno-type FIS. Based on lattice theory novel interpretations are introduced and, in addition, a tunable metric distance d_K between FINs is shown. Several of the proposed advantages are demonstrated experimentally.