Fuzzy Inference System (FIS) Extensions Based on Lattice Theory

A Fuzzy Inference System (FIS) typically implements a function f : R → T, where the domain set R denotes the totally-ordered set of real numbers, whereas the range set T may be either T = R (i.e. FIS regressor) or T may be a set of labels (i.e. FIS classifier), etc. This work considers the complete lattice (F,1) of Type-1 Intervals’ Numbers, or INs for short, where an IN F can be interpreted as either a possibility distribution or a probability distribution. In particular, this work concerns the matching degree (or satisfaction degree, or firing degree) part of a FIS. Based on an inclusion measure function σ : F×F→ [0, 1] we extend traditional FIS design towards implementing a function f : F → T with the following advantages: (1) accommodation of granular inputs, (2) employment of sparse rules and (3) introduction of tunable (global, rather than solely local) nonlinearities as explained in the manuscript. New theorems establish that an inclusion measure σ is widely (though implicitly) used by traditional FISs typically with trivial (i.e., point) input vectors. A preliminary industrial application demonstrates the advantages of our proposed schemes. Far-reaching extensions of FISs are also discussed. Index Terms – Fuzzy inference system (FIS), fuzzy lattice reasoning (FLR), granular computing, inclusion measure, fuzzy interval, industrial dispensing, intervals’ number (IN), lattice computing (LC)