Abstract
This work substantiates novel perspectives and tools for analysis and design of fuzzy inference systems (FIS). It is shown rigorously that the cardinality of the set F of fuzzy numbers equals ℵ1, hence a FIS can implement “in principle” ℵ2 functions, where and ℵ1 is the cardinality of the set R of real numbers; furthermore, a FIS is endowed with a capacity for local generalization. A formulation in the context of lattice theory introduces a tunable metric distance d K between fuzzy numbers. Implied advantages include: (1) an alleviation of the curse-of-dimensionality problem, regarding the number of rules, (2) a capacity to cope rigorously with heterogeneous data including (fuzzy) numbers and intervals and (3) a capacity to introduce systematically useful non-linearities. Extensive evidence from the literature appears to corroborate the proposed novel perspectives. Computational experiments demonstrate the utility of the proposed tools. A real-world industrial application is also described.
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