Given a lattice (L,=), its dual lattice, symbolically (L,=)? or (L,=?) = (L,=), is a lattice with the inverse order relation (=).
Published in Chapter:
Neural/Fuzzy Computing Based on Lattice Theory
Vassilis G. Kaburlasos (Technological Educational Institution of Kavala, Greece)
Copyright: © 2009
|Pages: 6
DOI: 10.4018/978-1-59904-849-9.ch181
Abstract
Computational Intelligence (CI) consists of an evolving collection of methodologies often inspired from nature (Bonissone, Chen, Goebel & Khedkar, 1999, Fogel, 1999, Pedrycz, 1998). Two popular methodologies of CI include neural networks and fuzzy systems. Lately, a unification was proposed in CI, at a “data level”, based on lattice theory (Kaburlasos, 2006). More specifically, it was shown that several types of data including vectors of (fuzzy) numbers, (fuzzy) sets, 1D/2D (real) functions, graphs/trees, (strings of) symbols, etc. are partially(lattice)-ordered. In conclusion, a unified cross-fertilization was proposed for knowledge representation and modeling based on lattice theory with emphasis on clustering, classification, and regression applications (Kaburlasos, 2006). Of particular interest in practice is the totally-ordered lattice (R,=) of real numbers, which has emerged historically from the conventional measurement process of successive comparisons. It is known that (R,=) gives rise to a hierarchy of lattices including the lattice (F,=) of fuzzy interval numbers, or FINs for short (Papadakis & Kaburlasos, 2007). This article shows extensions of two popular neural networks, i.e. fuzzy-ARTMAP (Carpenter, Grossberg, Markuzon, Reynolds & Rosen 1992) and self-organizing map (Kohonen, 1995), as well as an extension of conventional fuzzy inference systems (Mamdani & Assilian, 1975), based on FINs. Advantages of the aforementioned extensions include both a capacity to rigorously deal with nonnumeric input data and a capacity to introduce tunable nonlinearities. Rule induction is yet another advantage.