Computational Intelligence

The intention of this article is to show how fiizzy set theory fits into classical topology. The basic concepts are filter and ideal bases and morphisms on these. Filter bases are used to define abstract distances. Then compositions and comparisons of filter and ideal bases are considered and uniform neighborhood measures between such bases are introduced. On filter bases homomorphisms and antimorphisms can be defmed as set fimctions, in particular, they can be given by the set extension of point functions. With these concepts, fuzzy set theory can be expressed in terms of topology. For some applications we consider contractive mappings, roundings, hierarchical filter bases ("pyramids"), adaptable networks. 1 Naive fuzzy set theory Classical general set theory started 1874 with the fundamental work of G. Cantor [1], was later on in many ways axiomized (e.g. the axiom systems of Russell, Zermelo-Fraenkel, von Neumarm-Bemays-Godel), and became the basis of several mathematical disciplines like algebra, topology, measure theory. In his approach to meastue theory, H. Lebesgue [2] introduced 1904 the "characteristic function" % : X -> {0, 1} c R+ of a subset X of a set S by %(s) = 1 if seX, and x(s) = 0 if S6S\X, and defined the measm-e of X by the integral J x(s)ds if existing. s In 1965 L. A. Zadeh [3] generalized the characteristic fimction x to any function f: S ^C c [0, 1] c R with 1 e C, 1 e f(S) (i.e. f is normalized) and considered this to be an extension of the set concept, named "fuzzy set", with the intuitive meaning that f(s), the "membership" function, expresses a weight or a measure of the membership of s in the fuzzy set F = (s, f(s))sgsThen numerous definitions for "union" u , "intersection" r\, "complement" C of fiizzy sets were invented as generalizations of classical set operations, preserving at least part of their classical properties. Let G = (s, g(s))sgs be another fuzzy set. Then for example FuG =def (s, max(f(s), g(s))), FoG =def (s, min(f(s), g(s))), CF = (s, (I-f(s)))ses, where in general FuCF ^ (s, l)sgs , FoCF ^ (s, 0)sgs • -1 For any normalized f and a e C, C[a] =def {c | ceC A c > a} , S[a] =def f (€[«]) E S denotes an "a-cut" of the fuzzy set. We notice the following structural properties: S[oc] and C[a] are in correspondence, for a -> 1 S[a] and C[a] are monotonously decreasing with non-empty limits. In case C = [0, 1], S = [a, b] c R, a < b, and for f integrable over S, to S[a] and to Qa] are assigned |(f(x)-a)dx and JSjyjdy S[a] C[a] respectively, both values are equal. Fuzzy set theory and generalizations of it have been introduced to express uncertainty, vagueness, approximating properties of objects in engineering science. On the other hand, topology is the earlier and well developed mathematical discipline dealing with neighborhoods and approximations. We are going to show the relationship between both. 2 Some set theoretical and topological concepts 2.1 Filters and ideals If ind: I -> S is an indexing of elements of S, we use the notation ind(i) = s(i) = S[i], s, = (i, S[i]). A, V denote the universal and the existential quantifier. Let there be given a lattice (Ŝ , <, D, U), Fl, U lattice meet and join, o the zero and e the unit element if in 3', and a non-empty subset ^ = {B^] \ keK} c: 3, the indexing bijective, with the following properties: AkeK (B[k] ^ o) A Ak',k"eK (Vk"'e K ((Bp, ] < B[k]) A (B[k , < B[k ]))). Then ^ is a "filter base" on 3. If in addition AkeKAL<e (B^ < L => L e ̂ ) then ^ is a "filter". We define l im^ = H ^ . The dual notions to filter base and filter are "ideal base" and "ideal". Filter bases were introduced by L. Vietoris in 1921 [4]. A filter base can also be an ideal base (see Fig. 1). If S is a non-empty set, then the above applies to the complete, atomic, boolean lattice (pow S, c , n, u, 0, S). For a filter base ^ = {Bptj | k e K} on pow S, B* =def lim.^ = n B[k]. For B* ^ <Z> (then we name ^ a "proper" filter base) the ksK neighborhood of any s e B, B =def U Br^i, to the elements of B* can be keK expressed by membership or non-membership of s in certain Bp ]̂: Let AseB ((K(s) =def {k | k e K A s e B[k]}) A K(S) =def K \ K(s)), ^^(s) =def {BM I k e K(s)}, ^>,(s) =def {BM I k e K(s)}. We have s e n B[k] r\ n CB[k], C the complement with respect to B. Let Kn,i„ (s) =def {k keK(s) keK(s) I k e K(s) A -nVk'6K(s) (B[k-] c B[k])}, Kn,ex(s) =d=f {k | k e K(s) A -.Vk'6K(s)(B[k] =1 B[k])}, tiien ^r^s) =def {B[k] | k e K ^ (s)}, ^..j^is) =def {B[k] I k 6 K max(s)}. General "distance" / "similarity" relations of s from /to s* e B* are tiien given by AseB (D^(s*, s) =def r̂Mnin(s) A DU(S*, S) =def ^wmax(s)). Dr̂ (s*, s) = Dn(s*, s') and D^(s*, s) = D^(s*, s") define equivalence relations s ~o s'and s ~u s". In particular, if ^ is itself a complete lattice then d^(s*, s) =def nD^(s*, s) e ^ and d^(s*, s) =def UD^j(s*, s) e ^ are functional in s and d^(s*, s) cz dn(s*, s). Dxial results hold for ^ being an ideal base. For illustration see Fig. 2.