Topological Foundation of Fuzziness

Rudolf F. Albrecht, Innsbruck

Theory of Fuzziness = Rediscovering of a Part of Classical Topology

1. MOTIVATION, HISTORY AND NAÏF APPROACHES

Need for a formal treatment of the limited accuracy of physical tools and measurements, of uncertain data causing uncertain functional results, y = f(x), approximated by .

development of computer technology stimulated numerical analysis and approximation theory.

Demand on error estimates, result inclusions, uncertainty measures.

1.1 Interval Arithmetic: R.F.Moore, 1965, University of Karlsruhe, 1966

f: X Í Rn ® R, computation of f(x) is replaced by computation of f(I(x)) (set extension of f), I(x) an n-dim. interval, containing x. f(I(x)) is i. g. NO interval and then is rounded to the smallest interval containing it.

c characteristic function

valuation C = {0, 1}

1.2 Fuzzy Set Theory: L.Zadeh, 1965

c "membership" function

valuation C = [0, 1]

c piecewise constant: generalization of interval arithmetic

Zadeh: a = inf [a, 1]® s, [0, 1] is not a complementary lattice!

"fuzzy set" = function c

"union" of 2 fuzzy sets f, g: (max {f(x), g(x)})xÎX

"intersection" of 2 fuzzy sets f, g: (min {f(x), g(x)})xÎX

"complement" cf of f: (1-f(x)) xÎX

union is not "1", intersection is not "0"

we: : [a, 1]® s Î pow [a, b], is a complementary lattice

for any f: S ®onto [0, 1], the particular neighborhood system {[a, 1] | 0£a£1} to 1 is mapped by to a neighborhood system (Û fuzzy set) of (1), [a, 1] is the valuation of ([a, 1]).

Intuitively:

{[a, 1] | 0£a£1} replaced by any complete lattice V = part. ordered set with unlimited union (least upper bound Ú) and intersection (greatest lower bound Ù) as set of valuations of neighborhoods Bk Ì S to a set B0 Ì S, k Î K. The "membership function" c, which is a point function, is replaced by any homomorphism f mapping the sets Bk into V, f(Bk) = vk Î V, Bk Í Bk´ Þ vk £ vk´.

Notational convention: I "index set", S "object set", "indexing" i: I ® S, i(i) = s[i],

si = (i, s[i]), (si)iÎI a "family" or parameterized set,

family in general has NO ordering

2. THE Topological Concepts

Given a set S, the set pow S of all subsets of S. (pow S, Í, È, Ç) is a complete Boolean lattice.

Let B[0] be a set Æ ¹ B[0] Í S of points which are all equivalent with respect to the admitted accuracy. Let B = {B[k] | B[k] Í S and k Î K} be a set of upper sets of B[0] with the properties

B[k´] Î B and B[k´´] Î B follows:

it exists a B[k] Î B with B[k] Í B[k´] and B[k] Í B[k´´]

we assume: B[0] = Ç B (denoted lim B), S = È B (support of B)

Such a family of subsets is a "filter base", introduced in topology » 1928, and serves as a neighborhood system to the points of B[0].

filter base B

B = {B[0], B[1], B[2], B[3], B[4], B[5]}, lim B = B[0],

DÇ(s*,s) = {B[3], B[4]}, DÈ(s*,s) = {B[1], B[2]}, if B is a complete lattice:

generalized distances: dÇ(s*,s) = {B[3]ÇB[4]}, dÈ(s*,s) = {B[1]ÈB[2]},

Inclusion: sÎ dÇ(s*,s)\dÈ(s*,s)

If any upper set of a set of B also belongs to B then B is a "filter"

Related topology developed > 1940

Given 2 filter bases B1, B2 on the same carrier, a comparison is: if for all B Î B1 exists a C Î B2 with C Í B then B2 is "finer" than B1.

B2 is finer B1.

Given a complete lattice (V , £), then any Í - homomorphism j: B ® V , i.e. from B[k] Í B[k´] follows j(B[k]) £ j(B[k´]), is a generalization of the "membership" function and is not a "point wise" function.

Setting v[k] = j(B[k]), then {(B[k], v[k]) | k Î K} is a "valuated" filter base which is the mathematical formulation and generalization of a "fuzzy set".

independent filter bases may have the same domain (V, £) for valuation

several filter bases B[k] on the same "carrier" S may have a "uniform structure", i.e. same (V, £), distance d(B[0], B´[0]) = d(B´[0], B[0]) (and some more properties).

d(B[0], B´[0]) ¹ d(B´[0], B[0])

3. SOME APPLICATIONS

3.1. Roundings

Given: B = {B[k] | B[k] Í S and k Î K countable} and U Í S, assume, it exits a least upper bound to U in B, then r: U  is an upward rounding. Dual: downward rounding.

upward rounding r: reals ® integers

3.2. Contractions

Given f: S ® S such that the iterates S[n] = fn(S), n = 1,2,... form a filter base F . lim F is a fix set (in particular a point).

expl. S = [0,2], f = 1/2 (x + e), e = 1/2, fn = x/2n + e , lim fn([0, 2]) = [0, 1]

3.3. Relationship to Probability Theory

prop. density fct. f, integrable (= measure on X), valuation of B by

Î [0, 1], = 1, antimorphism, not reversible!

p.d.f. f ® membership fct. of a fuzzy set, measure as valuation of a filter base

NO prob. dens. fct.¬ membership fct. of a fuzzy set,

3.4. Interval- / Fuzzy Arithmetic, x Å y = z

3.5. Fuzzy Logic

(p1, p2, ....pn), pi "propositions", valuated by vi Î V, (V, £, Ù, Ú) a Boolean lattice, composite proposition p =def ((p1, v1), (p2, v2), ...(pn, vn)), wanted a valuation of p.

Defined: "logic function" l: (v1, v2, ...vn)  v Î V, homomorphism or antimorphism

Expls. V = ({0 < 1}, max, min), l = Ù, Ú, ØÙ,ØÚ.

V = ({0 < w1 < w2 < .. < 1}, max, min), multi valued logic

V = non-negativ reals È {¥} with + , ´,

l = max(v1, v2, ...vn), (v1 + v2 + ...+vn), (v1 + v2 + ...+vn)/n,

(v1 ´ v2 ´ ... ´ vn), (v1 ´ v2 ´ ... ´ vn)1/n

exact

tolerance domain

actuel

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