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
6