volume 2, number 3, pages 285–305 - mathematical … journal of c 2006 institute for scientiﬁc...

INTERNATIONAL JOURNAL OF c© 2006 Institute for ScientificINFORMATION AND SYSTEMS SCIENCES Computing and InformationVolume 2, Number 3, Pages 285–305

AFS FUZZY LOGIC SYSTEM AND ITS APPLICATIONS TOMODEL AND CONTROL

XIAODONG LIU1,2, TIANYOU CHAI1, WEI WANG2

Abstract. Many research results of AFS (Axiomatic Fuzzy Set) theory and

its applications have been published and reported since Liu proposed it in

[20] in 1995. In this paper, an over review of AFS theory is done by both

theory analysis and illustrate examples to explain the abstract notations and

theorems in order to elicit the potential applications and the further research

topics. Many well-known datasets are applied to test the application algorithms

and the results show that AFS fuzzy logic system offers a far more flexible and

powerful framework for representing human knowledge and studying the large-

scale intelligence systems in real world applications.

Key Words. AFS structures, AFS algebras, AFS fuzzy logic, clustering, pat-

tern recognitions, hitch diagnoses, fuzzy decision trees, fuzzy identifications.

1. Introduction

The notion of a fuzzy set has been introduced by Zadeh [66] in order to formalizethe concept of gradedness in membership degree, in connection with the represen-tation of human knowledge. However, a fuzzy set is a rather abstract notion. Fuzzysets are useful for many purposes and membership functions do not mean the samething at the operational level in each and every context. As a consequence, when wescan the fuzzy set literatures, there is no uniformity in the interpretation of what amembership grade means. This situation has caused many a critique by fuzzy setopponents, and also many a misunderstanding within the field itself. Most nega-tive statements expressed in the literature turn around the question of interpretingand eliciting membership grades. Didier Dubois and Henri Prade pointed in [8]:“However, beyond the success of fuzzy logic in engineering problems, it seems thatthe condition for an improved recognition of fuzzy sets by the scientific communityis that the various semantics of fuzzy sets be articulated in a clear way. It seemsto be a crucial step in order to start considering the basic question of a measure-ment theory for membership function, a topic which only very few fuzzy researchscholars have considered.” In other fuzzy theories, the membership functions of thefuzzy sets, which are often given manually by human intuition. And the fuzzy logicoperators, which are implemented by t-norms, t-conorms and negation operatorschosen from infinite kinds of options in advance, are independent of the distributionof the original data. The large-scale intelligence systems in real-world applicationsare usually very complex, containing a large number of concepts. Therefore, it is

Received by the editors January 1, 2004 and, in revised form, April 22, 2004.

2000 Mathematics Subject Classification. 93A30.This work is supported in parts by the National Science Foundation in China under Grant

60174014, 60575039, 60534010 and by the National Key Basic Research Development Program ofChina under Grant 2002CB312200. The author would also like to acknowledge contributions of

all the collaborators who got involved in research works in this article.

285

286 X.D. LIU, T. CHAI AND W. WANG

impossible or difficult to define the membership functions and to choose suitablelogic operators from infinite kinds of options. In addition, different logic operatorchoices may also lead to different results for the same data-set.

In order to deal with the above discussed problems, AFS (Axiomatic FuzzySet) theory was firstly proposed in [20, 22] in 1995. In [18, 21], the mathematicalproperties of AFS algebras and AFS structures have been extensively investigatedand discussed, and the fuzzy theory based on AFS algebras and AFS structures hasbeen initially established. Then the topological structures on the AFS algebras andAFS structures were obtained in [19], and the preliminary combinatorical propertiesof AFS structures have been discussed in [27]. In essence, the AFS frameworkprovides an effective tool to convert the information in the training examples anddatabases into the membership functions and their fuzzy logic operations, and themembership functions and their logic operations are impersonally and automaticallydetermined by the consistent algorithms based on the distribution of the originaldata. AFS theory is based on AFS structures, a special kind of combinatoricsobjects [11], and AFS algebra, a family of completely distributive lattices. Recently,AFS theory has been developed further and applied to fuzzy clustering analysis[26], fuzzy classifiers designs [42], pattern recognition and hitch diagnoses [27, 33,62], fuzzy cognitive maps [28], concept representations [29, 39, 63], fuzzy decisiontrees [30], fuzzy identification of systems [31], credit rating analysis [32], fuzzyinference rule extraction and attribute reduction [34, 40, 42, 61, 67] and the furtheralgebra properties of AFS algebras have been obtained in [68, 69]. These theorystudies and their applications show that AFS framework is a new approach toknowledge representations and inference that is essential to any intelligence systems,and AFS theory offers a far more flexible and powerful framework for representinghuman knowledge and studying the large-scale intelligence systems in real worldapplications.

2. Survey of AFS Theory

For over one decade, we have been systematically researching into the mem-bership functions of fuzzy concepts and their fuzzy logic operations. The theorydeveloped not only has been deeply involved many mathematical theories such astopology, algebra, combintorics, measure,...,etc, but also has been applied to somereal problems ranging from pattern recognition to identification of systems.

2.1. AFS algebra. In [18, 20, 21, 22], the author has defined a family of com-pletely distributive lattices, the AFS algebras, and applied the AFS algebras tostudy the lattice value representations of fuzzy concepts.

Definition 1. ([18]) Let X1, ..., Xn,M be n + 1 non-empty sets. Then the setEX1...XnM∗ is defined as

EX1...XnM∗ = {∑

i∈I(u1i...uniAi)|Ai ∈ 2M , uri ∈ 2Xr ,r = 1, 2, ..., n, i ∈ I, I is a non-empty indexing set}.

In the case n = 0,

EM∗ = {∑

i∈I Ai|Ai ∈ 2M , i ∈ I, I is a non-empty indexing set}.Where

∑i∈I is just a symbol meaning that element

∑i∈I(u1i...uniAi) is com-

posed of terms (u1i...uni)Ai, uri ⊆ Xr, Ai ⊆ M, r = 1, 2, ..., n, i ∈ I separatedby “+” .

∑i∈I(u1i...uniAi) and

∑i∈I(u1p(i)...unp(i)Ap(i)) are the same elements of

EX1...XnM∗ if p is a bijection from I to I.

AFS FUZZY LOGIC SYSTEM AND ITS APPLICATIONS TO MODEL AND CONTROL 287

Definition 2. ([18]) Let X1, ..., Xn,M be n + 1 non-empty sets. A binary relationR on EX1...XnM∗ is defined as follows: ∀

∑i∈I(u1i...uniAi) ,

∑j∈J(v1j ...vnjBj)

∈ EX1...XnM∗,[∑i∈I(u1i...uniAi)

]R

[∑j∈J(v1j ...vnjBj)

]⇐⇒

(i) ∀(u1i...uni)Ai (i ∈ I), ∃(v1h...vnh)Bh (h ∈ J) such that Ai ⊇ Bh, uri ⊆ vrh,1 ≤ r ≤ n;

(ii) ∀(v1j ...vnj)Bj (j ∈ J), ∃ (u1k...unk)Ak (k ∈ I) such that Bj ⊇ Ak, vrj ⊆ urk,1 ≤ r ≤ n.

It is obvious that R is an equivalence relation. Thus EX1...XnM∗/R, the quo-tient set is well-defined and denoted as EX1...XnM . By

∑i∈I(u1i...uniAi) =∑

j∈J(v1j ...vnjBj), we mean that∑

i∈I(u1i...uniAi) and∑

j∈J(v1j ...vnjBj) areequivalent under the equivalence relation R.

Theorem 1. ([18]) Let X1, ..., Xn,M be n+1 non-empty sets. Then (EX1...XnM ,∨, ∧) forms a completely distributive lattice under the binary operations ∨,∧ definedas follows: ∀

∑i∈I(u1i...uniAi),

∑j∈J(v1j ...vnjBj) ∈ EX1...XnM,

(1)∑i∈I

(u1i...uniAi) ∨∑j∈J

(v1j ...vnjBj) =∑

k∈ItJ

(w1k...wnkCk),

(2)∑i∈I

(u1i...uniAi) ∧∑j∈J

(v1j ...vnjBj) =∑

i∈I,j∈J

[(u1i ∩ v1j ...uni ∩ vnj)(Ai ∪Bj)] ,

where ∀k ∈ I t J (disjoin union of indexing sets I and J), Ck = Ak, wrk = urk

if k ∈ I and Ck = Bk, wrk = vrk if k ∈ J, r = 1, 2, ..., n. (EX1...XnM,∨,∧) iscalled the EIn+1 (expanding n + 1 sets X1, ...Xn,M) algebra over X1, ...Xn andM . X1...Xn∅ and ∅...∅M are the maximum and minimum element of EX1...XnMrespectively.

We first explain EI algebra using the semantic signification represented by theelements in EM . For the sake of simplicity, let M be a set of some crisp attributeson X.

Example 1. Let X = {x1, x2, ..., x5} be a set of 5 persons, M = {m1,m2, ...,m8},where m1 = Chinese, m2 = American, m3 = Japanese, m4 = male, m5 =female, m6 = professor, m7 = engineer, m8 = lawyar, i.e., each attribute mi isa crisp concept. Suppose there exists the following table:

Table1 —Crisp attribute descriptionsChinese American Japanese male female prof. eng. law.

x1 yes no no yes no no yes yesx2 no yes no no yes yes no nox3 no yes no yes no no yes nox4 yes no no yes no yes no nox5 no no yes yes no yes yes no

Many new attributes can be generated by Boolean conjunction and disjunction ofthe attributes in M . For example α = (m1 ∧m4)∨ (m2 ∧m5 ∧m6) with a semanticsignification gives: “Chinese males or American female professors”; β = (m1 ∧m4) ∨ (m2 ∧ m5 ∧ m6) ∨ (m1 ∧ m4 ∧ m8) with a semantic signification impliesthat: “Chinese males or American female professors or Chinese male lawyers”;ν = (m5 ∧m6) ∨ (m5 ∧m8) with a semantic signification then indicates: “female


professors or female lawyers”. The new attributes α, β and ν can be represented bythe elements in EM as α = {m1, m4}+ {m2, m5, m6}, β = {m1, m4}+ {m2, m5,m6}+ {m1, m4, m8}, and ν = {m5, m6}+ {m5, m8}.

In general, M is a set of fuzzy or crisp attributes, for∑

i∈I Ai ∈ EM , Ai ex-presses a new attribute generated by the “and” of all attributes in Ai, i ∈ I and∑

i∈I Ai expresses a new attribute generated by the “or” of all attributes expressedby Ai, i ∈ I. Even if there are fuzzy attributes in M,

∑i∈I Ai has definite semantic

signification like crisp attributes α, β and ν we discussed above. Representing at-tributes generated by the attributes in M in the form

∑i∈I Ai ∈ EM can not only

preserve the semantic signification, but also avoid to define the fuzzy logic opera-tors “and”, “or” in advance. We can apply EI algebra EM to study the intrinsicof fuzzy concepts and their logic operations. We continue to study Example 1. ByDefinition 2, we can verify that

β = {m1,m4}+ {m2,m5,m6}+ {m1,m4,m8} = {m1,m4}+ {m2,m5,m6} = α.

From Table 1, we can verify that for x ∈ X, x satisfies α if and only if it satisfiesβ. In other words, term {m1,m4,m8} in β is redundant. In general, M is a set offuzzy or crisp concepts, for α, β ∈ EM . If α = β, then the ordinary fuzzy sets (i.e.,the membership degrees in the interval [0, 1]) or L-fuzzy sets (i.e., the membershipdegrees in a lattice L) representing fuzzy concepts α and β should be identical. Inthe following, we call each element in EM a fuzzy or crisp concept. The semanticsignifications of concepts “α or ν” and “α and ν” can be simply denoted as α ∨ νand α ∧ ν respectively. with (1), (2), we have

α ∨ ν = {m1,m4}+ {m2,m5,m6}+ {m5,m6}+ {m5,m8}= {m5,m6}+ {m5,m8}+ {m1,m4}.

α ∧ ν = {m1,m4,m5,m6}+ {m2,m5,m6}+ {m1,m4,m5,m8}+{m2,m5,m6,m8}

= {m1,m4,m5,m6}+ {m2,m5,m6}+ {m1,m4,m5,m8}.In general, for any fuzzy concepts α, β ∈ EM, no matter how to define their

membership functions and fuzzy logic operators, fuzzy concepts α∨β and α∧β arethe fuzzy logic “or”, “and” of α, β, respectively. α ≥ β in lattice EM implies thatno matter how to define the membership functions for α, β, the membership degreeof any x belonging to α should be larger than or equal to that of x belonging to β.In [18], the author has proved that EI algebra has a more general algebra structurethan Boolean algebra. For M a set of few fuzzy or crisp concepts, a great numberof fuzzy concepts can be expressed by the elements in EM and the fuzzy logicoperations of them can be implemented by the operations ∨,∧ of the completelydistributive lattice (EM,∨,∧). If attribute set M has n elements, then there aremore than

∑ni=1(2

Cin − 1) elements in EM with Cr

n = n!(n−r)!r! and each element

has a definite semantic signification and can be comprehend. The complexity ofhuman concepts is directly resulted from the logic combinations of a few concepts.As long as we can determine the fuzzy logic operations of the few attributes in M,the fuzzy logic operations of all concepts in EM can also be determined. Therefore,not only the accuracy of membership functions and their fuzzy logic operations canbe improved compared with the fuzzy logic defined by the t-norm and negativeoperators, but also the complexity of determining membership functions and theirlogic operations for the complex fuzzy concepts will be alleviated. It is obviousthat the simpler the concepts are in M , the more accurate and convenient for usto determine the membership degrees and the fuzzy logic operations for the fuzzyconcepts in EM.


In [68, 69], the expressions of both ∗-irreducible elements (or ∧-irreducible el-ements) in molecular lattice (EM,≤) and (EX1...XnM,≤) are developed, whichare

{∑x∈A

{x}|A ∈ 2M}

{∑x∈A

u1(x)...un(x){x}|A ∈ 2M , ur(x) ∈ 2Xr , r = 1, 2, ..., n}

The standard minimal family of∑

i∈I Ai ∈ (EM,≤) is{∪i∈ISi} ∪ {Ai|i ∈ I},

where Pi = {B ∈ 2M |B ⊇ Ai}, Si ⊆ Pi (i ∈ I). The standard minimal family of∑i∈I(u1i...uniAi) ∈ (EX1...XnM,≤) is

{∪i∈ISi} ∪ {(u1i...uniAi)|i ∈ I},where Pi = {(v1...vnB)|B ⊇ Ai, vr ⊆ uri, r = 1, 2, ..., n}, Si ⊆ Pi (i ∈ I). It is alsoproved that neither molecular lattice (EM,≤) nor molecular lattice (EX1...XnM,≤) is a fuzzy lattice, but (SEM,≤) and (SEX1...XnM,≤) as sub algebras of EMand EX1...XnM are a fuzzy lattice, where

SEM = {∑i∈I

Ai|Ai ∈ 2M − {∅}, i ∈ I, I is any indexing set}.

SEX1...XnM = {∑i∈I

u1i...uniAi|Ai ∈ 2M − {∅},

i ∈ I, uri ∈ 2Xr , r = 1, 2, ..., n, I is any indexing set}.

Remark 1. : Although a conversely ordered mapping on (SEX1...XnM,≤) hasbe established in [68], a conversely ordered mapping on (SEX1...XnM,≤), n ≥ 1,which can be applied to the AFS algebra representations of fuzzy concepts, has notbe found. This is very interested and useful problem to AFS theory.

2.2. AFS structures. In our opinion, ordinary fuzzy sets, L-fuzzy sets or crispsubsets of some universe of discourse X are all representing forms of fuzzy conceptsor crisp concepts. We regard ordinary fuzzy sets or L-fuzzy sets as different rep-resenting forms of fuzzy concepts. By saying fuzzy sets of X we mean all kinds ofrepresenting forms for fuzzy concepts. The fuzzy sets and crisp subsets on X canbe explained as the following:

For a fuzzy set ζ on universe of discourse X, any x ∈ X, either x belongsto ζ at some degree or does not belong to ζ at all. While for a crisp subsetA of X, any x ∈ X, either x belongs to A or does not belong to A at all.

Based on this statement, both a fuzzy set and a crisp subset on X can berepresented by a binary relation R on X through comparing degrees of each pairof x, y belonging to the concept without a need to define their membership degreesin advance.

Definition 3. Let ζ be any concept on the universe of discourse X. Rζ is called abinary relation (i.e., Rζ ⊂ X ×X) of ζ if Rζ satisfies: x, y ∈ X, (x, y) ∈ Rζ ⇔ xbelongs to ζ at some degree and the degree of x belonging to ζ is larger than orequals to that of y, or x belongs to ζ at some degree and y does not at all.

In real world applications, the degrees of a pair x and y belonging to conceptζ can be obtained through human intuitions without representing the degrees by[0, 1] or lattice in advance. Hence we can use the “degrees of belonging” insteadof membership degrees in Definition 3. We should notice that (x, x) ∈ Rζ implies


that x belongs to ζ at some degree and that (x, x) /∈ Rζ implies that x does notbelong to ζ at all.

Definition 4. ([16]) Let X be a set and R be a binary relation on X. R is calleda preference relation on X if it satisfies the following conditions:

D4-1. ∀x ∈ X, (x, x) ∈ R.D4-2. If (x, y) ∈ R, (y, z) ∈ R, then (x, z) ∈ R, x, y, z ∈ X.D4-3. For any x, y ∈ X, either (x, y) ∈ R or (y, x) ∈ R.

In [16], Kim K. H. mentioned that preference relations can be applied to studyhuman concepts. By Definition 3, we know in some cases, for concept ζ, thereexists x ∈ X such that (x, x) /∈ Rζ (i.e., x does not belong to ζ at all). Althoughpreference relations are very simple and have very good mathematical properties,D4-1 is too strict to represent ordinary concepts.

Definition 5. ([26]) Let X be a set and R be a binary relation on X. R is calleda sub-preference relation on X if for x, y, z ∈ X, x 6= y, R satisfies the followingconditions:

D5-1. If (x, y) ∈ R, then (x, x) ∈ R;D5-2. If (x, x) ∈ R and (y, y) /∈ R, then (x, y) ∈ R;D5-3. If (x, y), (y, z) ∈ R, then (x, z) ∈ R;D5-4. If (x, x) ∈ R and (y, y) ∈ R, then either (x, y) ∈ R or (y, x) ∈ R.

In addition, ζ is called a simple concept or simple attribute on X if Rζ is a sub-preference relation on X. Otherwise ζ is called a complex concept or a complexattribute on X.

It is obvious that a preference relation is a sub-preference relation and any crispattribute is a simple concept. For example, let X be a set of persons and cars.x, y ∈ X, x is a person and y is a car. If we consider concept “beautifull”,then the degrees of x, y belonging to “beautifull” are incomparable although bothx and y belong to “beautifull” at some degree, i.e., (x, x), (y, y) ∈ Rbeautifull,(x, y) /∈ Rbeautifull. This implies that D5-4 is not satisfied and “beautifull” is acomplex concept on X. Let ζ be a simple concept on X,

Tζ = {x ∈ X|(x, y) ∈ Rζ ,∀y ∈ X}(3)Fζ = {x ∈ X|(x, x) /∈ Rζ},Mζ = X − Tζ − Fζ .

For each element in Tζ , its degree belonging to concept ζ is 1 or maximum elementof lattice L if concept ζ is represented by an ordinary fuzzy set or a L-fuzzy set.For each element in Fζ , its degree belonging to concept ζ is 0 or minimum elementof lattice L if concept ζ is represented by an ordinary fuzzy set or a L-fuzzy set.The elements in Mζ belong to concept ζ at different degrees in the open interval(0,1) if concept ζ is represented by an ordinary fuzzy set. The degrees of elementsin Mζ belonging to concept ζ form a linear ordered chain in lattice L if concept ζis represented by a L-fuzzy set ( D5-4). Concept ζ is a crisp concept if Mζ = ∅.

Definition 6. ([29]) Let X = {x1, x2, ..., xn} and Rζ be the binary relation ofconcept ζ on X. Boolean matrix Bζ = (bij) is called the Boolean matrix of conceptζ if bij = 1 ⇔ (xi, xj) ∈ Rζ . Concept ζ is called an atomic fuzzy concept if thereexist a permutation Boolean matrix P such that

PBζP′ =

J11 J12

O21 J22

O31 O32

,


where J11, J12, J22, O21, O31, O32 are sub-block with appropriate orders and each el-ement in J11, J12, J22 is 1 and each element in O21, O22, O32 is 0.

From Definition 6, we can easily verify that each atomic fuzzy concept is a simpleconcept. Similarly, X is also divided into three classes Tζ , Fζ ,Mζ (refer to (3)) byan atomic fuzzy concept ζ. Each element in Tζ , Fζ is the same as in the situationof ordinary simple concepts. Compared with ordinary simple concept, the degreesof all elements in Mζ belonging to atomic fuzzy concept ζ are equal to one value in(0,1) or one element in lattice L. While for ordinary simple concept γ, the degrees ofelements in Mγ belonging to γ are numbers in (0,1) or form a linear ordered chainof lattice L. Using the results of [24, 29], we can prove that for a simple concept γ,if Bγ > Bζ , where ζ is an atomic fuzzy concept, then concept γ must be a crispconcept. This implies that the atomic fuzzy concepts are the simplest among thesimple concepts except crisp one.

Theorem 2. ([23]) Let M be a set. ∀∑

i∈I Ai ∈ EM, if the operator “′” is definedas follows

(4) (∑i∈I

Ai)′ = ∧i∈I(∨a∈Ai{a′}),

then “′” is an order-reversing involution on EI algebra EM .

If a′ is the negation of simple concept a ∈ M, then “′” is a negation operator,i.e., (

∑i∈I Ai)′ is the negation of fuzzy concept

∑i∈I Ai and (EM,∨, ∧,′ ) is called

an AFS logic System. The algorithm of obtaining a′ can be found in [23].

Definition 7. ([18, 21]) Let X, M be two sets and 2M be the power set of M,τ : X ×X → 2M . (M, τ,X) is called an AFS structure if τ satisfies the followingconditions:

AX1: ∀(x1, x2) ∈ X ×X, τ(x1, x2) ⊆ τ(x1, x1);AX2: ∀(x1, x2), (x2, x3) ∈ X ×X, τ(x1, x2) ∩ τ(x2, x3) ⊆ τ(x1, x3).

In addition, X is called universe of discourse, M is called an attribute set and τ iscalled a structure.

In the real world applications, for any x, y ∈ X, τ(x, y) are subsets of M ,

τ(x, y) = {m|m ∈ M, (x, y) ∈ Rm},

That is, for any m ∈ τ(x, x), x belongs to attribute m in some degree and for anym ∈ τ(x, y), the degree of x belonging to m is larger than or equal to that of y.The AX1 implies that for each concept m ∈ M, if the degree of x belonging tom is larger than or equal to that of y, x must belong to m at some degree (e.g.,µm(x) > 0 or the lattice degree of x belonging to m is not the minimum element oflattice L). AX2 implies that for each concept m ∈ M , the degrees of the elementsin X belonging to m satisfy transitive law.

In the following, we always suppose that each concept in M is simple concepton X. We can verify that (M, τ,X) is an AFS structure if τ is defined as

τ(xi, xj) = {m|m ∈ M, (xi, xj) ∈ Rm}, xi, xj ∈ X.

For an AFS structure (M, τ,X), if we define fτ (x, y) = τ(x, y) ∩ τ(y, y), ∀(x, y) ∈X ×X, then (M , fτ , X ×X)[27] is a system which is a main mathematical objectof combinatorics [11]. The following Example 2 demonstrates how to establish anAFS structure according to original data and facts.


Example 2. Let X = {x1, x2, ..., x10} be a set of 10 persons. M = {m1,m2, ...,m10},where m1 = age, m2 = height high, m3 = weigh, m4 = salary high, m5 = morefortune, m6 = male, m7 = female, m8 = hair black, m9 = hair white,m10 = hair yellow. About the universe of discourse X and the attribute set M ,we have the following Table 2 and sub-preference relations expressed by the chains.x = y in the chain implies the degrees of x and y belonging to the attribute areequal, instead of x and y being the same element in X.

hair black m8: x7 > x10 > x4 = x8 > x2 = x9 > x5 > x6 = x3 = x1;hair white m9: x6 = x3 = x1 > x5 > x2 = x9 > x4 = x8 > x10 > x7;hair yellow m10: x2 = x9 > x4 = x8 = x5 > x10 > x6 = x3 = x1 = x7.

Table2−Descriptions of Attributesage height weigh salary fortune male femalem1 m2 m3 m4 m5 m6 m7

x1 20 1.9 90 1 0 y nx2 13 1.2 32 0 0 n yx3 50 1.7 67 140 34 n yx4 80 1.8 73 20 80 y nx5 34 1.4 54 15 2 y nx6 37 1.6 80 80 28 n yx7 45 1.7 78 268 90 y nx8 70 1.65 70 30 45 y nx9 60 1.82 83 25 98 n yx10 3 1.1 21 0 0 n y

From Table 2 and the sub-preference relations, we can verify that each conceptm ∈ M is a simple concepts and the structure τ can be defined as follows: For anyxi, xj ∈ X, i 6= j,

τ(xi, xi) = {m|m ∈ M,xi possesses attribute m at some degree },τ(xi, xj) = {m|m ∈ M,xi ≥m xj},

where xi ≥m xj implies that xi possesses attribute m in some degree and the degreeof xi belonging to m is larger than or equal to that of xj. For example

τ(x1, x1) = {m1,m2,m3,m4,m6,m8,m9,m10}.τ(x4, x7) = {m1,m2,m6,m9,m10}.

(M, τ,X) is the mathematical abstraction of the complicated relationships amongthe ten persons considering the attributes in M . This implies that the informationcontained in Table 2 and the sub-preference relations are represented by (M, τ,X).

In [27], the following basic combinatorical properties of the AFS structures areobtained: For any AFS structure (M, τ,X), there exist Vk ⊆ M , Uk ⊆ X, k ∈ Ksuch that

(5) (M, τ,X) = ⊕k∈K(Vk, τUk×Uk, Uk),

where Vk 6= ∅, Uk 6= ∅, Vq ∩ Vl 6= ∅, Uq ∩ Ul 6= ∅, q 6= l, q, l ∈ K, system(Vk, fτUk×Uk

, Uk × Uk) is the maximal connected component of system (M,fτ , X)(refer to [11]). In practice, the complexity of real world application systems can begreatly reduced by the formula (5). In [33], an algorithm of decomposing an AFSstructures into the form as (5) is proposed and is applied to hitch diagnoses of themarine diesel engines.


Remark 2. : We should notice that any raw data in real world applications areindeed the discrete mathematical objects. Thus the applications of combinatoricstheory, which is a power tools to solve the discrete mathematical problems, to in-vestigate the AFS structures as an abstraction of the raw data will obtain manyinterested combinatorical interpretations of the distribution of the raw data and themathematical models contained in the raw data can be found, presented and analyzedby the modern combinatorics.

Theorem 3. ([24, 29]) Let X = {x1, x2, ..., xn} be a set and M be the set of allatomic fuzzy concepts on X and (M, τ,X) be an AFS structure in which ∀x, y ∈X, τ(x, y) = {m ∈ M |(x, y) ∈ Rm}.Then for any concept γ on X, there exists afuzzy concept

∑i∈I Ai ∈ EM, such that Bγ = B∑

i∈I Ai,where B∑

i∈I Ai= (bij),

bij = 1 ⇔ ∃k ∈ I, τ(xi, xj) ⊇ Ak.

Theorem 3 implies that for any concept γ on X, there exists a fuzzy concept∑i∈I Ai ∈ EM such that their binary relations are equal, in other words, fuzzy

concept∑

i∈I Ai is a representation of concept γ.In real world applications, complex fuzzy concepts are compounded by some

simple concepts. In Example 2 credit, unlike hair black, hair white, hair yellow,..., etc., can not be described by a sub-preference relation, hence it is a compoundof other simple concepts such as male, female, salary high, more fortune, age,...,etc. We suppose

credit: “salary high age males or more fortune age males or salary high femalesor more fortune females”

= {m6, m4, m1} +{m6, m5, m1}+ {m7, m4}+ {m7, m5} ∈ EMAbout complex concept credit, although different persons may have different as-sumptions, there exists a definite element in EM which expresses each assumptionof credit. In general, Theorem 3 ensures that any concept on X can be representedby an element in EM for an appropriately selected simple concept set M.

Remark 3. : We should notice that by Theorem 3 and [63], we know that the setof elements in EM, which can represent a given concept on X, indeed is a sub EIalgebra of EM . It may obtain interested results, if the relationships between thesub EI algebras and the concepts are investigated further. This approach can findthe algebra characteristics of each concept. In [24], the author has developed somebasic tools for further discussion of this problem.

In the following, we recall how to determine the ordinary fuzzy sets or L-fuzzysets for the fuzzy concepts in EM based on the AFS structure (M, τ,X) and theAFS algebras.

2.3. AFS algebra represented fuzzy sets. In [25], the authors give an exhaus-tive study and discussion for various kinds of AFS algebra representations of fuzzyconcepts. In the following, we recall a kind of L-fuzzy sets proposed in [18] whosemembership degrees are represented by the elements in lattices EXM .

Theorem 4. ([18]) Let (M, τ,X) be an AFS structure. For u ⊆ X, A ⊆ M , wedefine the symbol

(6) A(u) = {y | y ∈ X, τ(x, y) ⊇ A for any x ∈ u}.For any given x ∈ X, if we define a map φx : EM → EXM as follows: For any∑

i∈I Ai ∈ EM,

φx(∑i∈I

Ai) =∑i∈I

Ai({x})Ai ∈ EXM,


then φx is a homomorphism from the lattice (EM,∨,∧) to the lattice (EXM,∨,∧).

For A ⊆ M, x ∈ X, A({x}) = {y|y ∈ X, τ(x, y) ⊇ A},which is the subset of X,and any y ∈ A({x}), the degree of x belonging to attribute A is larger than or equalto that of y since τ(x, y) ⊇ A. By Theorem 4, we know that for any given concept∑

i∈I Ai ∈ EM, we get a map∑

i∈I Ai : X → EXM , ∀x ∈ X,

(∑i∈I

Ai)(x) =∑i∈I

Ai({x})Ai ∈ EXM.

Since (EXM,∨,∧) is a lattice, then map∑

i∈I Ai is a L-fuzzy set (with membershipdegrees in lattice EXM ). Thus

∑i∈I Ai is a L-fuzzy set on X and the membership

degree of x (x ∈ X) belonging to fuzzy set∑

i∈I Ai is∑

i∈I Ai({x})Ai ∈ EXM . If∑i∈I Ai({x})Ai ≥

∑i∈I Ai({y})Ai in lattice EXM , then the degree of x belonging

to the fuzzy set∑

i∈I Ai is larger than or equal to that of y. For fuzzy sets α =∑i∈I Ai, β =

∑j∈J Bj ∈ EM, the fuzzy sets α ∨ β and α ∧ β are logic “or” and

“and” of the L-fuzzy sets α and β respectively. “′” is the negation of the fuzzyconcepts in EM . Thus, (EM,∨,∧,′ ) is a fuzzy logic system. In [18], the authorhas proved that if each attribute in M is a crisp attribute then (EM,∨,∧,′ ) isreduced to a Boolean logic system.

In [18, 21, 25], the representations of fuzzy concepts based on raw data have beeninvestigated within the framework of AFS (Axiomatic Fuzzy Set) theory, includingEI2, EI3 algebra representations. In [25], first, the E#I algebra, which is also acompletely distributive lattice and a new family of the AFS algebras, is proposed.Secondly, two kinds of E#I algebra representations of fuzzy concepts are derivedin detail. In order to represent the membership functions of fuzzy concepts in theinterval [0,1], the norms of the AFS algebras are defined and studied. Finally, therelationships of various representations with their advantages and drawbacks areanalyzed.

In [19], using topological molecular theory[58], the topological structures on Xinduced by the topological molecular lattices generated by some fuzzy sets in EMhave been obtained. This kind topology on X is determined by the original dataand the chosen fuzzy sets in EM and is an abstract geometry relations among theobjects in X. What are the interpretations of the special topological structureson the AFS structures directly obtained by a given dataset through the differentialdegrees between objects in X and the fuzzy similarity relations on X determined bythe topology? With the topological space on X induced by the fuzzy concepts, thepattern recognition problems of ordinary datasets like Example 2 can be studied.We know that human can classify, cluster and recognize the objects in the ordinarydataset X without any metric in Euclidean space. What is human recognition basedon if X is not a subset of some metric space in Euclidean space? For example, ifyou want to classify all your friends into two classes {close friends} and {commonfriends}. The criteria/metric you are using in the process is very important thoughit may not be based on the Euclidean metric.

Remark 4. : The important unsolved problems are :1. For what kinds of an AFS structure (M , τ , X), there exists a norm ||.||

of the AFS algebra such that for any membership function µ : X → [0, 1], ∃γ ∈EM, µ(x) = ||γ(x)||, ∀x ∈ X.

2. How can we applied the topological structures on an AFS structures induced bysome concepts in EM to establish the measurement for the norms of AFS algebrasor metric for the recognition problems?


3. The applications of the AFS fuzzy logic

In this section, by the introduction for the applications of AFS theory to fuzzyclustering analysis [25], fuzzy classifier designs [42], fuzzy cognitive maps [28], fuzzydecision trees [30], fuzzy identification of systems [31], credit rating analysis [32],the potential applications and the further research topics of AFS theory are elicited.

3.1. Fuzzy Clustering. In [26], the authors propose a fuzzy clustering algorithmbased on a new fuzzy validation index. Compared with the current fuzzy clusteringalgorithm, the new algorithm has the following advantages:

1. The attributes of objects in it can be various data types or sub-preferencerelations, even human intuition descriptions.

2. The distance function and objective function are not required, and the clusternumber or the class label need not be given beforehand.

3. Each class is described by a fuzzy set in EM , which is the AFS fuzzy logicalcompound of the simple attributes on some features with definite semantic signi-fication and determines the degree of each pattern belonging to this class. Sincethe new fuzzy clustering algorithms imitate the clustering process in which humanclusters a set of objects by some given fuzzy concepts and attributes, hence thisresearch is a new approach to knowledge representations and inference that is essen-tial to any intelligence systems. It offers a far more flexible and powerful frameworkfor representing human knowledge and studying the large-scale intelligence systemsin real world applications.

Recently, we have applied the fuzzy clustering algorithm and the new fuzzyvalidation index proposed in [26] to the well known iris-data which is the machine-learning database at University of California, Irvine via an anonymous ftp server(ftp://ftp.ics.uci.edu/pub/machine-learning-databases/). We first cluster the Iris-data just based on the order descriptions of the attributes, in stead of the numericaldescriptions of the attributes, and the clustering accuracy rate is 96.67%. Usingthe function kmeans in MATLAB toolbox for the iris-data, which is based on thewell known k-mean clustering algorithm, the clustering accuracy rate is 89.33%.And Using the function fcm in MATLAB toolbox for the iris-data, which is basedon the well known fuzzy c-mean clustering algorithm, the clustering accuracy rateis also 89.33%. Both k-mean and fuzzy c-mean algorithms can only be appliedto the datasets with numerical attributes. The examples further to show that theproposed fuzzy validation index is quite accurate to describe the clarity of theclustering results.

A new design of fuzzy classifiers based on the fuzzy descriptions of the clusterslearned by the fuzzy clustering algorithm proposed in [26] can be proposed asfollows. Let X be the set of training samples and M be the set of some fuzzyconcepts on the features. X is parted into l classes or patterns, C1, C2, ... , Cl.Let (M, τ,X) be the AFS structure of dataset X. Based on the training dataset X,learn the fuzzy descriptions ζCi , i = 1, 2,...,l, through the fuzzy clustering analysiswith the AFS structure (M, τ,X) and Λ = {{m}|m ∈ M}. Refer to the givenclasses C1, C2, ... , Cl, find appropriate parameters to optimize the accurate rate ofthe clustering results obtained by the above fuzzy clustering algorithm. Let fuzzydescriptions ζCi

, i = 1, 2,...,l, yield the best accurate rate of the clustering. Thefuzzy classifier is denoted as {ζCi , |i = 1, 2, ..., l}. For each test or new sampley /∈ X, y is classified to class or pattern Ck, if µζCk

(y) = max1≤i≤l{µζCi(y)}.


Using Iris-data, we take 10 experiments. In each experiment, 60% samples ofeach classes (i.e., total 90 samples) are randomly selected as training samples andthe other 40% samples (i.e., total 60 samples) are served as test samples.

Table1 Number of incorrectly classified test samples in the 10 experimentsNo.experiement 1 2 3 4 5 6 7 8 9 10Number errors 1 2 0 0 0 1 2 1 1 3

3.2. Fuzzy classifier desgins. In [42], the author proposed the following fuzzyclassifier design algorithm:

Let X be the set of training samples and there are n features to describe thesamples. The training sample are labelled by c classes, which are X1, X2, ..., Xc,i.e., X =

⋃1≤i≤c Xi, Xi ∩ Xj = ∅, i 6= j. M be a set of simple attributes on

X (crisp or fuzzy attributes), (M, τ,X) be the AFS structure established by theoriginal data. Λ ⊆ M, Λ is a set of simple concepts which are selected to designthe classifier.

In the following, we describe the design method:Step1:Establish a semi-cognitive field (M, τ,X, S) and give the weigh function ρm,for

each m ∈ M by the original information or data (refer to [26]).Step2:Select Λ ⊆ M,Λ is a set of fuzzy sets selected to design the classifier. (Λ)EI is

the sub EI algebra generated by Λ.Step 3:Given small positive numbers ε > 0, δ > 0,for each i = 1, 2, ..., c, find the fuzzy

set ζXi ∈ (Λ)EI , such that

(7) ζXi :∑

y∈Xi

µζXi(y) = max

ξ∈F δε

{∑

y∈Xi

µξ(y)},

where

EδΛ = {γ|γ ∈ (Λ)EI ,∀y ∈ X −Xi, µγ(y) < δ},(8)

F δε = {ξ|ξ ∈ Eδ

Λ,∀y ∈ Xi, µξ(y) ≥ µ∨b∈Λ b(y)− ε}.(9)

ζXiis the fuzzy description for class Xi, i = 1, 2, ..., c. δ is a parameter to control

the extent of fuzzy set ζXi to distinguish x ∈ Xi and y /∈ Xi. ε is a parameter tocontrol the degree of each training sample x (x ∈ Xi) belonging to ζXi .

Step 4:For each testing sample or new pattern s, we estimate the degree of s belonging

to the fuzzy set ζXi, i = 1, 2, ..., 3. Although s /∈ X, we can estimate the EII algebra

degree of s belonging to the fuzzy set ζXi by the AFS structure (M, τ,X). For eachsimple concept m ∈ M, by the weight function ρm, one can get ρm(s) ∈ R+.Foreach A ⊆ M, define

A({s}) = {x|x ∈ X,∀m ∈ A, ρm(x) ≤ ρm(s)},A({s}) = X − {x|x ∈ X,∀m ∈ A, ρm(s) < ρm(x)}.

For any fuzzy set∑

i∈I Ai ∈ EM,the EII algebra degree of s belonging to the fuzzyset

∑i∈I Ai is denoted as (

∑i∈I Ai)(s), then the lower bound and upper bound

of (∑

i∈I Ai)(s) in lattice EXM are∑

i∈I Ai({s})Ai,∑

i∈I Ai({s})Ai respectively.


If the membership degree of s belonging to the fuzzy set∑

i∈I Ai is denoted asµ∑

i∈I Ai(s) ∈ [0, 1],then

M(∑i∈I

Ai({s})Ai) ≤ µ∑i∈I Ai

(s) ≤M(∑i∈I

Ai({s})Ai).

Here, we simply let µζXi(s) = M(

∑i∈I Ai({s})Ai) be the membership degree of s

belonging to class i, i = 1, 2, ..., c, where ζXi=

∑i∈I Ai ∈ EM , (about M(.) refer

to [25, 26]). For each sample s, s is determined to be the sample in the class i ifµζXi

(s) = maxk{µζXk(s)}.The classifier is denoted as C = (ζX1 , ζX2 , ..., ζXc

), whereζXi ∈ EM , i.e., C is a vector on EI algebra EM .

We apply the desgin algorithm to the well-known Wine Recognition Data at Uni-versity of California, Irvine via an anonymous ftp server (ftp://ftp.ics.uci.edu/pub/machine-learning-databases/). We take 10 experiments. In each experiment, 60% samplesof each classes (i.e., total 106 samples) are randomly selected as training samplesand the other 40% samples (i.e., total 72 samples) are served as test samples.

Table 2 Number of misclassified samples in 10 experiencesi-th experience 1 2 3 4 5 6 7 8 9 10number of misclassification 2 4 2 4 3 5 2 5 1 1

In [32], the fuzzy classifier desgin algorithm also be applied to credit rating anal-ysis. For the Credit Data at University of California, Irvine via an anonymousftp server (ftp://ftp.ics.uci.edu/pub/machine-learning-databases/). 10 experimentswere taken. In each experiment, 60% samples of each classes are randomly selectedas training samples and the other 40% samples are served as test samples.

Table 3. Correct rate of 10 experiments

i-th experience 1 2 3 4 5 6 7 8 9 10Correct rate 66% 72% 54% 72% 60% 70% 62% 60% 66% 50%

Remark 5. The theory analysis of the above fuzzy clustering algorithms usingAFS theory, measure theory, combinatorics and topological theory is an interestedresearch topic both in view of mathematics and real world applications. The math-ematical proofs of the stability of the above fuzzy clustering algorithms remain anunsolved important theory problem and the beautiful mathematics results are expect-ing to be achieved.

For the design of high accurate fuzzy classifiers, we should study how to selectfuzzy attribute set Λ ⊆ EM to optimize the accurate rate of the proposed algorithms.

The above approaches to fuzzy clustering analysis and fuzzy classifier designsshow that AFS theory offers a far more flexible and effective means for data mining,knowledge discovery and representation and the intelligence systems in real-worldapplications. Compared with popular fuzzy algorithms, such as c-means fuzzyalgorithm and k-nearest-neighbor fuzzy algorithm, the proposed algorithms aremore simple and understandable, the data types of the attributes can be variousdata types or sub-preference relations, even descriptions of human intuition, andthe distance function and the class number need not be given in advance.


3.3. Fuzzy decision trees. Today, in the mass storage era, knowledge acqui-sition constitutes a major knowledge engineering bottleneck. There are variousapproaches aimed at the alleviation of this problem such as decision trees, induc-tive trees, rule-based architectures. The decision trees gained popularity because oftheir conceptual transparency. The well developed design methodology comes withefficient design techniques supporting their construction, cf. [48-51]. The decisiontrees generated by these methods were found useful in building knowledge-basedexpert systems. Due to the character of continuous attributes as well as variousfacets of uncertainty one has to take into consideration, there has been a visibletrend to cope with the factor of fuzziness when carrying out learning from exam-ples in the case of tree induction. In a nutshell, this trend gave rise to the nameof fuzzy decision trees and has resulted in a series of development alternatives.The incorporation of fuzzy sets into decision trees enables us to combine the un-certainty handling and approximate reasoning capabilities of the former with thecomprehensibility and ease of application of the latter. This naturally enhancesthe representative power of decision trees with the knowledge component inherentin fuzzy logic, leading to their higher robustness, noise immunity, and substantialapplicability within uncertain/imprecise contexts. Fuzzy decision trees assume thatall domain attributes or linguistic variables have pre-defined fuzzy terms for eachfuzzy attribute. Those could be determined in a data driven manner.

In fuzzy decision trees, every node except for their root comes equipped with afuzzy set that can be represented as a conjunction of several fuzzy linguistic terms,where the conjunctions are implemented by some standard operators encounteredin fuzzy logic. It becomes apparent that to a significant extent the fuzzy decisiontrees are pre-determined by the membership functions of the fuzzy terms and thefuzzy logic operators. In [30], the authors focused on the generation of fuzzy sets(membership functions) and underlying logic operators, which eliminates poten-tial subjective bias and puts this development in terms of the AFS theory whileexploiting available experimental data. The authors also show how the resultingconstructs are used in the design of the fuzzy decision tree and illustrate the pro-posed construct by showing numeric examples and comparing the results with otheravailable. The examples help us reveal and emphasize the main advantages of thetrees involving their ability to cope with the experimental data, capture its detailsat the desired level of granularity and required specificity of interpretation. Theapproaches to fuzzy decision trees in [30] show the role of the AFS framework inthe design and analysis of fuzzy decision trees and the role of the AFS in the deter-mination of fuzzy sets (membership functions) and logic operators. It was stressedthat this technique takes advantage of the available experimental data. The de-sign algorithm offers a comprehensive interpretation of the tree. In particular, theAFS formalism helped us come up with endowing the classification results withmeaningful confidence levels.

Remark 6. : The further investigations of the AFS fuzzy decision trees with thelarge dataset and the comparisons with the C4.5 (refer to [51]) should be done inthe future study.

3.4. Fuzzy cognitive maps and fuzzy identification of systems. Currently,the advanced digital technology has made the digitized data easy to capture andcheap to store. However, raw digital data is rarely of useful in practice and thecapability to extract information from raw data is extremely important for decisionsupport and understanding dynamic phenomenon of the data source [17]. Thishas prompted the need for intelligent data analysis methodologies or called soft


computing methodologies, which could discover information from raw data effec-tively [5]. Recently various soft computing methodologies have been proposed tomeet the challenges and the main constituents of them include fuzzy logic, neuralnetworks, genetic algorithms, and rough sets [44]. The aim of researches on softcomputing is to provide more intelligent and robust systems which could providehuman-interpretable, low cost and sub-optimal solutions for some important appli-cations. Specifically, fuzzy models can model complex systems with prudent anduser-oriented sifting of data, qualitative observations and calibration of common-sense rules in an attempt to establish meaningful and useful relationships amongsystem variables [47]. Therefore, there is a growing indisputable role of fuzzy tech-nology in the realm of data mining [64]. The use of fuzzy techniques has beenconsidered to be one of the key components in artificial intelligence systems be-cause of its affinity with human knowledge representations [43].

Neural networks have the desirable feature that little knowledge about a pro-cess is required to successfully apply a network to the problem of interest, thoughsome domain specific knowledge is known to be beneficial [65]. Thus, they are typ-ically regarded as “black box” techniques with a few technical key issues to handle.This approach often leads to nice practical solutions in a relatively short periodand effort, since extensive modelling is not needed which is otherwise unavoidableconventionally. However, sufficient data and training are required for better per-formance [12].

Many researchers have been investigating different types of problems with regardto static data via using different fuzzy/neural approaches for quite a long time [43,47, 64]. Recently, interest in temporal data mining [54] has been increasing and agrowing number of implemented systems are using an enhanced temporal under-standing to explore aspects of behavior associated with the implicit time-varyingnature of the universe. A theoretical framework for temporal knowledge discoverywas proposed by Al-Naemi in [1] and a full discussion of some issues involved withtemporal knowledge discovery appeared in [52]. Currently the studies in temporalknowledge discovery include Apriori-like discovery of association rules [6], evolutionand maintenance of static association rules [7], template-based mining for sequences[15], sequence mining [59] and classification of temporal data [53]. Fuzzy logic hasalso been applied to study temporal knowledge discovery such as dynamic datamining [4] and mining changes in association rules [2]. All these approaches mainlyfocus on how to discover useful knowledge from database in stead of modellingthe dynamic pattern representing the discovered temporal knowledge. As to largedatabases, a large amount of temporal knowledge (hundreds and thousands associa-tion rules, sequences of events,. . . ,etc) often posits difficulty for human’s capabilityof understanding. It is very important that the discovered temporal knowledgeshould be integrated into a dynamic model which not only can be convenientlysimulated by computers and systematically studied by many powerful mathemati-cal tools, but also can be very expediently understood and utilized by users.

The modelling of complex systems requires new methods that can utilize theexisting knowledge (e.g., temporal and static knowledge) and human experience.Fuzzy cognitive map (FCM) [14] based on fuzzy logic theory is an illustrativecausative representation, the description and modelling of complex systems. FCMdraws a causal representation, which intends to model the behavior of any systemand is an interactive structure of fuzzy concepts, each of which interacts with therest showing the dynamics and different aspects of the behavior of the system [14,9]. Recently FCM has been widely applied to knowledge representations and variety


of intelligent fields to study a variety of practical problems such as real environ-mental management [46], geographic information systems [56] and process control[55].

The approaches used to construct the FCM have great potential to model adynamic system [3]. In [13], Kosko pointed out that it is very difficult to build largescale intelligence systems based on the current FCM constructions since currentFCM only involves a small number of nodes and arrows (6–20 nodes or arrows, see[3, 46]). Also the current dominant emphasis for the study of FCM is in using theFCM rather than its construction and also the construction of FCM is mainly basedon human experts who can observe and know the operation of the system and itsbehavior under different circumstances, while ignores the discovered rich temporalknowledge. The large amount of temporal knowledge discovered from the databaseis important information which objectively and truthfully reflects the nature ofdynamic complex systems. Applying this temporal knowledge to construct FCMhas two significant benefits: One is that large amount of knowledge discovered fromhuge databases can be integrated into a model which not only can be systematicallystudied by many powerful mathematical tools, but also can be very expedientlyunderstood and utilized by users. The other is that large-scale intelligence systems,which are beyond human expert’s capability to observe and understand, can bedeveloped via constructing large dimensional FCM. Therefore, the investigation ofconstructing FCM is very important.

Remark 7. : The following problems are interested research topics.1. Establish metric space based on the topological structures induced by the

involved fuzzy concepts in the AFS framework, propose measure for membershipfunctions and apply AFS fuzzy logic to find and represent the static and temporalknowledge contained in complex data systems.

2. In the framework of AFS theory, by integrating the huge amount of knowledgediscovered from the database and expert experiences into a model, construct FCMwhich can model the dynamic patterns contained in complex data systems.

Recall that the simple concepts play important roles in the AFS structures, whichare like the role of basis in linear vector space, and all the complex concepts canbe expressed by their fuzzy logic compositions. Therefore, in order to effectivelydiscover and correctively represent the static and temporal knowledge in database,we need to define appropriate measures for simple concepts first as did in our recentstudy for static database in Section 3.1 and 3.2. Then, with the property of AFSalgebra, the membership functions and their logic operations for any concepts inX can be obtained automatically based on the original data. Though a measureis given in our recent study based on the assumption that the data distributionis Gaussian, we still need to define the appropriate measure for simple conceptsbased on assumptions for other type data distributions in the future study. Moreimportantly, if we do not have any knowledge for the data distributions, we needto develop a measure for the simple concepts via a learning approach with theprocedure of estimating the data distribution from the training dataset and thendefining the appropriate measure based on the estimation. Here many learningtechniques can be used such as neural networks, genetic algorithm and supportvector machine [10]. Though several choices for the measure, such as norm, ..., etcare available for the simple concepts through this process, we need to testify theeffectiveness via the real database and select the appropriate one which can mimichuman recognition process.


In order to develop pattern recognition algorithm for temporal database, weneed to describe the temporal knowledge such as rules, event sequence and dynamictrends in AFS structure. One common representation is like following. With α, β ∈EM , the fuzzy rule is usually stated as “if α, then β”. Then, the total knowledgediscovered in the database can be viewed as a map Ψ from Λ to Γ, where Λ,Γ ⊆ EM . For α ∈ Λ, “if α, then β” can be represented by “if α, then Ψ(α)”, whereΨ(α) = β. Then the rule can be represented by a mathematical map in this case.It is obvious that if all the discovered knowledge are consistent and non-conflictive,then for α, β ∈ Λ, and α ≥ β in the lattice EM , one should have Ψ(α) ≥ Ψ(β) inEM . Mathematically, Ψ should be a GOH [58] (General Order Homomorphism)from sub AFS algebra generated by Λ to that generated by Γ. We can construct themap with a given database. With the possible constructions of GOH Ψ, topologicalmolecular lattices and the AFS algebras, we not only can study how to exclude theconflictive knowledge, but also can reduce and abstract the discovered knowledge.Then, it is possible that the total static and temporal knowledge discovered in ahuge database can be abstracted to be a GOH Ψ between two sub AFS algebras.The process of constructing GOH Ψ is also a process of excluding conflictive data.Then, the GOH Ψ can be studied and analyzed in the topological molecular latticesgenerated by Λ and Γ. This abstracts the temporal relationship into an algebraicrelationship between Λ and Γ. Mathematically, this is an algebraic problem andhow to define a GOH Ψ properly for given Λ and Γ in a database will help solveknowledge discovering and representing problems.

In [45], Thierry Marchant proposed a formal definition of a cognitive map de-pending on the concept of fuzzy implication. Thus in the framework of fuzzy logic,a node is a logical proposition and a link is an implication. Along this direction,we initially construct FCM directly from database in which the fuzzy implica-tion is implemented by AFS fuzzy logic in [28]. The basic idea is as follows: LetX = {x1, x2, ..., xt} be the set of sample data and Λ be a set of fuzzy concepts onX which describe the dynamic changes of the sample data. The following rule canbe defined: For some α ∈ Λ, δ > 0 and x ∈ X, y ∈ Y , if both µα(x) and µα(y)are larger than the given bound δ and also µα(x) < µα(y). Then we define thedata sample y happens after x. In other words, we define the change trend fromsmaller membership degree to large membership degree. The core issue is to definethe fuzzy concept set Λ which can describe the temporal data changes properly.

In general, to construct FCM for the knowledge discovered in temporal data canbe described to find the weighting along the link from Λ to Γ with the mappingGOH Ψ, where Λ, Γ ⊆ EM , such that for each α ∈ Λ, if state is α, then it willchange to Ψ(α). For each x ∈ X, find the fuzzy description ζx ∈ (Λ)EI where (Λ)EI

is the sub AFS algebra generated by Λ. In this case, the nodes of FCM are fuzzyconcepts ζx, x ∈ X and the weight of the link ζx to ζy is defined as

µζx(x)× (µΨ(ζx)∧ζy

(y)− µΨ(ζx)∧ζy(x)),

where µζ(x) is the membership degree of x belongs to the concept ζ. This impliesthat the possibility/degree of x changing to y is determined by the degree of xbelonging to the concept ζx and the difference of the degree of y and x belonging tofuzzy concept Ψ(ζx)∧ζy. If the difference is positive, then x changes to y. Otherwisey changes to x. Remember that there are many possible concepts in (Λ)EI and ifwe calculate all possible links in (Λ)EI , it will have a computationally dimensionaldisaster. We need to prune some links with some optimization techniques as did in[57] depending on the data structure and requirements of the designed system.


Remember that we construct FCM as previously discussed with the mappingGOH Ψ. We will study mathematical relations between mathematical propertiesof GOH Ψ and the nature of FCM based on AFS fuzzy logic. This study can helpus study the errors between the FCM and the real data it represents.

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Xiaodong Liu, received the B.S. and the M.S. degrees in mathematics

from Northeastern Normal University in 1986 and Jilin University, Jilin,in 1989, P. R. China respectively, and the Ph.D. degree in control theory

and control engineering from Northeastern University, Shenyang, P. R.

China in 2003.

He is currently a Professor in Research Center of Information and

Control, Dalian University of Technology, Dalian, and Research Center

of Automation, Northeastern University, Shenyang P. R. China, a Guest

Professor of the ARC Research Center of Excellence in PIMCE, Curtin

University of Technology, Australia. He was a Senior Visiting Scien-

tist in Department of Electrical and Computer Engineering, University

of Alberta, Edmonton Canada in 2003 and Visiting Research Fellow in

Department of Computing, Curtin University of Technology, Perth Aus-

tralia in 2004. He has been a Reviewer of American Mathematical Re-

viewer since 1993. He has proposed AFS theory and a coauthor of three

books. His research interests include algebra rings, combinatorics, topology molecular lattices,

AFS (axiomatic fuzzy sets) theory and its applications, knowledge discovery and representations,

data mining, pattern recognition and hitch diagnoses, analysis and design of intelligent control

systems. Dr. Liu is a recipient of the 2002 Wufu-Zhenhua Best Teacher Award of the Ministry of

Communications of People’s Republic of China.


Tianyou Chai, received Ph. D. degree in Industrial Automation

from Northeastern University of Technology, China, in 1985. He is pro-

fessor and head of the Research Center of Automation, Northeastern

University, China. He was the Member of IFAC’s Technical Committee

and the Chairman of IFAC’s Coordinating Committee on Manufacturing

and Instrumentation during 1996-1999. He was elected as the Member of

Chinese Academy of Engineering in 2003. His research interests include:

adaptive control, multi-variable intelligent decoupling control, optimiz-

ing control, integrated automation of process industry.

Wei Wang, obtained the Bachelor, Master Degree and PhD in Indus-

trial Automation from Northeastern University, China, in 1982, 1986 and1988 respectively. He is presently professor and director of Research Cen-

ter of Information and Control, Dalian University of Technology, China.

Previously he was a post-doctor at the Division of Engineering Cybernet-ics, Norwegian Science and Technology University (1990-1992), professor

and vice director of Research Center of Automation, Northeastern Univer-sity, China (1995-1999), vice director of the National Engineering Research

Center of Metallurgical Automation (1995-1999), and a research fellow at

the Department of Engineering Science, University of Oxford (1998-1999).His research interests are in adaptive control, predictive control, robot-

ics, computer integrated manufacturing systems, and computer control of

industrial process. He is the author of the book entitled Generalized Pre-dictive Control Theory and its Applications published by Science Publish-

ing House, China. He has published over 100 papers in international and domestic journals and

conferences. He is now a member of IFAC Technical Committee of Cost Oriented Automationand a member of IFAC Technical Committee of Mining, Mineral and Metal Processing.

1Research Center of Automation, Northeastern University, Shenyang 110004, China.

2Research Center of Information and Control, Dalian University of Technology, Dalian, 116024,

P. R. of ChinaE-mail : [email protected]

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