An Axiom Foundation for Uncertain Reasonings in Rule ... - CiteSeerX

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An Axiom Foundation for Uncertain Reasonings in Rule-Based Expert Systems: NT-Algebra Xudong Luo1 and Chengqi Zhang2 Department of Computer Science and Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong Special Administrative Region, P. R. China E-mail: [email protected] 2 School of Computing and Mathematics, Deakin University, Geelong, Victoria 3217, Australia E-mail: [email protected] 1

Received 00 January 1999 Revised 00 January 1999 Accepted 00 January 1999

Abstract. This paper identi es an axiom foundation for uncertain rea-

sonings in rule-based expert systems: a near topological algebra (NTalgebra for short), which holds some basic notions hidden behind the uncertain reasoning models in rule-based expert systems. According to the basic means of topological connection in an inference network, an NT-algebraic structure has ve basic operators, i.e. AND, OR, NOT , Sequential combination and Parallel combination, which obey some axioms. An NT-algebraic structure is de ned on a near-degree space introduced by the authors, which is a special topological space. The continuities of real functions, of fuzzy functions and functions in other senses can be uniformly considered in the framework of a near-degree space. This paper also proves that EMYCIN's and PROSPECTOR's uncertain reasoning models correspond to good NT-algebras. Moreover, the existence of any nite NT-algebraic structure is constructively proved. Compared to other related works, the NT-algebra as an axiom foundation has the following characteristics: (1) various cases of assessments for uncertainties of evidence and rules are put into a uni ed algebraic structure; and (2) major emphasis has been placed on the basic laws of the propagation for them in an inference network, especially the continuity of propagation operations and the relationships between propagation operations. Keywords: Uncertainty, expert system, algebra, topology, fuzzy set.

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1 Introduction In the real world, both evidence and the relationships between evidence and hypotheses may not always be certain. These facts have led the designers of expert systems to abandon the pursuit of logical completeness in favor of developing eective heuristic ways to exploit the fallible, but valuable, judgemental knowledge that human experts bring to particular classes of problems. Based on the above reason, some uncertain reasoning models applied in rule-based expert systems have been proposed. Although they are quite dierent in appearance, what they simulate is an ability equivalent to human experts' uncertain reasonings. For this reason, these models imply that there should be a common structure to satisfy some necessary common conditions. Thus, it is possible to establish an axiom foundation for them. If we have such an axiom, we can hold common notions in dierent models, develop the appropriate models for special applications, and examine the relationships between them, e.g. the transformation relationship between dierent models in a distributed expert system, etc. In fact, some researchers have already done some signi cant work in this area, but their attempts fail to provide satisfactory answers to the problem. In this paper, an axiom foundation for uncertain reasonings in rule-based expert systems has been identi ed. Corresponding to the topological connection in an inference network in rule-based expert systems, there are ve basic ways to propagate assessments for uncertainties of evidence and rules. The formulae for ve ways of propagation satisfy some axioms abstracted from some common laws of uncertain reasonings of human beings. Thus, the operators corresponding to these ve ways and a set of assessments for uncertainties can constitute an algebraic structure, called a near topological algebra (NT-algebra for short). On human beings' uncertain reasoning, uncertainties of evidence and rules are probably assessed in terms of numbers, intervals, fuzzy numbers, fuzzy intervals, even general possibility distributions, and so on. All of them constitute a partial ordered structure, respectively. As a result, such an NT-algebraic structure needs to be de ned on a partial ordered structure. In the real world, the operators in an NT-algebra are continuous with respect to each of their parameters owing to the continuity of human beings'uncertain reasonings. For this reason, it is necessary to de ne a sort of space on the domain of an NT-algebraic structure, referred to as a near-degree space. As a result, such an algebraic structure is de ned on a near-degree space and composed of operations corresponding to basic basic ways of propagation for uncertainties through an inference network. Finding a general framework for uncertain reasonings is an important issue in the study of uncertain reasonings. Some researchers have done some work on this aspect. In light of the model of the EMYCIN uncertain reasoning, Heckerman [10] presented some axioms which the Sequential and Parallel combinations should satisfy. On interval representation of uncertainty, Driakov [5] presented an axiom system which AND; OR; NOT; the Sequential and Parallel combinations

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should satisfy. With regard to EMYCIN's uncertain reasoning model, Wang [14] suggested some descriptive conditions which operations of propagation for uncertainty through an inference network should satisfy. In other words, all of them considered the generality but only suitable for a few special cases. Although the work of Zhang in [17] is more general than those mentioned previously, he is mainly concerned with parallel combinations. In fuzzy mathematics, some researchers [4,16,11] studied various operations of AND, OR and NOT and presented an axiom foundation for them which re ects part of the law of uncertain reasoning by a human being. In Zheng's PhD dissertation [18], he suggested a uniform model for numerical uncertain reasonings. However, his model appears to abstract some considerations only in the assessments for the initial evidence and the nal hypothesis in an inference network, but not in the propagation for the uncertainties of the initial evidence and the rules in an inference network. It is the propagation that is more important. Hajek et al. [8] have made a study of an algebraic structure corresponding to an uncertain reasoning model like EMYCIN's and PROSPECTOR's, but they had considered the parallel combination operation rather than the others, and their investigation does not present any ultimate foundations of uncertain reasoning in rule-based expert systems [9]. In short, this paper presents NT-algebra as an axiom foundation for uncertain reasonings in rule-based expert systems, in which various cases of assessments for uncertainties of evidence and rules as well as the basic laws of the propagation for them in an inference network are taken into account. The remainder of the paper is organized as follows. Section 2 introduces the concepts of a near-degree space and of a continuous function on it, and explores its relationships to a metric space and to a topological space. Next, in Section 3 we present an algebraic structure, NT-algebra, as an axiom foundation for uncertain reasonings. Such a structure corresponds to a model of uncertain reasonings in a rule-based expert system, and exposes some of the laws of uncertainty assessments and propagation in an inference network. In turn, this is followed by two sections in which we prove that EMYCIN's [13,12] and PROSPECTOR's [6] uncertain reasoning models correspond to good NT-algebras, and the constructive existence of any nite NT-algebra. Finally, Section 6 summarizes this paper.

2 The Concept of Near-Degree Space In reality, the uncertainties of evidence or rules may be assessed in terms of numbers, intervals, fuzzy numbers, fuzzy intervals, possibility distributions or elements in some partial ordered structure. Thus, in order to make an abstract study of uncertain reasonings, it is natural that the concept of continuity of real functions, one of fuzzy functions and one of functions in other cases need to be placed under a uniform framework, which should not be too abstract to reveal appropriate characters. For this reason, we shall introduce the concept of a near-

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degree space. Oriented to the uncertain reasonings of human beings, rather than physical objects, a near-degree space should be dierent from a metric space. Evidently, if the assessment for uncertainty of proposition A is nearer to that of proposition C than that of proposition B, then the near-degree between A and C should be greater than that between B and C. This particular rule is dierent from the rule of triangle inequality in a metric space. Nevertheless, the dierence-degree corresponding to the near-degree should still satisfy the triangle inequality.

2.1 Near-Degree Space and N-Continuity De nition 1 Let L be a non-empty set and let  be a partial ordering on L2. If there is a map  : L2 ! [0; 1] which satis es: 8x; y; z 2 L 1. (x; x) = 1, 2. (x; z)  (y; z) , (x; z)  (y; z), 3. (x; z)  (y; x) + (z; y) ? 1,

then  is called a near-degree on L with respect to , 0 = 1 ?  is called a dierence-degree on L with respect to , (L; ; ) is called a near-degree space, and (x; y) and 0 (x; y) are called near-degree and dierence-degree between x and y, respectively.

Clearly, a dierence-degree satis es the triangle inequality and a near-degree satis es the commutative law. Example 1. Let L = fa1; a2; a3; a4g, and let a partial ordering  be de ned on

L2 by the following matrix:

01 3 2 41 BB 3 1 2 5 CC : @2 1 1 6A

4783 That is, (ai ; aj )  (ak ; aj ) , eij  ekj , here eij is the ith row and jth column element in the matrix above and ekj is the kth row and jth column element in the above matrix. It is easy to prove that a near-degree  on L with respect to  can be de ned by the following matrix:

0 1 0:4 0:4 0:3 1 BB 0:4 1 1 0:28 CC : @ 0:4 1 1 0:28 A 0:3 0:28 0:28 1

That is, (ai ; aj ) = aij , here aij is the ith row and jth column element in the above matrix. Thus, (L; ; ) is a near-degree space.

De nition 2 Let (L; ; ) be a near-degree space.

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1. Let x 2 L, " 2 (0; 1), then the set

B (x; ") = fy 2 Lj(x; y) > "g is called a "-open ball with center x. 2. A subset G of L is called a -open set , 8x 2 G; 9" 2 (0; 1); B (x; ")  G. If confusion does not occur, such an open ball is brie y denoted as B(x; ") and such a -open set is called an open set. The following theorem implies any open ball B(x; ") is an open set by the above de nition. Theorem 1. 8y 2 B(x; "); 9"0 2 (0; 1) such that B(y; "0 )  B(x; "). Proof. Let "0 = " + 1 ? (x; y). 8z 2 B(y; "0 ), we have (y; z) > "0 , thus, (y; z) > " + 1 ? (x; y), that is, (y; z) + (x; y) ? 1 > ". By De nition 1 and considering that a near-degree satis es the commutative law, we have (x; z) > (y; z) + (x; y) ? 1, thus, (x; z) > ", namely, z 2 B(x; "). Therefore, B(y; ")  B(x; "). 2

De nition 3 Let (L1; 1; 1) and (L2; 2; 2) be two near-degree spaces, and f : L 1 ! L2 . 1. f is N -continuous at point x 2 L1 , 8" 2 (0; 1); 9 2 (0; 1); 8y 2 L1 , if 1(x; y) >  , then 2(f(x); f(y)) > ". 2. f is N -continuous , f is N -continuous at every point in L1 . If confusion does not occur, a N-continuous function is brie y said to be continuous. Example 2. In Example 1, if a function ' : L ! L is de ned as: a1 ! a 2 ; a 2 ! a 2 ; a 3 ! a 3 ; a 4 ! a 3 ; then this function is continuous at point a1 . In fact, 8" 2 (0; 1), let  = 0:45, if (a1; ai ) > , then i = 1, thus, ('(a1 ); '(ai )) = ('(a1 ); '(a1 )) = 1 > ". Example 3. In Example 1, if a function ' : L ! L is de ned as: a1 ! a 1 ; a 2 ! a 3 ; a 3 ! a 4 ; a 4 ! a 1 ; then this function is not continuous at point a2 . In fact, let " = 0:4; 8 2 (0; 1), though (a2 ; a3) = 1 > ; ('(a2 ); '(a3 )) = (a3 ; a4) = 0:28 < ". The following two theorems are similar to the corresponding ones on metric space. Theorem 2. Let (L1 ; 1; 1) and (L2 ; 2; 2 ) be two near-degree spaces, then the function f : L1 ! L2 is continuous , if G is a 2-open set, then f ?1 [G] is a 1-open set.

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Proof. ()) Suppose that f is continuous and G is a 2-open set. If f ?1 [G] = ;, obviously it is a 1-open set. If f ?1 [G]@ = ;, then 8x 2 f ?1 [G]; f(x) 2 G,thus 9" 2 (0; 1), such that B(f(x); ")  G. Since f is continuous at point x, that is, 9 2 (0; 1), if y 2 B(x; ), then f(y) 2 B(f(x); ")  G, thus y 2 f ?1 [G]. As a result, B(x; )  f ?1 [G], thus f ?1 [G] is a 1-open set. (() Let x 2 L1 and " 2 (0; 1). By Theorem 1, B(f(x); ") is a 2-open set and it follows the assumption that f ?1 [B(f(x); ")] is a 1-open set containing point x, thus there exists  2 (0; 1) such that B(x; )  f ?1 [B(f(x); ")], that is, 8y 2 L1 , if (x; y) > , then y 2 B(x; )  f ?1 [B(f(x); ")],thus (f(x); f(y)) > ", that is, f is continuous at point x.

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Theorem 3. Let (L; ; ) be a near-degree space and let T = fG  LjG is a  ? open setg; then T is a topology on L, called near-degree topology induced by . The space (L; T ) is still denoted as (L; ; ).

Proof. (1) Clearly, ; and L 2 T . (2) If G; H 2 T ; then G \ H 2 T . In fact, if G \ H = ; 2 T ; otherwise, let x 2 G \ H, then 9"1 ; "2 2 (0; 1); B(x; "1)  G; B(x; "2)  H. Thus, let " = maxf"1 ; "2 g, then B(x; ")  G \ H. As a result, G \ H 2 T . (3) If A T , then [A2 T . In fact, if [A= ; 2 T ; otherwise, let x 2 [A, then 9G 2A; x 2 G,thus 9" 2 (0; 1); B(x; ")  G  [A. As a result, [A2 T . 2 This theorem implies that a near-degree space is a topological space.

2.2 A Near-Degree Space of Fuzzy Sets In this subsection, we prove that in a sense the concept of a near-degree in fuzzy mathematics is a special case of the concept de ned in Sec. 2.1. The de nition of a near-degree of two fuzzy sets is as follows [15]: De nition 4 Let $(X) be the set of all fuzzy sets on a domain X . The function  : $2(X) ! [0; 1] is called an f -near-degree on $(X), if ~ A) ~ = 1; 8A~ 2 $(X); 1. (A; ~ ~ ~ A); ~ 8A; ~ B~ 2 $(X); 2. (A; B) = (B; ~ ~ ~ ~ C) ~  (B; ~ C) ~. 3. 8A; B; C 2 $(X), if A~  B~  C~ , then (A; The following theorem is almost obvious. Theorem 4. Let the map  : $2(X) ! [0; 1] be a f -near-degree on $(X). If ~ C~ 2 $(X); (A; ~ C) ~  (A; ~ B) ~ + (B; ~ C) ~ ? 1, then ($(X); ; ) is a near8A;~ B; ~ C) ~  (B; ~ C) ~ , degree space, here  is a partial-ordering on $(X), de ned as (A; ~ ~ ~ A  B  C: If there is no danger of ambiguity,($(X); ; ) is brie y denoted as ($(X); ).

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Example 4. The semantics of such a linguistic term as probable and very probable

can be provided by a trapezoid fuzzy number N~ = (a; b; ; ) de ned by [1]: 80 if x < a ? , > > < 1 (x ? a + ) if x 2 [a ? ; a], if x 2 [a; b], N (x) = 1 1 (b + ? x) if x 2 [b; b + ], > > : 0 if x > b + . Let the set of all fuzzy numbers in the above form be denoted as L~ , and de ne the map ~ : L~ 2 ! [0; 1] as

~ (m; ~ n~ ) = 1 ? maxfja ? cj; jb ? dj; j(a ? ) ? (c ? )j; j(b + ) ? (d + )jg; here m~ = (a; b; ; ); n~ = (c; d; ; ). We easily show that ~ is a f-near-degree on L~ , thus (L~ ; ~ ) is a near-degree space. [2] gives the addition and subtraction operations on L~ as follows: m~ + n~ = (a + c; b + d;  + ; + ); m~ ? n~ = (a ? d; b ? c;  + ; + ): For the two operations above, when one of their parameters is a constant, both of them are continuous with respect to another parameter on the near-degree space (L; ). In fact, by letting n~ 0 = (c0; d0; 0 ; 0) be a constant, we can easily verify ~ (m~ + n~ 0 ; m~ 0 + n~ 0 ) = ~ (m; ~ m~ 0 ); here m~ 0 = (a0 ; b0; 0; 0); hence 8" 2 (0; 1); let  = ", if ~ (m; ~ m~ 0 ) > , then 0 ~ (m~ + n~0 ; m~ + n~ 0) > ", that is, m~ + n~ 0 is continuous at any point m. ~ This also applies for the case of the subtraction operation. This example implies how the concept of the continuity of fuzzy function is placed under the framework of a near-degree space.

2.3 The Near-Degree Space Induced by a Metric Space The following obvious theorem gives a way in which a near-degree space can be induced by a metric space. Theorem 5. Suppose that (L; ) is a metric space. Let the function g : [0; 1) ! [0; 1) be a continuous and strictly increasing map, and g(0) = 0, and additionally de ne g(1) ! 1. And let (x; y) = 1 ? g((x; y)), and a partial ordering  on L2 is de ned as (x; z)  (y; z) , (x; z)  (y; z). If 8x; y; z 2 L; g((x; z))  g((x; y)) + g((y; z)); then (L; ; ) is a near-degree space, said to be induced by (L; ), and g is called its induced function.

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The following theorem implies how the conception of continuity on a metric space is placed in the framework of a near-degree space. Theorem 6. Let (L; ) be a metric space and let (L; ; ) be a near-degree space induced by it. If the function f : L ! L is continuous on (L; ), then it is also continuous on (L; ; ). Proof. Let the induced function of (L; ; ) be g, that is, (x; y) = 1 ? g((x; y)): Suppose that f : L ! L is continuous on (L; ). Thus, 8" 2 (0; 1), let "0 = g?1(1 ? "); 9 0 > 0, if (x; y) <  0, then (f(x); f(y)) < g?1 (1 ? "). Therefore, 8" 2 (0; 1), let  = 1 ? g( 0 ), if (x; y) > , then (f(x); f(y)) > ", that is, f is continuous on (L; ; ). 2

3 The De nition and the Background of NT-Algebra An NT-algebra is an algebraic structure de ned on a partially ordered structure, with ve operations which are continuous with respect to some of their parameters on a near-degree space. This algebraic structure is abstracted from uncertain reasonings, and re ects some common laws of assessments for uncertainty and of their propagation through an inference network. Let (L; ; ) be a near-degree space and let (L; L ; ?; >) be a partially ordered structure with the maximum element > and the minimum element ?. The function f continued on (L; ; ) is said to be continuous on L.

3.1 AND Operation

Let the I operation 'I on L correspond to AND operation of propositions. We know that for a set of propositions E1; : : :; En, the compositions obtained by applying the operation AND in any order are equivalent. Logically, the assessments for uncertainty of the equivalent compositions of propositions should be equal. Therefore, the operation I should satisfy associative law and commutative law because the result of this operation is independent of the order. For two propositions, if the assessment of the uncertainty of one is constant and that of the other varies continually, then the assessment of the uncertainty of their composition by the AND operation should vary continually. So, 'I is continuous on L. For two propositions, if the assessment of the uncertainty of one is constant and that of the other increases, then the assessment of the uncertainty of their composition by the AND operation should not decrease. That is, 'I is monotonic and does not decrease on L. For two propositions, if one is completely believed to be true, then the assessment of the uncertainty of their composition by the AND operation should be equal to that of the other. Therefore, 8x 2 L; 'I (x; >) = x. Therefore, formally we have

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De nition 5 The map 'I : L2 ! L is called an I operation, if the following conditions hold: 1. 2. 3. 4. 5.

'I is continuous on L for each parameter; 'I is monotonic and does not decrease for each parameter; 8x; y 2 L; 'I (x; y) = 'I (y; x); 8x 2 L; 'I (x; >) = x; 8x; y; z 2 L; 'I (x; 'I (y; z)) = 'I ('I (x; y); z).

3.2 OR Operation Let the U operation 'U on L correspond to the OR operation of propositions. Its background is similar to that of the I operation and is therefore omitted here.

De nition 6 The map 'U : L2 ! L is called a U operation, if the following conditions hold: 1. 2. 3. 4. 5.

'U is continuous on L for each parameter; 'U is monotonic and does not decrease for each parameter; 8x; y 2 L; 'U (x; y) = 'U (y; x); 8x 2 L; 'U (x; ?) = x; 8x; y; z 2 L; 'U (x; 'U (y; z)) = 'U ('U (x; y); z).

3.3 NOT Operation Let the C operation 'C on L correspond to the NOT operation of proposition. When the assessment for uncertainty of a proposition varies continually, the assessment for uncertainty of its negation should vary continually also. Thus, 'C is continuous on L. When the assessment for uncertainty of a proposition increases, the assessment for uncertainty of its negation should decrease strictly. So, 'C decreases strictly. We know that negating the negation of a proposition is equivalent to being certain of it. The assessment for uncertainty of the negation of the negation of a proposition should be equal to that of this proposition. Then, 8x 2 L; 'C ('C (x)) = x. Therefore, formally we have

De nition 7 The map 'C : L ! L is called a C operation, if the following conditions hold:

1. 'C is continuous on L; 2. 'C decreases strictly; 3. 8x 2 L; 'C ('C (x)) = x.

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3.4 The Operation of Sequential Combination From the rule E ! H along with its strength (y1 ; : : :; yn), and the evidence E along with the assessment x for its uncertainty, the procedure of drawing H along with the assessment for its uncertainty is called the operation of Sequential combination. Sometimes, this procedure is relative to the unit eE of E and the unit eH of H. In an NT-algebra, the operation S corresponds to this kind of Sequential combination. If we are in ignorance of assessment for uncertainty of the premise of a rule, then this premise has no eect upon the assessment of the uncertainty of the conclusion of the rule. This means that as long as the assessment for uncertainty of the premise is equal to its unit eE , no matter what value the strength of the rule takes, the uncertainty of the conclusion is always equal to the unit eH . As a result, 'S needs to satisfy 'S (eE ; y1; : : :; yn; eE ; eH ) = eH : If the evidence E is independent of the hypothesis H, no matter what value the assessment of the uncertainty of E is equal to, it has no eect upon that of hypothesis H. This means that as long as the strength of the rule E ! H is equal to the unit (ers1 , : : :; ersn ), no matter what value the assessment for uncertainty of E is equal to, that of H is always equal to its unit eH . Correspondingly, 'S (x; ers1 ; : : :; ersn ; eE ; eH ) = eH : If the strength of a rule takes a constant and the assessment for uncertainty of the premise of the rule increases, that of the conclusion of the rule should not decrease. Consequently, 'S (x; d1; : : :; dn; eE ; eH ) is monotonic and does not decrease. As a rule, if the assessment for uncertainty of the premise or every parameter of its strength varies continually, the change of that of its conclusion should be continuous. Accordingly, 'S (x; y1; : : :; yn ; eE ; eH ) is continuous on L with respect to x and each yi (1  i  n), respectively. Therefore, formally we have

De nition 8 The map 'S : Ln+3 ! L is called an S operation, if there are units eE ; ers1 ; : : :; ersn ; eH 2 L such that the following conditions hold: 1. 'S (x; y1 ; : : :; yn ; eE ; eH ) is continuous on L with respect to x and each yi (1  i  n), respectively; 2. for any constants d1; : : :; dn 2 L, 'S (x; d1; : : :; dn; eE ; eH ) is monotonic and does not decrease; 3. 8y1 ; : : :; yn 2 L; 'S (eE ; y1 ; : : :; yn; eE ; eH ) = eH ; 4. 8x 2 L; 'S (x; ers1 ; : : :; ersn ; eE ; eH ) = eH .

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3.5 The Operation of Parallel Combination If m assessments for uncertainty of the hypothesis H are obtained respectively from m rules E1 ! H; : : :; Em ! H, which represent respectively the dierent support degrees for hypothesis H from the dierent sources E1; : : :; Em , then the operation for nding the combined assessment for uncertainty of hypothesis H is called an operation of Parallel combination. On applying an operation of Parallel combination, with the the increase in the assessment xi of the uncertainty of hypothesis H obtained from Ei ! H, the combined assessment of the hypotheses should not decrease. Consequently, 'mP (x1 ; : : :; xm ; e0 ) does not decrease with respect to every parameter xi (1  i  m), respectively. On applying an operation of Parallel combination, with the continuous change of the assessment xi of the uncertainty of the hypothesis H obtained from Ei ! H, the combined assessment of the hypotheses should vary continually. Consequently, 'mP (x1; : : :; xm ; e0) is continuous with respect to every parameter xi (1  i  m), respectively. If some source Ei does not provide any information to hypothesis H, it should have no eect upon hypothesis H. That is, 'mP (e0 ; x2; : : :; xm ; e0) = 'mP ?1 (x2 ; : : :; xm ; e0); '2P (e0 ; x2; e0) = x2 : If we apply a parallel operation on a set of assessments, from dierent sources, for uncertainty of the same hypothesis, its combined assessment should be independent of the order of the operation. Namely, 'mP (x1 ; : : :; xm ; e0) = 'mP (x01; : : :; x0m ; e0); where x01; : : :; x0m is any permutation of x1 ; : : :; xm . Therefore, formally we have De nition 9 The map 'mP : Lm+1 ! L is called a P operation, if there is the unit e0 2 L such that the following conditions hold: 1. 'm P (x1; : : :; xm ; e0) is continuous on L for each parameter xi (1  i  m),

respectively; 2. 'm P (x1; : : :; xm ; e0) is monotonic and does not decrease for each parameter xi (1  i  m), respectively; m?1 3. 8x2; : : :; xm 2 L; 'm P (e0 ; x2; : : :; xm ; e0) = 'P (x2; : : :; xm ; e0); 2 4. 8x 2 L; 'P (e0 ; x; e0) = x; m 0 0 0 0 5. 8x1; : : :; xm 2 L; 'm P (x1 ; : : :; xm ; e0 ) = 'P (x1; : : :; xm ; e0), where x1; : : :; xm, is any permutation of x1; : : :; xm .

3.6 NT-Algebra De nition 10 The 9-tuple (L; 'I ; 'U ; 'C ; 'S ; 'mP; L ; ?; >) is called a near topological algebra on (L; ; ), NT-algebra on L for short.

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In this section, our discussion mainly refers to uncertain reasoning models like those used by EMYCIN [13] and PROSPECTOR [6]. We have known that in a rule-based knowledge base, every rule is in a form of IF E THEN H That is, E!H where E is a Boolean combination of E1; : : :; En, which can be obtained by three ways: AND; OR and NOT. The three operators I, U and C are the operations of combining assessments for uncertainty of propositions combined by the three ways, respectively. The two basic structures of an inference network are as follows: E ! H 1 ! H2 ! : : : ! H n E1 ! H

%-

E2 : : : E n The procedure of uncertain reasoning is actually that of propagation for uncertainty through an inference network. How to propagate? Clearly, to answer this question is to make it clear that the way of propagation for uncertainty is along these two basic structures above. The S and P operators are their abstract descriptions. Therefore, an NT-algebra structure corresponds to a model of uncertain reasoning.

De nition 11 If the following conditions hold: 'C ('U ('C (a); 'C (b))) = 'I (a; b); 'mP ('S (a1 ; b1; : : :; bn; eE ; eH ); : : :; 'S (am ; b1 : : :; bn; eE ; eH ); eH ) = 'S ('mP (a1 ; : : :; am ; eE ); b1; : : :; bn; eE ; eH ); m?1 2 3. 'm P (x1; : : :; xm ; e0) = 'P (x1 ; 'P (x2; : : :; xm ; e0); e0 ); with eE ; eH ; e0 being constants, and the other parameters being any elements in L, then this NT-algebra is called a perfect NT-algebra. If the second condi1. 2.

tion above holds, then it is called a distributive NT-algebra. If the rst and last conditions hold, then it is called a good NT-algebra.

In the above de nition, we suggest three relationships. The rst and last ones are easily understood. Now let us explain the second of relationship in the case m = 2. Suppose, now, that in the knowledge base there are the three following rules: E1 ! E; E2 ! E; E ! H: Let the value of strength of rule E ! H be (b1; : : :; bn), the unit of E be eE and the unit of H be eH . Evidently, there are two ways of deriving the assessment of the uncertainty of hypothesis H from the three rules above in an inference:

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(1) At moment t1 , the available information enables us to derive E and the assessment, a1 , for its uncertainty from E1 ! E, but does not enable us to derive E from E2 ! E. Then, we continually infer to draw the hypothesis H and the assessment, 'S (a1 ; b1; : : :; bn; eE ; eH ), for its uncertainty from E ! H. Up to the moment t2 (> t1 ), we get some new information enabling us to derive E and the assessment, c1 , of its uncertainty from E2 ! E. In turn, again we derive the hypothesis H and another assessment, 'S (c1; b1; : : :; bn; eE ; eH ), for its uncertainty from E ! H. Finally, by the P operation we obtain '2P ('S (a1; b1; : : :; bn; eE ; eH ); 'S (c1; b1; : : :; bn; eE ; eH ); eH ); i.e. the combined assessment of the uncertainty of hypothesis H from dierent sources. This is similar to depth- rst. (2) At the same time, the available information enables us to derive E and the two assessments, a2 and c2 , for its uncertainty from E1 ! E and E2 ! E, respectively. Later we obtain the combined assessment, '2P (a2; c2; eE ), for the uncertainty of proposition E. Finally, we derive the hypothesis H and the assessment 'S ('2P (a2 ; c2; eE ); b1; : : :; bn; eE ; eH ) for its uncertainty from E ! H. This is similar to breadth- rst. Suppose that a1 = a2 = a; c1 = c2 = c. The nal result of the assessment of the uncertainty of the hypothesis H may be independent of the two inference ways above. Formally, '2P ('S (a; b1; : : :; bn; eE ; eH ); 'S (c; b1; : : :; bn; eE ; eH ); eH ) = 'S ('2P (a; c; eE ); b1; : : :; bn; eE ; eH ): In many other cases, notably in the case of distributed expert systems, we need the above assumption. Suppose that the information, for E2 ! E, of one node expert system ES1 is dependent upon another node expert system ES2 . If the obtained information is enough to derive E from E1 ! E, there is no need to wait for ES2 to send the information for using E2 ! E, ES1 can also derive hypothesis H from E ! H. Once ES1 obtains the information for using E2 ! E from ES2 , it derives E, and then does H from E ! H again. Of course, it is possible that the obtained information by ES1 is enough for ES1 to derive E both from E1 ! E and from E2 ! E. Later in the next moment ES1 derives the hypothesis H from E ! H. Thus, in order to guarantee the nal results in these cases are the same, in the distributed expert systems, the uncertain reasoning model used by ES1 should satisfy the above equality.

4 NT-Algebras of EMYCIN and PROSPECTOR In this section, we give two examples to prove that some uncertain reasoning models correspond to NT-algebra structures. In fact, we use two very famous

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systems, i.e. the EMYCIN model [13,12] and the PROSPECTOR model [6,7], as examples. The EMYCIN (Essential MYCIN) expert systems shell is the MYCIN expert system [3] without its data. EMYCIN has been successfully used in many practical systems. In EMYCIN, the method of dealing with uncertainty is the certainty factor model. (Here we call it the EMYCIN model.) In this model, a certainty factor is used to express the degree of belief or disbelief. It takes a value on [?1; 1]. Concretely, when it takes a value on [?1; 0), the value expresses the degree of disbelief. In particular, ?1 indicates absolutely false. When it takes a value on (0; 1], the value expresses the degree of belief. In particular, 1 indicates absolutely true. When it takes 0, it means no idea about belief or disbelief. In an inference network, the propagation of certainty factors follow operations as listed in Theorem 7 below, which states that the EMYCIN model corresponds to an NT-algebra structure. PROSPECTOR [7] was designed to predict potential mineral deposits. In PROSPECTOR, the approach towards handling uncertainty is the subjective Bayesian method. (Here we call it the PROSPECTOR model.) In this model, uncertainties are measured by probability. However, the operations of propagation for uncertainties are not strictly consistent with probability theory. These operations are listed in Theorem 8 below, which states that the PROSPECTOR model corresponds to an NT-algebra structure. Readers can readily verify the following two theorems for themselves.

Theorem 7. (EMYCIN NT-algebra) Let 1. 'I : [?1; 1]2 ! [?1; 1] be given by 'I (x; y) = minfx; yg; 2. 'U : [?1; 1]2 ! [?1; 1] be given by 'U (x; y) = maxfx; yg; 3. 'C : [?1; 1] ! [?1; 1] be given by 'C (x) = ?x; 4. 'S : [?1; 1]4 ! [?1; 1] be given by 'S (x; y; eE ; eH ) = y  maxf0; xg; m+1 ! [?1; 1] be given recursively by 5. 'm P : [?1; 1] 8 x + x ? x x if x ; x  0, < 1 x 2+x 1 2 1 2

'2P (x1; x2; e0 ) = : 1?min1fjx12j;jx2 jg if x1x2 < 0, x1 + x2 + x1x2 if x1; x2  0, 'mP (x1 ; : : :; xm ; e0) = '2P (x1; 'mP ?1 (x2; : : :; xm ); e0): Then ([?1; 1]; 'I ; 'U ; 'C ; 'S ; 'm P ; ; ?1; 1) is a good NT-algebra de ned on the near-degree space ([?1; 1]; ; ), here (x; y) = 1 ? jx ? yj and  is de ned as (x; z)  (y; z) , jx ? z j  jy ? z j, and eE  ers  eH  e0  0.

Theorem 8. (PROSPECTOR NT-algebra) Let 1. 'I : [0; 1]2 ! [0; 1] be given by 'I = minfx; yg; 2. 'U : [0; 1]2 ! [0; 1] be given by 'U = maxfx; yg; 3. 'C : [0; 1] ! [0; 1] be given by 'C (x) = 1 ? x; 4. 'S : [0; 1]5 ! [0; 1] be given by

(

 + eH ? x if 0  x < e , 'S (x; y1 ; y2; eE ; eH ) = e + e E?eH (x ? e ) if e  x  E1, E E H 1?eE

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where

 = (f (yf2()y?21))eeHH +1 ; = (f (yf1()y?11))eeHH +1 ;

here f : [0; 1) ! [0; 1) is a 1-1 map increasing strictly and continually, f(0) = 0 and f(1) ! 1; m+1 ! [0; 1] be given by 5. 'm P : [0; 1]

Q

e00  ( mi=1 ( ex0i )) m 'P (x1 ; : : :; xm ; e0) = ; Q 1 + e00  ( mi=1 ( ex0i )) 0

0

0

0

here x0i = 1?xixi ; i = 1; : : :; m; e00 = 1?e0e0 : Then ([0; 1]; 'I; 'U ; 'C ; 'S ; 'm P ; ; 0; 1) is a good NT-algebra de ned on the neardegree space ([0; 1]; ; ), here (x; y) = 1 ? jx ? yj and  is de ned as (x; z)  (y; z) , jx ? z j  jy ? z j.

Note that in the PROSPECTOR's uncertain reasoning model, the strength of a rule E ! H is a pair (LS; LN), de ned as LS = PP((EEj:jHH)) ; LN = PP((::EEj:jHH)) ; which take values in [0; 1). From the two equations above, by using Bayesian formula P(AjB) = P (APjB(B)P) (A) , we can rewrite the above formulas as follows: LS = PP ((:HHjEjE)P)P(:(HH )) ; LN = PP ((:HHj:j:EE)P)P(:(HH )) : Thus, we can easily derive P(H jE) = (LSLS?1)PP ((HH))+1 ; P(H j:E) = (LNLN?1)PP ((HH))+1 ; which take values in [0; 1]. Hence, we need ; and a function f : [0; 1) ! [0; 1) in 'S , which satis es y1 = f ?1 (LS); y2 = f ?1 (LN). Thus for rule E ! H; ers1 = ers2 = f ?1 (1); eE = P(E) and eH = P(H). Evidently, eE , eH , ers1 and ersn can be dierent according to dierent rules. In the operation 'mP , for the rules: E1 ! H; E2 ! H; : : :; Em ! H; the unit

e0 = 1 ?P(H) P(H) :

Obviously, this unit can vary with hypothesis H.

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5 The Existence of Finite NT-Algebra Let (L; L ; ?; >) be a nite linear order structure, where L is an ordered relationship on L = fa0; : : :; an?1g, de ned as ai?1 L ai (1  i  n ? 1), and ? and > are minimum and maximum elements, respectively. Lemma 1 Let  : L2 ! [0; 1] be given by (ai ; aj ) = 1 ? ji ?n j j ; then (L; ; ) is a near-degree space, here  is a partial ordering on L2, de ned as: 8ai ; aj ; ak 2 L; (ai; ak )  (aj ; ak ) , jj ? kj  ji ? kj: Lemma 2 Any function f : L ! L is continuous on the near-degree space (L; ; ). Proof. 8ai 2 L, since L is a nite set, 9a 2 L such that 8ak (@ = ai ) 2 L; (ai ; a)  (ai; ak ): Let  = (ai ; a) + 1?(2ai;a) . 8" 2 (0; 1); if (ai ; aj ) > ; then (ai ; aj ) = 1, and noticing that (ai ; aj ) = 1 if and only if i = j, thus, (f(ai ); f(aj )) = 1 > "; that is, f is continuous at any point ai :2

Theorem 9. Let 1. 'I : L2 ! L be given by 'I (ai ; aj ) = minfai ; aj g; 2. 'U : L2 ! L be given by 'U (ai ; aj ) = maxfai; aj g; 3. 'C : L ! L be given by 'C (ai ) = an?i?1; 4. 'S : L4 ! L be given by 'S (ai ; aj ; eE ; eH ) = al ,where n c)(j ? b n c) l = d (i ?nb?2 b n c ? 1 2 e + b n2 c; 2 m+1 ! L be given by 5. 'm P :L 'mP (ai1 ; : : :; aim ; e0 ) = al ;

where

m X l = maxf?b n2 c; minf (ik ? mb n2 c); n ? b n2 c ? 1)gg + b n2 c: k=1

Then (L; 'I ; 'U ; 'C ; 'S ; 'm P ; L ; ?; >) is an NT-algebra de ned on the neardegree space (L; ; ), and eE = eH = ers = e0 = b n2 c.

Lemma 1 and Theorem 9 in this section can be easily proved.

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6 Summary and Conclusions This paper introduced the concepts of a near-degree space and of a continuous function on it. A near-degree space is abstracted from the continuity of propagations for uncertainties in an uncertain reasoning, under the uniform framework of which the concepts of continuities of both real and fuzzy functions are placed. In this paper, we have explored the relationship between a metric space and a near-degree space, and proved that a near-degree space is a topological space. This paper has presented an algebraic structure, NT-algebra, as an axiom foundation for uncertain reasonings in a rule-based expert system, which exposes some of the laws of uncertainty assessments and propagation in an inference network. In this paper, we also proved that EMYCIN's and PROSPECTOR's uncertain reasoning models correspond to a good NT-algebra, respectively. Moreover, the existence of any nite NT-algebra is constructively proved. In short, this paper shows an axiom foundation for uncertain reasoning based on near-degree space and N-continuity. It will have applications in some elds, such as expert systems, fuzzy control systems and fuzzy decision-making. Particularly, NT-algebra as a general framework for uncertain reasoning has special signi cance for research on common laws of interaction among intelligent agents with dierent uncertain reasoning models in a distributed multi-agent environment. In fact, we can expose these laws by studying the relationships of dierent NT-algebra structures. Therefore, it is worth investigating further the relationships of dierent NT-algebra structure under the background of a distributed multi-agent environment.

Acknowledgment The authors would like to thank the anonymous referees for their comments. This research is supported by a grant from the Australian Research Council (A49530850).

References 1. P. P. Bonissone. A fuzzy set based linguistic approach: Theory and applications. In: J. I. Oren, C.M. Shub, P. F. Roth (eds.), Proc. 1980 Winter Simulation Conference, 1980, pp. 99{111. 2. P. P. Bonissone, K. S. Decker. Selecting uncertainty calculi and granularity: An experiment in trading-o precision and complexity. In: L. N. Kanal, J. F. Lemmer (eds.), Uncertainty in Arti cial Intelligence, North Holland, 1986, pp. 217{247. 3. B. G. Buchanan, E. H. Shortlife. Rule-Based Expert Systems: The EMYCIN Experiments of the Stanford Heuristic Programming Projects, Addision-Wesley, 1984. 4. Y. Cheng. An approach to fuzzy operators (I), Fuzzy Mathematics 2(2), 1{9, 1982. 5. D. Driakov. A calculus for belief-interval representation of uncertainty. In: Uncertainty in Knowledge-Based Systems, Springer-Verlag, 1987, pp. 205{216. 6. R. O. Duda, P. E. Hart, N. J. Nillson. Subjective Bayesian methods for rule-based inference systems. In: AFIPS Conference Proceedings 45, AFIPS Press, 1976, pp. 1075{1082.

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7. R. O. Duda, P. E. Hart, N. J. Nilsson, R. Reboh, J. Slocum, G. Sutherland. Development of a Computer-Based Consultant for Mineral Exploration, SRI Report, Stanford Research Institute, Menlo Park, CA, October, 1977. 8. P. Hajek, J. J. Valdes. Algebraic foundations of uncertainty processing in rule-based expert systems (group-theoretic approach), Computer and Arti cial Intelligence 9(4), 325{344, 1990. 9. P. Hajek, J. J. Valdes. A generalized algebraic approach to uncertainty processing in rule-based expert systems (Dempsteroids), Computer and Arti cial Intelligence 10(1), 29{42, 1991. 10. O. Heckerman. Probabilistic interpretations for Mycin's certainty factors. In: L. N. Kanal, J. F. Lemmer (eds.), Uncertainty in Arti cial Intelligence, North Holland, 1986, pp. 167{196. 11. G. J. Kli, T. A. Folger. Fuzzy Sets, Uncertainty and Information, Prentice-Hall, 38{51, 1988. 12. W. V. Melle. A domain-independent system that aids in constructing knowledgebased consultation programs, PhD Dissertation Report STAN-CS-80-820, Computer Science Department, Stanford University, CA, 1980. 13. E. H. Shortie, B. G. Buchanan. A model of inexact reasoning in medicine, Mathematics Biosciences 23, 351{379, 1975. 14. S. Wang. An application of bases formula in expert system, Computer Research and Development 24(6), 55{56, 1987. 15. R. Zhao, X. Chen. The universal de nition of near-degree for the fuzzy subsets and the grade of imprecision, J. Xi'an Jiaotong University 15(6), 21{27, 1981. 16. W. Zhang, H. Le. The structure of the norm system on fuzzy sets, J. Engineering Mathematics 1(1), 55-62, 1984. 17. C. Zhang. Cooperation under uncertainty in distributed expert systems, Arti cial Intelligence 56, 21{69, 1992. 18. F. Zheng. A common reasoning model: The theory and method about numeral reasonings, PhD thesis, Department of Computer Science, Jilin University, China, 1989.

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Xudong Luo received received a PhD degree in Computer Sci-

ence from the University of New England, Australia in 1999. He is currently a post-doctorate research fellow at the Chinese University of Hong Kong, China. His research interests center around the study of Arti cial Intelligence, especially in the area of uncertain reasonings in a cooperative multi-agent environment. He (as author or co-author) has published one book and more than 60 research papers in journals and proceedings. Dr. Luo is a member of Australian Computer Society. He has served as a referee for several international conferences and journals.

Chengqi Zhang received a PhD in Computer Science from the University of Queensland, Brisbane, Australia in 1990. Since then, he has been a lecturer at the University of New England, Australia, and was promoted to Associate Professor in 1998. He is currently Associate Professor at Deakin University, Australia. Prof. Zhang's research interests are related to multi-agent systems, in particular cooperation under uncertainty. He is the author or co-author of over one hundred research papers. He is a member of the Australian National Committee for Arti cial Intelligence and Expert Systems and a senior member of the IEEE Computer Society as well as a member of American Association for Arti cial Intelligence (AAAI). He has served as a member of Program Committees for many international and national conferences, such as PRICAI'94, PRICAI'96, ICMAS'96, ICMAS'98, etc.