outputs and the cooperative knowledge base (CKB) to generate the principles of coop- eration which are some observations (elements of hypothesis space < ; ;\;.
Horizontal Cooperation under Uncertainties in Distributed Expert Systems Chengqi Zhang and Yuefeng Li School of Computing and Mathematics Deakin University, Geelong VIC 3217, AUSTRALIA E-mail: fchengqi; yuefengg@deakin:edu:au
Abstract
In this paper, a model to do horizontal cooperation under uncertainties in distributed expert systems is presented. This model uses synthesis of solutions under uncertainties and decision to ful ll horizontal cooperation. Firstly, a Boolean algebra is used to represent the hypothesis space of synthesis of solutions in the model. Then the generalization of evidence theory is used as the mathematical model to synthesize all of conclusions which come from other expert systems. Finally, a competing mechanism is introduced to make decisions.
1. Introduction A distributed expert system (DES) consists of expert systems (ESs) which are linked by Internet. It di ers from the more general area of distributed processing because it is concerned with distributing control as well as data, and it can involve extensive cooperation between expert systems [7]. Some approaches have been done on cooperation in DESs [7]. These approaches to cooperation concentrate mainly on scheduling of tasks and ESs under certainty, or in other words, the organization of ESs to cooperatively solve a problem under certainty. In [7], Zhang classi es the cooperation among ESs in a DES into four types: horizontal cooperation, hierarchical cooperation, recursive cooperation and hybrid cooperation. Horizontal cooperation and hierarchical cooperation are the very basic kinds of cooperation among ESs, because there are only two possibilities of relationships among ESs, dependence or independence, which correspond to hierarchical cooperation or horizontal cooperation, respectively. This paper is concerned only with horizontal cooperation. In particular, it deals with cooperation under uncertainties in a DES. In order to do this, it is necessary to decide what the real principle of horizontal cooperation is, in the sense that horizontal cooperation means that each expert in a DES can obtain solutions to problems without depending on other experts. It is argued that the principle is cooperation and competing, which means that not only does the model consider every result of all expert systems (cooperation, the rst step in horizontal cooperation), it can also select a solution which is one of the best alternatives and a delegate of common goods in the set of the best alternatives (competing, the second step). Synthesis of solutions with uncertainties is considered as an appropriate method to accomplish the rst step of horizontal cooperation. Up to now, there have been several methods of synthesizing uncertainties of solutions in cooperative problem solving
[3,6,8,11], and the most typical method is the default method of solution synthesis o ered in HECODES [8]. However, these models are often criticized for their robustness. This drawback stems mainly from the fact that an appropriate mathematical model is not used. Another drawback of these models is that they can not capture the cooperative knowledge in application elds when various information draw from every expert system; instead, they do the cooperation with a few parameters and simple arithmetic expressions. In order to use a robust method, we use a Boolean algebra to describe the hypothesis space when solutions in a DES is synthesized [9]. This method does not increase the complexity in computing (and which can also deal with ignorance), In this research, we select the generalization of evidence theory [2,5] as a basic mathematical model to handle horizontal cooperation under uncertainties in a DES. For cooperation, a qualitative [1] approach to capture the cooperative principles of application domains, which can direct ego-modify belief for each expert system, and a synthesis mapping to do the rst step in horizontal cooperation are presented [9]. In Section 3, we present the associated algorithm for the cooperation. After getting the result (a mass function) of synthesis of solutions under uncertainty in a DES, the next step is to make a decision. By using mass functions which include all conclusions made by all ESs, the rst e ort is to give the de nition of the expected utilities of hypotheses. Then, the de nition of conclusion distances which describe the di erence degrees between conclusions (a method of competing) is given, and it is proved that the de nition satis es the demands of distance functions [10]. In Section 4, based on this kind of competing mechanism, we present an algorithm to make decisions.
2. The structure of horizontal cooperation The structure of horizontal cooperation model under uncertainties in distributed expert systems is shown in Figure 1. There are two steps in this structure: cooperation (the rst step in horizontal cooperation: ego-modi ed belief and synthesis of solutions [9]), and competing (the second step: rst level decision and second level decision [10]). In the cooperation, the supervisor of expert systems, SES, rst receives all various information draw from every expert system (ES1; : : : ; ESn). Then it uses these outputs and the cooperative knowledge base (CKB) to generate the principles of cooperation which are some observations (elements of hypothesis space < ; [; \; ; ; > [2]), and it sends the principles of cooperation to corresponding expert systems. These principles of cooperation can direct ego-modifying belief for each expert system, and expert systems can give their nal output m1 ; : : : ; mn to SES. After that, SES synthesizes the solutions (m1 ; : : : ; mn) given by several expert systems. The result of the cooperation is a mass distribution m [2,5] on Boolean algebra hypothesis space , which synthesize one ES or several ESs opinions. Based on m, in the competing, SES selects the target set, and decides which is the delegate in the target set. 0
ES 1
ES n
...
SES m1
m2
...
mn
Synthesize Mass function
m First decision
CKB
Target set Second decision Result
Figure 1: The structure of horizontal cooperation model
3. Synthesis of solutions under uncertainties The cooperation in a DES can be described as the following algorithm. There are two main procedures in this algorithm: ego-modi ed belief and synthesis of solutions (more detail can be found in [9]).
Algorithm SSU (ES; E; ; P )
/* ES is a set of expert systems, for every expert system e, its output is a probability or mass function me on set U (e). Here U (e) is the hypothesis space of e and it is a subset of (how to get U (e) see [9]). is the hypothesis space of Boolean algebra. Ee, the element of E , is an observation obtained by expert system e from the principles of cooperation. P is a probability function on ES , P (feg) is the weight or authority given by human experts. */ P 1. FOR e 2 ES DO /* we suppose B:B Ee = me(B ) > 0 for all e 2 */ G(e) = fX j 9Y 2 U (e) such that X = Ee \ Y g; 2. FOR e 2 ES DO /* computing every revised mass functions */ ( se( j e) = 0; FOR A 2 Ue DO P se(A j e) = C U (e):C Ee =A me(C ); W = PA;A G(e) se(A j e); FOR A 2 G(e) DO se(A j e) = W1 se(A j e) ); S 3. G = e ES G(e); 4. m() = 0; /* Using synthesis mapping to synthesize the solutions */ FOR A 2 G DO m(A) = Pe ES;A G(e) se(A j e)P (feg); RETURN(m) 2 \
2
6
\
2
2
2
2
In the rst step of algorithm SSU, we use G(e) to represent all intersections of observation Ee and hypotheses of U (e). The second step computes the revised mass
functions, and the last two steps obtain a mass function m as the result of the synthesis of solutions.
4. Making decisions Based on the mass distribution m obtained by the cooperation, the competing mechanism [10] can be described as the following algorithm. Algorithm MD (ES; ; m; card; ) /* ES is a set of expert systems, for every expert system e, it has a granule collection G(e), its all conclusions. is the hypothesis space of Boolean algebra. The mass function m is the result of synthesis of solutions, its focal elements are Utility(A1 ); : : : ; Utility(AF ); and card is a cardinality function on . is a given value, [0 : F ] is an array. */ 1. FOR i=1 to F DO /* Computing all expected utilities of conclusions */
Utility(Ai ) =
F X k=1
"
#
card(Ai \ Ak ) m(A ) ; card(Ai) card(Ak ) k
2. k=1; N=0; [0]=1.0; FOR i=1 to F DO IF Utility(Ai ) THEN ( j=k-1; WHILE Utility(Ai ) > Utility( [j]) DO j=j-1; [j+1]=Ai; k=k+1; N=N+1 ); IF N=0 THEN RETURN(); IF N=1 OR N=2 THEN RETURN([1]); /* Selecting the rst level targets f[1]; [2]; : : : ; [N ]g, which satisfy Utility([1]) : : : Utility([N ]) */ 3. FOR i=1 to N DO FOR j=i to N DO h (Ak C ) m(A )i); ( C = ([i] \ [j ] ) [ ([i] \ [j ]); D[i; j ] = PFk=1 card k card(Ak ) /* Computing the up triangle of the conclusion distance matrix D*/ 4. FOR i=2 TO N DO FOR j=1 TO i-1 DO D[i,j]=D[j,i]; /* Computing the down triangle of the conclusion distance matrix D*/ 5. k=1; sum=0; /* Selecting the delegate of common goods*/ FOR i=1 TO N DO sum=sum+D[1,i]; FOR i=2 TO N DO ( S=0; FOR j=1 TO N DO S=S+D[i,j]; IF S
