Refer the clusters union just cover the agent set

11  21  24  VCH 3  32  33  31  32  33  

Community  Dynamic�Mediator�Holon  
11  12  13 

31 


22  23  24  
12  13  14  21  22  23  
Machine Logical� Group and Associated� Order and Product Information 

Taskdriven Machine�Groups Identified by�GTbased methods  m1  m2 m3  n1  n2  n3  p1  p2  p3  p4 

q1  q2  q3 



The dynamic characteristics of the eventdriven holon community become more complicated as the population grows. In the next section, we present an approach for automatic grouping into holonic clusters depending on the assigned task. This approach, due to its strong mathematical foundation, should be applicable to large multiagent systems.
3.6 


3.6.1 
As already noted in Section 3.3.2, and illustrated by examples in Sections 3.3.6 and 3.5.3, agents can dynamically be contracted to a problemsolving group (cluster), through the virtual clustering mechanism. In the following, it is shown how agents can automatically be selected for such holonic clusters, using a new theoretical approach.
To model the multiagent system (MAS), we will use set theoretical concepts that extend to fuzzy set theory. Consider the set of all agents in the MAS. As already mentioned, in our metamorphic architecture, clusters and partitions or covers can change any time during the MAS evolution, according to a global strategy which aims to reach a goal.
>  b or b  >  a or it exists c so that a  >  c and b  >  c,  

b. If a cluster is defined by another axiom:
• the agents a and b are in the same cluster if a > b or b > a,
then, when a > c, b > c and no relation exists between a and b, the pairs {a,c} and {b,c} belong to different clusters, but c belongs to two clusters at the same time. In this case, clusters could overlap and their union is just a cover of the agents set.
3.6.2  {  Pm  }  m∈1, M  a set of  

Denote by �N =  {  an  }  n∈1, N  the set of N ≥ 1 agents acting as an MAS and by � = 
M ≥ 1 partitions of �N, that seem to occur during the MAS evolution toward its goal. Notice that the number of all possible partitions covering �N, denoted by �N, increases faster with N than the number of all possible clusters (which is 2N), as proves Theorem 1 from Appendix A. For example, if N = 12, then �12 = 4,213,597, whereas the number of all clusters is only 212 = 4,096.
©2001 CRC Press LLC