Hierarchical star or rooted tree networks have been implemented and have any desirable characteristics, including excellent performance, conceptual simplicity, and suitability for optical fiber implementations. However, they have a single point of vulnerability to catastrophic failure (the master or root hub) as well as vulnerabilities to single failures of links or hubs high in the hierarchy, which could disable large portions of the network. Redundant network components are added to eliminate this vulnerability and to make an economic but extremely robust and fault-tolerant network, while preserving the advantages of previous hierarchical networks.

A Moore automaton A = (A, X,Y,S, A) can be obtained in two steps: first we consider the triplet (A, X, 6) called a semiautomaton and denoted by S — and then we add the components Y and A which concern the output functioning. Our approach is: S is supposed to be fixed, we vary A in any possible way, and among the resulting automata we want to separate the simple and the nonsimple ones from each other. This task is treated by combinatorial methods. Concerning the efficiency of the procedure, we note that it uses a semiautomaton having |A|(|A| + l ) /2 states.