Products of finite state machines with full coverage

Abstract

Given a collection of finite state machines, M i , with the same input alphabet, let M be the product machine, M = ΠM j . In general, not every state in M is reachable. A natural question is whether there are any inherent limits to the number of reachable states in a system that is the product of many small finite state machines. This note constructs a family of product machines M where the number of states is doubly exponential in the number of states in any individual machine M i and every product state is reachable. Products of finite state machines such as discussed in this note occur when analyzing large collections of independently designed telecommunications services. These examples raise the possibility that product finite state machines modeling systems of independently designed services may have different characteristics from finite state machines modeling communications protocols. Consequently, analyzing collections of telecommunications services may require new heuristic methods.