A characterization of Thompson digraphs

A nite-state machine is called a Thompson machine if it can be constructed from a regular expression using Thompson's construction. We call the underlying digraph of a Thompson machine a Thompson digraph. We characterize Thompson digraphs and, as one application of the characterization, we give an algorithm that generates an equivalent regular expression from a Thompson machine in time linear in the number of states. Although the construction is simple, it is novel in that the usual constructions of equivalent regular expressions from nite-state machines produce regular expressions that have size exponential in the size of the given machine, in the worst case. The construction provides a rst step in the construction of small expressions from nite-state machines.