Tours in Machines and Digraphs

A tour in a machine is a shortest input sequence taking the machine from some initial state, through all of its remaining states and back again into its initial state. A best upper bound for tour length is found for two types of machines: n-state sequential machines with unrestricted input alphabet and n-state sequential machines with a two-letter input alphabet. The problem of finding a best upper bound for length of tours in machines is restated and solved using the language of the theory of directed graphs. The solutions to the above special cases restated in this language seem obvious but require a nontrivial proof of their status as solutions.