On the Shannon Covers of Certain Irreducible Constrained Systems of Finite Type

A construction of Crochemore, Mignosi and Restivo in the automata theory literature gives a presentation of a finite-type constrained system (FTCS) that is deterministic and has a relatively small number of states. This construction is thus a good starting point for determining the minimal deterministic presentation, known as the Shannon cover, of an FTCS. We analyze in detail the Crochemore-Mignosi-Restivo (CMR) construction in the case when the list of forbidden words defining the FTCS is of size at most two. We show that if the FTCS is irreducible, then an irreducible presentation for the system can be easily obtained by deleting a prescribed few states from the CMR presentation. By studying the follower sets of the states in this irreducible presentation, we are able to explicitly determine the Shannon cover in some cases. In particular, our results show that the CMR construction directly yields the Shannon cover in the case of an irreducible FTCS with exactly one forbidden word, but this is not in general the case for FTCS's with two forbidden words.