On the Construction of Sequential Machines from Regular Expressions

Methods have been described [1], [2] for constructing sequential machines directly from regular expressions. In the construction, every machine is assumed to have an input terminal, an output terminal and a special starting input terminal. The methods described [1], [2] involve a recursive procedure. For example, if machines M
E
and M
F
realize the regular expressions E and F respectively, then the machine realizing the concatenation, R = EF, is obtained [1], [2] by connecting the output of M
E
to the starting terminal of M
F
. In this paper it is shown that a submachine generated by the algorithm in the middle of the synthesis procedure cannot be replaced in general, by an equivalent machine. It is felt that the proofs of the original algorithms [1], [2] are rather brief and do not point out the difficulties which might arise if proper precautions are not taken. To remedy this situation, this paper examines the construction in detail. The notion of recurrent realization of a regular expression is introduced, and a theorem is proved that the construction is valid if the proper precautions are taken.