Rose Gene F.. Output completeness in sequential machines. Proceedings of the American Mathematical Society , vol. 13 (1962), pp. 611–614.

GENE F. ROSE. Output completeness in sequential machines. Proceedings of the American Mathematical Society, vol. 13 (1962), pp. 611-614. The author pursues a problem of Ginsburg (as found, for example, in XXVIII 175(7)). Given a complete sequential machine (deterministic finite automaton), let g(p, X) be the output word that results when the input word X is applied to a finite automaton in state p. A set P of states is output complete if, for every output word Y, there exists a p in P and an input word X such that g(p, X) = Y. P has proper degree d if the above is true for all Y of length d or less, but not d -\1. The author proves that "any output incomplete set of c distinct states has proper degree at most 2 — 2~ — 1," and goes on to show that this bound cannot be improved upon. He thus sharpens Ginsburg's observation that there exists a finite bound which depends in some way on the number of states, the number of input values, and the number of output values. From the author's sharpened result, it follows that there is a decision procedure for output completeness. ROBERT MCNAUGHTON