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On rational solution of the state equation of a finite automaton 
International Journal of Mathematics and Mathematical Sciences
Volume 11 (1988), Issue 2, Pages 355-364
http://dx.doi.org/10.1155/S0161171288000420

On rational solution of the state equation of a finite automaton

R. Chaudhuri and H. Höft

Department of Computer Science, Eastern Michigan University, Ypsilanti 48197, MI, USA

Received 21 January 1987; Revised 25 June 1987

Copyright © 1988 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove that the necessary and sufficient condition for the state equation of a finite automaton M to have a rational solution is that the lexicographical Gödel numbers of the strings belonging to each of the end-sets of M form an ultimately periodic set. A method of determining the existence of a rational solution of the state equation is also given.