A new proof for the decidability of D0L ultimate periodicity

L systems were originally introduced by A. Lindenmayer to model the development of simple filamentous organisms [6, 7]. The challenging and fruitful study of these systems in the 70s and 80s created many new results and notions [9]. In this paper we consider the important problem of recognizing ultimately periodic D0L sequences. Let A be a finite alphabet and denote the empty word by ε . A D0L system is a pair (h,u), where h : A ∗ →A ∗ is a morphism and u is a finite word over A . The language of the D0L system is L(h,u) = {hi(u) | i ≥ 0} and the limit set limL(h,u) consists of all infinite words w such that for all n there is a prefix of w longer than n belonging to L(h,u). Clearly, if the limit set is non-empty, then one can effectively find integers p and q such that hp(u) is a proper prefix of hp+q(u) and