NIon~shelle Itiz Mh. Math. 83, 265--278 (1977) Maihematik 9 by Springer-Verlag 1977 Finite Procedures for Sofic Systems By Ethan M. Coven, Middletown, Conn., and Michael E. Patti, Baltimore, Yd. (Received 19 July 1976) Abstract A sofia system is a symbolic flow defined by a finite semigroup. We exhibit finite procedures, involving only the defining semigroup, for answer- ing certain questions about a sofia system and for constructing certain subshifts of finite type associated with a sofia system. Introduction Sofia systems, the class of symbolic flows which are images of subshifts of finite type, were introduced by ~u [10]. They have been studied further by the authors [1] and by FISCHER [2], [3]. Aside from whatever intrinsic interest they may have, sofia systems are important because they provide easily analyzed models for some properties of Axiom A diffeomorphisms, the latter also being images of subshifts of finite type. Furthermore, sofia systems appear naturally in language theory. Let us call a language L dynamical provided L is the set of words of some symbolic flow XL. L. W. GooDwu (private communication) has pointed out that a dynamical language L is regular (see [6]) if and only if XL is a Sofia system. In this paper, we adopt the spirit, although not the letter, of [10] in declaring that a sofia system is a symbolic flow defined by a finite semigroup. We show that there are finite procedures, in- volving only the defining semigroup, for answering certain questions about a sofia system and for constructing certain maps and sub- shifts of finite type associated with a sofia system. As an example of the point of view of this paper, consider the following well-known result in the theory of subshifts of finite Monatshefteflit Mathematik,Bd. 83/4 18