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1. Motivation. Many variants of the notion of automaton have appeared in the literature. We find it convenient here to adopt the notion of E. F. Moore [7]. Inasmuch as Rabin-Scott [9] adopt this notion, too, it is convenient to refer to [9] for various results presumed here. In particular, Kleene's theorem [5, Theorems 3, 5] is used in the form in which it appears in [9]. It is often perspicacious to view regular expressions, and this notion is used in the sense of [3]. In general, we are concerned with the problems of automatically designing an automaton from a specification of a relation which is to hold between the automaton's input sequences and determined output sequences. These "design requirements" are given via a formula of some kind. The problems with which we are concerned have been described in [1]. With respect to particular formalisms for expressing "design requirements" as well as the notion of automaton itself, the problems are briefly and informally these: (1) to produce an algorithm which when it operates on an automaton and a design requirement produces the correct answer to the question "Does this automaton satisfy this design requirement?", or else show no such algorithm exists; (2) to produce an algorithm which operates on a design requirement and produces the correct answer to the question "Does there exist an automaton which satisfies this design requirement?", or else show no such algorithm exists; (3) to produce an algorithm which operates on a design requirement and terminates with an automaton which satisfies the requirement when one exists and otherwise fails to terminate, or else show no such algorithm exists. Interrelationships among problems (1), (2), (3) will appear in the paper [1]. This paper will also indicate the close connection between problem (1) and decision problems for truth of sentences of certain arithmetics. The paper [1 ] will also make use of certain results concerning weak arithmetics already obtained in the literature to obtain answers to problems (1) and (3). Thus

In ''Monadic Computation and Iterative Algebraic Theories'' by Calvin C. Elgot,the notion ''iterative theory'' (more fully, ''ideal theory closed under conditional iteration'') is introduced and applied to the study of computational processes. The main point of the present paper is to show the existence (in a constructive sense) of free iterative theories. The main complication is the fact that in an iterative theory I the ''iteration'' operation is not defined for all elements of I. Were it not for this complication, the existence of free iterative theories would follow from general algebraic considerations (extended to many-sorted algebras). Actually we sketch two proofs of the existence of free iterative theories. One argument follows as much as possible general algebraic lines and is given a linguistic flavor in order to emphasize the concreteness of the ideas involved. The second argument depends upon ''normal descriptions'': a morphism in the free iterative theory being an equivalence class of normal descriptions.