A new hierarchy of elementary functions

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0. Introduction. Let C be the class of all functions on nonnegative integers into nonnegative integers for which there is a mechanical procedure for obtaining the values from the arguments. Such functions are called computable. C will contain functions whose computations are of very different complexity. The searclh for a good measure of the computational complexity of a function has been stuLdied recently by several logicians, computer scientists and even statisticians. One method of obtaining such a measure is the method of hierarchies. This method consists of giving a sequence Ho, H1, H2, * * *, of classes of computable functions, such that, for all i>0, HfiCH+j and we have some reason to believe that a function which belongs to Ii+, but not to Hi is of a greater computational complexity than any function in Hi. Then a measure of the computational complexity of a function f can be taken to be the least i such that fEHi. (Such hierarchies have been studied by Axt [1], Cleave [2], Grzegorczyk [4] and Ritchie [8].) Ritchie [8] provides a hierarchy Fo, F1, F2, * e . Fo is taken to be the class of all functions computable by a finite automaton and, for all i_0, Fi+1 is defined to be the class of functions computable by Turing machines which use in their computations an amount of tape bounded by a function in Fi. It is shown by Ritchie [8, Theorem 3, p. 148] that U 0 Fi is exactly the class of elementary functions in the sense of Kalmar (i.e. the class 83 in Grzegorczyk [4]). Since this class contains most simple arithmetic functions the hierarchy is of considerable interest. (See ?4 for an application.) This paper contains two main results. The first (Theorem 1) is a characterization of Ritchie's classes in a new and neat way. This usually allows us to locate an elementary function of interest in the hierarchy more simply and efficiently than it could be done using the results of Ritchie [8]. Secondly, for various reasons (some of which are explained on p. 143 especially in footnote 8 in [8]), Ritchie encodes his functions in binary notation on the Turing machines. However, encoding in the literature is usually unary. In order to make use of this available information, it is desirable to have a hierarchy of elementary func-