in Cambridge, Massachusetts (U.S.A.)I) 1. Introduction There have been numerous attempts to classify recursive functions into hierarchies. A well known example is that of GRZEGORCZYK who defines a hierarchy & ' I c b1 c c 6 s c . . -of the primitive recursive functions based on a form of ACKERMANN'S function [14].a) The classes of the GRZEGORCZYK hierarchy are closed under a variety of operations such as limited recursion and composition. They are intimately related to the computational complexity of functions, measured for example by the amount of time or space required to compute functions with TURING machines, and the classes can also be characterized syntactically by depth of nesting of primitive recursion [lo], [as], [19], [20], [23]. Several extensions of the GRZEGORZCYK hierarchy through the multiply recursive functions of PETER have been proposed [28], [ah], [ll]. Several refinements of the hierarchy have been considered as well [26], [9], [16]. I n these refinements certain closure properties of the original GRZEOORCZYK classeg are sacrificed ; in particular the refined classes are not closed under composition of functions. I n this paper we define elementary-honest classes of recursive functions which retain all of the closure properties of the original GRZEGORCZYK classes including closure under composition. Elementary-honest classes are the functions elementary in ffor recursive functions f whose graphs correspond to elementary predicates. We then show that between any two GRZEGORCZYK classes containing the elementary functions there are densely ordered elementary-honest classes and infinitely many such classes which are setwise incomparable.The existence of densely ordered chains of classes emphasizes the fact that ordinal length of a hierarchy is not a particularly significant measure of either the extent or the refinement of a hierarchy. One can embed a hierarchy of any countable order type into a dense chain. (A similar observation about primitive recursive degrees is made by FEFERMAN [ 131 .) Moreover, the existence of incomparable classes suggests that efforts to construct totally ordered hierarchies of recursive functions are not likely to lead to natural classifications.
Grzegorczyk classes and Turing machinesWe assume the reader is familiar with the definition of the elementary functions 6' (which equals GRZEGORCZYR'S class as), and their closure properties under such operations as composition of functions, explicit transformation, limited recursion, limited sum and limited product [14]. We also assume familiarity with an arithmetization of TURING machines such as is carried out by R. W. RITCHIE, or DAVIS [26], [12]. 1) This research was supported in part by NSF Grant GP-2880, ARPA grant SD-146, and 2) We use L L C" to denote proper inclusion, and ''2" for improper inclusion. DSR 79457 under Project MAC.
