Hierarchies of Primitive Recursive Functions

Abstract

then the definition of 97 is of degree max ( p ) q) + 1 . l) l ) We omit from consideration the schema for recursion without pareymeters. It can be accommodated into our arguments by simplified versions of the arguments for (V), or it can be omitted as a primitive schema without affecting the degrees or levels of the functiohs. OECARLES PARSONSA function g~ is of degree p if it has a definition of degree p but none of degree < p . Let . @ be the class of functions of degree $ p .We will say that a function a, is explicitly definable from functions of a class &'if it can be expressed by a term constructed from variables and symbols for functions in S. The argument of [4], pp. 220-221, shows that if 2 contains all the functions U; and is closed under (IT), then &' is closed under explicit definition.A more sophisticated criterion of the complexity of a definition is as follows: Primitive recursion represents an essential increase in complexity over explicit definition because of the element of iteration it introduces, so that if a, is defined by (V), the computation of a, for a given argument requires that X be computed for arguments which themselves depend on prior computations of X . We might thus seek to segregate the functions which are "iterated" in a primitive recursion from those which are not.If g~ is defined by (V) with y E @' , X E 9, then there are terms 6 and t composed of variables 0, S , and symbols for functions defined by primitive recursion of degrees s p and $ q respectively, such that (v*) cp(0, $) = 2($), (% stands for a sequence of variables such as yl, . . . , yn) .We say that an occurrence of a function symbol is iterated in (V*) if it is an oecurrence in t(x, z , y) with z within its scope.We now define an application of (I), (11) or (111) to be of level 0; if a, is defined by (IV) where the definitions of X , yl, . . ., w2 have levels p , q l , . . ., q l , the defini-1) If levels are defined as suggested in the footnote on p. 358, this shows tp E Ymax(p~2), which suffices for the proof of theorem 2.1.

Function operators spanning the arithmetical and the polynomial hierarchy

“…This technique goes back to the early sixties of the past century when degrees of primitive recursive functions with respect to the operator pr were studied. Related results and references can be found, e.g., in [3,19,22,24]. In order to avoid confusions with the Turing, or other degrees, throughout this paper we shall denote nesting degrees (with respect to arbitrary operators ω) as (ω-)levels.…”

Ranking Primitive Recursions: The Low Grzegorczyk Classes Revisited

“…Coerced recurrence seems to be more powerful than stratified recurrence. To restrict it, Leivant uses an approach similar to Parson's earlier definition [20] p. 358: one separately determines whether or not the step function uses the critical term. Leivant defined a hierarchy by putting the predicative recurrence rules at levels 0 and 1, together with the Parsonlike rules at levels 2 and above [12] §4.…”

“…This simple definition has the advantage of coinciding with the Grzegorczyk hierarchy E r at and above the elementary functions: E r+1 = D r for r ≥ 2, where D r are the primitive recursive derivations with degree at most r. As discussed by Clote [6], the characterization for r ≥ 3 was shown by Schwichtenberg [23], and later the case of r = 2 was shown by Müller [15]. Another possible classification was given prior to Müller's result by Parsons [20], who referred to whether or not the step function (i.e. the function h in f (x + 1) = h(x, f (x))) accesses the critical value.…”

Plain Bases for Classes of Primitive Recursive Functions

“…The existence of a finite basis for E has been shown for the first time in [19], a basis of nineteen functions for E has been introduced in [17] and a basis of seven functions for the unary elementary functions has been introduced in [14]. All of these papers are based on more or less complex codings, so that the bases obtained comprehend some uncommon functions.…”