Über Darstellungen der Elementaren Funktionen II

Rödding, Dieter

doi:10.1007/bf01977247

Cited by 2 publications

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“…The argument of theorem 2.1 then shows that every elementary function is lower elementary in 2". This is also given by several of the normal forms of [8] and [9]. The converse is obvious.…”

Section: ) I N [Lo] Skolem Calls "Lower Elementary Functions" the Fmentioning

confidence: 89%

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Hierarchies of Primitive Recursive Functions

Parsons¹

1968

Mathematical Logic Qtrly

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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.

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“…The argument of theorem 2.1 then shows that every elementary function is lower elementary in 2". This is also given by several of the normal forms of [8] and [9]. The converse is obvious.…”

Section: ) I N [Lo] Skolem Calls "Lower Elementary Functions" the Fmentioning

confidence: 89%

“…It seems very unlikely that 81 = or even 8 0 2 P. (IV). Then the functions Y, , Y; clearly belong to C (9).…”

Section: Explicit Definitionmentioning

confidence: 99%