## Characterizing the Grzegorczyk hierarchy by safe recursion (1999)

### BibTeX

@MISC{Wirz99characterizingthe,

author = {Marc Wirz and Marc Wirz},

title = {Characterizing the Grzegorczyk hierarchy by safe recursion},

year = {1999}

}

### OpenURL

### Abstract

We show how the charaterization of the polytime functions by Bellantoni and Cook [1] can be extended to characterize any stage of the Grzegorzyk hierarchy above the second, thus proposing an answer to a problem posted by Clote [3]. This is done by allowing an arbitrary fixed number of distinct positions for variables instead of only two as in the original work of Bellantoni and Cook. As turned out after writing down this paper, comparable results were also proved by Bellantoni and Niggl [2]. Keywords: Recursion theory, Complexity theory, Grzegorzcyk. 1 Introduction Bellantoni and Cook [1] characterized the polytime functions by distinguishing between two sorts of arguments of functions, called normal and safe arguments. Recursion is only allowed over normal arguments, whereas the recursively computed values must be inserted in a safe position, and function composition is defined accordingly. Related tiering notions also appeared elsewhere, e.g. in Leivant [5] or Simmons [10]. Clote [...

### Citations

179 | A new recursion-theoretic characterization of the poly-time functions
- Bellantoni, Cook
- 1992
(Show Context)
Citation Context ...athematik, Universität Bern Neubrückstrasse 10, CH–3012 Bern, Switzerland wirz@iam.unibe.ch November 22, 1999 Abstract We show how the charaterization of the polytime functions by Bellantoni and C=-=ook [1]-=- can be extended to characterize any stage of the Grzegorzyk hierarchy above the second, thus proposing an answer to a problem posted by Clote [3]. This is done by allowing an arbitrary fixed number o... |

57 |
Ramified recurrence and computational complexity I: Word recurrence and poly-time
- Leivant
- 1994
(Show Context)
Citation Context ...guments, whereas the recursively computed values must be inserted in a safe position, and function composition is defined accordingly. Related tiering notions also appeared elsewhere, e.g. in Leivant =-=[5]-=- or Simmons [10]. Clote [3] gives a short review of some recent results and rises the problem of relating these concepts to the Grzegorczyk hierarchy (Problem 3.102). This paper proposes an answer to ... |

38 |
Classes of predictably computable functions
- Ritchie
- 1963
(Show Context)
Citation Context ...h we will prove below. However, Bellantoni and Cook characterized the polytime functions, rather than the second level E 2 of the Grzegorczyk hierarchy which equals linear time by a result of Ritchie =-=[7]. -=-This is, of course, due to the fact they don’t use primitive recursion, but recursion on binary notation. If their definitions are adapted for unary notation of integers, their class would correspon... |

37 | Computation models and function algebras
- Clote
- 1999
(Show Context)
Citation Context ...tion of the polytime functions by Bellantoni and Cook [1] can be extended to characterize any stage of the Grzegorzyk hierarchy above the second, thus proposing an answer to a problem posted by Clote =-=[3]-=-. This is done by allowing an arbitrary fixed number of distinct positions for variables instead of only two as in the original work of Bellantoni and Cook. As turned out after writing down this paper... |

10 | and K-H Niggl. Ranking primitive recursions: The low Grzegorczyk classes revisited
- Bellantoni
- 1999
(Show Context)
Citation Context ...ct positions for variables instead of only two as in the original work of Bellantoni and Cook. As turned out after writing down this paper, comparable results were also proved by Bellantoni and Niggl =-=[2]-=-. Keywords: Recursion theory, Complexity theory, Grzegorzcyk. 1 Introduction Bellantoni and Cook [1] characterized the polytime functions by distinguishing between two sorts of arguments of functions,... |

8 |
Rekursionszahlen und die Grzegorczyk-Hierarchie. Archiv fur mathematische Logik und Grundlagenforschung, 12:85{97
- Schwichtenberg
- 1969
(Show Context)
Citation Context ...erarchy (Problem 3.102). This paper proposes an answer to that question. Many characterizations of the Grzegorczyk hierarchy are based on controlling the depth of nested recursions, as e.g. in [6] or =-=[9]-=-. Even its usual definition can be seen from this angle: If we ignore the instances of bounded recursion as negligible then the functions in the n + 1-st level of the Grzegorczyk hierarchy are exactly... |

7 |
The Realm of Primitive Recursion. Archive for Mathematical Logic, 27:177–188
- Simmons
- 1988
(Show Context)
Citation Context ...s the recursively computed values must be inserted in a safe position, and function composition is defined accordingly. Related tiering notions also appeared elsewhere, e.g. in Leivant [5] or Simmons =-=[10]-=-. Clote [3] gives a short review of some recent results and rises the problem of relating these concepts to the Grzegorczyk hierarchy (Problem 3.102). This paper proposes an answer to that question. M... |

3 | A foundational delineation of computational feasability - Leivant - 1991 |

3 |
Hierarchies of primitive recursive functions
- Parsons
- 1968
(Show Context)
Citation Context ...czyk hierarchy (Problem 3.102). This paper proposes an answer to that question. Many characterizations of the Grzegorczyk hierarchy are based on controlling the depth of nested recursions, as e.g. in =-=[6]-=- or [9]. Even its usual definition can be seen from this angle: If we ignore the instances of bounded recursion as negligible then the functions in the n + 1-st level of the Grzegorczyk hierarchy are ... |

2 |
Subrecursion: Functions and Hierarchies, vol. 9 of Oxford Logic Guides
- Rose
- 1984
(Show Context)
Citation Context ...ovided that f(0, x) = g(x) f(y + 1, x) = h(y, x, f(y, x)), f(y, x) ≤ j(y, x). We conclude this section recalling some properties of the functions En. Proofs, when not straightforward, can be found i=-=n [8]. Remark 1 For-=- all n ≥ 1: i) En(x) ≥ x + 1 ii) En(x + 1) ≥ En(x) + 1 3siii) En(x) ≥ 2x iv) En(x) ≥ x 2 v) En(x) + En(y) ≤ � En(x + y), if x, y > 0 En(x + y) + 2, else vi) If f is in E n+1 then there i... |