Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

Sequences Having an Effective Fixed-Point Property

T. H. Payne
Transactions of the American Mathematical Society
Vol. 165 (Mar., 1972), pp. 227-237
DOI: 10.2307/1995883
Stable URL: http://www.jstor.org/stable/1995883
Page Count: 11
  • Read Online (Free)
  • Download ($30.00)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Sequences Having an Effective Fixed-Point Property
Preview not available

Abstract

Let $\alpha$ be any function whose domain is the set $N$ of all natural numbers. A subset $B$ of $N$ precompletes the sequence $\alpha$ if and only if for every partial recursive function (p.r.f.) $\psi$ there is a recursive function $f$ such that $\alpha f$ extends $\alpha \psi$ and $f\lbrack N - \operatorname{Dom} \psi \rbrack \subset B$. An object $e$ in the range of $\alpha$ completes $\alpha$ if and only if $\alpha^{-1}\lbrack \{e\} \rbrack$ precompletes $\alpha$. The theory of completed sequences was introduced by A. I. Mal'cev as an abstraction of the theory of standard enumerations. In this paper several results are obtained by refining and extending his methods. It is shown that a sequence is precompleted (by some $B$) if and only if it has a certain effective fixed-point property. The completed sequences are characterized, up to a recursive permutation, as the composition $F\varphi$ of an arbitrary function $F$ defined on the p.r.f.'s with a fixed standard enumeration $\varphi$ of the p.r.f.'s. A similar characterization is given for the precompleted sequences. The standard sequences are characterized as the precompleted indexings which satisfy a simple uniformity condition. Several further properties of completed and precompleted sequences are presented, for example, if $B$ precompletes $\alpha$ and $S$ and $T$ are r.e. sets such that $\alpha^{-1} \lbrack \alpha \lbrack S \rbrack \rbrack \neq N$ and $\alpha^{-1} \lbrack \alpha T \rbrack \rbrack \neq N$, then $B - (S \cup T)$ precompletes $\alpha$.

Page Thumbnails

  • Thumbnail: Page 
227
    227
  • Thumbnail: Page 
228
    228
  • Thumbnail: Page 
229
    229
  • Thumbnail: Page 
230
    230
  • Thumbnail: Page 
231
    231
  • Thumbnail: Page 
232
    232
  • Thumbnail: Page 
233
    233
  • Thumbnail: Page 
234
    234
  • Thumbnail: Page 
235
    235
  • Thumbnail: Page 
236
    236
  • Thumbnail: Page 
237
    237