Sequences having an effective fixed-point property

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 $f[N - \operatorname {Dom} \psi ] \subset B$. An object e in the range of $\alpha$ completes $\alpha$ if and only if ${\alpha ^{ - 1}}[\{ e\} ]$ 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 ${\alpha ^{ - 1}}[\alpha [S]] \ne N$ and ${\alpha ^{ - 1}}[\alpha [T]] \ne N$, then $B - (S \cup T)$ precompletes $\alpha$.