Extremal numberings and fixed point theorems

Abstract

We consider so‐called extremal numberings that form the greatest or minimal degrees under the reducibility of all A‐computable numberings of a given family of subsets of N$\mathbb {N}$, where A is an arbitrary oracle. Such numberings are very common in the literature and they are called universal and minimal A‐computable numberings, respectively. The main question of this paper is when a universal or a minimal A‐computable numbering satisfies the Recursion Theorem (with parameters). First we prove that the Turing degree of a set A is hyperimmune if and only if every universal A‐computable numbering satisfies the Recursion Theorem. Next we prove that any universal A‐computable numbering satisfies the Recursion Theorem with parameters if A computes a non‐computable c.e. set. We also consider the property of precompleteness of universal numberings, which in turn is closely related to the Recursion Theorem. Ershov proved that a numbering is precomplete if and only if it satisfies the Recursion Theorem with parameters for partial computable functions. In this paper, we show that for a given A‐computable numbering, in the general case, the Recursion Theorem with parameters for total computable functions is not equivalent to the precompleteness of the numbering, even if it is universal. Finally we prove that if A is high, then any infinite A‐computable family has a minimal A‐computable numbering that satisfies the Recursion Theorem.

On computable numberings of families of Turing degrees

“…(cf., e.g., Ershov's and Mal'tsev's works [1,4,5]), and they and their generalizations have been studied really intensively for the last two decades (cf., e.g., papers by Arslanov, Badaev, Goncharov, Jain, Nessel, Sorbi, etc. [6][7][8][9][10][11]). In addition, the theory of complete numberings allows to consider many important results of Muchnik [12], Myhill [13], Rogers [14], and Smullyan [15] from a general position, which previously seemed unrelated to each other.…”

On non-principal arithmetical numberings and families

“…Another key property of the Gödel numbering is that for any partially computable function ψ there exists a computable function f such that, for every x, W f (x) = W ψ(x) if ψ(x) converges, and W f (x) = ∅ otherwise. This property called by Mal'tsev [5,6] the completeness (with respect to ∅) is also actively studied in the theory of numberings (cf., e.g., [1,[7][8][9][10][11][12][13]) and was used by Ershov [14] to prove Kleene's recursion theorems in arbitrary (not necessarily computable) numberings (i.e. surjective mappings from N onto nonempty countable sets).…”

Two theorems on minimal generally-computable numberings

“…Гончаровым [20] эта проблема была решена для Σ 0 n -вычислимых семейств, где n 2, а в [28] это решение было обобщено на A-вычислимые семейства для произвольного оракула A высокой степени. В [29] было доказано, что если оракул A вычисляет невычислимое в.п. множество, то любое A-вычислимое семейство обладает хотя бы одной минимальной A-вычислимой нумерацией.…”