arXiv subjects

Dávid Natingga

Publications and source records attributed to Dávid Natingga.

2 recordsLinked to original sources

arXiv2019

α degrees as an automorphism base for the α-enumeration degrees

Selman's Theorem in classical Computability Theory gives a characterization of the enumeration reducibility for arbitrary sets in terms of the enumeration reducibility on the total sets: $A \le_e B \iff \forall X [X \equiv_{e} X \oplus \overline{X} \land B \le_{e} X \oplus \overline{X} \implies A \le_{e} X \oplus. \overline{X} ]$. This theorem implies directly that the Turing degrees are an automorphism base of the enumeration degrees. We lift the classical proof to the setting of the $α$-Computability Theory to obtain the full generalization when $α$ is a regular cardinal and partial results for a general admissible ordinal $α$.

math.LO↗

arXiv2019

Kalimullin Pair and Semicomputability in $α$-Computability Theory

We generalize some results on semicomputability by Jockusch \cite{jockusch1968semirecursive} to the setting of $α$-Computability Theory. We define an $α$-Kalimullin pair and show that it is definable in the $α$-enumeration degrees $\mathcal{D}_{αe}$ if the projectum of $α$ is $α^*=ω$ or if $α$ is an infinite regular cardinal. Finally using this work on $α$-semicomputability and $α$-Kalimullin pairs we conclude that every nontrivial total $α$-enumeration degree is a join of a maximal $α$-Kalimullin pair if $α$ is an infinite regular cardinal.

math.LO↗