Some properties of classes of minimal numberings of arithmetical set families
Sh. D. Nodirov, M. Kh. Faĭzrahmanov · Mathematics and Theoretical Computer Science · 2024
We prove that for each u ⩾ 2 the class of all single-valued Σ0 ucomputable numberings of any infinite family of total functions is effectively infinite and the class of all its Σ0 u-1-computable numberings is generated by the downward closure with respect to the reducibility of the set of all infinite direct sums of uniformly Σ0 u-1-computable sequences of its single-valued numberings. It is established that if u > 2, then the class of all Σ0 u-computable numberings of any infinite family is generated by infinite direct sums of uniformly Σ0 u-computable and uniformly Σ0 u-minimal sequences of its numberings.