Conjunctive reducibilities and completeness

Irakli Chitaia; Keng Meng Ng; Roland Omanadze; Andrea Sorbi

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arXiv · 2606.00845

Conjunctive reducibilities and completeness

Irakli Chitaia·Keng Meng Ng·Roland Omanadze·Andrea Sorbi

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Abstract

In this article we study the notion of completeness for conjunctive reducibilities. We investigate the relationship between $c$-completeness and $r$-completeness of computably enumerable (c.e.) sets with respect to various strong reducibilities $\le_r$. By using simplicity properties of sets, we prove that there exist c.e. sets that are simultaneously $Q$-complete and $bd$-complete, yet fail to be $c$-complete. Similarly, there exist c.e. sets that are simultaneously $Q$-complete and $bwtt$-complete (respectively, $btt$-complete) but not $c$-complete. Furthermore, we study two restrictions of $c$-reducibility, namely $c_1$- and $c_{1,N}$-reducibility, and show that they are distinct on the c.e. sets. Nevertheless, we prove that the notions of completeness for $c$, $c_1$, and $c_{1,N}$ coincide.

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BibTeX RIS

Irakli Chitaia, Keng Meng Ng, Roland Omanadze, Andrea Sorbi. 2026-05-30. Conjunctive reducibilities and completeness. https://arxiv.org/abs/2606.00845

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