Variants of Solovay reducibility
Ivan Titov
TL;DR
The paper investigates variants of Solovay reducibility for arbitrary reals, contrasting the original rational-translation formulation with real-translation and monotone variants. It proves that reductions defined via $ $-translation functions on rationals induce corresponding reductions via real-valued translations, and shows that the monotone variant is strictly stronger than the original on general reals. It further connects these reducibilities to randomness notions, establishing upward-closure properties for Martin-Löf and Schnorr randomness under the open and local variants, respectively, and demonstrates separations (e.g., computable vs non-left-c.e. reals) to illuminate the landscape of these notions. The results consolidate a coherent hierarchy among Solovay-type reducibilities and their total/monotone variants, with implications for left-c.e. reals and total Solovay reducibility in relation to Martin-Löf and Schnorr randomness. These findings advance understanding of how algorithmic randomness interacts with generalized reducibility concepts.
Abstract
Outside of the left-c.e. reals, Solovay reducibility is considered to be behaved badly [10.1007/978-0-387-68441-3]. Proposals for variants of Solovay reducibility that are better suited for the investigation of arbitrary, not necessarily left-c.e. reals were made by Rettinger and Zheng [10.1007/978-3-540-27798-9_39], and, recently, by Titov [10.11588/heidok.00034250] and by Kumabe and co-authors [10.4115/jla.2020.12.2; 10.3233/COM-230486]. These variants all coincide with the original version of Solovay reducibility on the left-c.e. reals. Furthermore, they are all defined in terms of translation functions. The latter translate between computable approximations in the case of Rettinger and Zheng, are monotone in the case of Titov, and are functions between reals in the case of Kumabe et al. In what follows, we derive new results on the mentioned variants and their relation to each other. In particular, we obtain that Solovay reducibility defined in terms of translation function on rationals implies Solovay reducibility defined in terms of translation functions on reals, and we show that the original version of Solovay reducibility is strictly weaker than its monotone variant. Solovay reducibility and its variants mentioned so far have tight connections to Martin-Löf randomness, the strongest and most central notion of a random sequence. For the investigation of Schnorr randomness, total variants of Solovay reducibility have been introduced by Merkle and Titov [10.48550/arXiv.2407.14869] in 2022 and, independently, by Kumabe et al. [10.3233/COM-230486] in 2024, the latter again via real-valued translation functions. In what follows, we show that total Solovay reducibility defined in terms of rational functions implies total Solovay reducibility defined in terms of real functions.