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Defining the Turing Jump

Richard A. Shore & Theodore A. Slaman

Bulletin of Symbolic Logic 7 (1):73-75 (2001)

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Levels of Uniformity.Rutger Kuyper - 2019 - Notre Dame Journal of Formal Logic 60 (1):119-138.details We introduce a hierarchy of degree structures between the Medvedev and Muchnik lattices which allow varying amounts of nonuniformity. We use these structures to introduce the notion of the uniformity of a Muchnik reduction, which expresses how uniform a reduction is. We study this notion for several well-known reductions from algorithmic randomness. Furthermore, since our new structures are Brouwer algebras, we study their propositional theories. Finally, we study if our new structures are elementarily equivalent to each other. Download Export citation Bookmark
The ω-Turing degrees.Andrey C. Sariev & Hristo Ganchev - 2014 - Annals of Pure and Applied Logic 165 (9):1512-1532.details In this paper we initiate the study of the ω-Turing reducibility between sequences of sets of natural numbers. We shall prove that the induced degree structure is an extension of the structure of the Turing degrees and that the two structures are closely connected, but different enough. Further we shall prove some definability results for the local theory of the newly defined structure. Download Export citation Bookmark
2-Minimality, jump classes and a note on natural definability.Mingzhong Cai - 2014 - Annals of Pure and Applied Logic 165 (2):724-741.details We show that there is a generalized high degree which is a minimal cover of a minimal degree. This is the highest jump class one can reach by finite iterations of minimality. This result also answers an old question by Lerman. Download Export citation Bookmark
A Reducibility Related To Being Hyperimmune-free.Frank Stephan & Liang Yu - 2014 - Annals of Pure and Applied Logic 165 (7-8):1291-1300.details The main topic of the present work is the relation that a set X is strongly hyperimmune-free relative to Y . Here X is strongly hyperimmune-free relative to Y if and only if for every partial X -recursive function p there is a partial Y -recursive function q such that every a in the domain of p is also in the domain of q and satisfies p (...) Download Export citation Bookmark
Degree structures: Local and global investigations.Richard A. Shore - 2006 - Bulletin of Symbolic Logic 12 (3):369-389.details The occasion of a retiring presidential address seems like a time to look back, take stock and perhaps look ahead.Institutionally, it was an honor to serve as President of the Association and I want to thank my teachers and predecessors for guidance and advice and my fellow officers and our publisher for their work and support. To all of the members who answered my calls to chair or serve on this or that committee, I offer my thanks as well. Your (...) Download Export citation Bookmark 4 citations
The automorphism group and definability of the jump operator in the $$\omega $$ ω -enumeration degrees.Hristo Ganchev & Andrey C. Sariev - 2021 - Archive for Mathematical Logic 60 (7):909-925.details In the present paper, we show the first-order definability of the jump operator in the upper semi-lattice of the \-enumeration degrees. As a consequence, we derive the isomorphicity of the automorphism groups of the enumeration and the \-enumeration degrees. Download Export citation Bookmark

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