Levels of uniformity. (English) Zbl 1441.03028

Kuyper, Rutger

Notre Dame J. Formal Logic 60, No. 1, 119-138 (2019).

Author’s abstract: 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.

Reviewer: Liang Yu (Nanjing)

MSC:

03D30	Other degrees and reducibilities in computability and recursion theory
03D32	Algorithmic randomness and dimension
03G10	Logical aspects of lattices and related structures
03B20	Subsystems of classical logic (including intuitionistic logic)

Keywords:

Medvedev degrees; Muchnik degrees; algorithmic randomness