On the relationship between the complexity, the degree, and the extension of a computable set
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We consider the equivalence relation A =c B (“A and B have the same time complexity”) ⇔ (for all time constructible f : A ∈ DTIME(f) ⇔ B ∈ DTIME(f)). In this paper we give a survey of the known relationships between this equivalence relation and degree theoretic and extensional properties of sets. Furthermore we illustrate the proof techniques that have been used for this analysis, with emphasis on those arguments that are of interest from the point of view of recursion theory. Finally we will discuss in the last section some open problems and directions for further research on this topic.
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