Abstract
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.
Keywords
- Time Complexity
- Polynomial Time
- Turing Machine
- Complexity Type
- Extensional Property
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Written under partial support by NSF-Grant CCR 8903398. Part of this research was carried out during a visit of the first author at the University. The first author would like to thank the Department of Computer Science at the University of Chicago for its hospitality.
Written under partial support by Presidential Young Investigator Award DMS-8451748 and NSF-Grant DMS-8601856.
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Maass, W., Slaman, T.A. (1990). On the relationship between the complexity, the degree, and the extension of a computable set. In: Ambos-Spies, K., Müller, G.H., Sacks, G.E. (eds) Recursion Theory Week. Lecture Notes in Mathematics, vol 1432. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086124
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DOI: https://doi.org/10.1007/BFb0086124
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