David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Journal of Symbolic Logic 64 (3):1037-1064 (1999)
We prove that any speedable computably enumerable set may be split into a disjoint pair of speedable computably enumerable sets. This solves a longstanding question of J.B. Remmel concerning the behavior of computably enumerable sets in Blum's machine independent complexity theory. We specify dynamic requirements and implement a novel way of detecting speedability-by embedding the relevant measurements into the substage structure of the tree construction. Technical difficulties in satisfying the dynamic requirements lead us to implement "local" strategies that only look down the tree. The (obvious) problems with locality are then resolved by placing an isomorphic copy of the entire priority tree below each strategy (yielding a self-similar tree). This part of the construction could be replaced by an application of the Recursion Theorem, but shows how to achieve the same effect with a more direct construction
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References found in this work BETA
Rod Downey & Michael Stob (1993). Splitting Theorems in Recursion Theory. Annals of Pure and Applied Logic 65 (1):1-106.
Frank A. Bäuerle & Jeffrey B. Remmel (1994). On Speedable and Levelable Vector Spaces. Annals of Pure and Applied Logic 67 (1-3):61-112.
J. C. E. Dekker (1971). Two Notes on Vector Spaces with Recursive Operations. Notre Dame Journal of Formal Logic 12 (3):329-334.
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