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Implicit measurements of dynamic complexity properties and splittings of speedable sets

Published online by Cambridge University Press:  12 March 2014

Michael A. Jahn
Michael A. Jahn*
Affiliation:
Digipen Institute of Technology, 5001 150th Ave. N.E., Redmond, WA 98052, USA E-mail: mjahn@digipen.edu
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Abstract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 3 , September 1999 , pp. 1037 - 1064
Copyright
Copyright © Association for Symbolic Logic 1999

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