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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Easy constructions in complexity theory: Gap and speed-up theorems

Author: Paul Young
Journal: Proc. Amer. Math. Soc. 37 (1973), 555-563
MSC: Primary 68A20; Secondary 02F15
MathSciNet review: 0312768
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Abstract: Perhaps the two most basic phenomena discovered by the recent application of recursion theoretic methods to the developing theories of computational complexity have been Blum's speed-up phenomena, with its extension to operator speed-up by Meyer and Fischer, and the Borodin gap phenomena, with its extension to operator gaps by Constable. In this paper we present a proof of the operator gap theorem which is much simpler than Constable's proof. We also present an improved proof of the Blum speed-up theorem which has a straightforward generalization to obtain operator speed-ups. The proofs of this paper are new; the results are not. The proofs themselves are entirely elementary: we have eliminated all priority mechanisms and all but the most transparent appeals to the recursion theorem. Even these latter appeals can be eliminated in some ``reasonable'' complexity measures. Implicit in the proofs is what we believe to be a new method for viewing the construction of ``complexity sequences."

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Keywords: Recursive function theory, computational complexity, speedup theorems, gap theorems, Blum theory
Article copyright: © Copyright 1973 American Mathematical Society