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 Blumâs 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 Constableâs 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 âreasonableâ complexity measures.
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