Easy Constructions in Complexity Theory: Gap and Speed-up Theorems

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." Unspecified notation follows Rogers [12]. Xi(pt is any standard indexing of the partial recursive functions. N is the set of all nonnegative integers. XiDi is a canonical indexing of all finite subsets of N: from i we can list Di and know when the listing is completed. Similarly, XiFt is a canonical indexing of all finite functions defined (exactly) on some initial segment {0, 1, 2, • • • , n}. X./bi is any Blum measure of computational complexity or resource. Specifically, for all i, domain Oi=domain fa, and the ternary relation <P¿(x)^jy is decidable (recursive). Intuitively, 0¿(x) might be the amount of time or space used by Turing machine /' or the ¿th ALGOL Received by the editors August 15, 1971 and, in revised form, May 8, 1972. A MS (MOS) subject classifications (1970). Primary 68A20, 02-02; Secondary 02F20, 94A30, 94-02.