Integer-valued martingales and cl-Turing reductions

Abstract

We prove that a sequence X is integer-valued random (IVR) if and only if for every cl-tt reduction Φ, there are only finitely many oracles that compute X via Φ. A well-known property of integer-valued supermartingales is that under any cone, there is a subcone where the values are constant. We extend this property and prove that for a special integer-valued martingale M, which starts with initial capital 1 and always bets 1 on bit “1” if possible, and for any integer-valued supermartingale f, under any cone there is a subcone such that f actually simulates M. Using this property, we prove that there exists a non-IVR sequence that is cl-computable by only countably many oracles, as a complement to the first result.