Resource-bounded martingales and computable Dowd-type generic sets

Abstract

Martin-Löf defined algorithmic randomness, the use of martingales in this definition and several variants were explored by Schnorr, and Lutz introduced resource-bounded randomness. Ambos-Spies et al. have studied resource-bounded randomness under timebounds and have shown that resource-bounded randomness implies resource-bounded genericity. While the genericity of Ambos-Spies is based on time-bound on finite-extension strategy, the genericity of Dowd, the main topic of this paper, is based on an analogy of the forcing theorem. For a positive integer r, an oracle D is r-generic in the sense of Dowd (r-Dowd) if the following holds: If a certain formula F on an exponential-sized portion of D is a tautology then a polynomial-sized subfunction of D forces F to be a tautology. Here, r is the number of occurrences of query symbols in F . The relationship between resource-bounded randomness and Dowd-type genericity has been not clear so far. We show that there exists a primitive recursive function t(n) with the following property: Every t(n)-random set is r-Dowd for each positive integer r. A proof is done by means of constructing resource-bounded martingales. In our former paper, we left an open problem whether there exists a primitive recursive set which is r-Dowd for every positive integer r. In our recent work, we give an affirmative answer to the problem. The main theorem of the present paper gives an alternative proof of the affirmative answer.