Randomness, Stochasticity, and Approximations

Polynomial time unsafe approximations for intractable sets were introduced by Meyer and Paterson [9] and Yesha [19] respectively. The question of which sets have optimal unsafe approximations has been investigated extensively, see, e.g., [1, 5, 15, 16]. Recently, Wang [15, 16] showed that polynomial time random sets are neither optimally unsafe approximable nor \Delta-levelable. In this paper, we will show that: (1) There exists a polynomial time stochastic set in E 2 which has an optimal unsafe approximation. (2). There exists a polynomial time stochastic set in E 2 which is \Delta-levelable. The above two results answer a question asked by Ambos-Spies and Lutz et al. [3]: Which kind of natural complexity property can be characterized by p-randomness but not by p-stochasticity? Our above results also extend Ville's [13] historical result. The proof of our first result shows that, for Ville's stochastic sequence, we can find an optimal betting strategy (prediction function) such that w..