Probabilistic Martingales and BPTIME Classes

We define probabilistic martingales based on randomized approximation schemes, and show that the resulting notion of probabilistic measure has several desirable robustness properties. Probabilistic martingales can simulate the "betting games" of [BMR+ 98], and can cover the same class that a "natural proof" diagonalizes against, as implicitly already shown in [RSC95]. The notion would become a full-fledged measure on bounded-error complexity classes such as BPP and BPE if it could be shown to satisfy the "measure conservation" axiom of [Lut92] for these classes. We give a sufficient condition in terms of simulation by "decisive" probabilistic martingales that implies not only measure conservation, but also a much tighter bounded error probabilistic time hierarchy than is currently known. In particular it implies BPTIME[O(n)] = BPP, which would stand in contrast to recent claims of an oracle A giving BPTIMEA [O(n)] = BPPA . This paper also makes ...

• Type: Conference
• Year: 1998
• Where: COCO
• Authors: Kenneth W. Regan, D. Sivakumar
• Keywords: Algorithms, COCO 1998, Measure Conservation, Probabilistic Martingales, Probabilistic Measure