From logarithmic advice to single-bit advice

Building on Barak's work (Random'02), Fortnow and Santhanam (FOCS'04) proved a time hierarchy for probabilistic machines with one bit of advice. Their argument is based on an implicit translation technique, which allow to translate separation results for short (say logarithmic) advice (as shown by Barak) into separations for a single-bit advice. In this note, we make this technique explicit, by introducing an adequate translation lemma.

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[2]  Lance Fortnow,et al.  Hierarchy theorems for probabilistic polynomial time , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[3]  Boaz Barak.  A Probabilistic-Time Hierarchy Theorem for "Slightly Non-uniform" Algorithms , 2002, RANDOM.