Two Sides of the Coin Problem

In the coin problem, one is given n independent flips of a coin that has bias β > 0 towards either Head or Tail. The goal is to decide which side the coin is biased towards, with high confidence. An optimal strategy for solving the coin problem is to apply the majority function on the n samples. This simple strategy works as long as β > Ω(1/ √ n). However, computing majority is an impossible task for several natural computational models, such as bounded width read once branching programs and AC0 circuits. Brody and Verbin [FOCS 2010] proved that a length n, width w read once branching program cannot solve the coin problem for β < O(1/(log n)3w). This result was tightened by Steinberger [CCC 2013] to O(1/(log n)w−2). As for the model of AC0 circuits, Aaronson [STOC 2010] proved that a depth d size s Boolean circuit cannot solve the coin problem for β < O(1/(log s)d+2). This work has two contributions: • We strengthen Steinberger result and show that any Santha-Vazirani source with bias β < O(1/(log n)w−2) fools length n, width w read once branching programs. In other words, the strong independence assumption in the coin problem is completely redundant in the model of read once branching programs. That is, the exact same result holds for a much more general class of sources. • We tighten Aaronson result and show that a depth d, size s Boolean circuit cannot solve the coin problem for β < O(1/(log s)d−1). Moreover, our proof technique is different and we believe that it is simpler and more natural. ∗Weizmann Institute of Science, Rehovot, Israel. {gil.cohen, anat.ganor}@weizmann.ac.il. Supported by an ISF grant and by the I-CORE Program of the Planning and Budgeting Committee. †Weizmann Institute of Science, Rehovot, Israel, and the Institute for Advanced Study, Princeton, NJ. [email protected]. Supported by an ISF grant, by the I-CORE Program of the Planning and Budgeting Committee and by NSF grant numbers CCF-0832797, DMS-0835373. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 23 (2014)