Srikanth Srinivasan
ABSTRACT
The δ-Coin Problem is the problem of distinguishing between a sequence of coin tosses that come up Heads with probability either 1+δ/2 or 1−δ/2. The computational complexity of this problem in various models has been studied in many previous works with various applications related to derandomization, hierarchy theorems, cryptography and meta-complexity.
In this paper, we construct improved small-depth explicit formulas for the coin problem. Specifically, we construct explicit formulas of optimal size exp(O(d(1/δ)d−1)) and information-theoretically optimal sample complexity O(1/δ2) (the sample complexity is the number of coin tosses supplied to the formulas) for this problem, as long as 1/δ ≥ dC· d for a large enough absolute constant C. Previous constructions of size-optimal AC0 formulas for the coin problem were either randomized (and hence non-explicit) or had a much worse sample complexity of (1/δ)Ω(d).
Our improved construction yields better Fixed-Depth Size Hierarchy theorems for uniform classes of small-depth circuits with AND, OR and ⊕ gates.
Our techniques deviate considerably from previous explicit constructions with non-trivial sample complexity due to Limaye, Sreenivasaiah, Venkitesh and the two authors (SICOMP 2021). While the approach there was to derandomize randomized formula constructions based on results of O’Donnell and Wimmer (ICALP 2007) and Amano (ICALP 2009), we instead look to derandomize a randomized circuit construction due to Rossman and Srinivasan (Theory of Computing 2019). This leads us to the problem of constructing certain pseudorandom graphs, which we do explicitly using ideas of Viola (Computational Complexity 2014) involving an iterative use of expander graphs. The constructions of these graphs, which are related to dispersers, may be independently interesting.
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Index Terms
- Optimal Explicit Small-Depth Formulas for the Coin Problem
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