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Revision #1 to TR17-091 | 26th April 2018 07:45
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#### Small bias requires large formulas

**Abstract:**
A small-biased function is a randomized function whose distribution of truth-tables is small-biased. We demonstrate that known explicit lower bounds on (1) the size of general Boolean formulas, (2) the size of De Morgan formulas, and (3) correlation against small De Morgan formulas apply to small-biased functions. As a consequence, any strongly explicit small-biased generator is subject to the best-known explicit formula lower bounds in all these models.

On the other hand, we give a construction of a small-biased function that is tight with respect to lower bound (1) for the relevant range of parameters. We interpret this construction as a natural-type barrier against substantially stronger lower bounds for general formulas.

**Changes to previous version:**
Corrected a mistake in the quoted high-probability shrinkage lemma; Generalized Proposition 2; fixed some inaccuracies in calculations and references.

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TR17-091 | 17th May 2017 06:37
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#### Small bias requires large formulas

**Abstract:**
A small-biased function is a randomized function whose distribution of truth-tables is small-biased. We demonstrate that known explicit lower bounds on the size of (1) general Boolean formulas, (2) Boolean formulas of fan-in two, (3) de Morgan formulas, as well as (4) correlation lower bounds against small de Morgan formulas apply to small-biased functions. As a consequence, any strongly explicit small-biased generator is subject to the best known explicit formula lower bounds in all these models.

On the other hand, we give a construction of a small-biased function that is tight with respect to lower bounds (1) and (2) for the relevant range of parameters. We interpret this construction as a natural-type barrier against substantially stronger lower bounds for general formulas.