Does Looking Inside a Circuit Help?

The Black-Box Hypothesis, introduced by Barak et al. [BGI12], states that any property of boolean functions decided efficiently (e.g., in BPP) with inputs represented by circuits can also be decided efficiently in the black-box setting, where an algorithm is given an oracle access to the input function and an upper bound on its circuit size. If this hypothesis is true, then P 6= NP. We focus on the consequences of the hypothesis being false, showing that (under general conditions on the structure of a counterexample) it implies a non-trivial algorithm for Circuit-SAT. More specifically, we show that if there is a property F of boolean functions such that F has high sensitivity on some input function f of subexponential circuit complexity (which is a sufficient condition for F being a counterexample to the Black-Box Hypothesis), then Circuit-SAT is solvable by a subexponential-size circuit family. Moreover, if such a counterexample F is symmetric, then Circuit-SAT ∈ P/poly. These results provide some evidence towards the conjecture (made in this paper) that the Black-Box Hypothesis is false if and only if Circuit-SAT is easy.