# Uniform derandomisation of circuit complexity classes

Let $\mathcal{C}$ be a complexity class and $\textrm{BP-}\mathcal{C}$ be the randomized counterpart of $\mathcal{C}$ defined in the same way as $\textrm{BPP}$ is defined with respect to $\textrm{P}$. More formally we provide polynomially many random bits and we accept an input iff the probability to accept is over $\frac{2}{3}$.

In a previous post, I asked if it was known whether the equality holds between $\mathcal{C}$ and $\textrm{BP-}\mathcal{C}$ for $\mathcal{C}$ a circuit complexity class. The answer is yes for all complexity classes expressive enough to compute Majority and for $\textrm{AC}^0$ for some other reason. Those results are however non-uniform and I would like to know:

1. Are uniform versions of those results studied or known? Any partial results ?

2. Do they imply long standing conjecture?

I believe that uniform derandomisation of $\textrm{P}/\textrm{poly}$ is exactly $\textrm{P}=\textrm{BPP}$ so I expect the answer to be "yes" but it is less clear to me what uniform derandomisation of small classes within the $\textrm{NC}$-hierarchy would imply.

• They imply circuit lower bounds? – Nikhil Mar 17 '17 at 10:21