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

It is known that for non-uniform circuits class we have $\textrm{BPAC}^0=\textrm{AC}^0$:


  Miklós Ajtai, Michael Ben-Or: A Theorem on Probabilistic Constant Depth Computations STOC 1984: 471-474


Are generalizations of this theorem known ? For instance, do we know if $\mathrm{BPNC}^1=\mathrm{NC}^1$ (still in the non-uniform setting)? This last question seems somehow non trivial to me since it seems plausible that for instance $s,t\textrm{-Connectivity}$ is in $\textrm{BPNC}^1$.

A relevant post on the subject: https://mathoverflow.net/questions/35184/use-of-randomness-in-constant-parallel-time
Most nonuniform complexity classes—$\mathrm{NC^1}$ included—are closed under the $\mathrm{BP}$ operator by the same argument as $\mathrm{BPP\subseteq P/poly}$: using the Chernoff–Hoeffding bound, the probability of error can be reduced below $2^{-n}$ by running $O(n)$ instances of the circuit with independent random bits in parallel, and taking a majority vote; then by the union bound, a sequence of random bits gives the correct result for all $2^n$ inputs of length $n$ simultaneously with nonzero probability, and in particular, there exists such a sequence. We can hardwire it into the circuit.

This argument applies to any class of circuits that is closed under taking majority of $O(n)$ parallel copies of a circuit, and fixing input gates to constants. In practice, this means any decent nonuniform class above $\mathrm{TC^0}$, as majority is computable in $\mathrm{TC^0}$.

The proof is more complicated for $\mathrm{AC^0}$, because this class does not compute the majority function. (While I haven’t seen the Ajtai and Ben-Or paper, I’d guess they use some sort of approximate majority.)