By Circuit Minimization, I am referring to the following decision problem.
Input: A bit string $x$ and a number $k$.
Question: Does there exist a Boolean Circuit $C$ of size at most $k$ that prints $x$?
By $C$ prints $x$, I mean that the following are satisfied:
(1) The circuit $C$ has $\lceil \log_2(\vert x \vert)\rceil$ inputs.
(2) Consider lexicographically ordering bit strings of length $\lceil \log_2(\vert x \vert)\rceil$. For each $0 \leq i < \vert x \vert$, let $w_i$ denote the $i$th bit string and $x_i$ denote the $i$th bit of $x$. We have $C(w_i) = x_i$.
Is Circuit Minimization $P$-hard under logspace reductions?
I know that others have asked if this and related problems are $NP$-hard, but I am wondering if it is hard for any smaller complexity classes such as $P$ or $L$ under appropriate kinds of reductions.
If this is an open problem, then what are the implications if it is $P$-hard?