Given the following language: $$ L=\left\{\,(\,C,\,\{v_i\}_{i=1}^m, \,x\,) \enspace :\enspace \substack{C(x) \text{ is a boolean circuit with } m \text{ gates} \\i\text{'th gate value is } v_i \text{ on input } x} \right\} $$ I need to find out to what known class $L$ is complete to with respect to logspace reductions.
I think that it is impossible to reduce $\text{CVAL}$ to $L$ because if we can compute gate's values $\{v_i\}_{i=1}^m$'s we could solve $\mathrm{CVAL}$ in logspace which would imply $\mathbf{P = LOG}$.
I also think that $L\in\mathbf{P}$ because we can calculate $C(x)$ and of course all the gates $\{v_i\}_{i=1}^m$ using BFS as in computing every boolean circuit value.
Please advise. Thank you.