# What is the best reduction we know from flavors of $SAT$ to $MCSP$?

Consider $$\mathcal P(n,m)$$ to be a class from the set $$\{k\mbox{-}\mathsf{SAT}(n,m),\mathsf{CIRCUITSAT}(n,m)\}$$ where $$k$$ is fixed, $$n$$ is number of variables and $$m$$ is number of clauses.

Denote $$\mathsf{MCSP}[s(t)]$$ to the class of problems which have a circuit of size $$s(t)$$ when input a truth table of size $$t$$

What is the best reduction that we know from a member in $$\mathcal P(n,m)$$ to $$\mathsf{MCSP}[s(t)]$$ when $$t=f(n)$$ for a function $$f:\mathsf{Z_{\geq0}}\rightarrow\mathsf{Z_{\geq0}}$$?

I am looking for the correct parametrization between the problems that we know (best $$s(t)=s(f(n))$$).

I know some reductions provide separation such as $$\mathsf{EXP}\neq\mathsf{ZPP}$$ or $$\mathsf{PSPACE}\neq\mathsf{ZPP}$$ or $$\mathsf{NP}\not\subseteq\mathsf{P/poly}$$. Literature says these are $$\mathsf p$$, $$\mathsf{LOGSPACE}$$ or $$\mathsf{AC}^0$$ reductions respectively and the question is what would the parameters of $$f$$ and $$s$$ be?