$P$-complete problems and Logspace reductions It is known that HORN-3SAT is complete for $P$ under Logspace many-one reductions ($&lt;_L^m$). 

This implies that $\bar{A} &lt;_L^m A$ for any $P$-complete problem $A$, where $\bar{A}$ means the complement of $A$. This immediately follows from that fact that $P$ is closed under complement and the $P$-hardness of $A$. 

My question is to find such Logspace many-one reduction $\bar{A} &lt;_L^m A$ for any $P$-complete problem $A$ (such as HORN-3SAT). The reduction must show the mapping between short certificates of $A$ and $\bar{A}$. 

EDIT: As David stated in his comment on the case of HORN-3SAT problem,  we know that Logspace function exists but it seems that it is hard to find explicitly.
The circuit value problem: input is a Boolean formula and a truth assignment to its variables; accept if, and only if, the input is well-formed and the formula evaluates to true under the given assignment. The complement is to accept iff the input is malformed or the formula is false.

The reduction is to check that the input is well-formed.  If so, output the negation of the formula, with the same truth assignment; if not, output either $X; X=\mathrm{t}$ or $X; X=\mathrm{f}$, depending on which way you're going.