What is the smallest class of reductions under which there is a $\mathsf{P}$-complete problem? It is common to define $P$-completeness with respect to log-space many-one reductions.

I am looking for a complexity class $C \subseteq \mathsf{L}$ such that 
there are $\mathsf{P}$-complete problems w.r.t. many-one $C$-reductions. 


  What is the smallest known many-one reduction class $C$ 
  such that HornSAT is complete for $\mathsf{P}$ under $C$-reductions?


The question was originally posted on CS with no answer.
It is easy to show that the Circuit Value Problem is complete for $\mathsf{P}$ 
under $\mathsf{AC^0}$ reductions (see  András's comment below).

For an easier example consider 
$$A = \{\langle M,x,y\rangle \mid \text{$M$ accepts $x$ in $|y|$ steps} \}$$

If a class of reductions $C$ contains constant functions, pairing of strings, and functions where the size of their output can bound any polynomial;
then $A$ is complete for $\mathsf{P}$ w.r.t. $C$.