Weakest reduction for P-completeness It is common to define $P$-completeness with respect to logspace many-one reductions. I am looking for a complexity class $C$ such that if $C=P$ then all problems in $P$ are $P$-complete under many-one $C$-reductions. 


  What is the weakest many-one reduction (computable in class $C$) for which the class of P-complete problems remains unchanged?


Note that $C$ is contained in $P$.
There is first-order reductions.

See Immerman's book, Chapter 3, Theorem 3.26 which states that $\mathrm{REACH}_a$ is complete for $\mathrm{P}$ via first-order reductions.

Also, see my question, for related information on $\mathrm{NP}$.