# 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 $P$.

• I don't understand your intro. $\mathrm{P}$ is closed under polytime reductions so, for any $\mathrm{C}$, $\mathrm{C=P}$ implies that every [non-trivial] problem in $\mathrm{P}$ is $\mathrm{P}$-complete under $\mathrm{C}$-reductions. – David Richerby Jun 1 at 11:02

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 $\mathrm{NP}$.
• Yes, it seems that weak reduction for $\mathrm{P}$-completeness is more cumbersome. $\mathrm{FO}+\mathrm{BIT}$ is known to capture uniform-$\mathrm{ACC}^0$, which in turn, has not yet been separated from $\mathrm{NC}^1$. See Lautemann et al.. – Thinh D. Nguyen Sep 19 '18 at 14:38