The usual many-one reduction involves Turing machines transforming an input, $w$, of some language to an input, $f(w)$, of some other language in polynomial time. (Where $w \in L_1 \iff f(w) \in L_2$).
But for $P$-completeness, logspace or NC reductions are often used instead (taken from wikipedia), as using polynomial time reductions would allow every problem in $P$ to be complete, besides the empty language and the full language.
But there is a stronger reduction, the "$O(1)$" reduction. Transforming an input $w$ to $f(w)$ in constant time. Are there any known problems that are complete for $P$ under these reductions? If there are, are they known?
As noted by John L in the comments, the specific model of Turing machine must be defined, as not all models are equivalent by a constant factor. Because I am looking for the strongest reduction, I am refering to single-tape one directional TM's when I am talking about reduction.
I am beginning to think that a reduction like this doesn't exist under many-one reductions, from this question I asked. So I think broadening the scope of this question to Turing reductions that run in constant time is beneficial.
