# Is there any problem that is R-complete and RE-complete

R-complete, i.e. it is an analogue to all recursive language can be reduced to that problem and also recursive? Or is there a really such definition?

RE-complete is described on wikipedia. But what kind of reductions is used? Since we are already out of the set R, does it still make sense to define by many-one reduction?

Wikipedia hints at the definition of RE-complete. A language $$A$$ is RE-complete if for every r.e. language $$L$$ there is a computable reduction $$f$$ such that $$x \in L$$ iff $$f(x) \in A$$. There are two features of this definition:
As for R-completeness, I am not aware of any such definition, and I don't expect one. Indeed, the class of reductions allowed has to be smaller than all computable functions, since otherwise the definition trivializes (all nontrivial computable languages would be R-complete). Suppose for definiteness that we consider reductions running in time $$t_1(n)$$, and let $$L$$ be some R-complete problem with respect to such many-one reductions. Then $$L$$ runs in time $$t_2(n)$$, and so we would get that all languages in $$R$$ can be computed in time $$t_1(n) + t_2(t_1(n))$$, which clearly isn't the case. The same kind of argument rules out oracle reductions and space-bounded reductions.