Reduction of two recursive languages

Question

I'd like to ask if this implication holds, I have no idea how to prove/disprove it.

A is regular B is recursive $\implies$ $ A \le_m B$

Answer 1

Rigorously, the answer is no. If the language $B$ is trivial (empty or complement of empty), no non-trivial language can be many-one reduced to it.

Practically, if $B$ is any non-trivial language, $A$ can even be recursive and the implication would still be true. In order to see why, note that the reduction $\leq_m$ has full power of recursive functions available to itself -- so it can find whether a particular word belongs to $A$ or not and, depending on that, produce either a fixed word from $B$ or a fixed word from complement of $B$. Thus, the structure of $B$ does not really matter if $A$ is recursive, as long as it can provide both "yes" and "no" answers for $\leq_m$.