# Can you reduce every decidable language to a regular language?

One of my previous questions on an exam was the following

Can you reduce a decidable language to a given regular language? (decidable language $$\leq _m$$ regular language). If so, does this mean that you can reduce every CFL to a regular language?

The reductions are all referring to many-one reductions. My intuition tells me that this is possible as all regular languages are also decidable.

In fact, every non-trivial language is $$\text{R}$$-hard. That is, every decidable language is reducible to every non-trivial language. Indeed, let $$A$$ be a decidable language, and let $$B$$ be a non-trivial language. A reduction from $$A$$ to $$B$$ operates as follows. On input $$x$$, check whether $$x\in A$$ (this can be done as $$A$$ is decidable), then:
• if $$x \in A$$, the reduction outputs $$y_{in}$$
• if $$x\notin A$$, the reduction outputs $$y_{out}$$
where $$y_{in}$$ and $$y_{out}$$ are constant words in $$B$$ and $$\overline{B}$$, respectively.