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Regular language is a language which can be accept by finite automata including deterministic and non-deterministic automata.

The statement, 鈥淚f $A{猢�}_{m}B$and B is a regular language, does that imply that A is a regular language?鈥� - is wrong. This is because the power of reduction function can be more the Deterministic Finite Automata (DFA).

For example, assume that there are two languages $A=\left\{0^n{1}^n\right|\text{for}n鈮�0\}$and $B=\left\{1\right\}$, both are over the function $鈭�=\left\{0,1\right\}$.

Let there be a function as:

$\text{f}\left(\text{w}\right)=\{\begin{array}{c}1;ifw鈭�A\\ 0;ifw鈭�A\end{array}$

Now we can see, that A is Context Free language, thus it must be decidable. So, this makes f as computable function.

We also observe that $w鈭�A$if and only if $f\left(w\right)=1$, this is only true if $f\left(w\right)鈭�B$.

Therefore, $A{猢�}_{m}B$ is regular language true but A is not regular language.