Show that the language L = {vwv : v, w ∈ {a, b}∗ , |v| = 3} is regular.

Answer 1

To show a language is regular, it is suffice to show that a regular expression can generate that language.

This regular expression does the job for the language L you specified:

(v1(a|b)*v1) | (v2(a|b)*v2) | ... (v8(a|b)*v8)

where v1 to v8 is aaa, aab, aba, abb, baa, bab, bba, bbb.

Notice that the string v before w and the string v after w have to be identical and have length of 3. The above regular expression obeys both, plus the string w can be anything as long as the vocabulary is in {a, b}.