Question

Let Ln be the language containing all words of the form ak where k is a multiple of n, and n≥0 (k is a multiple of n if there is a natural number x such that k = x∗n). For example, the language L2 = {ε, aa, aaaa, aaaaaa, ...}. Give a proof outline that for every n≥0, Bn is regular.

Answer 1

Solution for the problem is provided below, please comment if any doubts:

To prove a language is regular we need to prove that there exists a DFA to accept the language.

Here Ln is a language where the language will contains all the strings a^k, where k is the multiple of n>=0.

That is the strings in the language are multiplies of the base string.

That is if “n=2”, then strings are multiplies of length two string “aa”. That is length 4, 6, etc.

For any such n, there exist a DFS with “n” states with start state as final state and a transition from “n”th state to first state.

The primitive DFA is as given below:

All the strings are multiplication of the primitive basic string of length “n”, thus lla string in the language will be accepted by a DFA.

That is the given language is a regular language.