This is a problem involving the theory of regular languages. I am stuck on this problem and do not know how to solve this type of problem.

Prove that the language $B_n = \{ a^k \mid k \text{ is a multiple of } n \}$ is a regular language for any $n \ge 1$.

Let me describe my thoughts thus far: It is easy to show that for $n=1$, we have $B_1 = \{a\}$.

In other words, a regular expression can easily be built for the value of $n=1$, and thus for any $k$.

  • $\begingroup$ sorry about that i was mixing up similar problems, i will change this asap $\endgroup$ – Musicode Feb 18 '14 at 1:09
  • $\begingroup$ $B_1 = \{a\}^*$. $\endgroup$ – Raphael Feb 18 '14 at 8:14

Since the multiples of $n$ are $0, n, 2n, 3n, \dots$, it's clear that $B_n=\{\epsilon, \mathtt{a}^n, \mathtt{a}^{2n}, \mathtt{a}^{3n}, \dots\}$, in other words the set of all strings that can be made by concatenating arbitrarily many copies of $\mathtt{a}^n$. I'll bet you can find a regular expression denoting that language, for any given value of $n$.

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Given a language L, and a pair of strings x and y, define a distinguishing extension to be a string z such that exactly one of the two strings xz and yz belongs to L. Define a relation R on strings by the rule that x R y if there is no distinguishing extension for x and y. It is easy to show that R is an equivalence relation on strings, and thus it divides the set of all strings into equivalence classes.

The Myhill–Nerode theorem states that L is regular if and only if R has a finite number of equivalence classes, and moreover that the number of states in the smallest deterministic finite automaton (DFA) recognizing L is equal to the number of equivalence classes in R. In particular, this implies that there is a unique minimal DFA with minimum number of states.

For the above problem, try using congruence relation and partition the set of all strings into k equivalence classes.

References : http://en.wikipedia.org/wiki/Myhill%E2%80%93Nerode_theorem

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  • 1
    $\begingroup$ In this case, Myhill–Nerode is massively, massively more complicated than just producing an automaton. $\endgroup$ – David Richerby Feb 18 '14 at 15:16

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