An arithmetic progression is a set of numbers {j + n · k : n = 0} for some natural numbers j and…

An arithmetic progression is a set of numbers {j + n · k : n ≥ 0} for some natural numbers j and k. So, for example, {7, 11, 15, 19, …} is an arithmetic progression, with j = 7 and k = 4. Define the set of word lengths for any language L to be the set of lengths of words in L; that is, {|w| :w ∈ L}. For example, the set of word lengths of (ab)∗ is the set of even numbers.

Prove that a set of natural numbers is a finite union of arithmetic progressions if and only if it is the set of word lengths of some regular language. Hint: The “only if ” direction is easier; given a finite union of arithmetic progressions, you need to construct a regular language in which the word lengths are exactly the numbers in the set. For the “if ” direction, start by assuming an alphabet with only one symbol, and then generalize.

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