Roots and Powers of Regular Languages

For a set H of natural numbers, the H-power of a language L is the set of all words pk where p ∈ L and k ∈ H. The root of L is the set of all primitive words p such that pn belongs to L for some n ≥ 1. There is a strong connection between the root and the powers of a regular language L namely, the H-power of L for an arbitrary finite set H with 0, 1, 2 ∉ H is regular if and only if the root of L is finite. If the root is infinite then the H-power for most regular sets H is context-sensitive but not context-free. The stated property is decidable.

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