Testing Periodicity

We study the string-property of being periodic and having periodicity smaller than a given bound. Let Σ be a fixed alphabet and let p,n be integers such that p ￡ \fracn2p\leq \frac{n}{2} . A length-n string over Σ, α=(α ₁,…,α _n ), has the property Period(p) if for every i,j∈{1,…,n}, α _i =α _j whenever i≡j (mod p). For an integer parameter g ￡ \fracn2,g\leq \frac{n}{2}, the property Period(≤g) is the property of all strings that are in Period(p) for some p≤g. The property Period( ￡ \fracn2)\mathit{Period}(\leq \frac{n}{2}) is also called Periodicity.