We have now de?ned classes of k-local languages for all k ≥ 2. Together, these classes form the Strictly Local Languages in general.
De?nition (Strictly Local Languages) A language L is strictly local (L ∈ SL) iff it is strictly k-local for some k.
Again, we can generalize the work we have done so far to establish properties of the class of strictly local languages as a whole.
Theorem 3 ((General) Suffix Substitution Closure) A language L is strictly local iff there is some k such that, for all strings u_{1}, v_{1}, u_{2}, and v_{2} in Σ* and all strings x in Σ^{k-1} :
u_{1}xv_{1} ∈ L and u_{2}xv_{2} ∈ L ⇒ u_{1}xv_{2} ∈ L.