Our primary concern is to obtain a clear characterization of which languages are recognizable by strictly local automata and which aren't. The view of SL2 automata as generators lets us do this by considering the characteristics of the tilings they build. Consider, for instance the situation in the top half of Figure 5, where there are two tilings u₁σv₁ and u₂σv₂ in which the symbol ‘σ' occurs. Clearly, after having built u₁σ we had the choice of continuing with either v₁ or with v₂. We had the same choice after having built u₂σ. Hence both of the tilings in the bottom half are constructable as well.

What this means for the strings, is that the question of whether we can extend a particular string to produce a longer string that is in the language depends only on the last symbol of that string.