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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.
We saw earlier that LT is not closed under concatenation. If we think in terms of the LT graphs, recognizing the concatenation of LT languages would seem to require knowing, while
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Theorem The class of recognizable languages is closed under Boolean operations. The construction of the proof of Lemma 3 gives us a DFA that keeps track of whether or not a give
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