Another way of interpreting a strictly local automaton is as a generator: a mechanism for building strings which is restricted to building all and only the
automaton as an inexhaustible set of tiles labeled with the pairs of symbols, infinitely many instances of each type. The generator starts by selecting any tile labeled with 'x' on its left half. It then proceeds by selecting any tile for which the left half symbol matches the symbol on the right half of the previously selected tile and placing it with its left half overlapping the right half of that previous tile. In this way, the sequence of tiles grows until some tile with 'x' on its right half is placed. The generated string is the sequence of exposed symbols, not including the beginning and end symbols. Generation is non-deterministic-at each step the choice of tile is restricted only by the right symbol of the previous tile. A derivation of the generator is just the sequence of choices it makes in assembling a string, a sequence of pairs of symbols. The language generated by the generator is the set of all strings assembled by any of its derivations.
It should be clear that every string assembled by a derivation of the generator will be accepted by the automaton: the computation of the automaton will check the same sequence of pairs as the derivation of the generator uses and each of those pairs will be in the lookup table, hence, the computation will accept. Similarly it should be clear that every string accepted by a computation of the automaton will be assembled by the corresponding derivation of the generator.