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The computation of an SL2 automaton A = ( Σ, T) on a string w is the maximal sequence of IDs in which each sequential pair of IDs is related by |-A and which starts with the initial con?guration of A on w: (p1,w1), where p1 . w1 = ?w?.
Since w is ?nite, the computation of A on w will be ?nite. Since it is required to be maximal, the last ID will be one that does not directly compute any other ID. This will either be of the form (σiσj) , wii where σiσj ∈ T, or of the form (σn?, ε), in which σn? ∈ T but all the input has been consumed. In the ?rst case we will say that the computation is rejecting (or that it crashes). In the second we will say that it is accepting. Note that we have adopted the convention that the automaton halts with FALSE as soon as it encounters a pair of symbols that are not in T.
These assumptions hold for addition, for instance. Every instance of addition has a unique solution. Each instance is a pair of numbers and the possible solutions include any third
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Lemma 1 A string w ∈ Σ* is accepted by an LTk automaton iff w is the concatenation of the symbols labeling the edges of a path through the LTk transition graph of A from h?, ∅i to
This close relationship between the SL2 languages and the recognizable languages lets us use some of what we know about SL 2 to discover properties of the recognizable languages.
While the SL 2 languages include some surprisingly complex languages, the strictly 2-local automata are, nevertheless, quite limited. In a strong sense, they are almost memoryless
how many pendulum swings will it take to walk across the classroom?
The Recognition Problem for a class of languages is the question of whether a given string is a member of a given language. An instance consists of a string and a (?nite) speci?cat
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