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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.
Sketch an algorithm for the universal recognition problem for SL 2 . This takes an automaton and a string and returns TRUE if the string is accepted by the automaton, FALSE otherwi
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.
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Kleene called this the Synthesis theorem because his (and your) proof gives an effective procedure for synthesizing an automaton that recognizes the language denoted by any given r
So we have that every language that can be constructed from SL languages using Boolean operations and concatenation (that is, every language in LTO) is recognizable but there are r
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