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The key thing about the Suffx Substitution Closure property is that it does not make any explicit reference to the automaton that recognizes the language.
While the argument that establishes it is based on the properties of a Myhill graph that we know must exist, those properties are properties of Myhill graphs in general and don't depend on the speci?cs of that particular graph. This lets us reason about the strings in an SL2 language without having to actually produce the automaton that recognizes it. Perhaps more importantly, it lets us establish that a particular language is not SL2 by supposing (counterfactually) that it was SL2 and showing that Suffx Substitution Closure would then imply that it included strings that it should not.
The k-local Myhill graphs provide an easy means to generalize the suffix substitution closure property for the strictly k-local languages. Lemma (k-Local Suffix Substitution Clo
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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
The path function δ : Q × Σ*→ P(Q) is the extension of δ to strings: Again, this just says that to ?nd the set of states reachable by a path labeled w from a state q in an
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All that distinguishes the de?nition of the class of Regular languages from that of the class of Star-Free languages is that the former is closed under Kleene closure while the lat
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Suppose A = (Q,Σ, T, q 0 , F) is a DFA and that Q = {q 0 , q 1 , . . . , q n-1 } includes n states. Thinking of the automaton in terms of its transition graph, a string x is recogn
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 le
The fact that SL 2 is closed under intersection but not under union implies that it is not closed under complement since, by DeMorgan's Theorem L 1 ∩ L 2 = We know that
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