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Intuitively, closure of SL2 under intersection is reasonably easy to see, particularly if one considers the Myhill graphs of the automata. Any path through both graphs will be a path through the intersection of the graphs (by which we mean the graph resulting by taking the intersection of the vertex sets and the intersection of the edge sets).
For the union, on the other hand, the corresponding construction won't work. An automaton built from the union of the two automata will still recognize all of the strings in L1 and all of the strings in L2, but it is likely to also recognize strings made up of adjacent pairs from L1 combined with adjacent pairs from L2 that aren't in either language. And, in fact, we can use Suffx Substitution Closure to show that there are languages that are the union of two SL2 languages that are not, themselves, SL2.
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The universe of strings is a very useful medium for the representation of information as long as there exists a function that provides the interpretation for the information carrie
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Proof (sketch): Suppose L 1 and L 2 are recognizable. Then there are DFAs A 1 = (Q,Σ, T 1 , q 0 , F 1 ) and A 2 = (P,Σ, T 2 , p 0 , F 2 ) such that L 1 = L(A 1 ) and L 2 = L(
We represented SLk automata as Myhill graphs, directed graphs in which the nodes were labeled with (k-1)-factors of alphabet symbols (along with a node labeled ‘?' and one labeled
Applying the pumping lemma is not fundamentally di?erent than applying (general) su?x substitution closure or the non-counting property. The pumping lemma is a little more complica
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De?nition (Instantaneous Description) (for both DFAs and NFAs) An instantaneous description of A = (Q,Σ, δ, q 0 , F) , either a DFA or an NFA, is a pair h q ,w i ∈ Q×Σ*, where
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