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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 ‘?') and the edges were labeled with individual alphabet symbols. The k-factors of the automaton could be recovered by appending the symbol on an edge to the factor of the node it is incident from. The key value of the graphs is the way that they capture the set of all computations of the automaton in a concise form: every computation of the automaton corresponds to a path through the automaton from ‘?' to ‘?' and vice versa. The su?x substitution closure property is, in essence, a consequence of this fact. All that is signi?cant about the initial portion of a computation is the node it ends on. All strings that lead to the same node are equivalent in the sense that any continuation that extends one of them to form a string that is accepted will extend any of them to form a string that is accepted, and any continuation that leads one of them to be rejected will lead any of them to be rejected.
In adapting this idea for LTk automata, we have to confront the fact that the last k - 1 symbols of the input are no longer enough to characterize the initial portion of a string. We now will also need the record of all k-factors which occurred in that initial portion. To accommodate this, we will extend the labeling of our nodes to include sets of k-factors. The node set will be pairs in which the ?rst component is a k - 1 factor (the last k - 1 symbols of the input) and the second component is a set of k-factors. At the initial node, not having scanned any of the input yet, we have seen no k-factors, that is, the initial set of k-factors is empty (∅). The label of the initial node, then is (?, ∅).
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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
design a turing machine that accepts the language which consists of even number of zero''s and even number of one''s?
Intuitively, closure of SL 2 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
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We now add an additional degree of non-determinism and allow transitions that can be taken independent of the input-ε-transitions. Here whenever the automaton is in state 1
Let L1 and L2 be CGF. We show that L1 ∩ L2 is CFG too. Let M1 be a decider for L1 and M2 be a decider for L2 . Consider a 2-tape TM M: "On input x: 1. copy x on the sec
Find the Regular Grammar for the following Regular Expression: a(a+b)*(ab*+ba*)b.
Theorem The class of ?nite languages is a proper subclass of SL. Note that the class of ?nite languages is closed under union and concatenation but SL is not closed under either. N
Another way of representing a strictly 2-local automaton is with a Myhill graph. These are directed graphs in which the vertices are labeled with symbols from the input alphabet of
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