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The class of Strictly Local Languages (in general) is closed under
• intersection but is not closed under
• union
• complement
• concatenation
• Kleene- and positive closure
Proof: For intersection, we can adapt the construction and proof for the SL2 case again to get closure under intersection for SLk. This is still not quite enough for SL in general, since one of the languages may be in SLi and the other in SLj for some i = j. Here we can use the hierarchy theorem to show that, supposing i < j, the SLi language is also in SLj . Then the adapted construction will establish that their intersection is in SL .
For non-closure under union (and consequently under complement) we can use the same counterexample as we did in the SL2 case:
To see that this is not in SLk for any k we can use the pair
which will yield abk-1 a under k-local suffix substitution closure.
For non-closure under concatenation we can use the counterexample
The two languages being concatenated are in SL2, hence in SLk for all k ≥ 2 but their concatenation is not in SLk for any k, as we showed in the example above.
As we are primarily concerned with questions of what is and what is not computable relative to some particular model of computation, we will usually base our explorations of langua
The class of Strictly Local Languages (in general) is closed under • intersection but is not closed under • union • complement • concatenation • Kleene- and positive
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(
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
The generalization of the interpretation of strictly local automata as generators is similar, in some respects, to the generalization of Myhill graphs. Again, the set of possible s
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
De?nition Instantaneous Description of an FSA: An instantaneous description (ID) of a FSA A = (Q,Σ, T, q 0 , F) is a pair (q,w) ∈ Q×Σ* , where q the current state and w is the p
Exercise Show, using Suffix Substitution Closure, that L 3 . L 3 ∈ SL 2 . Explain how it can be the case that L 3 . L 3 ∈ SL 2 , while L 3 . L 3 ⊆ L + 3 and L + 3 ∈ SL
build a TM that enumerate even set of even length string over a
proof of arden''s theoram
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