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We have now de?ned classes of k-local languages for all k ≥ 2. Together, these classes form the Strictly Local Languages in general.
De?nition (Strictly Local Languages) A language L is strictly local (L ∈ SL) iff it is strictly k-local for some k.
Again, we can generalize the work we have done so far to establish properties of the class of strictly local languages as a whole.
Theorem 3 ((General) Suffix Substitution Closure) A language L is strictly local iff there is some k such that, for all strings u1, v1, u2, and v2 in Σ* and all strings x in Σk-1 :
u1xv1 ∈ L and u2xv2 ∈ L ⇒ u1xv2 ∈ L.
Suppose G = (N, Σ, P, S) is a reduced grammar (we can certainly reduce G if we haven't already). Our algorithm is as follows: 1. Define maxrhs(G) to be the maximum length of the
For example, the question of whether a given regular language is positive (does not include the empty string) is algorithmically decidable. "Positiveness Problem". Note that
This was one of the ?rst substantial theorems of Formal Language Theory. It's maybe not too surprising to us, as we have already seen a similar equivalence between LTO and SF. But
turing machine
Construct a Mealy machine that can output EVEN or ODD According to the total no. of 1''s encountered is even or odd.
Can you say that B is decidable? If you somehow know that A is decidable, what can you say about B?
how many pendulum swings will it take to walk across the classroom?
To see this, note that if there are any cycles in the Myhill graph of A then L(A) will be infinite, since any such cycle can be repeated arbitrarily many times. Conversely, if the
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The Universality Problem is the dual of the emptiness problem: is L(A) = Σ∗? It can be solved by minor variations of any one of the algorithms for Emptiness or (with a little le
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