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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 complicated-rather than just the single universal quanti?er ("for all languages L") and single existential quanti?er ("there exists n"), we have a nest of alternating quanti?ers (denoting "for all" as ∀ and "there exists" as ∃):
(∀L)[L regular ⇒
(∃n)[
(∀x)[x ∈ L and |x| ≥ n ⇒
(∃u, v,w)[x = uvw and
|uv| ≤ n and
|v| ≥ 1 and
(∀i ≥ 0)[uviw ∈ L]]]]].
Just as with the lemmas for the local languages, we will approach this as an adversary game. Our proof will consist of a strategy for showing that L fails to satisfy the pumping lemma. Our choices are the "for all"s; the "there exists"s are our adversary's choices. There are just a few more rounds in this game than there were in the lemmas for the local languages. The key things are being clear about which are our choices and which are the adversary's and making sure that our strategy accounts for every legal choice the adversarymight make.
The game starts with our choice of the L we wish to prove to be non regular. Our adversary then chooses some n, we choose a string x ∈ L of length at least n, etc. We win if, at the end of this process, we can choose i such that uviw ∈ L. Of course, our strategy at each step will depend on the choices our adversary has made.
What we end up with is a proof by contradiction. For instance:
To show that Lab = {ajbj| j ≥ 0} is not regular.
The class of Strictly Local Languages (in general) is closed under • intersection but is not closed under • union • complement • concatenation • Kleene- and positive
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Since the signi?cance of the states represented by the nodes of these transition graphs is arbitrary, we will allow ourselves to use any ?nite set (such as {A,B,C,D,E, F,G,H} or ev
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explain turing machine .
This close relationship between the SL2 languages and the recognizable languages lets us use some of what we know about SL 2 to discover properties of the recognizable languages.
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
design a turing machine that accepts the language which consists of even number of zero''s and even number of one''s?
i have some questions in automata, can you please help me in solving in these questions?
Automaton (NFA) (with ε-transitions) is a 5-tuple: (Q,Σ, δ, q 0 , F i where Q, Σ, q 0 and F are as in a DFA and T ⊆ Q × Q × (Σ ∪ {ε}). We must also modify the de?nitions of th
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