Question
Show that the set of palindromes over $\{0,1\}$ is not regular using the pumping lemma given in Exercise 22 . [Hint: Consider strings of the form $0^{N} 10^{N} . ]$
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Step 1: Assume that the set of palindromes over $\{0,1\}$ is regular. Show more…
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Show that the set $\left\{0^{2 n} 1^{n} | n=0,1,2, \ldots\right\}$ is not regular using the pumping lemma given in Exercise $22 .$
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Show that the set $\left\{1^{n^{2}} | n=0,1,2, \ldots\right\}$ is not regular using the pumping lemma from Exercise 22 .
One important technique used to prove that certain sets are not regular is the pumping lemma. The pumping lemma states that if $M=\left(S, I, f, s_{0}, F\right)$ is a deterministic finite-state automaton and if $x$ is a string in $L(M),$ the language recognized by $M,$ with $l(x) \geq|S|,$ then there are strings $u, v,$ and $w$ in $I^{*}$ such that $x=u v w, l(u v) \leq|S|$ and $l(v) \geq 1,$ and $u v^{i} w \in L(M)$ for $i=0,1,2, \ldots$ Prove the pumping lemma. [Hint: Use the same idea as was used in Example $5 . ]$
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