3. These languages are not regular. For each, list three strings that would work in a Pumping
Lemma proof. Then, use one of them to show the language is not regular. But not a.
a. L = {ww | w Î {a, b}*}
b. L = {anba2n | n >= 0}
c. {w Î S* | w contains more a’s than b’s}.
a)L = {ww | w Î {a, b}*
Assume L is regular. From the pumping lemma there exists an n such that every w Î L such that |w| >=n can be represented as x y z with |y| not =0 and |xy| <= n. Let us choose an bn an bn . Its length is 4n >= n. Since the length of xy cannot exceed n, y must be of the form a for some k > 0. From the pumping lemma an+1bnanbnmust also be in L but it is not of the right form since the middle of the string would be in the middle of the b which prevents a match with the beginning of the string. Hence the language is not regular.
3. These languages are not regular. For each, list three strings that would work in a...
John Doe claims that the language L, of all strings over the alphabet Σ = { a, b } that contain an even number of occurrences of the letter ‘a’, is not a regular language. He offers the following “pumping lemma proof”. Explain what is wrong with the “proof” given below. “Pumping Lemma Proof” We assume that L is regular. Then, according to the pumping lemma, every long string in L (of length m or more) must be “pumpable”. We...
Pumping lemma s. (7+5 points) Pumping lemma for regular languages. In all cases, -a,b) a) Consider the following regular language A. ping length p 2 1. For each string s e pumping lemma, we can write s -xy, with lyl S p, and s can be pumped. Since A is regular, A satisfies the pumping lemma with pum A, where Is] 2 p, by the a) Is p 3 a pumping length for language 4? (Yes/No) b) Show that w...
6.[15 points] Recall the pumping lemma for regular languages: Theorem: For every regular language L, there exists a pumping length p such that, if s€Lwith s 2 p, then we can write s xyz with (i) xy'z E L for each i 2 0, (ii) ly > 0, and (iii) kyl Sp. Prove that A ={a3"b"c?" | n 2 0 } is not a regular language. S= 6.[15 points] Recall the pumping lemma for regular languages: Theorem: For every regular...
Prove that the following are not regular languages. Just B and F please Prove that the following are not regular languages. {0^n1^n | n Greaterthanorequalto 1}. This language, consisting of a string of 0's followed by an equal-length string of l's, is the language L_01 we considered informally at the beginning of the section. Here, you should apply the pumping lemma in the proof. The set of strings of balanced parentheses. These are the strings of characters "(" and ")"...
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(9 pts 3 pts each) For each of the following languages, name the least powerful type of machine that will accept it, and prove your answer. (Hint: a finite state automata is less powerful than a pushdown automata, which in turn is less powerful than a Turing Machine.) For example, to prove a language needs a PDA to accept it, you would use the Pumping Lemma to show it is not regular, and then build the PDA or CFG that...
(9 pts 3 pts each) For each of the following languages, name the least powerful type of machine that will accept it, and prove your answer. (Hint: a finite state automata is less powerful than a pushdown automata, which in turn is less powerful than a Turing Machine.) For example, to prove a language needs a PDA to accept it, you would use the Pumping Lemma to show it is not regular, and then build the PDA or CFG that...
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