# hw3_1 - ∈ L and w ∈ LL } Prove that if L is a regular...

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Finite Automata and Theory of Computation September 13, 2007 Handout 7: Homework 3 Professor: Moses Liskov Due: September 20, 2007, in class. Note: Some of these problems will take some hard thinking! You should start on this problem set early. Problem 1 Given the following NFA, determine a regular expression for the language it accepts by following the proof from class. Show all your steps. ε b c a 0 1 0 0,1 d 1 0 Problem 2 If L is a language, deﬁne ADDIN ( L ) to be the language { w | w = xσz : xz L,σ Σ } . Prove that if L is a regular language, then ADDIN ( L ) is also regular. Problem 3 Prove that the following languages are not regular by using the pumping lemma. (a) { 0 n 1 m 0 n | m > n } . (b) { w | w = uv with | u | = | v | , and where the number of 0s in v is less than the number of 1s in u } . (Note that if w = uv with | u | = | v | then u is the ﬁrst half of w and v is the second half.) Problem 4 If L is a language, deﬁne F ( L ) to be the language { w | w /

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Unformatted text preview: ∈ L and w ∈ LL } Prove that if L is a regular language, then F ( L ) is also regular. 7-1 Problem 5 (optional) An all-NFA is a 5-tuple ( Q, Σ ,δ,q ,F ) that operates exactly as an NFA does with one crucial diﬀerence: an all-NFA will accept a string w if every valid computation on w ends in an accepting state. (By contrast, an ordinary NFA will accept a string w if there exists a valid computation on w that ends in an accepting state.) Prove that the class of languages accepted by all-NFAs is the class of regular languages. Problem 6 (optional) If L is a language, let L 1 2-be the set of all ﬁrst halves of strings in L . That is, L 1 2-= { x |5 y : | x | = | y | and xy ∈ L } . Prove that if L is regular, L 1 2-is also regular. 7-2...
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## hw3_1 - ∈ L and w ∈ LL } Prove that if L is a regular...

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