Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 34.3, Problem 6E
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Present a language B such that B is complete for the class of Turing-decidable languges under ≤m reductions.
Suppose L is a language over {a,b}, and there is a fixed integer k such that for every x ∈ Σ*, xz ∈ L for some string z with |z| ≤ k. Does it follow that there is an FA accepting L? Why or why not?
Prove that the language L1={} is not regular language with the pumping lemma.
Chapter 34 Solutions
Introduction to Algorithms
Ch. 34.1 - Prob. 1ECh. 34.1 - Prob. 2ECh. 34.1 - Prob. 3ECh. 34.1 - Prob. 4ECh. 34.1 - Prob. 5ECh. 34.1 - Prob. 6ECh. 34.2 - Prob. 1ECh. 34.2 - Prob. 2ECh. 34.2 - Prob. 3ECh. 34.2 - Prob. 4E
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- handwritten solution If a language L ⊆ Σ∗ is recognized by a FA, then there is an NFA M = (K,Σ,δ,s0,F) with |F|= 1 such that L = L(M).arrow_forwardIt is well-known that regular languages are closed under the following operations: union, complement, and intersection. It is also well-known that all finite languages are regular. For each of the operations, union, intersection and complement, are finite languages closed under them? If yes, prove it. Else provide couter examples.arrow_forwardA language L over an alphabet Σ is co-finite, if Σ∗ \L is empty or finite. LetCOFNFA = {〈N 〉|N is a NFA accepting a co-finite language}.Show that COFNFA is decidable.arrow_forward
- Use the pumping lemma to prove that the language L over {a,b,c}*, given by L = {wcv: |wa = |v|b} is not regular.arrow_forwardLet L be any regular language over {a, b, c}. Show how the Pumping Lemma can be used to demonstrate that in order to determine whether or not Lis empty, we need only test at most (3n - 1)/2 strings.arrow_forwardUse a closure operation to reduce the language L of all even-lengthed strings of as and bs where both halves of the string end with an a, that is L = {w0aw1a | w0,w1 ∈ {a,b}∗ ∧ |w0| = |w1|}, to a simpler language L0 which must also be regular if L is regular. Use the pumping lemma to show L0 is not regular.arrow_forward
- Question 1 Let LC * be a regular language over some alphabet Σ. The Kleene-closure L* of L is given by {V1V2 • Vn | n ≥ 0, V1, V2, . . . , Vn € L}. (a) Suppose DFA D = (Q, Σ, 8, qo, F) satisfies L(D) = L. Construct an NFA N = (Q', Σ, 8', %, F') such that L(N) = L*. L* (b) Prove by induction on n that each word v₁v2 Un with n ≥ 0, V₁, V2, ..., Un E L is accepted by the NFA N proposed in item (a). (c) Prove that each word accepted/recognized by the NFA N proposed in item (a) is a word of the language L*.arrow_forwardSuppose L is a non empty regular language, over Σ, such that every w E L satisfies |w] =k, for some ke N. Show that there cannot exist a DFA M = (E, Q, 8, q0, F), where |Q| = k.arrow_forwardSuppose that A is a language such that λ /∈ A. Let w be a string of length k . Show that there exists a natural i such that for every natural j > i, every string in Aj is longer than k. Explain how this fact can be used to decide whether w is in A⋆arrow_forward
- What does the Pumping lemma for regular languages say? Select one: O Every "long enough" word in language L can be written in the form uv'w. O If a language L contains a "long enough" word z=uvw, then L contains infinitely many words of the form uv w. O Every word that can be written with a regular expression, can be also "pumped". O For an arbitrary word z=uvw in language L, there exist infinitely many words of the form uv'w, that are also in L.arrow_forwardLet A be the language of binary strings of even length that are also palindromes. That is, A is a language over the alphabet S={0,1} and win A if and only if the length of w is even and w is equal to the reverse of w. Use the pumping lemma for regular languages to prove that A is not regular.arrow_forwardGiven a language L over the alphabet Σ, define f(L) = {vo :v € L, o Σ, and v=uo for some u € [*} Prove that f(L) is regular if L is regular.arrow_forward
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