Introduction to Computational Complexity - Homework 2 | CS 530, Assignments of Computer Science

Download Introduction to Computational Complexity - Homework 2 | CS 530 and more Assignments Computer Science in PDF only on Docsity! 1.5em CS530: Introduction to Computational Complexity Xiang-Yang Li Homework 2 Due: Oct 2nd, 2008 YanWei Wu Problem 1: Sipser problem 1.45 (page 90). Let A/B = {w | wx ∈ A for some x ∈ B}. Show that if A is regular and B = Σ∗ (B is a language containing all possible strings), then A/B is regular. Problem 2: Sipser exercise 2.4 and 2.6 (page 128). Given contex free grammmars that generate the following languages. In all parts, the alphabet Σ is {0, 1}. 1. {w | w starts and ends with the same symbol} 2. {w | the length of w is odd} 3. {w | w = wR}. Here wR is the reverse of w. 4. The complement of the language {0n1n | n ≥ 0}, i.e., {w | w not in format 0n1n}. 5. {w | w has equal number of 0s and 1s}. 6. {0m1n | m 6= n, and 2m 6= n} Problem 3: Sipser problem 2.20 (page 130). Let A/B = {w | wx ∈ A for some x ∈ B}. Show that if A is context-free and B is regular language, then A/B is context-free. Problem 4: Given pushdown automatas that generate the following languages. You have to explicitly explain the meaings of the states used in your automata. 1. {w | w has equal number of 0s and 1s}. Here the alphabet Σ = {0, 1}. 2. {aibjck | i = j, or j = k, i, j, k ≥ 0}. Here the alphabet Σ = {a, b, c}. Problem 5: Assume that you are given the DFA automata MA = (QA, ΣA, qa, FA) for a regular language A and the DFA automata MB = (QB ,ΣB , qb, FB) for a regular language B. Construct a pushdown automata for the following language using MA and MB . A ¦B = {xy | x ∈ A and y ∈ B and |x| = |y|}. Here |x| is the length of the string x. Problem 6: Prove that the following languages are not context-free. 1. L1 = {w | N(w, a) = N(w, b) = N(w, c)}. Here the alphabet Σ = {a, b, c} and N(w, a) denotes the number of a’s in the string w. 2. L2 = {w | N(w, a) is a prime number}. Here the alphabet Σ = {a, b}. 2-1