a1 - Hint ²ollow the proo± that √ 2 is not a rational...

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Homework 1 CS 321 Due Date: 10/6/10, 2 PM Note: The homeworks should be your own work. You can discuss the homeworks orally with your peers, however. You are allowed to use the internet sources only when you are explicitly asked to do so. In such cases, you should cite the sources and answer in your own words (don’t copy verbatim which would be considered plagiarism.) All questions carry equal weight. Please see the TA and the instructor during the office hours to get more help. 1. Search the web and find 3 different applications of finite state automata other than digital logic design, neural networks, and scanners. Give a brief description or example of each application. 2. Section 1.1. Problem 11. Show that S 1 × ( S 2 S S 3 ) = ( S 1 × S 2 ) S ( S 1 × S 3 ). Note: You need to show that every member of the left hand side is a member of the right hand side and vice versa. 3. Section 1.1. Problem 26. Show by induction on n that n X i =1 1 i 2 2 - 1 n (1) 4. Section 1.1. Problem 31. Show that 3 is not a rational number.
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Unformatted text preview: Hint: ²ollow the proo± that √ 2 is not a rational number in the book. 5. Section 1.2. Problem 1. Use induction on n to show that | u n | = n | u | ±or all srings u and integers n ≥ 0. Hint: Use the ±act that u i +1 = u i u and u = λ . 6. Section 1.2. Problem 8. Show that ( L 1 L 2 ) R = L R 2 L R 1 . Hint: Use the ±ollowing ±act (problem 2) as given. ²or any strings u,v over some alphabet ( uv ) R = v R u R . Note that u R is the reverse o± string u . ( L ) R is the set o± reverses o± strings in L . 7. Section 1.2. Problem 9. Show that ( L * ) * = L * ±or all languages L . 8. Section 1.2. Problem 11(b). ²ind a grammar ±or ∑ = { a,b } that generates the sets o± all strings with at least one a . 9. Section 1.2. Problem 14(b). Give a grammar to generate the language L 2 = { a n b 2 n : n ≥ } . 10. Give a grammar to generate the language L * 2 , where L 2 is defined in the previous problem. 1...
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