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hw4f03-sol-post - CS 373 Fall 2003 NMG SOLUTION*post...

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CS 373 – Fall 2003 - NMG ****** SOLUTION****post**** September 27, 2003 Homework Assignment 4 for chapter 4, PROB 9 DROPPED Due Friday, 10/3/03 After class Problems related to chapter 4: General directions: 1. Clearly justify all statements in your proofs 2. If using a counter example to disprove, give a SPECIFIC example 3. For pumping lemma questions, Use the text book format of a r game against an opponentr and all steps must be explained. EXPLICITLY SPECIFY: a. Pumping lemma integer m which starts off the proof. Although you cannot assume a specific value for the integer, show how your knowledge of its existence is used. “m”, must be explicitly shown in your formulas for the pumped string. b. One specific string which you choose c. Form of the partition "the opponent is capable of choosing" as a result of your string choice. d. A value of i which you will use to pump up/down. P1. (10 points) Prove or disprove: If L 1 and L 2 are non-regular languages, then L 1 L 2 is non- regular. P2 . (10 points) Show that if L 1 and L 2 are languages over {a, b} where L 1 is regular and L 2 is non-regular, then L 1 L 2 may be either regular or non-regular, depending on L 1 and L 2 . P3. (15 points total ) Drill Time!. Use the pumping lemma for regular languages to show that the following languages are not regular (assume contains he symbols used in the definitions of the languages): a. (5 points) L = {a 3i b i : i 1} … a warm-up exercise! b. (5 points) L = (a n b m : n > m } c. (5 points) Show that L = {a g b h c h : g > 0 and h > 0 . P4 . (10 points) Prove that the language: L = { a n b j : n/j is an integer, and n 0, j > 0 } is not regular. P5 . (10 points) Use the pumping lemma for regular languages to show that the language below is not regular. L = {a n^3 : n > 0 } where n^3 means n 3 (problem doing double superscript). Hint: Consider the binomial expansion of (m+1) 3 . P6. (10 points) Find a regular language which is a proper subset of the language: L = {a n b n : n 0}. CS 373 Fall 2003 Solution key page 1
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P7. (15 points) Without using the pumping lemma prove that L = {a n b m : n m} is not regular.
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