CS 3613: Theoretical Foundations of Computing
Spring 2009
Test 2
Student Name:
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1. The examination contains 4 problems. You have 90 minutes for 40 points.
   
   
2. Show all important steps in your work. Your answers will be graded on its correctness and clarity.
   
   

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1. [10 points]
   1. [6 points] Let L = { aⁱ b^j c^k | i, j, k ³ 0, i + k = j }. Prove that L is not regular by using the pumping lemma for regular languages.
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
   2. [4 points] Using the closure properties of regular languages (regularity-preserving operators) to show that L is not regular.
      
      
   
   
   
2. [10 points] Give context-free grammars generating the following languages.
   Notes: No conversion from pushdown automaton to context-free grammar is accepted. For each grammar, give a brief and precise interpretation of each variable (that is, the set of strings derived from the variable).
   1. [5 points] L₁, the set of all non-palindromes over alphabet { a, b }. (A palindrome over an alphabet S is a string over S that reads the same backward and forward; what is a non-palindrome?)
      
      
      
   2. [5 points] L₂ = { a^m bⁿ c^p d^q | m, n, p, q ³ 0, m + n = p + q }. Note that the condition m + n = p + q does not necessarily imply that m = q and/or n = p.
      
      
   
   
   
3. [10 points] Consider the following language:
   
   

   L = { z Î { a, b }* | z is of the form x y x for some x and y with |x| ³ 1 }.

   For examples: The strings aaa (with x = a and y = a) and abababa (with x = a and y = babab, or x = aba and y = b) are in L, but the string aaab is not in L.
   Decide if L is regular. Prove your answer.
   
   
   
4. [10 points] Give the 6-tuple definition of a pushdown automaton for the language:
   
   

   L = { a, b }* - { w w | w Î { a, b }* }.

   Note: A direct construction of a pushdown automaton is required. Give a brief and precise interpretation of the states and transitions of your machine.
   
   

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