CS 3613: Theoretical Foundations of Computing
Spring 2008
Final Examination
Student Name:
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1. The examination contains 4 problems. You have 110 minutes for 70 points.
   
   
2. Show all important steps in your work. Your answers will be graded on its correctness and clarity.
   
   

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1. [15 points] Prove, or disprove by giving a counter-example, each of the following statements.
   1. [5 points] "For arbitrary languages L₁, L₂, and L₃ over an alphabet S, (L₁ ÇL₂) L₃ = (L₁ L₃) Ç(L₂ L₃)."
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
      
   2. [5 points] "If L^* is a regular language, then L is a regular language."
      
      
      
   3. [5 points] (Note: For a context-free grammar G = (V, S, P, S), a variable A Î V is called terminating if A Þ^* w for some w Î S^*.)
      "For a context-free grammar G = (V, S, P, S), if A and B are two variables of V with A Þ^* a₁ B b₁ and B Þ^* a₂ A b₂ for some a₁, b₁, a₂, b₂ Î V^*, then A is terminating if and only if B is terminating."
      
      
   
   
   
2. [20 points] Consider the following language:
   
   

   K = { x $ y | x and y are in {a, b}*, and y is not a permutation of x }.

   Example strings in K are a a a b $, a $ a a a b and b a b a $ a a a b; example strings not in K are a b b a, a b b b a $ a b b b a, and a b b b a $ b a b a b.
   1. [10 points] Prove that the language K is not regular.
      
      
      
   2. [10 points] Construct a context-free grammar that generates the language K. Briefly and precisely describe the strings generated by each variable in the grammar.
      
      
   
   
   
3. [20 points] For each of the following languages, either:
   • prove that the language is context-free by a direct construction of pushdown automaton accepting it (no conversion from context-free grammar; give brief and precise interpretations of the states and transitions in the machine), or
     
     
   • prove that the language is not context-free by using the pumping lemma.
     
     
   1. [10 points] { aⁱ b^j c^k | i, j, k ³ 1, and j £ max{ i, k } }.
      
      
      
   2. [10 points] { aⁱ b^j c^k | i, j, k ³ 1, and j ³ max{ i, k } }.
      
      
   
   
   
4. [15 points] Consider the following language:
   
   

   L = { x Î {a, b}* | #a (x) = #b (x) }.

   (Note: #_u (v) denotes the number of occurrences of a substring u in a string v.)
   Give the formal 7-tuple definition of a deterministic Turing machine that accepts/decides the language L. Briefly and precisely comment on the states and transitions in the machine. You may assume that an input string is left-delimited with $.
   
   

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