CS 3613: Theoretical Foundations of Computing
Spring 2008
Test 1
Student Name:
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1. The examination contains 3 problems. You have 75 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. [20 points]
   1. [4 points] Recall that the reversal of a string w over an alphabet S can be defined inductively on |w|:
      
      

      wr = ì í î e if w = e a ur if w = u a for some u Î S* and a Î S

      Use the inductive definition of reversal and mathematical induction to show that for all strings x and y over an alphabet S, (x y)^r = y^r x^r.
      
      
      
   2. [8 points] Let the alphabet S = { a, b }. Recall that a string x Î S^* is called a palindrome if x = x^r (x reads the same backward as forward).
      1. [5 points] Consider the following binary relation R on S^* as follows: For all x, y Î S^*, x R y if and only if |x| = |y| and x y^r is a palindrome.
         1. [4 points] Show that R is an equivalence relation on S^*.
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
            
         2. [1 point] For an arbitrary x Î S^*, describe the equivalence class [ x ]_R (the equivalence class of R containing x). Justify your answer.
            
            
            
            
            
            
            
            
            
            
         
         
         
      2. [3] Consider the following binary relation R¢ on S^* as follows: For all x, y Î S^*, x R¢y if and only if x y^r is a palindrome. Is R¢ an equivalence relation on S^*? Justify your answer.
         
         
         
      
      
      
   3. [5 points] Let L be a language over an alphabet S such that L ¹ Æ, L ¹ { e}, and L² = L. Prove that:
      1. [2 points] e Î L, and
         
         
      2. [3 points] L is not a finite language (that is, |L| is not finite).
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
         
      
      
      
   4. [3 points] Consider the language:
      

      L = { x Î {a, b}* | x has exactly one occurrence of the substring aa }.

      Justify your solution by properly structuring and annotating your regular expression.
      
      
      
   
   
   
2. [10 points]
   1. [5 points] Give the state-transition diagram of a deterministic finite automaton that accepts the following language:
      
      

      { x Î {a, b}* |   |#a(u) - #b(u)| £ 2 for every prefix u of x },

      where #_c (u) denotes the number of occurrences of the symbol c in the string u. Give a brief interpretation of the states in your construction.
      
      
      
   2. [5 points] Let S = { a, b }, M = (Q, S, d, q₀, A) be a deterministic finite automaton, and
      
      

      EM = { x | x Î L(M), and |x| is even }.

      Show that E_M is regular by giving explicitly the 5-tuple definition of a deterministic finite automaton accepting E_M. Give a brief interpretation of the states in your construction.
      
      
   
   
   
3. [10 points] Let S be the alphabet of 3 symbols: S = { a, b, c }.
   1. [5 points] Consider the following language:
      

      L1 = { w Î S* | there exists a symbol of S not appearing in w }.

      Give the formal 5-tuple definition and the state-transition diagram of a nondeterministic finite automaton with at most 4-states (with or without e-transitions) that accepts L₁. Notes: A brief and precise interpretation of the states of your machine is required, and no credit will be given to finite automaton with more than 4 states.
      
      
      
   2. [5 points] Consider the following language:
      

      L2 = { w Î S* | w has a substring of length 3 containing each of the symbols of S}.

      Give the state-transition diagram of a nondeterministic finite automaton (with or without e-transitions) with at most 8 states that accepts L₂. Notes: A brief and precise interpretation of the states of your machine is required, and no credit will be given to finite automaton with more than 8 states.
      
      
   
   
   

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