CS 3613: Theoretical Foundations of Computing CS 3613: Theoretical Foundations of Computing
Spring 2008
Test 2
Student Name:
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  1. The examination contains 3 problems. You have 75 minutes for 50 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] Consider the following language:
    L
    =
    { ai bj | i ³ 0, j ³ 0, and i ¹ j }.
    1. [9 points] Construct a pushdown automaton accepting the language L. Brief and precise interpretation of the states and the transitions of your machine is required.

    2. [6 points] Construct a context-free grammar generating the language L. Give the interpretation of the variables used in your grammar.






























    3. [5 points] Is L regular? Support your answer by giving a proof.



  2. [15 points] Prove, or disprove by giving a counter-example, each of the following statements. A correct answer without justification will not receive any credit.
    1. [3 points] If L* is a regular language, then L is a regular language.

    2. [3 points] If Li is a context-free language for i = 0, 1, ¼, then Èi = 0¥ Li is a context-free language.

    3. [9 points] The language { x xr x | x Î {a, b }* } is context-free. Note: wr denote the reversal of the string w.




  3. [10 points] Consider the language:

    L = { an bn c m | n, m ³ 1, and n \not = m }.
    Place the language (a + b + c)* - L in one of the following classes: "regular", "context-free but not regular", "not context-free". Prove your answer. A correct answer without justification will not receive any credit.



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