CMPSCI 250: Introduction to Computation

Final Exam

David Mix Barrington

8 May 2012

Directions:

• Answer the problems on the exam pages.
• There are four problems for 125 total points. Actual scale is A = 105, C = 68.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• In case of a numerical answer, an arithmetic expression like "32× 26 + 13 × 41" need not be reduced to a single integer.

  Q1: 20 points
  Q2: 50 points
  Q3: 25 points
  Q4: 30 points
Total: 125 points

Here are several definitions used on the exam.

• Remember that natural always means "non-negative integer".

  This exam's boolean logic problem involves four atomic propositions and three compound propositions:

  • DB means "Duncan is barking".
  • HF means "Duncan heard a fox".
  • HO means "Duncan heard an owl".
  • IN means "Duncan imagined a noise".
  • Statement 1 is "If Duncan is not barking, then he heard an owl if and only if he imagined a noise."
  • Statement 2 is "¬(¬HF ∨ IN) → DB".
  • Statement 3 is "If Duncan heard an owl, then he is barking, and if he is not barking, then either he heard a fox or he imagined a noise or both."

• Here are two properties that a formal language X might or might not have, expressed in predicate logic where the type of the variables is "string". Since I can't think of good descriptive names for these properties, I have named them after my dogs.
  • X has the Cardie property if ∃u:∀v: uv ∈ X.
  • X has the Duncan property if ∀u:∃v: uv ∈ X.

• A death state of a DFA is a non-final state that has all its arrows to itself. That is, if d is the state, δ(d, a) = d for all letters a in the alphabet. A victory state in a DFA is a final state that has all its arrows to itself.

• The function g from naturals to naturals is defined by the rules g(0) = 1 and g(n + 1) = 2g(n) + 2. So g(1) = 4, g(2) = 10, and g(3) = 22.

• The function h from naturals to naturals is defined by the following recursive pseudo-Java code:

  
        public natural h (natural n) {
           if (n == 0) return 0;
           if ((n % 2) == 1) return 1;
           return 2 * h(n/2);}
       

• The λ-NFA M has state set {1, 2, 3, 4}, start state 1, final state set {4}, and transitions (1, a, 2), (1, b, 3), (2, b, 2), (2, λ, 4), and (3, a, 4). It looks like this:

  
  
          /---\ b
          \   /
           \ V     
        /-->(2)--\
     a /          \ lambda
      /            V
  >(1)           ((4))
      \            ^
     b \          / a
        \-->(3)--/
  

• Question 1 (20): This boolean logic problem deals with the atomic propositions and the Statements 1, 2, and 3 defined above.
  • (a, 5) Translate statements 1 and 2 into symbols, and translate Statement 2 into English.

  • (b, 15) From Statements 1, 2, and 3, prove that Duncan is barking. You may use either truth tables or propositional proof rules. (Hint: The similar "murder mystery" made heavy use of Proof by Cases. You might also try Proof by Contradiction.)

• Question 2 (50): These questions all deal with the Cardie and Duncan properties of a formal language, defined above.

  • (a, 5) Give English descriptions of the Cardie property and the Duncan property.

  • (b, 10) Prove, using quantifier proof rules, that the language a^*b^* does not have the Duncan property.

  • (c, 10) A given language might or might not have either of those properties. Give four regular expressions, representing:
    • A language with neither property
    • A language with the Cardie property but not the Duncan property
    • A language with the Duncan property but not the Cardie property
    • A language with both properties

    You do not have to justify your answers, but make sure you indicate which regular expression is an answer to which of the four parts.

  • (d, 15) Prove that a regular language has the Cardie property if and only if its minimal DFA has a victory state. (I don't need an inductive proof, but make sure you argue carefully from all the relevant definitions.)

  • (e, 10) Suppose you are given a DFA M as a labeled directed graph. Explain how you could use our search algorithms on the directed graph to determine whether the language L(M) has the Duncan property.

• Question 3 (25): These problems refer to the functions g and h defined above.
  • (a, 10) Prove, by ordinary induction on naturals, that for any natural n, g(n) = 3(2ⁿ) - 2.
  • (b, 10) Prove, by strong induction on naturals, that for any positive natural n, the number h(n) divides n.
  • (c, 5) Describe the output of the function h(n) in terms of the prime factorization of the input n.

• Question 4 (30): Here is the inevitable question on Kleene's Theorem constructions.
  • (a, 5) Using the construction from lecture, build a λ-NFA from the regular expression ab^* + ba. (This λ-NFA is equivalent to, but different from, the λ-NFA M given above. We will use M for the rest of the problem, to make your life easier.
  • (b, 10) Build an ordinary NFA N that is equivalent to the given λ-NFA M, so that L(N) = L(M).
  • (c, 10) Using the Subset Construction, build a DFA D such that L(D) = L(N).
  • (d, 5) Show the steps of the State Elimination algorothm to get the regular expression ab^* + ba from D.

Last modified 3 June 2012