CMPSCI 250: Introduction to Computation

Final Exam

David Mix Barrington

9 May 2013

Directions:

• Answer the problems on the exam pages.
• There are six problems, some with multiple parts, for 120 total points. Actual scale was A = 105, C = 75.
• Some useful definitions precede the questions below.
• No books, notes, calculators, or collaboration.
• In case of a numerical answer, an arithmetic expression like "2¹⁷ - 4" need not be reduced to a single integer.

  Q1: 10 points
  Q2: 20 points
  Q3: 20 points
  Q4: 30 points
  Q5: 20 points
  Q6: 20 points
Total: 120 points

Definitions Used on the Exam:

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

• The scope of any quantifier is always to the end of the statement it is in.

• Questions 1 and 2 use a set of dogs A = {b, c, d} that includes exactly the three named (and distinct) dogs Biscuit, Cardie, and Duncan. We define the following four unary predicates for any dog x in the set A:
  • M(x) means "x is muddy"

  • P(x) means "x went in the pond"

  • S(x) means "x went in the swamp"

  • W(x) means "x is wet"

• The four statements of Question 1 are:
  • (I) Either Cardie or Biscuit, but not both, went in the pond, and if Cardie is not muddy, then Biscuit is muddy.
  • (II) ∀x: W(x) ↔ (S(x) ∨ P(x))
  • (III) Every muddy dog went in the swamp, and every dog who went in the pond is not muddy.
  • (IV) ¬∀z: W(z)

• A state s in an ordinary NFA is defined to be reachable if there exists a string w such that there is a w-path from the start state to s.

• The λ-NFA N has state set {1, 2, 3, 4}, start state 1, final state set {2}, and the following five transitions: (1, b, 2), (1, b, 3), (2, a, 4), (3, λ, 1), and (4, λ, 3). It looks like this, with "L" representing "λ":

  
             b
      >(1) ----> ((2))
       |^         |       
      b||L        |a       
       ||         |       
       V|         V       
       (3) <---- (4)
            L
  
  (Diagram fixed 8 December 2013 -- it had (4, a, 2) instead of (2, 1, 4).) 
  

  Let n be any positive integer. Define the DFA D_n to have alphabet {a}, state set {0, 1, 2,..., n-1}, start state 0, final state set {1}, and transition function δ(i, a) = (i+1) % n where "%" is the Java modular division operator.

• Question 1 (10): Translate each statement as indicated, using the set of dogs {Biscuit, Cardie, Duncan} and the four predicates given above.

  • (a, 3) (to symbols) (Statement I) Either Cardie or Biscuit, but not both, went in the pond, and if Cardie is not muddy, then Biscuit is muddy.

  • (b, 2) (to English) (Statement II) ∀x: W(x) ↔ (S(x) ∨ P(x))

  • (c, 3) (to symbols) (Statement III) Every muddy dog went in the swamp, and every dog who went in the pond is not muddy.

  • (d, 2) (to English) (Statement IV) ¬∀z: W(z)

• Question 2 (20): These questions use the definitions, assumptions, and predicates from Question 1, which are also listed above.
  • (a, 5) Convert Statement IV to an equivalent statement beginning with an existential quantifier (∃). What conclusion can you draw from this statement by the Existential Instantiation rule?

  • (b, 5) What conclusion about Duncan can you draw from Statement III by the Universal Instantiation rule?

  • (c, 10) Prove that if Statements I, II, III, and IV are all true, then Duncan is not muddy. You may use English rather than symbols if you are convincing and include all necessary steps of the argument.

• Question 3 (20): Here are two induction proofs involving finite-state machines.
  • (a, 10) Prove by induction on all positive integers k that if M is any ordinary NFA with k states, and M has fewer than k - 1 transitions, then there exists a state of M that is not reachable. ("Reachable" is defined above.)

  • (b, 10) Let N be the λ-NFA defined above. Prove by induction for all natural numbers i that the string b(ab)ⁱ is in the language L(N).

• Question 4 (30): Here is a series of constructions implementing parts of the proof of Kleene's Theorem. If you cannot get the answer to an earlier part of the question, you can still get some credit for demonstrating the construction of a later part on another appropriate example.
  • (a, 10) Construct an ordinary NFA N' with the same language as the λ-NFA N given above.

  • (b, 10) Use the Subset Construction to construct a DFA D with the same language as N and N'.

  • (c, 10) Use the state elimination method (starting from N, N', or D -- your choice) to get a regular expression for the language L(N).

• Question 5 (20): This question involves some number theory and the family of DFA's D_n defined above.
  • (a, 10) Prove that the set of numbers {14, 27, 95} are pairwise relatively prime. You may either use the Euclidean Algorithm or argue carefully from the prime factorizations of these numbers. Make it clear that you understand the relevant definitions.

  • (b, 10) Using the Chinese Remainder Theorem, describe a regular expression for the language L(D₁₄) ∩ L(D₂₇) ∩ L(D₉₅). Explain your reasoning carefully.

• Question 6 (20): These ten questions involve specific vocabulary from the course. For the first five, parts (a)-(e), identify the give term. For the second five, parts (f)-(j), explain the difference between the two terms.
  • (a, 2) prime factorization of a positive integer

  • (b, 2) inductive step of an ordinary induction proof

  • (c, 2) Myhill-Nerode Theorem

  • (d, 2) ∅^*

  • (e, 2) construction of an λ-NFA for R^*, given a λ-NFA for R

  • (f, 2) → and ↔

  • (g, 2) proof by contradiction and proof by contrapositive

  • (h, 2) tree and forest

  • (i, 2) r.e.-NFA and λ-NFA

  • (j, 2) ∅ and λ

Last modified 8 December 2013