CMPSCI 250: Introduction to Computation

Final Exam

David Mix Barrington

16 December 2010

Directions:

• Answer the problems on the exam pages.
• There are four problems for 130 total points. Actual scale is A=102, C=66.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.

  Q1: 35 points
  Q2: 35 points
  Q3: 40 points
  Q4: 20 points
Total: 130 points

Question 1 (35): Here are three situations in which you are to construct induction proofs.

• (a, 10) Define a function f from naturals to naturals by the rules f(0) = 0 and, for n > 0, f(n) = n - f(n-1). Prove that for all odd n, f(n) = (n+1)/2, and that for all even n, f(n) = n/2. (You can use ordinary induction on n and proof by cases, or separate inductions for odd and even n.)

• (b, 15) Define a function g from naturals to naturals as computed by the following pseudo-Java method:

  
               natural g (natural n) {
                  if (n == 0) return 0;
                  return n + g(n/2);}
       

  where "/" is Java integer division, with no remainder, so that for example "7/2" is 3. Prove that for all positive naturals n, the method terminates on input n and outputs a number g(n) which is less than 2n. (You should use strong induction.)

• (c, 10) Define a function h from strings to strings (over the alphabet {a, b, c,..., z}) by the following rules:

  • h(λ) = λ

  • If w is a string and a is a letter, h(wa) = h(w^R)a where "w^R" denotes the reversal of w (w written backwards).

  Compute the string h("cardie").

  Prove that for any string w, h(w) is a string whose length is the same as the length of w. (You can use either induction on strings or ordinary induction on the length of w.)

• Question 2 (35): This problem concerns a set C of customers at Zane's Noodle Bowl, and the set V of vegetables that a customer may order for their soup there. Let R ⊆ C × V be the relation such that the pair (c,v) is in R if and only if customer c orders vegetable v. (Equivalently, R(c, v) is the predicate "customer c orders vegetable v".) Also, S is a binary relation on customers, that is, a subset of C × C, so that S(c, c') is translated into English as "c and c' are similar".

  • (a, 10) Translate the following statements as indicated, using the predicates R and S, constant symbols w and k of type "customer" for "William" and "Kate", and constant symbols of type "vegetable" for "bean sprouts" and "tofu".

    • (I, to English) ∀c: R(c,b) → R(c,t)

    • (II, to symbols) Either William or Kate, or both, ordered bean sprouts, but Kate did not order tofu.

    • (III, to English) ∀c: ∀c':S(c,c') ↔ (∀v: R(c, v) ↔ R(c', v))

  • (b, 10) Prove, assuming that statements I and II are true, that William ordered tofu. (This involves propositional logic on the four statements R(w, b), R(w, t), R(k, b), and R(k, t), along with two uses of the Specification Rule for quantifiers.)

  • (c, 15) Prove, using quantifier proof rules and the definition in Statement III, that S is an equivalence relation.

• Question 3 (40): Let Σ be the alphabet {a, b} and let R be the regular expression (a+b)(a+ba+bb)^*. Let M be a λ-NFA with state set {1, 2, 3, 4}, start state 1, final state set {4}, and transitions (1, a, 2), (1, b, 2), (2, a, 2), (2, b, 3), (3, a, 2), (3, b, 2), and (2, λ, 4).

  • (a, 5) Argue convincingly that L(M) = L(R). You may want to use some of the constructions we covered in lecture, or you may want ot argue informally first that L(R) ⊆ L(M) and then that L(M) ⊆ L(R).

  • (b, 10) Using the construction from lecture on M, build an ordinary NFA N such that L(N) = L(M).

  • (c, 10) Using the Subset Construction, build a DFA D such that L(D) = L(N).

  • (d, 5) Either prove that your D is minimal or find a minimal DFA D' such that L(D) = L(D').

  • (e, 5) Find a DFA D'' such that L(D'') is the complement of L(D'), that is, such that for any string w, (w ∈ L(D'')) ↔ (w ∉ L(D')).

  • (f, 5) Using the construction from lecture, find a regular expression R' such that L(R') = L(D'').

• Question 4 (20): If u and v are any two strings in {a, b}^*, we say that u and v are anagrams if and only if v can be made by rearranging the letters of u. That is, u and v must have the same number of a's, and u and v must have the same number of b's. (For example, the strings "aabbbaa" and "baaaabb" are anagrams.) Define the language X, over the alphabet {a, b, c}, to be the set {ucv: u and v are anagrams}. (Remember that u and v must be strings in {a, b}^*. For example, the string "aabbbaacbaaaabb" is a member of X.)

  • (a, 10) Is there a regular expression denoting X? Prove your answer.

  • (b, 10) Argue informally that the language X is Turing decidable. (Remember that in designing a Turing machine, you can make the alphabet as large as you want as long as it is finite.)

Last modified 21 December 2010