University of Toronto Scarborough. Aids allowed: None... Duration: 3 hours.

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1 University of Toronto Scarborough CSC B36 Final Examination 12 December 2017 NAME: (circle your last name) STUDENT NUMBER: Do not begin until you are told to do so. In the meantime, put your name and student number on this cover page and read the rest of this page. Aids allowed: None.... Duration: 3 hours. There are 9 pages and each is numbered at the bottom. Make sure you have all of them. Write legibly in the space provided. Use the backs of pages for rough work; they will not be graded. 1. / / / / 8 5. / / 5 7. / 10 Total / 80 Fall 2017 cscb36 Final Exam Page 1 of 9

2 1. [10 marks total; 5 for each part] For the purpose of this question, we define a real number x to be positive rational if and only if there exist positive integers p,q such that x = p q. Let PR be the smallest set such that Basis: 2017 PR. Induction Step: If x,y PR, then (i) 1 x PR and (ii) x+y PR. (a) Use structural induction to prove that every number in PR is positive rational. Remember to use proper proof structure when proving that a number is positive rational. (b) Let p and q be arbitrary positive integers. Informally explain why p q PR. Fall 2017 cscb36 Final Exam Page 2 of 9

3 2. [20 marks] Given a list L of integers, we define E(L) to be the number of even integers in L. For example, if L is [4,8,8,9,6,3,4], then E(L[: 3]) is 3 and E(L[3 :]) is 2. Loosely speaking, the program below takes a list L and returns the number of even integers in the second half of L minus the number of even integers in the first half of L. Furthermore, the variables n and h do not change values after they are initialized. So you may treat them as constants. They are there mainly to save you from having to repeatedly write len(l) and len(l). Use methods from class to prove the program below correct with respect to its given specification. Precondition: L is a list of integers. Postcondition: Return E(L[h :]) E(L[: h]), where h = len(l) 2. For example, if L is [4,8,8,9,6,3,4], then h is 3 and 2 3 = 1 is returned. DiffNumEvens(L) 1 n = len(l); h = n 2 ; 2 while i < n: j = h; i = 0; s = 1; t = 0 3 if L[j] is even: 4 t = t+s 5 i = i+1; s = s; j = j +s i 6 return t 2 [more space available on next page...] Fall 2017 cscb36 Final Exam Page 3 of 9

4 [... additional space for question 2] Fall 2017 cscb36 Final Exam Page 4 of 9

5 3. [12 marks total; 6 for each part] For each of the following implications, state whether it holds for arbitrary regular expressions R, S and T. Justify your answers. You will receive no credit without correct justification. (a) If R S S R, then RS SR. (b) If R ST, T SR and SRT STR, then RT TR. Fall 2017 cscb36 Final Exam Page 5 of 9

6 4. [8 marks] Let L = {1 i 0 j : i,j N and j < i < 2j}. Use the Pumping Lemma to prove that L is not regular. Fall 2017 cscb36 Final Exam Page 6 of 9

7 5. [15 marks total] Let L = {1 i 0 j : i,j N and j < i < 2j}. This is the same language as in question 4. (a) [1 mark] Give the shortest string in L. No justification is required. (b) [7 marks] Using as few productions as possible, give a CFG for L and explain why it is correct. (c) [7 marks] Give a PDA that accepts L and explain why it is correct. Fall 2017 cscb36 Final Exam Page 7 of 9

8 6. [5 marks] We define a ternary boolean function called AdjacentOnes by the following truth table. x y z AdjacentOnes(x, y, z) Use the given truth table to write a DNF formula and a CNF formula that represent AdjacentOnes. No justification is required. Your DNF formula: Your CNF formula: Fall 2017 cscb36 Final Exam Page 8 of 9

9 7. [10 marks total] Consider a first-order language with binary predicate B and equality predicate =. Recall from class that a binary predicate B on a domain D can be represented by a directed graph, where each node in the graph represents an element in D and an arrow from node x to node y is used to indicate B(x,y) = 1. Put another way, node x pointing to node y means B(x,y) is true. Here are 3 formulas along with the statements they express. F 1 x y B(x,y) Every node points to some node. F 2 x y B(y,x) Every node is pointed to by some node. F 3 x y (B(x,y) B(y,x)) No node points to a node that points back to it. (a) [4 marks] Write formulas F 4 and F 5 to express the following statements. F 4 : Some node points to more than one node. F 5 : Some node is pointed to by more than one node. (b) [3 marks] For this part we let our domain be D = {i : 0 < i n, where i,n N}. I.e., D = {1,2,,n}. Use the pigeonhole principle to explain why when restricted to domain D, (F 1 F 4 ) logically implies F 5. (c) [3 marks] Prove that (F 1 F 2 F 3 F 4 ) does not logically imply F 5. Fall 2017 cscb36 Final Exam Page 9 of 9

University of Toronto Scarborough. Aids allowed: None... Duration: 3 hours.

University of Toronto Scarborough CSC C63 Final Examination 20 August 2015 NAME: STUDENT NUMBER: Do not begin until you are told to do so. In the meantime, put your name and student number on this cover