Module Contact: Dr Pierre Chardaire, CMP Copyright of the University of East Anglia Version 1

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1 UNIVERSITY OF EAST ANGLIA School of Computing Sciences Main Series UG Examination 2015/16 COMPUTING PRINCIPLES CMP-4002B Time allowed: 2 hours. Answer EIGHT questions. All questions carry equal weight. Notes are not permitted in this examination. Do not turn over until you are told to do so by the Invigilator. CMP-4002B Module Contact: Dr Pierre Chardaire, CMP Copyright of the University of East Anglia Version 1

2 Page 2 1. In World War II the German High Command used a cipher system based on the XOR operation. This operation, denoted by, has the following truth table: x y x y T T F T F T F T T F F F The cipher was implemented by Lorenz rotor machines. The same machine with the same settings could be used for both enciphering and deciphering because (x y) y = x. (1) (a) Give a DNF expression for x y. (b) Simplify x x. (c) Simplify x F. (d) Show that is an associative operation, i.e. that (x y) z = x (y z). (e) Show that equation (1) holds by using the results of questions 1b, 1c and 1d. 2. (a) Convert the binary number into decimal by [10 marks] using Horner s method. (b) The two s complement binary representation of an integer is What is the decimal representation of this integer? CMP-4002B Version 1

3 Page 3 3. (a) Give all steps necessary to convert the decimal number [10 marks] into binary, providing a result with 10 digits in the mantissa. (b) What is the two s complement binary representation of the decimal integer 99 on a machine that uses 8-bit words to store signed integers? 4. (a) The following truth table specifies the boolean function, output(p, q, r) p q r output F F F T F F T T F T F T F T T F T F F F T F T T T T F F T T T F (i) Express output as a DNF. (ii) Use the Quine-McCluskey algorithm to simplify the DNF obtained. (b) Show that the well-formed formula (A C = B) (A = C) = (A = B) is a tautology by (i) transforming it into a DNF (ii) and applying the dual Davis-Putnam procedure to prove its [4 marks] [4 marks] validity. CMP-4002B Version 1 TURN OVER

4 Page 4 5. An NFSA with starting state S and final state F is specified by the following transition table: a b A {A} {F} F {A,F} {A} S {A,S} {} (a) Provide the transition table, starting state and final states of the [10 marks] DFSA obtained by transforming the NFSA. (b) Which strings of length less than or equal to four are accepted by the NFSA? 6. Consider the language L accepted by the following automaton: (a) Describe L in words. (b) Demonstrate the state elimination technique seen in the lectures to [10 marks] find a regular expression that specifies L. CMP-4002B Version 1

5 Page 5 7. Let U = {x Z : 3 x < 2} be the universal set. (a) List the elements in the set A = {x U : x > 1}. (b) List the elements in the set B = {x U : x is not a multiple of 2 }. (c) Illustrate the sets U,A,B and C = {x U : x is even } in a Venn diagram. (d) List the elements of the power set P(B). (e) List the elements of A A. (f) List the elements in A C. (g) List the elements in B C 8. (a) For each of the following mappings state which are functions and whether the functions are injective, surjective or bijective. (i) f : Z Z where f (x) = x + 1 (ii) g : R R where f (x) = 3x (iii) h : Z Z where f (x) = x 3 (b) Given the functions m : R R where m(x) = 2x 2 n : R R where n(x) = 2x 2 x + 1 (i) determine the composite function m n. (ii) find the inverse function m 1 of m. CMP-4002B Version 1 TURN OVER

6 Page 6 9. Table 1 gives the ages of the members of two groups, Group A and Group B. Group A Group B Table 1: Members Ages (a) Calculate five figure summaries for each of the groups. (b) Sketch approximate box plots of the five figure summaries in order to compare the relative ages of the two groups. (c) Calculate the standard deviation of Group A to two decimal places. 10. Consider the set A = {a,b,c,d} and the relation defined on A A by setting R = {(a,a),(a,b),(b,c),(c,d),(b,a),(c,c)}. (a) State, with reasons, whether R is: (i) reflexive (ii) transitive (iii) symmetric (b) List the elements of A A that must be added to R to make up its reflexive closure. (c) List the elements of A A that must be added to R to make up its symmetric closure. (d) List the elements of A A that must be added to R to make up its transitive closure. (e) Draw a Hasse diagram for the partial order relation R such that {(X,Y ) R : max(x) > max(y )} on the set A = {{1},{0,2},{1,3},{4},{2,4}}. END OF PAPER CMP-4002B Version 1