Question: SECTION C 9 Semantics of Programming wLanguages (a) The integer expressions E of a programming language are given by E ::= n | X | E | E + E where n ranges over integer constants and X ranges over identifiers. Explain the principle of structural induction for proving that some property (E) holds for all integer expressions E. [5 marks] (b) Taking states to be finite partial functions mapping identifiers to integer constants, define a relation E, s n giving the result n (if any) of evaluating integer expression E in state s. [7 marks]

(c) Use structural induction to prove that if E, s n1 and E, s n2 both hold, then n1 = n2. [Hint: Consider the property (E) given by s, n1, n2((E, s n1) & (E, s n2) n1 = n2).] [7 marks] (d) What property of the pair E, s ensures that there is some n for which E, s n holds?

) Write java program that takes both n (as input) and n integer inputs and add the given values to a defined array. , write program that asks the user for a positive integer number. If the user enters anything else, the program would ask the user to re-enter a number or enter -1 to quit. If it is a positive number, the program then prints out a multiplication table that goes up to that number.

Write \ program that asks for a number and then prints as many lines as the user inputs. Each line contains as many pairs of characters ("*#") as the number of this line.

(a) Give precise definition of the complexity class NP and of NP-completeness.

(b) For any natural number k, the problem k-colourability is defined as the following decision problem. Given a graph G = (V, E), is there a mapping : V {c1, . . . , ck} such that if (u, v) E, then (u) 6= (v)? (i) Explain why, for each k, the problem k-colourability is in the class NP. [2 marks] (ii) For what values of k is the problem k-colourability decidable in polynomial time? Why? [2 marks] (iii) For which values of k is the problem k-colourability NP-complete? Give a brief indication how this might be proved. [2 marks] (c) The company Fon-X runs a mobile 'phone service. It has 2000 'phone masts stationed across the country. The frequency spectrum assigned to the company is split into 20 bands. Each mast is to be assigned a frequency band in such a way that masts within 50 miles of each other do not share the same frequency band. (i) What is the relationship between this problem and k-colourability? [2 marks] (ii) What can you say about the complexity of the problem Fon-X is trying to solve? [3 marks] (d) Fon-X solved its frequency assignment problem by an exhaustive search algorithm, which took a week to run. The company has just doubled in size through a merger. It intends to repeat the frequency assignment on 4000 masts, setting aside two weeks for the task. As a consultant, write a short note to the company explaining what you think of the idea, and suggesting any alternatives you think might be better.

Question:

Both institutions use relational databases. Write brief notes to alert the database administrators to the difficulties that they may encounter. [8 marks] SECTION C 9 Semantics of Programming Languages (a) The integer expressions E of a programming language are given by E ::= n | X | E | E + E where n ranges over integer constants and X ranges over identifiers. Explain the principle of structural induction for proving that some property (E) holds for all integer expressions E. [5 marks] (b) Taking states to be finite partial functions mapping identifiers to integer constants, define a relation E, s n giving the result n (if any) of evaluating integer expression E in state s. [7 marks]

(c) Use structural induction to prove that if E, s n1 and E, s n2 both hold, then n1 = n2. [Hint: Consider the property (E) given by s, n1, n2((E, s n1) & (E, s n2) n1 = n2).] [7 marks] (d) What property of the pair E, s ensures that there is some n for which E, s n holds?

) Write java program that takes both n (as input) and n integer inputs and add the given values to a defined array. , write program that asks the user for a positive integer number. If the user enters anything else, the program would ask the user to re-enter a number or enter -1 to quit. If it is a positive number, the program then prints out a multiplication table that goes up to that number.

Write \ program that asks for a number and then prints as many lines as the user inputs. Each line contains as many pairs of characters ("*#") as the number of this line.

(a) Give precise definition of the complexity class NP and of NP-completeness.

(b) For any natural number k, the problem k-colourability is defined as the following decision problem. Given a graph G = (V, E), is there a mapping : V {c1, . . . , ck} such that if (u, v) E, then (u) 6= (v)? (i) Explain why, for each k, the problem k-colourability is in the class NP. [2 marks] (ii) For what values of k is the problem k-colourability decidable in polynomial time? Why? [2 marks] (iii) For which values of k is the problem k-colourability NP-complete? Give a brief indication how this might be proved. [2 marks] (c) The company Fon-X runs a mobile 'phone service. It has 2000 'phone masts stationed across the country. The frequency spectrum assigned to the company is split into 20 bands. Each mast is to be assigned a frequency band in such a way that masts within 50 miles of each other do not share the same frequency band. (i) What is the relationship between this problem and k-colourability? [2 marks] (ii) What can you say about the complexity of the problem Fon-X is trying to solve? [3 marks] (d) Fon-X solved its frequency assignment problem by an exhaustive search algorithm, which took a week to run. The company has just doubled in size through a merger. It intends to repeat the frequency assignment on 4000 masts, setting aside two weeks for the task. As a consultant, write a short note to the company explaining what you think of the idea, and suggesting any alternatives you think might be better.

(a) Write down sim

(a) Write down sim