SCHOOL OF COMPUTING, MATHEMATICAL AND INFORMATION SCIENCES Semester 2 Examinations 2008/2009. CI311 Specification and Refinement

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1 SCHOOL OF COMPUTING, MATHEMATICAL AND INFORMATION SCIENCES Semester 2 Examinations 2008/2009 CI3 Specification and Refinement Time allowed: Answer: THREE hours TWO questions, each in a SEPARATE book Items permitted: there is no restriction on material that students may take into the examination Items supplied: Summary of Notations from Logic & Discrete Mathematics Marks for whole or part questions are indicated in ( ) brackets CI3/2008/2009 Pages: 7 Date: 2 Apr 2009

2 . This question is on UML and OCL, the Object Constraint Language. The following class diagram represents a partial model of an ordering system that keeps track of orders made by customers and process them. The system also records the items and their prices. Each item has a description and uniquely identified by a number. Each customer has a name, a credit rating, and is uniquely identified by a number. The credit rating for a customer takes one of the following values: good or bad. Each order is assigned a unique id, and its entry date is recorded. An order consists of one or more items with the ordered quantity. An order may have many shipments, where each shipment has a date and consists of one ore more shipment items. OrderingSystem customers Customer number : Integer name : String creditrating: String orders Order id : String datemade : Date price() : Money filled() : Boolean orders shipments Shipment shipmentdate : Date.. orderitems OrderItem quantity : Integer price() : Money filled() : Boolean Item number : Integer description : String price : Money shipmentitems shipmentitems ShipmentItem quantity : Integer.. items Date isbefore(d : Date) isafter(d : Date) equals (d : Date) Question continues on the next page. CI3/2008/2009 Page of 7 Date: 2 Apr 2009

3 Question continued (a) Formalise the following informal requirements as OCL invariants: i. Each order within the system is uniquely identified by its id. (2) ii. The date of each order must be before the dates of all its shipments. (2) iii. An order is filled if all its items have been shipped. (2) iv. For each order, the quantity ordered for each item is greater than or equal to the sum of all the quantities for that item shipped so far. (2) (b) Write in English, and then formally specify in OCL, some other constraints that you think might be sensible for the ordering system. (7) (c) Formally specify the following query operations using OCL: i. context Order::filled() : Boolean which returns true if the order is filled and false otherwise. An order is filled if all its items have been shipped to the customer. (3) ii. context OrderingSystem::getCustomer(n : Integer) : Customer which returns the customer with number n. (4) iii. context OrderingSystem::getOutstandingOrders(n : Integer): Set(Order) which returns the set of outstanding (i.e. not filled) orders of a customer given the customer s number n. (4) iv. context OrderingSystem::getNoOrders(n : Integer, d, d2 : Date) : Integer which returns the number of orders for customer with number n between the dates d and d2. (4) (d) Specify the following update operations as OCL preconditions and postconditions. i. context OrderingSystem::addCustomer(n : Integer, m : String, r : String) which adds a customer with number n, name m and credit rating r. (6) ii. context OrderingSystem::makeOrder(c : Customer, i : String, d : Date, b : Bag(Item)) which creates an order with id i, for customer c, on date d from a bag of items b. iii. (context Order:: addorderitem(i : Item, q : Integer) which adds the item i with quantity q to the order. (6) (8) CI3/2008/2009 Page 2 of 7 Date: 2 Apr 2009

4 2. This question is on abstract modelling and Object-Oriented Formal Specification. A first step towards developing a generalized (and persistent) Patient Record System is to model the known Patient Population, specified as class PP[ P, I ]: A Patient Population includes an initially-empty set of patients (as uniquely identified by elements of a given type P). Each patient has some associated information (of type I), and is either alive or dead. PP Patient : P Info : Patient I {Alive, Dead}: part Patient Patient = (a) Two queries, leaving its state unchanged, are to be provided for the class PP: Current information i for patient p can be shown at any time. Whether patient p is or is not alive can be shown as a boolean test t. PP?info(p i) PP?isAlive(p t) Give a simple formal specification for each query, guided by its documentation. (6) (b) All possible changes-of-state at this level are specified by the following events: PP!NewPatient(i p) i : I ; p : P \Patient Info p = i ; p Alive PP!UpdateInfo(p, i) p : Patient ; i : I i Info p Info p = i PP!Death(p) p : Alive p Dead Document each event informally, using simple but precise natural language. (9) Question 2 continues on the next page. CI3/2008/2009 Page 3 of 7 Date: 2 Apr 2009

5 Question 2 continued (c) At a later development stage, the overall Health-Care System might then be introduced by extending PP. This is specified as class HCS[ P, I, S, H ]: The Health-Care System supports its Patient Population by means of some hierachical set of services and nested sub-services (with unique identifiers from type S). It also maintains a central register of health-professionals (with unique identifiers from type H), and their current associations with any service or sub-service; those who have at least one such association are said to be active. All of the services for which each patient is enrolled are maintained at this level as well. HCS PP[ P, I ] Service : S SubService := dom NestIn NestIn : Service Service NestIn + id Service = HProf : H Assoc : HProf Service Active := dom Assoc Roll : Patient Service Service = ; HProf = Express the expanded state-specification for HCS visually: in diagrammatic form. Identify all sets defined by its state-invariant; explain why they are initially empty. (20) (d) Several new queries will now be required for this class, e.g. the following: HCS?subServices(s S) I A I J 5 HCS?associations(h S) ) I I? D 5 Informally document each of these queries, once again using natural language. (6) (e) A number of events will obviously be required for HCS, e.g. the following: A new sub-service s nested in n can be defined, provided its name is unique. A new health professional can be registered, and given unique identifier h. Some patient p can be newly enrolled for (sub-)service s, provided they are not already enrolled there. HCS!NewSubService(s, n) HCS!RegisterHProf ( h) HCS!EnrolPatient(p, s) Formally specify each event in diagrammatic form, guided by its documentation. (9) CI3/2008/2009 Page 4 of 7 Date: 2 Apr 2009

6 3. This question is on the complexity of algorithms. (a) Suppose that we have an unordered list L and we want to find the largest and the smallest elements in L (this is called the min&max problem). A naive approach simply traverses the list once, keeping the current maximum and minimum in variables, comparing each element to both of these variables as we proceed. i. Refine this solution, writing pseudocode (or java, if you prefer) which presents this solution in more detail. (6) ii. Compute the exact time complexity of your solution with respect to the number of comparisons made. (3) iii. What is the time complexity of this naive solution in terms of the big-o notation? iv. Is there difference between worst case and best case time complexity? Explain. () (2) Question 3 continues on the next page. CI3/2008/2009 Page 5 of 7 Date: 2 Apr 2009

7 Question 3 continued (b) The following is a description of a recursive solution to the min&max problem: find-min&max-of L: i. if L consists of one element, then set MIN and MAX to it; if L consists of two elements, then set MIN to be the smaller of them and MAX to be the larger; ii. otherwise do the following: A. split L into two halves L and L 2 ; B. call find-min&max of L, placing returned values in MIN and MAX; C. call find-min&max of L 2, placing returned values in MIN2 and MAX2; D. set MIN to smaller of MIN and MIN2; E. set MAX to larger of MAX and MAX2; iii. return with MIN and MAX. Let C(N) denote the number of comparisons required by the recursive man&max routine on lists of length N. i. How many comparisons are made for N = and for N = 2? (2) ii. If N > 2 how many comparisons are carried out? You may assume N is even if you wish to. (4) iii. Hence show that the recurrence relation is C() = C(2) = ; C(N) = 2C( N 2 ) + 2. (3) iv. Use these relations to compute C(N) for N = 4, 8. (2) v. Prove, by induction or otherwise, that C(N) = 3N 2 recurrence relation. 2 is a solution to the (8) vi. What is the time complexity of this solution in terms of the big-o notation? () Question 3 continues on the next page. CI3/2008/2009 Page 6 of 7 Date: 2 Apr 2009

8 Question 3 continued (c) Suppose we have three sorting algorithms Alg, Alg 2 and Alg 3 that require 3n 2 n +, 2nlog 2 (n) + 3 and 2 n2 + n comparisons respectively to sort an unordered list of n items in the worst case. i. Recall that if there exist constants N, K such that for all n > N we have that f(n) Kg(n) then we say that f(n) = O(g(n)). Give a simple big-o expression for the worst-case time complexity of Alg, Alg 2 and Alg 3. Provide a constant of proportionality K and a corresponding number N. (6) ii. Which of the three algorithms would you be likely to choose if you were considering input sizes of length at most 00? (4) iii. Which of the three algorithms would you be likely to choose if you were considering input sizes of length at most 0,000? Would you ever alter your choice for any larger values of n? (4) iv. Which of the three algorithms are you unlikely to ever choose, based on this information? Suggest a possible reason that one might still choose it, despite this information. (4) CI3/2008/2009 Page 7 of 7 Date: 2 Apr 2009