solved: please follow instructions A reinforcement learning problem has…

  

please follow instructions A reinforcement learning problem has…please follow instructions A reinforcement learning problem has states {s1, . . . , sn}, actions {a1, . . . , am}, reward function R(s, a) and next state function S(s, a). (a) Give a general definition of a policy for such a problem. [1 mark] (b) Give a general definition of the discounted cumulative reward and the corresponding optimal policy for such a problemWhat sources of conventional earth time might be used by computer systems? How would you estimate bounds on the accuracy of time received from such a source? [4 marks] (c) What constraint does distributed inter-process communication (IPC) impose on the clock values of the communicating parties? [1 mark] (d) Outline one clock synchronisation protocol that satisfies this constraint. [4 marks] (e) For each of the cases of IPC illustrated below, give the vector clock values that message receiving and delivery modules could maintain for each process. (i) P P P 0 1 2 (ii) P P P 0 1 2 (iii) P P P 0 1 2 (iv) P P P 0 1 2 [6 marks] (f ) Define “causal order” of message delivery. In which, if any, of (i) to (iv) above is causal order violated at the message receiving module? [3 marks] 2 CST.2001.9.3 2 VLSI Design (a) Give transistor level designs for 2-input NAND gates using static and dynamic CMOS, explaining how the latter is controlled by a clock. [8 marks] (b) Comment on the relative merits of static and dynamic logic for evaluating more complicated combinatorial functions, including transistor count, wiring complexity, speed and implications for cascaded logic. [4 marks] (c) Consider the following circuit for a dynamic latch: d c q Specify the input and output signals, and explain its operation carefully. [8 marks] 3 [TURN OVER CST.2001.9.4 3 Digital Communication II Write notes about four of the following types of Access Network. You should discuss any special challenges posed by each transmission medium, describe the current technology, and comment on any commercial factors pertinent to its deployment. (a) Analogue telephone modems. (b) Cable modems. (c) Digital Subscriber Line (xDSL). (d) Fixed wireless. (e) Optical fibre to the home. (f ) Satellite. [5 marks each] 4 CST.2001.9.5 4 Advanced Graphics and HCI (a) Which usability evaluation techniques are most appropriate to the following situations? For each, say why this is so. (i) An expert user performing a familiar task. (ii) An expert user designing a novel solution. (iii) A new user using a system for the first time. [6 marks] (b) Sketch three screens of a prototype for an online newspaper that can be customised to users’ interests. (i) How could a development team test the usability of this prototype cheaply and quickly? Describe some basic procedures that could be applied during testing. (ii) Contrast the benefits of doing a Cognitive Walkthrough evaluation to those of the procedure you have just described, in the context of this system. (iii) Give an example of one stage of the Cognitive Walkthrough process, as it would apply to one of the screens you have sketched. [14 marks] 5 [TURN OVER CST.2001.9.6 5 Business Studies (a) What is meant by the terms supply curve and demand curve, and what is significant about the point where they cross? [5 marks] (b) What is meant by the terms cost curve and break-even point? Draw a diagram to show the relation between them and the demand curve. How would you attempt to establish a demand curve in practice? [5 marks] (c) The cost and demand schedules for a particular product are given in the following table. What price should the manufacturer set? [5 marks] Volume, k Unit Cost Unit Price 1 11.8 19 2 11.6 18 3 11.4 17 4 11.2 16 5 11.0 15 6 10.8 14 7 10.6 13 8 10.4 12 9 10.2 11 10 10.0 10 (d) Discuss the economic impact of the advent of the Internet. [5 marks] 6 CST.2001.9.7 6 Types (a) Define the typing relation of the polymorphic lambda calculus (PLC). [5 marks] (b) For each of the following PLC typing judgements, state whether or not there is a type t that makes the judgement provable. Justify your answer in each case. (i) ` ?x : ?a(a)(?ß(x ß)) : t [3 marks] (ii) ` ?a(?x : a(?ß(x ß))) : t [3 marks] (iii) ` ?x : t (?a(x(a ? a)(x a))) : t ? ?ß(ß) [3 marks] (iv) ` ?x : t (?a(x(a ? a)(x a))) : t ? ?a(a ? a) [3 marks] (v) ` ?a(?x : t (x(a ? a)(x a))) : ?a(a ? a) [3 marks] 7 Optimising Compilers (a) Sketch an algorithm for instruction scheduling. Be careful to specify the nature of your initial and final data-structures; also give the worst-case running time of the algorithm and a program which causes this worst-case running time to be achieved. [10 marks] (b) Explain the conflicts between register allocation, common sub-expression elimination and instruction scheduling on a machine where register usage should be minimised (for example to reduce procedure entry save/restore code). [5 marks] (c) Describe Static Single Assignment (SSA) form and explain how it can help to produce better results in both compilation and decompilation. [5 marks] 7 [TURN OVER CST.2001.9.8 8 Artificial Intelligence (a) Why does search in game-playing programs always proceed forward from the current position rather than backward from the goal? [5 marks] (b) Most game-playing programs do not save search results from one move to the next. Instead, they usually start completely afresh whenever it is the machine’s turn to move. Why? [5 marks] (c) Consider the following game tree, and assume that the first player is the maximising player: A B C D E F G H I J K L M N O P Q R S T U V W X Y 2 3 8 5 7 6 0 1 5 2 8 4 10 2 (i) Which move should the first player choose? [3 marks] (ii) Assuming that nodes are searched left-to-right using the alpha-beta algorithm, which nodes would not be examined? [7 marks] 8 CST.2001.9.9 9 Neural Computing (a) (i) Many classes of artificial neural networks learn from data by forming a lower dimensional parametric representation, or mapping, that resembles the process of curve-fitting. Explain this idea in reference to least-squares error minimisation or statistical regression. [4 marks] (ii) Explain why increasing the complexity of a model may cause a neural network’s error in the training phase to become smaller and smaller, but its generalisation in the validation phase to become worse and worse. How would you expect the optimal complexity of a neural network model to depend on the amount of data? [6 marks] (b) Answer each of the following short questions: (i) What is the approximate capacitance of nerve cell membrane, in microFarads per cm2 , and what functional parameters of neural activity are determined by this? [2 marks] (ii) Approximately what range of voltages does a nerve cell membrane move through during the course of generating a neural impulse, and what determines this range? [2 marks] (iii) What is the rˆole of positive feedback in nerve impulse generation? [2 marks] (iv) From which organ does the retina develop embryologically, and to what cells elsewhere in the body are the retinal photoreceptors most closely related? [2 marks] (v) What causes the refractory deadtime of about 1 msec after each nerve impulse, and what is its consequence? [2 marks] 9 [TURN OVER CST.2001.9.10 10 Information Theory and Coding (a) (i) Construct an efficient, uniquely decodable binary code, having the prefix property and having the shortest possible average code length per symbol, for an alphabet whose five letters appear with these probabilities: Letter Probability A 1/2 B 1/4 C 1/8 D 1/16 E 1/16 [4 marks] (ii) How do you know that your code has the shortest possible average code length per symbol? [2 marks] (b) (i) For a string of data of length N bits, what is the upper bound for its Minimal Description Length, and why? [2 marks] (ii) Comment on how, or whether, you can know that you have truly determined the Minimal Description Length for a set of data. [2 marks] (c) Suppose you have sampled a strictly bandlimited signal at regular intervals more frequent than the Nyquist rate; or suppose you have identified all of the zero-crossings of a bandpass signal whose total bandwidth is less than one octave. In either of these situations, provide some intuition for why you now also have knowledge about exactly what the signal must be doing at all points between these observed points. [3 marks] (d) (i) Explain how autocorrelation can remove noise from a signal that is buried in noise, producing a clean version of the signal. For what kinds of signals, and for what kinds of noise, will this work best, and why? What class of signals will be completely unaffected by this operation except that the added noise has been removed? Begin your answer by writing down the autocorrelation integral that defines the autocorrelation of a signal f(x). [5 marks] (ii) Some sources of noise are additive (the noise is just superimposed onto the signal), but other sources of noise are multiplicative in their effect on the signal. For which type would the autocorrelation clean-up strategy be more effective, and why? [2 marks] 10 CST.2001.9.11 11 Numerical Analysis II (a) Taylor’s theorem states that if x ? [a, b] and f ? C N+1[a, b] f(x) = TN (a) + 1 N! Z x a f (N+1)(t)(x – t) N dt where TN (a) = f(a) + (x – a)f 0 (a) + (x – a) 2 2! f 00(a) + · · · + (x – a) N N! f (N) (a). Prove Taylor’s theorem. [6 marks] (b) Peano’s theorem states that if a quadrature rule integrates polynomials of degree N exactly over an interval [a, b] then the error in integrating f ? C N+1[a, b] can be expressed as E(f) = Z b a f (N+1)(t)K(t) dt where K(t) = 1 N! Ex[(x – t) N + ]. Explain the notation E(f), Ex and (x – t) N + . [4 marks] (c) Use Taylor’s theorem to prove Peano’s theorem. [8 marks] (d) Under what additional condition may the simplified formula E(f) = f (N+1)(?) (N + 1)! E(x N+1) be applied? [2 marks] 11 [TURN OVER CST.2001.9.12 12 Specification and Verification II Suppose definitions of D1(a, r, p, q), D2(q, b, r, s) and D3(p, s, c) are given. (a) Using ? and ? write down a definition of a predicate D such that D(a, b, c) defines the relation between a, b and c when D1, D2 and D3 are connected as in the following diagram. [4 marks] D1 D2 D3 a b p q r s c Suppose now that D1, D2 and D3 are defined in terms of functions f1, f2, f3, f4 and f5 by D1(a, r, p, q) = (p = f1 a r) ? (q = f2 a) D2(q, b, r, s) = (r = f3 q b) ? (s = f4 q b) D3(p, s, c) = (c = f5 p s) (b) Write down an equation expressing c in terms of a and b. [4 marks] (c) Show the logical steps needed to derive the equation from the definition of D. [12 marks] 13 Computer Vision Give three examples of problems in computer vision which are formally ill-posed. In each case explain how one or more of Hadamard’s criteria for well-posed problems has failed to be satisfied. Illustrate how addition of ancillary constraints or assumptions, even metaphysical assumptions about how the world behaves, enable one to convert the ill-posed problem into a well-posed problem. Finally, discuss how the use of Bayesian priors can perform this function.. [5 marks] (c) Give an expression for the optimal policy in terms of R, S and the discounted cumulative reward, and show how this can be modified to produce the Q-learning algorithm. [7 marks] In a simple reinforcement learning problem, states are positions on a grid and actions are up and right. The only way an agent can receive a reward is by moving into one of two special positions, one of which has a reward of 10 and the other of -100. -100 s1,1 s1,2 s5,3 10 s4,3 Here, states are labelled by their grid coordinates. Derive the steady-state distribution for the general birth-death process considered in part (a). What are the conditions for the steady-state distribution to exist? [4 marks] (c) Describe the M/M/1 queue and give a stochastic model for the number N of customers present. Find the steady-state distribution for N and state the conditions for it to exist. [4 marks] (d) Derive the mean and variance of N. [4 marks] (e) State Little’s law and use it to derive the mean time spent in the M/M/1 queue under steady state conditions. [2 marks] (f ) Discuss what is meant by the traffic intensity for an M/M/1 queue and explain what happens to the distribution of the number of customers present as the traffic intensity increases towards one. [4 marks] 3 (TURN OVER) CST.2012.9.4 3 Computer Vision (a) Briefly define each of the following concepts as it relates to vision: (i) active contours and energy-minimising snakes [2 marks] (ii) Hadamard’s criteria for well-posed problems [2 marks] (iii) the hermeneutical cycle [2 marks] (iv) reflectance map [2 marks] (v) Bayesian prior and its role in visual inference [2 marks]Derive the steady-state distribution for the general birth-death process considered in part (a). What are the conditions for the steady-state distribution to exist? [4 marks] (c) Describe the M/M/1 queue and give a stochastic model for the number N of customers present. Find the steady-state distribution for N and state the conditions for it to exist. [4 marks] (d) Derive the mean and variance of N. [4 marks] (e) State Little’s law and use it to derive the mean time spent in the M/M/1 queue under steady state conditions. [2 marks] (f ) Discuss what is meant by the traffic intensity for an M/M/1 queue and explain what happens to the distribution of the number of customers present as the traffic intensity increases towards one. [4 marks] 3 (TURN OVER) CST.2012.9.4 3 Computer Vision (a) Briefly define each of the following concepts as it relates to vision: (i) active contours and energy-minimising snakes [2 marks] (ii) Hadamard’s criteria for well-posed problems [2 marks] (iii) the hermeneutical cycle [2 marks] (iv) reflectance map [2 marks] (v) Bayesian prior and its role in visual inference [2 marks] (b) Detecting, classifying, and recognising human faces is a longstanding goal in computer vision. Yet because the face is an expressive social organ, as well as an object whose image depends on identity, age, pose and viewing angle, and illumination geometry, many forms of variability are all confounded together, and the performance of algorithms on these problems remains very poor. Discuss how the different kinds and states of variability (e.g. same face, different expressions; or same identity and expression but different lighting geometry) might best be handled in a statistical framework for generating categories, making classification decisions, and recognising identity. In such a framework, what are some of the advantages and disadvantages of wavelet codes (Haar or Gabor) for facial structure and its variability? [10 marks] 4 CST.2012.9.5 4 Denotational Semantics Given a closed PCF term F of type nat ? nat and a function f : N ? N, say that F represents f if F(succn (0)) ?nat succf(n) (0) holds for all n ? N. (a) What is the soundness property of the denotational semantics of PCF? Use it to show that if f is not a constant function (that is, f(m) 6= f(n) for some m 6= n), then the denotation JFK : N? ? N? of any F that represents f is the strict function that equals f when restricted to N. [4 marks] (b) If f is a constant function (f(n) = c for all n, say), give, with justification, two PCF terms that represent it and that are not contextually equivalent. [5 marks] (c) Consider the PCF term G def = fix(fn x : nat ? nat .fn y : nat . if zero(F y) then y else x(succ(y))) where F represents a function f : N ? N with the property that f(n) = 0 holds for infinitely many n ? N. Let F : (N? ? N?) ? (N? ? N?) be the continuous function whose least fixed point is JGK. Show by induction on k that for all k, n ? N F k (?)(n) = ( least m such that n = m < n + k and f(m) = 0 ? if no such m exists. [4 marks] (d) State the adequacy property of the denotational semantics of PCF and Tarski's Fixed Point Theorem for continuous functions on a domain. Use them to deduce that the term G in part (c) represents the function µf : N ? N that maps each n ? N to the least m = n such that f(m) = 0. [7 marks] 5 (TURN OVER) CST.2012.9.6 5 Digital Signal Processing (a) Make the following statements correct by changing one word or number. (Negating the sentence is not sufficient.) (i) The z-transform of a sequence shows on the unit circle its discrete-time cosine transform. [1 mark] (ii) Delaying a sequence by two samples corresponds in the z-domain to multiplication with z 2 . [1 mark] (b) Consider a causal digital IIR filter of order 2, operated at a sampling frequency of 48 kHz, where the impulse response {hn} has (for n > 2) the shape of a sine wave of frequency 8 kHz (amplitude and phase do not matter). (i) Where in the z domain can you place two zeros and two poles to achieve such an impulse response {hn} in the time domain? [4 marks] (ii) Write down the z transform of {hn} as a rational function (with those zeros and poles). [6 marks] (iii) Provide the constant-coefficient difference equation that describes the time-domain behaviour of that filter. [4 marks] (iv) How can you use such a filter design to digitally generate an 8 kHz sinewave sampled at 48 kHz with very little computational effort? [4 marks] 6 CST.2012.9.7 6 Information Theory and Coding (a) Prove that the information measure is additive: that the information gained from observing the combination of N independent events, whose probabilities are pi for i = 1 . . . N, is the sum of the information gained from observing each one of these events separately and in any order. [3 marks] (b) What is the shortest possible decodable code length, in bits per average symbol, that could be achieved for a six-letter alphabet whose symbols have probabilities 1 2 , 1 4 , 1 8 , 1 16 , 1 32 , 1 32 ? [3 marks] (c) Consider Shannon’s third theorem, the Noisy Channel Coding Theorem, for a continuous communication channel having bandwidth W Hertz, perturbed by additive white Gaussian noise of power spectral density N0, and average transmitted power P. (i) Is there any limit to the capacity of such a channel if you increase its signal-to-noise ratio P N0W without limit? If so, what is that limit? [4 marks] (ii) Is there any limit to the capacity of such a channel if you can increase its bandwidth W in Hertz without limit, but while not changing N0 or P? If so, what is that limit? [3 marks] (d) For each of the four classes of signals in the left table below, identify its characteristic spectrum from the right table. (“Continuous” here means supported on the reals, i.e. at least piecewise continuous but not necessarily everywhere differentiable. “Periodic” means that under multiples of some finite shift the function remains unchanged.) Give your answer just in the form 1-A, 2-B, etc. Note that you have 24 different possibilities. [4 marks] Class Signal Type 1. continuous, aperiodic 2. continuous, periodic 3. discrete, aperiodic 4. discrete, periodic Class Spectral Characteristic A. continuous, aperiodic B. continuous, periodic C. discrete, aperiodic D. discrete, periodic (e) Define the Kolmogorov algorithmic complexity K of a string of data. What relationship is to be expected between the Kolmogorov complexity K and the Shannon entropy H for a given set of data? Give a reasonable estimate of the Kolmogorov complexity K of a fractal, and explain why it is reasonable. [3 marks] 7 (TURN OVER (b) Detecting, classifying, and recognising human faces is a longstanding goal in computer vision. Yet because the face is an expressive social organ, as well as an object whose image depends on identity, age, pose and viewing angle, and illumination geometry, many forms of variability are all confounded together, and the performance of algorithms on these problems remains very poor. Discuss how the different kinds and states of variability (e.g. same face, different expressions; or same identity and expression but different lighting geometry) might best be handled in a statistical framework for generating categories, making classification decisions, and recognising identity. In such a framework, what are some of the advantages and disadvantages of wavelet codes (Haar or Gabor) for facial structure and its variability? [10 marks] 4 CST.2012.9.5 4 Denotational Semantics Given a closed PCF term F of type nat ? nat and a function f : N ? N, say that F represents f if F(succn (0)) ?nat succf(n) (0) holds for all n ? N. (a) What is the soundness property of the denotational semantics of PCF? Use it to show that if f is not a constant function (that is, f(m) 6= f(n) for some m 6= n), then the denotation JFK : N? ? N? of any F that represents f is the strict function that equals f when restricted to N. [4 marks] (b) If f is a constant function (f(n) = c for all n, say), give, with justification, two PCF terms that represent it and that are not contextually equivalent. [5 marks] (c) Consider the PCF term G def = fix(fn x : nat ? nat .fn y : nat . if zero(F y) then y else x(succ(y))) where F represents a function f : N ? N with the property that f(n) = 0 holds for infinitely many n ? N. Let F : (N? ? N?) ? (N? ? N?) be the continuous function whose least fixed point is JGK. Show by induction on k that for all k, n ? N F k (?)(n) = ( least m such that n = m < n + k and f(m) = 0 ? if no such m exists. [4 marks] (d) State the adequacy property of the denotational semantics of PCF and Tarski's Fixed Point Theorem for continuous functions on a domain. Use them to deduce that the term G in part (c) represents the function µf : N ? N that maps each n ? N to the least m = n such that f(m) = 0. [7 marks] 5 (TURN OVER) CST.2012.9.6 5 Digital Signal Processing (a) Make the following statements correct by changing one word or number. (Negating the sentence is not sufficient.) (i) The z-transform of a sequence shows on the unit circle its discrete-time cosine transform. [1 mark] (ii) Delaying a sequence by two samples corresponds in the z-domain to multiplication with z 2 . [1 mark] (b) Consider a causal digital IIR filter of order 2, operated at a sampling frequency of 48 kHz, where the impulse response {hn} has (for n > 2) the shape of a sine wave of frequency 8 kHz (amplitude and phase do not matter). (i) Where in the z domain can you place two zeros and two poles to achieve such an impulse response {hn} in the time domain? [4 marks] (ii) Write down the z transform of {hn} as a rational function (with those zeros and poles). [6 marks] (iii) Provide the constant-coefficient difference equation that describes the time-domain behaviour of that filter. [4 marks] (iv) How can you use such a filter design to digitally generate an 8 kHz sinewave sampled at 48 kHz with very little computational effort? [4 marks] 6 CST.2012.9.7 6 Information Theory and Coding (a) Prove that the information measure is additive: that the information gained from observing the combination of N independent events, whose probabilities are pi for i = 1 . . . N, is the sum of the information gained from observing each one of these events separately and in any order. [3 marks] (b) What is the shortest possible decodable code length, in bits per average symbol, that could be achieved for a six-letter alphabet whose symbols have probabilities 1 2 , 1 4 , 1 8 , 1 16 , 1 32 , 1 32 ? [3 marks] (c) Consider Shannon’s third theorem, the Noisy Channel Coding Theorem, for a continuous communication channel having bandwidth W Hertz, perturbed by additive white Gaussian noise of power spectral density N0, and average transmitted power P. (i) Is there any limit to the capacity of such a channel if you increase its signal-to-noise ratio P N0W without limit? If so, what is that limit? [4 marks] (ii) Is there any limit to the capacity of such a channel if you can increase its bandwidth W in Hertz without limit, but while not changing N0 or P? If so, what is that limit? [3 marks] (d) For each of the four classes of signals in the left table below, identify its characteristic spectrum from the right table. (“Continuous” here means supported on the reals, i.e. at least piecewise continuous but not necessarily everywhere differentiable. “Periodic” means that under multiples of some finite shift the function remains unchanged.) Give your answer just in the form 1-A, 2-B, etc. Note that you have 24 different possibilities. [4 marks] Class Signal Type 1. continuous, aperiodic 2. continuous, periodic 3. discrete, aperiodic 4. discrete, periodic Class Spectral Characteristic A. continuous, aperiodic B. continuous, periodic C. discrete, aperiodic D. discrete, periodic (e) Define the Kolmogorov algorithmic complexity K of a string of data. What relationship is to be expected between the Kolmogorov complexity K and the Shannon entropy H for a given set of data? Give a reasonable estimate of the Kolmogorov complexity K of a fractal, and explain why it is reasonable. [3 marks] 7 (TURN OVERA possible sequence of actions (sequence 1) is shown by solid arrows, ending with a reward of 10 being received, and another (sequence 2) by dashed arrows ending with a reward of -100. (d) Assume that all Q values are initialised at 0. (i) Explain how the Q values are altered if sequence 1 is used twice in succession by the Q-learning algorithm. [4 marks] (ii) Explain what further changes occur to the Q values if sequence 2 is then used once by the Q-learning algorithm. [3 marks] 3 (TURN OVER) CST.2012.7.4 3 Bioinformatics (a) Considerable recent Bioinformatics research has focused on phylogenetics. (i) What is the motivation for this work? [1 mark] (ii) Describe with the aid of examples two different techniques for phylogeny. In each case discuss the issues of complexity and performance. [4 marks each] (b) Considerable recent Bioinformatics research has focused on structure prediction from sequence data. (i) Describe how you would build a hidden Markov model (HMM) to identify membrane segments in aminoacid sequences. [6 marks] (ii) How you would assess the sensitivity and specificity performance of your HMM? [5 marks] 4 CST.2012.7.5 4 Business Studies (a) Distinguish between a profit and loss statement and cash flow statement. [5 marks] (b) A company is proposing to build a low cost single board computer that will sell for £30 direct from its web page. The bill of materials (BoM) costs for components are £15/unit, while manufacturing and other costs are estimated as 33% of sale price. Components can only be bought in lots of 10,000 at a time and must be paid for 1 month before first use. The company estimates sales for the first six months as a ramp for 0 in the first month increasing by 1000/month to 5000 units in month 6. (i) Draw up a cash flow estimate for the first six months of operation. Ignore VAT, bank and other charges. [5 marks] (ii) How much working capital will be required? [5 marks] (iii) In month 5 the company is offered the opportunity to sell 50,000 units but at a price of £25/unit. Should the company take this opportunity? Justify your answer. [5 marks] 5 (TURN OVER) CST.2012.7.6 5 Comparative Architectures (a) How can specialising a processor design for a specific application domain help to reduce its power consumption? [7 marks] (b) For many applications chip-multiprocessors consume less power when compared to a superscalar uniprocessor design of equal performance. Why is this the case? [7 marks] (c) Why does a vector processor offer a particularly energy efficient solution to execute some types of program? [6 marks] 6 CST.2012.7.7 6 Denotational Semantics (a) If D and D0 are domains, explain what is the function domain D ? D0 ; give its partial order and least element, and explain how least upper bounds of chains are calculated in it. [4 marks] (b) An element d of a domain D is said to be isolated if for all countable chains x0 v x1 v x2 v . . . in D with d v F n=0 xn, there exists i = 0 with d v xi . We write K(D) for the subset of isolated elements. Given domains D and D0 and elements d ? D and d 0 ? D0 , let [d, d0 ] : D ? D0 be the function mapping each x ? D to d 0 if d v x and to ? otherwise. (i) Prove that [d, d0 ] is monotone. [2 marks] (ii) Prove that if f : D ? D0 is monotone, then [d, d0 ] v f if and only if d 0 v f(d). [2 marks] (iii) Prove that if d ? K(D), then [d, d0 ] is an element of the function domain D ? D0 . [3 marks] (iv) Prove that if both d ? K(D) and d 0 ? K(D0 ), then [d, d0 ] is an isolated element of the function domain D ? D0 . [3 marks] (v) Now suppose that every element of D is the least upper bound of some countable chain of isolated elements and the same is true for D0 . Show that each element f of the function domain D ? D0 is the least upper bound of the subset F def = {[d, d0 ] | d ? K(D) & d 0 ? K(D0 ) & d 0 v f(d)}. [6 marks] 7 (TURN OVER) CST.2012.7.8 7 Hoare Logic In this question we consider a semantics of FOR-commands in which FOR V :=E1 UNTIL E2 DO C is defined to be equivalent to V :=E1; WHILE V = E2 DO (C; V :=V + 1) (a) How does this semantics of FOR-commands differ from the one given in the lectures? [4 marks] (b) The following FOR-rule is similar to one proposed by John Wickerson: ` P ? R[E1/V ] ` R ? V >E2 ? Q ` {R ? V = E2} C {R[V +1/V ]} ` {P} FOR V :=E1 UNTIL E2 DO C {Q} Assuming the semantics of FOR-commands given above, derive this Wickersonstyle FOR-rule from the standard axioms and rules of Hoare logic. [10 marks] (c) Is the FOR-axiom: ` {P ? E2 Computer ScienceEngineering & TechnologyNetworking COMPUTER S 2011

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