Explain how to derive a sequence of transformations to achieve the overall effect of performing a 2D rotation about an arbitrary point.

Discuss the problems of providing tractable models of transistors suitable for hardware verification by formal proof. Compare and contrast at least two different models. Illustrate your discussion with concrete examples of transistor circuits. [20 marks] 10 Complexity For each of the following statements state whether the claim made is true, false or if more information is needed before a judgement can be made. Give one-sentence justifications of your assertions. (a) Sorting a list of numbers into ascending order is an NP problem. (b) Sorting a collection of programs into order so that the ones that finish quickly come before those that run for a long time is an NP-complete problem. (c) To be NP-complete is to be as difficult as any solvable problem can be. (d) Any NP problem can be solved (on an ordinary computer) in polynomial space and exponential time. (e) The problem of determining whether a k-clique is present in a graph is known to be NP-complete. Therefore for large graphs and large values of k it will always be impossible (in practice) to find such a clique even if it is known that one exists. (f ) For the purposes of complexity theory each of the cost functions n log n, n 1.573 and n! counts as polynomial growth.

Explain Turing's Thesis. [5 marks] (a) What is meant by saying that a Turing machine has searching states? Show that any Turing machine computation can be effected by a machine with searching states, equivalent in the sense that the head movements are identical and the same symbols are written to the tape. [5 marks] (b) Show that, subject to suitable encoding, any computation can be carried out by a Turing machine having only two states.

) Explain the terms Work Conservation and max-min fairness, in the context of packet switching. [8 marks] (b) Outline the operation of two work-conserving queueing schemes that provide max-min fairness, and two (simpler) ones that do not. [8 marks] (c) Give at least two main implementation costs associated with implementations of fairness in packet switched routers. [4 marks] 3 Security (a) Describe the Bell-LaPadula security policy. [6 marks] (b) Describe the Chinese Wall security policy. [6 marks] (c) To what extent is the Chinese Wall policy an extension of Bell-LaPadula? [6 marks] (d) Are either of these policies relevant to digital rights management? [2 marks 4 Advanced Graphics (a) We want to find the first intersection point between an arbitrary ray and a sphere of arbitrary radius at an arbitrary position in space. (i) List and define all of the parameters required to specify the geometry of the ray and the sphere. [2 marks] (ii) Give an algorithm which returns the desired intersection point (if it exists) and the appropriate normal vector at the intersection point. [5 marks] (b) Describe a method which converts an arbitrary sphere to a triangle mesh at a desired resolution. The desired resolution is specified as a desired number of triangles, D. Your method should produce a number of triangles, N, which is within an order of magnitude of D: D/10 < N < 10D. [4 marks]

Let N(t) denote the number of events in the time interval [0, t] for a (homogeneous) Poisson process of rate , ( > 0). (a) State the necessary properties on N(t) that define a (homogeneous) Poisson process of rate . [4 marks] (b) By dividing the interval [0, t] into equal length sub-intervals show that N(t) is a Poisson random variable with mean t. [4 marks] (c) Let X1 denote the time of the first event and for n > 1 let Xn denote the elapsed time between the (n 1)th and the nth events of the Poisson process. Determine the distribution of X1 and the joint distribution of X1 and X2. [4 marks] (d) Let Sn = Pn i=1 Xi denote the time of the nth event. Derive the probability density function of the random variable Sn(t). [4 marks] (e) Give an algorithm to generate the first T time units of a (homogeneous) Poisson process of rate . [4 marks] 6 Specification and Verification I (a) Explain the difference between a variant and an invariant. Briefly describe what they are used for. [4 marks] (b) State and justify the verification conditions for the total correctness of WHILE commands. [6 marks] (c) (i) Devise a precondition P that makes the following specification true. [P] WHILE IN DO SUM := SUM+(2I); I := I+1 [SUM = N(N+1)] [2 marks] (ii) Devise and justify annotations for this specification that yield provable verification conditions. [8 mark

Without giving a detailed mathematical derivation, discuss how this result may be used to give the Fast Fourier Transform algorithm. Discuss the advantages of the algorithm compared with direct evaluation of the DFT. [5 marks] Discuss briefly the use of window functions in discrete spectrum analysis. [3 marks] The generalised Hamming window function is defined by w(n) = (1 ) cos(2n/N) for 0 6 n < N = 0 otherwise where 0 6 6 1. Obtain an expression for the DFT of this window function. [6 marks] 1 [TURN OVER CST.93.8.2 2 Digital Communication II Define the layers of the OSI Reference Model and briefly describe their primary functions. [7 marks] Using this model, describe the significant differences between the protocols of a traditional file transfer mechanism and a distributed file system. [9 marks] How is security supported by the two mechanisms? [4 marks] 3 Computer System Modelling The state diagram for a Markov chain showing transition rates is shown below. Solve for state occupancy probabilities. 4 6 3 2 a b c [5 marks] The steady state distribution for the number of jobs in an M/M/1 queue, k, is pk = (1 ) k k = 0, 1, 2, . . . where = / < 1. Here and are the mean arrival rate and mean service rate respectively. Find the first and second moments of this distribution and hence verify that the variance of the number of the jobs in the system is given by (1 ) 2 [10 marks] What does this result show about the predictability of system performance at high loads? [5 marks] 2 CST.93.8.3 4 Developments in Technology EDSAC (1949) was the first practical stored-program computer in operation. Although it had no subroutine jump instruction, a method for using subroutines was devised by David Wheeler. Quote and explain the instruction sequences for subroutine entry and return. [12 marks] Describe the corresponding instruction sequences that you would use in a modern processor. [8 marks] SECTION B 5 Designing Interactive Applications Distinguish the terms needs analysis and requirements analysis. Provide an example to illustrate the difference. [3 marks] What is a strong requirement? Provide counter-examples and explain why each example is not a strong requirement. [4 marks] What role does a functional specification play in a requirements specification? Give a one-sentence example. [3 marks] The receptionist at a small research laboratory is required to field incoming messages and make sure that they reach the recipient in a timely manner. Some messages arrive by word of mouth, others by phone, courier, e-mail or FAX. There are about 100 recipients, most of whom are researchers. They spend a large proportion of their time in meetings of one sort or another, some of which are held in offices, the remainder in conference rooms. The receptionist endeavours to avoid interrupting important meetings unnecessarily. It is proposed to build a system based upon Active Badge technology to improve message handling activities in the laboratory. Each member of staff wears an active badge. An existing Location Server provides client applications with up-to-date information about the location and movements of each active badge. Sketch out the methods you would employ to establish the users' needs for the proposed system. Describe the categories of information you would pass on to the designer and illustrate each with one or two examples. [10 marks]

The 2d die is also biased, and has specific numbers: its distribution is Pr(n) = qn with n four, 5, 6, 7, 8, nine. Evil Robot flips the coin. If he gets a head then he rolls the first die, otherwise he rolls the second. He then tells you the outcome. You see most effective the quantity obtained and not anything else. He does this m times, so you have a look at a sequence of m numbers in the variety 1 to 9. Your goal is to estimate p and the distributions of each die, given the m numbers. In the subsequent, n is the vector of m determined numbers (n1 nm) T , is the set of parameters p, p1, . . . , p6, this fall, . . . , q9 and we define q = 1 p. (a) Write down an expression for the distribution Pr(nwherein n 1, . . . , 9. [2 marks

This question offers with stochastic approaches N(t), t zero where N(t) represents the number of activities within the time c programming language [0, t]. (a) (i) Define a Poisson technique N(t), t 0 of price > zero. [2 marks] (ii) Show that N(t) Pois(t) for every constant t > 0. You may additionally use the end result that limn(1 x/n) n = e x with out evidence. [4 marks] (iii) Let X1 be the time of the first occasion of the Poisson procedure N(t). Show that X1 Exp(). (iv) Now given that N(t) = 1 derive the distribution of the time of the single occasion in [0, t]. [4 marks] (b) Suppose that events of a Poisson method of rate are independently selected at random with chance p > 0. Show that the method of selected occasions is likewise a Poisson manner and establish its price. [2 marks] (c) Describe how your result from component (b) can be used to simulate a nonhomogeneous Poisson method whose rate characteristic (t) is such that (t) for all t 0. [6 marks] 4 CST2.2018.8.5 4 Computer Vision (a) Explain how every of the subsequent equations or expressions can be used for detecting and estimating visible motion in a spatio-temporal photograph series I(x, y, t). Include on your solution the call used to explain every of those preferred classes of motion extraction models: (i) I(x, y, t) t = v I(x, y, t)

(a) A processor's most important memory is usually carried out the usage of DRAM. (i) Describe a regular DRAM cellular. [2 marks] (ii) Show, with the useful resource of a diagram, how DRAM is organised, making reference to devices, ranks, banks and arrays. [4 marks] (iii) Describe the difference between an open-web page and closed-page row-buffer coverage and the types of get entry to styles they advantage. [2 marks] (b) The MOSI cache coherence protocol provides a new owned (O) nation to the fundamental MSI protocol.When a cache protecting a line in M nation snoops a read request from every other cache, it transitions to O nation and forwards the facts to the requestor. Subsequent snoops for read requests also are fulfilled by this owner cache. An owned line is dirty and simplest one cache can maintain a line in O kingdom at any time. (i) Describe the difference between cache coherence and reminiscence consistency. [2 marks] (ii) Draw a country transition diagram for the MOSI protocol, the use of a new motion Forward to suggest facts being forwarded from one cache to some other. [6 marks] (iii) Draw a table showing how the country of a line in one cache limits the states the same line will have in a one of a kind cache. [2 marks] (iv) Give two benefits of adding this greater owned country to the simple MSI protocol.

(a) At the bottom degree, what is the number one purchaser of electrical energy in digital logic today? Give a components for the predicted electricity or energy use for a CMOS gate. [2 marks] (b) A matrix (a 2-dimensional array) is saved on-chip in static RAM. What primary factors make a contribution to the time and energy had to transpose it? [4 marks] (c) Assume now a square matrix is to be held in DRAM. (i) When may it's beneficial to store a couple of-copies of a given matrix in distinctive DRAM banks? [1 mark] (ii) When might or not it's beneficial to keep a couple of-copies of the matrix (or every other example records structure) in one DRAM bank? [2 marks] (iii) One manner to avoid transposing a matrix is truly to keep an annotation that it's been transposed and to then switch over the row and column arguments for every operation. Why would possibly bodily performing the transpose in the end advantage performance? Where would the annotation be held? [2 marks] (d) A computation operates on rectangular matrices of length one zero five one zero five . The internal loop, to be elevated in hardware, has the subsequent simple shape: for (int i= ...) for (int j= ...)

DD[i, j] = ff(SS[i-1, j], SS[i, j-1])

(i) Are there any loop-carried dependencies? What does this suggest for overall performance optimisation? [1 mark] (ii) If the DRAM timings are 11-11-eleven, which means row activation, column activation and writeback each take eleven clock cycles, estimate roughly the minimum time for a naive implementation of the computation. Assume a simple linear facts format. State all in addition assumptions. [6 marWhich one, and why? [4 marks] (b) Complete the code below to implement the signed distance field function for a finite line segment with hemispherical end-caps (Figure 1) of arbitrary start point, end point, and radius. [4 marks] float lineSegment(vec3 p, vec3 start, vec3 end, float radius) { // [YOUR CODE HERE] } float getSdf(vec3 p) { return lineSegment( p, vec3(-PI, 0, 0), vec3(PI, 0, 0), 0.5); } (c) Implement a version of getSdf() that doubles the height of your line segment and translates it by 0.5 along the Z axis, to be centred at (0, 0, 0.5) (Figure 2). [4 marks] (d) Implement a version of getSdf() that warps the original line segment into a sine wave sin(X) (Figure 3). [4 marks] (e) Modify getSdf() to render the sine wave model subtracted from the taller model (Figure 4). [4 marks] 6 CST1.2018.7.7 Figure 1 Figure 2 Figure 3 Figure 4 Figure 1: A finite cylinder of radius 0.5 centred at (0, 0, 0) with hemispherical end-caps, starting at (, 0, 0) and ending at (, 0, 0). Figure 2: The original finite cylinder has been enlarged to double its height on the Y axis and has been translated in Z so that it is now centred at (0, 0, 0.5). Figure 3: The original finite cylinder has been warped with a sine wave. Its centre remains at (0, 0, 0) and its endpoints remain centred around (+/, 0, 0), but in between its central axis falls to Y = 1 and rises to Y = 1. Figure 4: The sine wave has been subtracted from the double-height cylinder. (Note: Ground plane shown at Y = 1 for illustration

7 Further Graphics (a) Write a GLSL function dartboard() which takes as input a texture co-ordinate texCoord which ranges from (0, 0) (1, 1), and returns the colours of the procedural texture for a black-and-white dartboard pattern of 16 squares around and 8 squares in radius (see F}?2 marks] (iv) the formula for Descartes' Theorem of Total Angle Deficit? [2 marks] (c) Consider a closed manifold surface with total angle deficit 4. (i) If your hypothetical surface has 20 vertices and 20 faces then how many edges must it have? [2 marks] (ii) Sketch a picture of your surface

answer the question clearly