Describe, in detail, how the heapsort algorithm works. [10 marks] Show that the worst-case cost of heapsort

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Describe, in detail, how the heapsort algorithm works. [10 marks] Show that the worst-case cost of heapsort is O(n log n). [6 marks] Would it be possible to implement a variant of heapsort based on a perfectly balanced ternary structure in which the children of node i are at positions 3i 1, 3i, and 3i + 1, and if so what would be the advantages and disadvantages of the new method?

Define the concepts of strong simulation and strong bisimulation between processes. [4 marks] Here are three vending machines, selling tea for 20p and/or coffee for 40p: T def = 20p.tea.T C def = 20p.20p.coffee.C CT def = 20p.(tea.CT + 20p.coffee.CT) "20p" is the user's action of inserting a coin; "tea", "coffee" are the user's actions of selecting a drink. (For simplicity, the delivery of a drink is not represented.) One might hope that CT can be made to behave like C or T respectively by blocking off (i.e. by restricting by ) the appropriate select button. But prove, by considering bisimulations or otherwise, that one of the following assertions is true, the other false: ( tea) CT C ( coffee) CT T [8 marks] In the case which is false, is there a simulation in one direction? Give a precise reason for your answer. [3 marks] Now redesign T, C and CT, in a way which still has reasonable behaviour for a machine selling tea and/or coffee, but so that both of the above assertions hold, and justify the claim.

Suppose that you are forbidden from using printf or Integer.toHexString, or indeed any other existing library way of doing it, but you still need to display a Java integer in hexadecimal as one to eight digits. For instance, you are to display the number 19 (decimal) as the string "13", and 1 must come outava method called toHex that takes an integer as its argument and returns the string form of the hexadecimal representation of that number. Explain clearly how your

You have been hired to design a mobile phone app for a brokerage firm, whose business is to help people to manage their investments. Their customers will use the app to check the prices of the shares and bonds that they own, sell them when they wish, and buy others. The customers' investment portfolios are typically worth six figures or more. You have already implemented industry standard mechanisms for the customer to authenticate to the app using the phone's biometric mechanisms, and for the app to authenticate itself to the broker's servers using TLS. Orders to sell investments, or to transfer sa brokerage to a bank account, also need a second authentication factor, as a requirement from the regulator. At present the brokerage sends an SMS message to the customer's phone. The firm wants a recovery mechanism to support users who have lost their phone, bought a new one, and downloaded a new copy of the app. Optographic keys, and send the public keys along with the phone's details to the broker, encrypted under the broker's TLS key. The recovery mechanism will authorise a new set of keys to be used to access an existing account and, together with the SMS second factor, to sell investments or withdraw the proceeds of sales. (a) The brokerage firm first suggests sending an email to any customer who needs to recover an account, with an activation code to type into the app. Discuss the security, usability and cost of this option. [5 marks] (b) Another suggestion is to rerun the initial 'know-your-customer' registration procedure which involves the customer sending the brokerage a photo of their passport or driving license plus two utility bills. Discuss the security, usability and cost of this option. [5 marks] (c) Does the SMS second factor do any useful work? Justify your answer. [5 marks] (d) What might you suggest that is better? Discuss its security, usability and cost. [5 mark

The abstract syntax of commands in a simple parallel programming language P is given by C ::= skip 2 where ie, be and X range over the syntactic categories of integer expressions, boolean expressions and program variables, respectively. The intended behaviour of C1 k C2 is that C1 and C2 are executed in parallel until they have both terminated. Hence atomic execution steps from C1 and C2 may be arbitrarily interleaved. The other command forms behave as usual. (a) Give a small-step transition semantics for P which derives statements of the form hC, Si hC 0 , S0 i, where S and S 0 are states. You may assume that rules for the evaluation of expressions have already been given. Comment briefly on your choice of what constitutes an atomic execution step. [9 marks] (b) The binary relation on commands is defined by C1 C2 S, S0 . hC1, Si hskip, S0 i hC2, Si hskip, S0 i. Show that is not a congruence. [5 marks] (c) Assuming that S(X) = S(Y ) = 0, describe the set of possible execution traces which are derivable in your semantics starting from the configuration hC, Si, where C is (X := 1) k (while X = 0 do Y := Y + 1). Why might one argue that this does not accurately reflect the behaviour of a reasonable implementation of the language?

This question concerns lexical grammars. (a) Tree Adjoining Grammars contain two types of elementary tree. (i) What are these trees called? [1 mark] (ii) If one were building a grammar for English which aspects of language do the two tree types model? [2 marks] (b) Provide a Tree Adjoining Grammar that can parse the string: students enjoy easy exams [5 marks] (c) Show how a parse for this string is constructed. Explain the operations. [5 marks] (d) Provide a Categorial Grammar that can parse the same sentence. [4 marks] (e) When children learn their first language they usually acquire nouns before verbs before modifiers. They also usually produce single word strings before moving on to longer strings. With reference to Tree Adjoining Grammars and/or Categorial Grammars propose some hypotheses for this. Justify your proposals.

A birdsong is composed of a sequence of these elements as follows: c c d c c e c c d c c d c c e (a) (i) Using byte pair encoding induce a context-free grammar for this sequence of birdsong. Show your workings and state what you have decided to do in the case of a tie. [6 marks] (ii) Draw the derivation tree that parses the birdsong using your induced grammar. [1 mark] (iii) What are the shortcomings of this method of grammar induction for natural languages? [3 marks] (b) Assuming that c, d and e are the only melodic elements available in the birdsong, and that the excerpt we are given is probabilistically representative of the birdsong in the wild, what is the average information produced per element? Provide relevant equations. [3 marks] (c) In Part (b) we assumed a 1st-order model of the birdsong. What assumption does this make about the sequence of elements? [1 mark] (d) Consider a 2nd-order model of the birdsong and calculate the conditional entropy. Provide relevant equations. [4 marks] (e) How can we calculate the entropy rate of birdsong? Provide relevant equations

Consider expressions represented using the following ML datatype: datatype exp = Var of string | Neg of exp | Divide of exp * exp (a) Using at least one union, define a type or types in the C language for conveniently storing such expressions. [5 marks] (b) Give efficient C code that checks whether two structures represent identical expressions. (Do not consider whether they might evaluate to the same result.) Explain how much of the input expressions is explored when they differ. [4 marks] (c) Given that a lot of expressions are to be rapidly generated and discarded, what considerations apply to storage management? Define and discuss at least 3 different approaches to storage management. [5 marks] (d) The substitution operation for an expression replaces all occurrences of one variable with another variable. Given that sub-expression trees are commonly shared over numerous expressions, explain a problem that could arise in the substitution operation. Explain the details of a solution to the problem by giving code or otherwise. [6 mark

(a) Derive an expression for the gradient Ei(w) wj for weight wj in an output node when Ei(w) is the error for the ith example Ei(w) = 1 2 (yi h(w; xi))2 , h(w; xi) is the output of the complete network for the ith example, and (a) = a. You need only derive the expression for the output node. [3 marks] (b) Derive an expression for the gradient Ei(w) wj for weight wj in an output node when (a) = 1/(1 + exp(a)) and the error for the ith example is Ei(w) = yi log h(w; xi) + (1 yi) log(1 h(w; xi)). You may use the fact that d(a)/da = (a)(1 (a)). You need only derive the expression for the output node. [7 marks] (c) In the standard backpropagation algorithm the central quantity of interest for each node N is = Ei(w)/a. It is proposed that, instead of using nodes in the form presented above, we introduce functions i and construct multilayer networks from nodes that compute z = (a) where a = Xn i=0 wii(z). Here, z T =

z0 z1 zn

and the functions i are fixed, having no further parameters. A multilayer perceptron is constructed from nodes of this kind. Give a detailed, general derivation of the formula for computing for a non-output node N in this network, assuming you know the values of for the nodes connected to the output of N. [10 marks

The standard linear regression model uses a hypothesis

h(x, w, b) = wT x + b

to fifit m examples ((x1, y1), . . . ,(xm, ym)) by minimizing the error

E(w, b) =

i=1

(yi h(xi, w, b))2 .

(a) Derive a gradient descent algorithm for training the linear regression model

described. [5 marks]

(b) In the application of interest, you believe that it is desirable to train such that

the learned parameters have ||w|| ' 1. Suggest a modifification to E(w, b) that

facilitates this, and derive the corresponding gradient descent training algorithm.

[5 marks]

(d) You are faced with a heuristic search problem, but the heuristics you have so far

developed are less effffective than desired. Suggest two ways in which supervised

machine learning might be used to develop a better heuristic, mentioning if

necessary any corresponding disadvantages of using the approach. You may

assume that a collection of problems to be solved by the heuristic search is

available. [6 marks]

2CST1.2021.6.3

Artifificial Intelligence

A Boolean satisfifiability problem has four variables, x1, x2, x3 and x4. A literal l

can be a variable or its negation, denoted l. The formula of interest, in conjunctive

normal form (CNF), is

f = (x2 x3) (x2 x3) (x1 x2 x4).

(1)

The aim is to fifind assignments to the variables such that f is true under the usual rules

for Boolean operations. This question addresses the use of more general constraint

satisfaction to solve this problem.

(a) Give a general description of a constraint satisfaction problem (CSP).

[3 marks]

(b) Explain how a Boolean satisfifiability problem in CNF form and with n variables

can be converted to a CSP, also having n variables and having a suitable

constraint for each clause. Illustrate your answer using the 4-variable formula f

in (1). [3 marks]

constraints can be converted to binary constraints. Provide a graph illustrating

the problem from (1) after it has been converted to a CSP with only binary

constraints. [4 marks]

(d) Explain, how forward checking works in the context of a general CSP. How does

this benefifit a CSP solver? [3 marks]

(e) Using the CSP equivalent you developed for (1), with only binary constraints,

demonstrate how forward checking works using the sequence of assignments

x1 = F, x2 = F, x4 = T. [5 marks]

(f ) How would you expect the solution obtained when applying forward checking to

be affffected if constraints were allowed to propagate more widely? [2 marks]

3CST1.2021.6.4

Complexity Theory

(a) Defifine the set of Boolean expressions 2CNF and the language 2SAT over them.

[2 marks]

(b) For a Boolean expression in 2CNF, let G() be the directed graph with vertices

the variables of and their negation, and with edges (a, b) if, and only if, there

is a clause (a b) or (b a) in . Note that an edge (a, b) is in G() if, and

only if, so is the edge (b, a).

Prove that a Boolean expression in 2CNF is unsatisfifiable if, and only if, there

is a variable x in such that there are paths from x to x and from x to x in

G(). [Hint: Recall that the proposition (P Q) is equivalently the implication

(P Q).] [12 marks]

use the fact that NL is closed under complementation. [6 marks]

4CST1.2021.6.5

Complexity Theory

(a) For a complexity class C, let co-C = { L | L C } and say that C is closed under

complementation whenever C = co-C.

Argue as to whether the following statements are true, false, or unknown.

(i) All deterministic time complexity classes are closed under complementation.

[3 marks]

(ii) All non-deterministic time complexity classes are closed under

complementation. [3 marks]

(b) For a mapping f : on an alphabet and a language L , defifine

f[L] = { f\ (w) | w L } where f\ (a1 an) = f(a1) f(an).

Prove that L NP implies f[L] NP. [4 marks]

Q: Given natural numbers m and n in N, and bits a

(k)

i,j

and bk in

{0, 1} for 1 k m and 1 i, j n, determine whether the system

of equations P 1i,jn a

(k)

i,j

xi xj = bk (1 k m) with unknowns

x1, . . . , xn has a solution in arithmetic modulo 2.

(i) Prove that Q is in NP. [3 marks]

(ii) By means of a polynomial-time reduction from the problem 3CNF, or

otherwise, prove that Q is NP-hard. [Hint: Note, for instance, that x = y

in the Boolean algebra {0, 1} if, and only if, xx + yy = 1 in arithmetic

modulo 2.] [7 marks]

You may use standard results provided that you state them clearly.

5CST1.2021.6.6

Computation Theory

(a) For each n, e N, let

(n)

denote the partial function Nn* N computed by the

and register R0 to store the result, if any.

Explain why for each m, n N there is a totally defifined register machine

computable function Sm,n : N 1+m N with the property that for all (e, ~x)

N 1+m and ~y Nn

)

m,n

(e,~x)(~y)

(m+n)

(~x, ~y) (1)

where denotes Kleene equivalence: for all z N, the left-hand side is defifined

and equal to z if and only if the right-hand side is defifined and equal to z.

Your explanation should make clear what assumptions you are making about

the relationship between numbers and register machine programs. [10 marks]

(b) Let f : N 1+m+n* N be a register machine computable partial function of 1+m+n

arguments for some m, n N.

(i) Why is the partial function f : N 1+m+n* N satisfying for all (z, ~x, ~y)

N 1+m+n

f(z, ~x, ~y) f(S1+m,n(z, z, ~x), ~x, ~y) (2)

(ii) By considering S1+m,n(e, e, ~x) where e is an index for the partial function

f in part (b)(i), prove that there is a totally-defifined register machine

computable function fifix f : Nm N with the property that for all ~x Nm

and ~y Nn

(n)

fifix

f(~x)(~y) f(fifix f(~x), ~x, ~y) (3)

[7 marks]

6CST1.2021.6.7

Computation Theory

A set A equipped with a binary operation @ : A A A is a combinatory algebra

if there are elements K, S A satisfying for all a, b, c A

@(@(K, a), b) = a

(1)

@(@(@(S, a), b), c) = @(@(a, c), @(b, c)) (2)

(a) Show that there is a binary operation on the set of equivalence classes of closed

-terms for the equivalence relation of -conversion that makes it a combinatory

algebra. [5 marks]

(b) Show that every combinatory algebra A contains an element I satisfying

@(I, a) = a

(3)

for all a A. [Hint: what does (2) tell us when a = b = K?] [2 marks]

given by the grammar

e ::= x | p aq | (ee)

where x is some fifixed symbol not in A and a ranges over the elements of A.

Given e A[x] and a A, let e[x := a] denote the element of A resulting from

interpreting occurrences of x in e by a, interpreting the expressions of the form

p a0 q by a0 and interpreting expressions of the form (ee0 ) using @.

(i) Give the clauses in a defifinition of e[x := a] by recursion on the structure of

e. [2 marks]

(ii) For each e A[x] show how to defifine an element xe A with the property

that

@(xe, a) = e[x := a] (7)

for all a A. [6 marks]

(d) Recall the usual encoding of Booleans in -calculus. Using Part (c)(ii), show

that in any combinatory algebra A there are elements True, False A and a

function If : A A A satisfying

@(If (a, b), True) = a

(11)

@(If (a, b), False) = b

(12)

for all a, b A

[5 marks]

7CST1.2021.6.8

Data Science

(a) Let xt be the number of new COVID infections on date t. We anticipate

approximately exponential growth or decay, xt+1 (1 + )xt , and we would

like to estimate from a dataset (x1, . . . , xT ).

(i) Find the maximum likelihood estimator for for the model

Xt+1 Poisson

Utilizing the data gave for the situation study, would you be able to assemble a Context graph, Level 0 Data Flow Diagram, and an ERD utilizing any outlining programming.

The 2020 Covid pandemic has affected numerous enterprises across the globe and Australia's organic product picking industry is no exemption. Natural product send out contributes fundamentally to Australia's exchange. Most natural products are by and large transitory and consequently natural product picking is something that should be done brilliantly and with care. In spite of the fact that Australia's organic product picking industry faces a few issues, one of them is absence of adequate labor. This issue, among many, has exasperated as of late because of COVID 19. To help the ranchers and the natural product picking industry, Zaf Soft, an Australian Information System supplier association has concocted an online arrangement that would assist with fruiting ranchers pick organic product well on schedule. The CEO of Zaf Soft, Rabee James has chosen to send off a site donesloching.com. The thought is basic, innovative and in particular it adds financial worth. donesloching.com would permit Australian ranchers and organic product pickers to enroll on this site. While ranchers can enlist separately, the site additionally permits bunch enrollment for organic product pickers. Natural product pickers are common individuals dwelling in Australia who wish to add to organic product picking. These individuals frequently wish to work/procure compensation or on occasion be essential for the organic product picking process for relaxation as it were. The enlistment cycle for the ranchers is basic. Ranchers give their contact subtleties, ranch address and above all; their business charge subtleties. This is went with a picture ID that is likewise given as a feature of the enlistment interaction. The organic product pickers give a lot of comparative subtleties in the event that they wish to do natural product picking for procuring an everyday, month to month or a time-based compensation. For this situation they additionally give bank subtleties while enlisting. As opposed to this, in the event that the natural product picker wishes to enroll for recreation just, bank subtleties and expense subtleties are not needed by the framework. Natural product picker's gathering enrollments is just conceivable assuming that the enlistment is for relaxation as it were. The site permits the natural product pickers to choose and plan their visits for natural product picking for a specific ranch; seven days ahead of time. Each mentioned plan is supported by the rancher and afterward the natural product pickers get endorsement warning by means of email or potentially as portable instant message (as solicitation by the natural product picker). Visits to the homestead without rancher's endorsement are not permitted and not upheld by the framework and the mentioned plan is auto declined in no less than multi week from the mentioned date if in any case not supported by the rancher. While mentioning for a timetable, the natural product pickers give subtleties of the fruit(s) that they need to pick. This is to guarantee the executives of wages, time, instrument and assets expected to play out the work action. Compensation differ with time, weight of organic product picked and the kind of organic product. The compensation is chosen according to the rancher's circumspection and is given by the rancher at the hour of enlistment. If there should be an occurrence of organic product pickers who are picking for recreation just, subtleties of organic product or organic products that they wish to pick is additionally significant on the grounds that this guarantees general and using time productively at the rancher's end. The framework likewise permits 'giving' also. With this usefulness, the ranchers conclude how much organic product they would gift (concerning weight, as a rate or proportion) to the natural product picker(s) who wish to just pick organic product for relaxation. This data is likewise given by the rancher at the hour of enlistment and comparable measure of organic product is given as gift by the rancher, on the homestead, to the natural product pickers, in view of how much natural product has been picked. The giving system likewise permits ranchers to make a job of gift administrator. The job of the gift administrator is simply to help the rancher. The gift supervisor is just liable for distributing gifts (natural product) according to the data given by the rancher at the hour of enrollment. The framework, subsequently, monitors how much natural product was talented and how much was picked. For the organic product pickers who are picking to acquire compensation, the data for example 'how much natural product is picked?', is significant for wage the executives reason. donesloching.com permits organic product pickers to drop their visits. It additionally permits, crossing out of enrollment for the two ranchers and organic product pickers. Essentially, the site permits both the gatherings to make changes to their enrollment as and when required. The site upholds a total entrance for the ranchers and the natural product picker.

The organic product picker's entrance permits them to see their past and impending exercises/schedules. They can likewise see subtleties of their record. For each visit to the homestead, they can add remarks, encounters and memory photos. Rancher's entry likewise permits them to see their past, current and impending homestead reservations. The ranchers can perceive how much natural product was gathered by each natural product pickers and what was gifted or paid to them. They can see nitty gritty remarks and encounters of each natural product picker for each booking. The rancher's entryway additionally permits them to choose photos of the natural product picking action for the site's photograph display. Just the photos that the organic products pickers permit to share can be shared by the ranchers. At last, the site additionally permits the ranchers to produce day to day, week after week and month to month reports to get quantitative understanding into entire organic product picking movement and the fruit(s) picked. Rabee James, the CEO of Zaf Soft, is sure that this site would be an enormous achievement. Zaf Soft designs to acquire from ranchers. Extremely negligible membership charges are charged from the ranchers at the hour of enrollment. donesloching.com permits month to month and yearly memberships. The membership charges are chosen by the administration of Zaf Soft and can be added to the framework by means of the administrator entry. The overseer is mindful of dealing with the general site's smooth working, information reinforcement, recuperation and up gradation.

18.3 For WEP, information integrity and statistics confidentiality are carried out using the RC4 stream encryption set of rules. The transmitter of an MPDU performs the subsequent steps, called encapsulation: 1. The transmitter selects an initial vector (IV) price. 2. The IV price is concatenated with the WEP key shared by transmitter and receiver to form the seed, or key input, to RC4. 3. A 32-bit cyclic redundancy take a look at (CRC) is computed over all the bits of the MAC information subject and appended to the facts field. The CRC is a not unusual blunders-detection code used in records link control protocols. In this example, the CRC serves as a integrity take a look at price (ICV). 4. The result of step three is encrypted using RC4 to form the ciphertext block. 5. The plaintext IV is prepended to the ciphertext block to shape the encapsulated MPDU for transmission. A. Draw a block diagram that illustrates the encapsulation manner. B. Describe the steps at the receiver give up to get better the plaintext and carry out the integrity check. C. Draw a block diagram that illustrates part b.

Consider the subsequent threats to Web security and describe how every is countered by using a particular characteristic of TLS. A. Brute-Force Cryptanalytic Attack: An exhaustive seek of the important thing area for a traditional encryption algorithm. B. Known Plaintext Dictionary Attack: Many messages will contain predictable plaintext, along with the HTTP GET command. An attacker constructs a dictionary containing each viable encryption of the regarded-plaintext message. When an encrypted message is intercepted, the attacker takes the component containing the encrypted known plaintext and appears up the ciphertext within the dictionary. The ciphertext must in shape towards an entry that changed into encrypted with the equal secret key. If there are several suits, every of those can be tried in opposition to the whole ciphertextual content to determine the right one. This assault is especially effective in opposition to small key sizes (e.G., forty-bit keys). C. Replay Attack: Earlier TLS handshake messages are replayed. D. Man-in-the-Middle Attack: An attacker interposes during key change, performing as the patron to the server and as the server to the patron. E. Password Sniffing: Passwords in HTTP or other application visitors are eavesdropped. F. IP Spoofing: Uses cast IP addresses to idiot a bunch into accepting bogus data.

kindly answer all the questions