Consider the Markov Chain, Xn, on the states i = 0, 1, 2, . . . with transition matrix given by pi,i1 = p i = 1, 2, . . . pi,i+1 = 1  p i = 0, 1, . . . p0,0 = p where 0 < p < 1. (i) Show that the Markov chain is irreducible. [2 marks] (ii) Show that the Markov chain is aperiodic. [2 marks] (iii) Find a condition on p to make the Markov chain positive recurrent and find the stationary distribution in this case. [6 marks] (b) Consider the PageRank algorithm. (i) Describe PageRank as a Markov chain model for the motion between nodes in a graph where the nodes correspond with web pages. [5 marks] (ii) Explain the main mathematical results that underpin the relevance of PageRank to a notion of web page importance. [5 marks]

(a) Let X be a random variable with finite m(|X| a) E(X2 ) a2 . [5 marks] (b) A study by a mobile phone operator shows that the expected number of simultaneous calls at a base station is 100. The actual number of simultaneous calls is a random variable, X, and so the base station is designed to handle a higher number of simultaneous calls up to a maximum of M = 150. (i) Use the Markov inequality to bound the probability that the station will receive more than 150 calls. [5 marks] (ii) Now suppose that we are given the additional information that the variance of the number of simultaneous calls is 50. Use the inequality from part (a)(ii) to give a second bound on the probability of exceeding 150 calls. [5 marks]

(a) State what is meant by a directed graph and a strongly connected component.

Illustrate your description by giving an example of such a graph with 8 vertices

and 12 edges that has three strongly connected components. [5 marks]

(b) Describe, in detail, an algorithm to perform a depth-fifirst search over such a

graph. Your algorithm should attach the discovery and fifinishing times to each

vertex and leave a representation of the depth-fifirst spanning tree embedded

within the graph. [5 marks]

(c) Describe an O(n) algorithm to discover all the strongly connected components

of a given directed graph and explain why it is correct. You may fifind it useful

to use the concept of the forefather (v) of a vertex v which is the vertex, u,

with highest fifinishing time for which there exists a (possibly zero length) path

from v to u. [10 marks]

2 Computer Design

(a) What is a data cache and what properties of data access does it exploit?

[5 marks]

(b) What is a direct mapped cache and under what conditions will it exhibit poor

performance? [5 marks]

(c) Under what circumstances might a word of data in main memory be

simultaneously held in two separate fifirst-level cache lines? [5 marks]

(d) A translation look aside buffffer is a specialised cache. What does it typically

store and why is it often a factor of 1000 smaller than a data cache? [5 marks]

2CST.2001.6.3

3 Digital Communication I

(a) Defifine the terms circuit and packet in the context of communication systems.

[5 marks]

(b) What sort of guarantee does circuit switching provide? [5 marks]

(c) What advantages does packet switching provide over circuit switching?

[5 marks]

(d) Which of frequency division multiplexing, time division multiplexing and code

division multiplexing lend themselves to circuit switching? Which to packet

switching? Explain why or why not in each case. [5 marks]

4 Computer Graphics and Image Processing

(a) Describe the z-buffffer polygon scan conversion algorithm. [10 marks]

(b) In ray tracing, once we have determined where a ray strikes an object, the

illumination at the intersection point can be calculated using the formula:

I = Iaka +X

i

Iikd(Li N) +X

i

Iiks(Ri V)n

Explain what real effffect each of the three terms is trying to model and explain

what each of the following symbols means, within the context of this formula:

I, Ia, i, Ii , ka, kd, ks,Li, N, Ri, V, n

[10 marks]

3

[TURN OVERCST.2001.6.4

SECTION B

5 Comparative Programming Languages

(a) Brieflfly explain the concept of coroutines as used in BCPL and outline

the effffect of the library functions createco(f, size), deleteco(ctpr),

callco(cptr, val) and cowait(val). [10 marks]

(b) Outline how you would design a coroutine to merge, in increasing order, two

infifinite streams of increasing integers supplied by two other coroutines.

[5 marks]

(c) Brieflfly outline how you would implement an analogous merging mechanism in

an object-oriented language, such as Java, that does not provide a coroutine

mechanism. [5 marks]

6 Compiler Construction

(a) Describe one possible structure (e.g. ELF) of an object fifile. Illustrate your

answer by considering the form of object fifile which might result from the

following C program.

int a = 1, b = -1;

extern int g(int);

extern int c;

int f() { return g(a-b) + c; }

It is not necessary to consider the exact instruction sequence, just issues

concerning its interaction with the object fifile format. [10 marks]

(b) Describe how a linker takes a sequence of such programs and produces an

executable fifile. [4 marks]

(c) Compare and contrast static and dynamic linking in a system using your object

fifile format. [6 marks]

4CST.2001.6.5

7 Prolog for Artifificial Intelligence

A weighted binary tree can be defifined using compound terms in the following way.

A node of the tree is represented by the term n(V, L, R), where V stands for the

value of the node, and L and R stand for the left and right branches, respectively.

A terminal node has the R and L components instantiated to the null list.

Given an input tree T, write a Prolog program that constructs a tree of the same

shape as T, but in which the value of each node has been set to the value of the

maximum value node in T.

[Note: Maximum marks are available only for programs that perform this task in

one recursive descent of the input tree, and which use no more than four clauses.]

[20 marks]

5

[TURN OVERCST.2001.6.6

8 Databases

The environmental agency is setting up an SQL database to monitor long-term

trends in the climate. Data are collected from observatories of a number of difffferent

kinds.

Flood risk is of particular concern. Each water authority measures river levels and

rates of flflow hourly at major points, and records reservoir levels daily.

In addition, the agency maintains weather stations both inland and at sea. These

record precipitation (rainfall etc.), temperature, sunshine, air pressure and wind.

Values of new precipitation, temperature, pressure, and wind speed and direction

are taken hourly; gusts of over 60 m.p.h. are noted whenever they occur.

Maximum and minimum temperature and pressure, the total number of hours of

sunshine and the total precipitation are recorded daily. Inland stations can be

grouped by water authority.

By default these primary data will be relegated to archive after 2 years. Selected

information is retained permanently in a data warehouse. This serves two purposes.

First, it holds monthly summary data consisting of the maximum (and minimum

as appropriate) day value for each statistic, together with the monthly totals of

sunshine and precipitation. The warehouse also keeps detailed information relating

to periods of extreme weather from the relevant observatories, with one or more

keywords describing the nature of the incident (flflood, blizzard, hurricane etc.) and

an optional comment.

Write notes to assist in the design of the schema for the relational data warehouse,

including any diagrams that you fifind helpful. Explain how your design will enable

meteorologists to fifind relevant past records, noting any assumptions that you make

about the nature of the data.

[You should not go into unnecessary detail about the structure of the primary

database. You may assume that expert meteorologists will select the data for the

warehouse.]

[20 marks]

6CST.2001.6.7

SECTION C

9 Semantics of Programming Languages

Write short notes on four of the following fifive topics.

(a) The relationship between three forms of operational semantics of the Language

of Commands (LC) given by

an evaluation relation h P, si hV, s0 i

a transition relation h P, si hP0 , s0 i

a transition relation between the confifigurations

h c, r, si of the

SMC-machine

(b) The notion of semantic equivalence of LC phrases and its congruence property.

(c) Call-by-name and call-by-value rules for evaluating function applications in the

Language of Functions and Procedures (LFP) and the relationship between the

evaluation relations for LFP based upon each of them.

(d) The notion of bisimilarity of two confifigurations in a labelled transition system.

(e) The rules defifining the possible labelled transitions of parallel composition

(P1|P2) and restriction ( c . P) in the Language of Communicating Processes

(LCP).

[5 marks each]

7

[TURN OVERCST.2001.6.8

10 Foundations of Functional Programming

The following are some concepts that have flflourished in the context of functional

programming but which have (so far) been less heavily used in main-stream

languages even when they have been available:

(a) polymorphic types

(b) type reconstruction

(c) higher-order functions

(d) lazy evaluation

(e) continuations

For each case give a brief explanation of the facility referred to, suggest a

circumstance in which it might be useful and comment on how immediately relevant

to non-functional languages it seems.

[4 marks per part]

8CST.2001.6.9

11 Logic and Proof

(a) In the context of clause-based proof methods, defifine the notion of pure literal

and describe what should be done if the set of clauses contains pure literals.

[3 marks]

(b) Use the Davis-Putnam method to discover whether the following set of clauses

is satisfifiable. If they are satisfifiable, show a satisfying interpretation.

{P, R} {P, R} {P, Q} {Q, R} {P, Q, R}

[6 marks]

(c) The three-fifingered inhabitants of the planet Triterra build base-3 computers.

A Triterran named Randal Tryant has found a way of verifying base-3

combinational logic. His Ordered Ternary Decision Diagrams (OTDDs) are

the same as a technology used on planet Earth except that all variables and

expressions range over the values 0, 1 and 2 instead of just 0 and 1.

(i) Describe how a full ternary decision tree can be reduced to an OTDD

without regard for effiffifficiency. [2 marks]

(ii) Sketch an effiffifficient algorithm to convert a ternary expression directly to an

OTDD without constructing the full decision tree. For a typical ternary

connective use modulo-3 multiplication, written as . [6 marks]

(iii) Demonstrate your algorithm by applying it to the ternary expression

((i i) j) 2. [3 marks]

9

[TURN OVERCST.2001.6.10

12 Complexity Theory

(a) Show that any language that can be accepted by a nondeterministic machine

in time f(n) can also be decided by a deterministic machine in space O(f(n)).

[4 marks]

(b) Show that any language that can be accepted by a nondeterministic machine

in space f(n) can also be decided by a deterministic machine in time

O(c(f(n)+log n) ), for some constant c. [6 marks]

(c) Explain what the above results tell us about the inclusion relationships among

the complexity classes:

NL, co-NL, P, NP, PSPACE and NPSPACE

[4 marks]

(d) It has been proved that the graph reachability problem is in co-NL. What

further inclusions can you derive among the above complexity classes using

this fact? Explain your answer. [6 ma

(a) Let X be a random variable with finite mean, E(X), and variance, Var(X), and let a > 0. (i) Show Markov's inequality that P(|X| a) E(|X|) a . [5 marks] (ii) Using Markov's inequality show that P(|X| a) E(X2 ) a2 . [5 marks] (b) A study by a mobile phone operator shows that the expected number of simultaneous calls at a base station is 100. The actual number of simultaneous calls is a random variable, X, and so the base station is designed to handle a higher number of simultaneous calls up to a maximum of M = 150. (i) Use the Markov inequality to bound the probability that the station will receive more than 150 calls. [5 marks] (ii) Now suppose that we are given the additional information that the variance of the number of simultaneous calls is 50. Use the inequality from part (a)(ii) to give a second bound on the probability of exceeding 150 calls. [5 marks]

(a) State what is meant by a directed graph and a strongly connected component.

Illustrate your description by giving an example of such a graph with 8 vertices

and 12 edges that has three strongly connected components. [5 marks]

(b) Describe, in detail, an algorithm to perform a depth-fifirst search over such a

graph. Your algorithm should attach the discovery and fifinishing times to each

vertex and leave a representation of the depth-fifirst spanning tree embedded

within the graph. [5 marks]

(c) Describe an O(n) algorithm to discover all the strongly connected components

of a given directed graph and explain why it is correct. You may fifind it useful

to use the concept of the forefather (v) of a vertex v which is the vertex, u,

with highest fifinishing time for which there exists a (possibly zero length) path

from v to u. [10 marks]

2 Computer Design

(a) What is a data cache and what properties of data access does it exploit?

[5 marks]

(b) What is a direct mapped cache and under what conditions will it exhibit poor

performance? [5 marks]

(c) Under what circumstances might a word of data in main memory be

simultaneously held in two separate fifirst-level cache lines? [5 marks]

(d) A translation look aside buffffer is a specialised cache. What does it typically

store and why is it often a factor of 1000 smaller than a data cache? [5 marks]

2CST.2001.6.3

3 Digital Communication I

(a) Defifine the terms circuit and packet in the context of communication systems.

[5 marks]

(b) What sort of guarantee does circuit switching provide? [5 marks]

(c) What advantages does packet switching provide over circuit switching?

[5 marks]

(d) Which of frequency division multiplexing, time division multiplexing and code

division multiplexing lend themselves to circuit switching? Which to packet

switching? Explain why or why not in each case. [5 marks]

4 Computer Graphics and Image Processing

(a) Describe the z-buffffer polygon scan conversion algorithm. [10 marks]

(b) In ray tracing, once we have determined where a ray strikes an object, the

illumination at the intersection point can be calculated using the formula:

I = Iaka +X

i

Iikd(Li N) +X

i

Iiks(Ri V)n

Explain what real effffect each of the three terms is trying to model and explain

what each of the following symbols means, within the context of this formula:

I, Ia, i, Ii , ka, kd, ks,Li, N, Ri, V, n

[10 marks]

3

[TURN OVERCST.2001.6.4

SECTION B

5 Comparative Programming Languages

(a) Brieflfly explain the concept of coroutines as used in BCPL and outline

the effffect of the library functions createco(f, size), deleteco(ctpr),

callco(cptr, val) and cowait(val). [10 marks]

(b) Outline how you would design a coroutine to merge, in increasing order, two

infifinite streams of increasing integers supplied by two other coroutines.

[5 marks]

(c) Brieflfly outline how you would implement an analogous merging mechanism in

an object-oriented language, such as Java, that does not provide a coroutine

mechanism. [5 marks]

6 Compiler Construction

(a) Describe one possible structure (e.g. ELF) of an object fifile. Illustrate your

answer by considering the form of object fifile which might result from the

following C program.

int a = 1, b = -1;

extern int g(int);

extern int c;

int f() { return g(a-b) + c; }

It is not necessary to consider the exact instruction sequence, just issues

concerning its interaction with the object fifile format. [10 marks]

(b) Describe how a linker takes a sequence of such programs and produces an

executable fifile. [4 marks]

(c) Compare and contrast static and dynamic linking in a system using your object

fifile format. [6 marks]

4CST.2001.6.5

7 Prolog for Artifificial Intelligence

A weighted binary tree can be defifined using compound terms in the following way.

A node of the tree is represented by the term n(V, L, R), where V stands for the

value of the node, and L and R stand for the left and right branches, respectively.

A terminal node has the R and L components instantiated to the null list.

Given an input tree T, write a Prolog program that constructs a tree of the same

shape as T, but in which the value of each node has been set to the value of the

maximum value node in T.

[Note: Maximum marks are available only for programs that perform this task in

one recursive descent of the input tree, and which use no more than four clauses.]

[20 marks]

5

[TURN OVERCST.2001.6.6

8 Databases

The environmental agency is setting up an SQL database to monitor long-term

trends in the climate. Data are collected from observatories of a number of difffferent

kinds.

Flood risk is of particular concern. Each water authority measures river levels and

rates of flflow hourly at major points, and records reservoir levels daily.

In addition, the agency maintains weather stations both inland and at sea. These

record precipitation (rainfall etc.), temperature, sunshine, air pressure and wind.

Values of new precipitation, temperature, pressure, and wind speed and direction

are taken hourly; gusts of over 60 m.p.h. are noted whenever they occur.

Maximum and minimum temperature and pressure, the total number of hours of

sunshine and the total precipitation are recorded daily. Inland stations can be

grouped by water authority.

By default these primary data will be relegated to archive after 2 years. Selected

information is retained permanently in a data warehouse. This serves two purposes.

First, it holds monthly summary data consisting of the maximum (and minimum

as appropriate) day value for each statistic, together with the monthly totals of

sunshine and precipitation. The warehouse also keeps detailed information relating

to periods of extreme weather from the relevant observatories, with one or more

keywords describing the nature of the incident (flflood, blizzard, hurricane etc.) and

an optional comment.

Write notes to assist in the design of the schema for the relational data warehouse,

including any diagrams that you fifind helpful. Explain how your design will enable

meteorologists to fifind relevant past records, noting any assumptions that you make

about the nature of the data.

[You should not go into unnecessary detail about the structure of the primary

database. You may assume that expert meteorologists will select the data for the

warehouse.]

[20 marks]

6CST.2001.6.7

SECTION C

9 Semantics of Programming Languages

Write short notes on four of the following fifive topics.

(a) The relationship between three forms of operational semantics of the Language

of Commands (LC) given by

an evaluation relation h P, si hV, s0 i

a transition relation h P, si hP0 , s0 i

a transition relation between the confifigurations

h c, r, si of the

SMC-machine

(b) The notion of semantic equivalence of LC phrases and its congruence property.

(c) Call-by-name and call-by-value rules for evaluating function applications in the

Language of Functions and Procedures (LFP) and the relationship between the

evaluation relations for LFP based upon each of them.

(d) The notion of bisimilarity of two confifigurations in a labelled transition system.

(e) The rules defifining the possible labelled transitions of parallel composition

(P1|P2) and restriction ( c . P) in the Language of Communicating Processes

(LCP).

[5 marks each]

7

[TURN OVERCST.2001.6.8

10 Foundations of Functional Programming

The following are some concepts that have flflourished in the context of functional

programming but which have (so far) been less heavily used in main-stream

languages even when they have been available:

(a) polymorphic types

(b) type reconstruction

(c) higher-order functions

(d) lazy evaluation

(e) continuations

For each case give a brief explanation of the facility referred to, suggest a

circumstance in which it might be useful and comment on how immediately relevant

to non-functional languages it seems.

[4 marks per part]

8CST.2001.6.9

11 Logic and Proof

(a) In the context of clause-based proof methods, defifine the notion of pure literal

and describe what should be done if the set of clauses contains pure literals.

[3 marks]

(b) Use the Davis-Putnam method to discover whether the following set of clauses

is satisfifiable. If they are satisfifiable, show a satisfying interpretation.

{P, R} {P, R} {P, Q} {Q, R} {P, Q, R}

[6 marks]

(c) The three-fifingered inhabitants of the planet Triterra build base-3 computers.

A Triterran named Randal Tryant has found a way of verifying base-3

combinational logic. His Ordered Ternary Decision Diagrams (OTDDs) are

the same as a technology used on planet Earth except that all variables and

expressions range over the values 0, 1 and 2 instead of just 0 and 1.

(i) Describe how a full ternary decision tree can be reduced to an OTDD

without regard for effiffifficiency. [2 marks]

(ii) Sketch an effiffifficient algorithm to convert a ternary expression directly to an

OTDD without constructing the full decision tree. For a typical ternary

connective use modulo-3 multiplication, written as . [6 marks]

(iii) Demonstrate your algorithm by applying it to the ternary expression

((i i) j) 2. [3 marks]

9

[TURN OVERCST.2001.6.10

12 Complexity Theory

(a) Show that any language that can be accepted by a nondeterministic machine

in time f(n) can also be decided by a deterministic machine in space O(f(n)).

[4 marks]

(b) Show that any language that can be accepted by a nondeterministic machine

in space f(n) can also be decided by a deterministic machine in time

O(c(f(n)+log n) ), for some constant c. [6 marks]

(c) Explain what the above results tell us about the inclusion relationships among

the complexity classes:

NL, co-NL, P, NP, PSPACE and NPSPACE

[4 marks]

(d) It has been proved that the graph reachability problem is in co-NL. What

further inclusions can you derive among the above complexity classes using

this fact? Explain your answer. [6 ma

(a) Let X be a random variable with finite mean, E(X), and variance, Var(X), and let a > 0. (i) Show Markov's inequality that P(|X| a) E(|X|) a . [5 marks] (ii) Using Markov's inequality show that P(|X| a) E(X2 ) a2 . [5 marks] (b) A study by a mobile phone operator shows that the expected number of simultaneous calls at a base station is 100. The actual number of simultaneous calls is a random variable, X, and so the base station is designed to handle a higher number of simultaneous calls up to a maximum of M = 150. (i) Use the Markov inequality to bound the probability that the station will receive more than 150 calls. [5 marks] (ii) Now suppose that we are given the additional information that the variance of the number of simultaneous calls is 50. Use the inequality from part (a)(ii) to give a second bound on the probability of exceeding 150 calls. [5 marks]

Describe an O(n log(n)) algorithm based on a variation of merge sort to fifind the

closest pair of a given set of points lying in a plane. You may assume that the set

of points is given as a linked list of (x, y) coordinates. [8 marks]

Carefully prove that your algorithm can never take longer than O(n log(n)).

[6 marks]

Modify, with explanation, your algorithm to fifind the pair of points with minimum

Manhattan distance. The Manhattan distance between points (x1, y1) and (x2, y2)

is |x1 x2| + |y1 y2|. [6 marks]

1

[TURN OVERCST.2000.6.2

2 Computer Design

Why are the following statements fallacies?

(a) MIPS is an accurate measure for comparing performance among computers.

[5 marks]

(b) A benchmark is a typical program which accurately predicts the performance

of all other applications. [5 marks]

(c) Complex instruction set computers minimise the semantic gap between

machine code and high-level languages, thereby making applications run more

quickly. [5 marks]

(d) Data caches always improve processor throughput. [5 marks]

3 Digital Communication I

Compare circuit switching and packet switching, paying attention to channel

characteristics and resource effiffifficiency. [7 marks]

What is wave division multiplexing (WDM)? Is it more like circuit switching or

packet switching and why? [7 marks]

Wave length conversion is the process, either optical or optical-electronic-optical,

of receiving a signal on one wavelength and transmitting on another.

How does wave length conversion ease the problem of routing optical carriers in a

network? [3 marks]

"The huge capacity of WDM systems will mean that IP becomes redundant."

Discuss. [3 marks]

2CST.2000.6.3

4 Computer Graphics and Image Processing

Give an algorithm for drawing the part of a circle which lies in the fifirst octant.

Assume that the circle has integer radius and is centered at the origin. Assume

that you have a function setpixel(x, y) which turns on pixel (x, y). [10 marks]

Derive a matrix, or a product of matrices, to perform a clockwise 2D rotation of

arbitrary angle, , about an arbitrary point, (xc, yc). [4 marks]

Provide an algorithm to ascertain whether the Bezier curve defifined by P1P2P3P4

lies within some tolerance, , of the straight line segment, P1P4, which joins the

Bezier curve's end points. Your algorithm must return false if the Bezier curve is

outside the tolerance; it must return true if the curve is well inside the tolerance;

it may return either true or false if the curve is inside, but not well inside, the

tolerance. [6 marks]

SECTION B

5 Comparative Programming Languages

Give a brief summary of the main syntactic constructs found in the programming

language Smalltalk. Other languages often have the conditional constructs

if-then-else and while. Show how these two constructs can be defifined in Smalltalk.

[8 marks]

Illustrate the use of Smalltalk by showing how you would defifine a method to

compute the factorial of an integer. [8 marks]

Although Smalltalk was originally designed to be an interpretive language, modern

implementations are dramatically more effiffifficient. Brieflfly outline what techniques

might have been used to make this improvement. [4 marks]

3

[TURN OVERCST.2000.6.4

6 Compiler Construction

Describe how a parse tree can be translated into a sequence of assembly language

instructions based on a pattern matching graph derived from a set of tree rewriting

rules where each rule has a cost and a corresponding fragment of code. Illustrate

your answer using the following rules:

Ri = Kk LDI Ri,Kk Cost 2

Ri = add(Ri,Kk) ADDI Ri,Kk Cost 3

Ri = add(Ri,Rj) ADD Ri,Rj Cost 3

Ri = add(Ri,add(Rj,Kk)) ADD Ri,Rj,Kk Cost 4

applied to the following parse tree:

add(K1,add(add(K2,add(K3,K4)),add(K5,K6)))

[15 marks]

Discuss the advantages and disadvantages of this approach to code generation.

[5 marks]

7 Prolog for Artifificial Intelligence

One of the regulations of the International Rugby Board (IRB) states that for a

player to be eligible to play for a given country, the player's father or mother or

grandfather or grandmother must have been born in that country. Assume that

there is a complete genealogical database consisting of Prolog clauses of the form

person(P, B, F, M), where P is a person's name, B is the country of P's birth, F is

their father's name and M is their mother's name. For example, the clause

person(bruce, australia, rhodri, bronwyn).

might appear in such a database. Further assume that names in the database are

constructed so as to refer uniquely to individuals. Write Prolog clauses defifining

the predicate eligible such that goals of the form eligible(P,C) succeed if and only if

person P is eligible to play for country C according to the above regulation.

[10 marks]

Given a list of players on a given country's team, defifine a predicate checkteam

that will check each member of the team for eligibility according to the eligible

predicate, and furthermore check that each player appears on the list only once.

The checkteam goal will fail if any player is ineligible or if any player is listed more

than once. [10 marks]

4CST.2000.6.5

8 Databases

Describe the basic architecture of the ODMG standard for Object Data

Management. [10 marks]

What support is provided for transactions? What locking modes are available, and

how are they used by the database runtime systems? [4 marks]

The query language OQL is recognised as a standard by the Object Management

Group (OMG). To what extent is it similar to SQL, and in what ways does it

diffffer? [6 marks]

SECTION C

9 Semantics of Programming Languages

What does it mean to say that two confifigurations of a labelled transition system

are bisimilar? [3 marks]

Describe a labelled transition system for a language of communicating processes

with input prefifixing (c(x). P), output prefifixing (

ch Ei . P), an inactive process (0),

choice (P + P0 ), parallel composition (P|P0 ) and channel restriction ( c . P). You

may assume there is a relation E n which defifines when an integer expression E

evaluates to an integer n. [7 marks]

For each of the following pairs of processes, say whether or not they are bisimilar.

Justify your answer in each case.

(a)

ch 1i .((

ch 2i . 0) + (

ch 3i . 0)) and (

ch 1i . ch 2i . 0) + (

ch 1i . ch 3i . 0) [4 marks]

(b) P and c .((c(x). 0)|(

ch 1i . P)), where c does not occur in P

[6 marks]

10 Foundations of Functional Programming

Give as simple a set of rules as you can for transforming lambda calculus to a form

where there are no bound variables mentioned, but where there are many instances

of the three standard combinator constants S, K and I. [6 marks]

Describe tree-rewrites suitable for reducing expressions written in terms of

combinators. [6 marks]

Explain how you might deal with the issue of keeping track of the values of bound

variables if you were to interpret lambda calculus directly. [8 marks]

5

[TURN OVERCST.2000.6.6

11 Logic and Proof

For each of the given pairs of terms, give a most general unififier or indicate why

none exists. (Here x, y, z are variables while a, b are constant symbols.)

)

[4 marks]

A standard unifification algorithm takes a pair of terms t1 and t2 and returns a

substitution such that t1 = t2. Show how this algorithm can be used to fifind

the unififier of several (n > 2) terms t1, t2, . . . , tn: a substitution such that

t1 = t2 = = tn. Indicate how the unififier is constructed from the unififiers

of n 1 pairs of terms. (Assume that all required unififiers exist and ignore the

question of whether the unififiers are most general.) [6 marks]

Prove using resolution the formula

This program extends the earlier "Online shopping cart" program. (Consider first saving your earlier program). (1) Extend the ItemToPurchase class per the following specifications: Private fields string itemDescription- Initialized in default constructor to "none" Parameterized constructor to assign item name, item description, item price, and item quantity (default values of 0),. (1 pt) .Public member methods setDescription) mutator & getDescription| accessor (2 pts) printltemCosto Outputs the item name followed by the quantity, price, and subtotal printltemDescription) - Outputs the item name and description Ex. of printitemCost) output Bottled Water 10 $1 = $10 Ex. of printitemCost) output: Bottled Water 10 @ $1 = $10 Ex. of printltemDescription) output: Bottled Water: Deer Park, 12 oz. (2) Create two new files: ShoppingCartjava - Class definition ShoppingCartManager.java Contains main0 method Build the ShoppingCart class with the following specifications. Note: Some can be method stubs (empty methods) initially, to be completed in later steps. Private fields String customerName - Initialized in default constructor to "none" String currentDate- Initialized in default constructor to "January 1, 2016"

kindly answer all the questions