lup

] (d) Show how a generating (or "mother") wavelet (x) can spawn a family of "daughter" wavelets jk(x) by simple shifting and scaling operations, and explain the advantages of representing continuous functions in terms of such a family of self-similar dilates and translates of a mother wavelet. [5 marks

(a) Let X be a random variable with finite mean = E(X) and finite variance 2 = Var(X). State and prove Chebyshev's inequality for the random variable X. You may assume Markov's inequality without prooffinite mean = E(X) such that fX(x) = 0 x 6 [, ] xfX(x) x [, ] where , and are non-negative real constants with < . Suppose that (Ai , Bi) for i = 1, 2, . . . , n is a sequence of independent and identically distributed 2-dimensional random variables where Ai and Bi are independent with marginal distributions Ai U[, ] and Bi U[0, ] for each i = 1, 2, . . Var(Zn) 1 4n . [5 marks] (ii) Using Chebyshev's inequality show that Zn converges in probability to the degenerate random variable with value /(( )). [5 marks] (iii) Describe an algorithm to estimate the mean of the random variable X. You may assume for the purpose of your algorithm that you have a function that returns random points of the given form (Ai , Bi). [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.)

h(x, y, x) and

h(y, z, u)

h(x, y, z) and

h(f(y), z, x)

h(x, y, b) and

h(a, x, y)

h(x, y, z) and

h(g(y, y), g(z, z), g(u, u))

[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

The complex form of the Fourier series is:

f(x) =

+

X

k=

ckei2kx

where ck is a complex number and ck = ck.

(a) Prove that the complex coeffiffifficient, ck, encodes the amplitude and phase

coeffiffifficients, Ak and k, in the alternative form:

f(x) =

+

X

k

=0

Ak cos(2kx k)

[10 marks]

(b) What is special about the case k = 0? [2 marks]

(c) Explain how the coeffiffifficients, ck, of the Fourier series of the periodic function,

f(x):

f(x) = f(x + T), x

can be obtained from the Fourier transform, FL(), of the related function,

fL(x):

fL(x) = f(x

),

T

2 6 x < T

2

0

,

otherwise

[8 marks]

2 Concurrent Systems

An interprocess communication environment is based on synchronous message

passing. A server is to be designed to support a moderate number of simultaneous

client requests.

Clients send a request message to the server, continue in parallel with server

operation, then wait for the server's reply message.

Discuss the design of the server's interaction with the clients. Include any problems

you foresee and discuss alternative solutions to them. [20 marks]

2CST.2001.4.3

3 Further Java

(a) Describe how mutual-exclusion locks provided by the synchronized keyword

can be used to control access to shared data structures. In particular you

should be clear about the behaviour of concurrent invocations of difffferent

synchronized methods on the same object, or of the same synchronized method

on difffferent objects. [6 marks]

(b) Consider the following class defifinition:

class Example implements Runnable {

public static Object o = new Object();

int count = 0;

public void run() {

while (true) {

synchronized (o) {

count ++;

}

}

}

}

Show how to start two threads, each executing this run method. [2 marks]

(c) When this program is executed, only one of the count fifields is found to

increment, even though threads are scheduled preemptively. Why might this

be? [2 marks]

(d) Defifine a new class FairLock. Each instance should support two methods, lock

and unlock, which acquire and release a mutual exclusion lock such that calls

to unlock never block the caller, but will allow the longest-waiting blocked

thread to acquire the lock. The lock should be recursive, meaning that the

thread holding the lock may make multiple calls to lock without blocking.

The lock is released only when a matched number of unlock operations have

been made.

You may wish to make use of the fact the Thread.currentThread() returns

the instance of Thread that is currently executing. [10 marks]

3

[TURN OVERCST.2001.4.4

4 Compiler Construction

Consider the following grammar giving the concrete syntax of a language:

E id

C E = E;

C {B}

C C repeatwhile E

C if E then C

C if E then C else C

B B C

B C

S C eof

where C repeatwhile E has the same meaning as do C while E in C or Java.

(a) List the terminals and non-terminals of this grammar and explain the

signifificance of S. [3 marks]

(b) Identify any ambiguities in the above grammar and rewrite it to remove them,

ensuring that your new grammar generates exactly the same set of strings.

[4 marks]

(c) Specify a suitable abstract syntax, for example by giving a type declaration

in a programming language of your choice, which might be used to hold parse

trees for this language. [3 marks]

(d) Give either a recursive descent parser or a characteristic fifinite state machine

(e.g. for SLR(1)) with associated parser for your grammar. Your parser need

not return a parse treeit suffiffiffices for your parser either to accept or to reject

the input string. [10 marks]

4CST.2001.4.5

5 Data Structures and Algorithms

(a) Outline how you would determine whether the next line segment turns left or

right during the Graham scan phase of the standard method of computing the

convex hull of a set of points in a plane. [5 marks]

(b) Describe in detail an effiffifficient algorithm to determine how often the substring

ABRACADABRA occurs in a vector of 106 characters. Your algorithm should be

as effiffifficient as possible. [10 marks]

(c) Roughly estimate how many character comparisons would be made when your

algorithm for part (b) is applied to a vector containing 106 characters uniformly

distributed from the 26 letters A to Z. [5 marks]

6 ECAD

(a) When designing clocked circuits there are times when asynchronous inputs

have to be sampled which may result in metastable behaviour in state holding

elements. How might metastability be avoided when sampling asynchronous

inputs? [5 marks]

(b) An optical shaft encoder (e.g. used on the internal rollers of a mechanical

mouse) consists of a disk with an evenly spaced alternating transparent and

opaque grating around the circumference. Two optical sensors are positioned

such that when one sensor is at the middle of an opaque region, the other

is at the edge. Consequently, the following Gray code sequence is produced,

depending upon the direction of rotation:

positive rotation

negative rotation

00

00

01

10

11

11

10

01

time

A shaft decoder module is required to convert the Gray code into an 8-bit

position. The 8-bit position should be incremented every time the input

changes from one state to another in a positive direction (e.g. from 00 to

01, or from 10 to 00). Similarly, the 8-bit position should be decremented

every time the input changes from one state to another in a negative direction

(e.g. from 00 to 10, or from 01 to 00).

Write and comment a Verilog module which performs the function of a shaft

decoder. [15 marks]

5

[TURN OVERCST.2001.4.6

7 Operating System Functions

(a) In the context of virtual memory management:

(i) What is demand paging? How is it implemented? [4 marks]

(ii) What is meant by temporal locality of reference? [2 marks]

(iii) How does the assumption of temporal locality of reference inflfluence page

replacement decisions? Illustrate your answer by brieflfly describing an

appropriate page replacement algorithm or algorithms. [3 marks]

(iv) What is meant by spatial locality of reference? [2 marks]

(v) In what ways does the assumption of spatial locality of reference inflfluence

the design of the virtual memory system? [3 marks]

(b) A student suggests that the virtual memory system should really deal with

"objects" or "procedures" rather than with pages. Make arguments both for

and against this suggestion. [4 and 2 marks respectively]

.

6CST.2001.4.7

8 Computation Theory

(a) Defifine precisely what is meant by the following:

(i) f(x1, x2, . . . xn) is a Primitive Recursive (PR) function of arity n.

[5 marks]

(ii) f(x1, x2, . . . xn) is a Total Recursive (TR) function of arity n. [3 marks]

(b) Ackermann's function is defifined by the following recursive scheme:

f(0, y) = S(y) = y + 1

f(x + 1, 0) = f(x, 1)

f(x + 1, y + 1) = f(x, f(x + 1, y))

For fifixed n defifine

gn(y) = f(n, y).

Show that for all n, y N,

gn+1(y) = gn(y+1)(1),

where h(k)(z) is the result of k repeated applications of the function h to initial

argument z. [4 marks]

(c) Hence or otherwise show that for all n N, gn(y) is a PR function. [4 marks]

(d) Deduce that Ackermann's function f(x, y) is a TR function. [3 marks]

(e) Is Ackermann's function PR? [1 mark]

7

[TURN OVERCST.2001.4.8

9 Numerical Analysis I

(a) What is meant by a symmetric positive defifinite matrix ? [3 marks]

(b) Verify that A = 2 1

1 2 is positive defifinite. [4 marks]

(c) The Choleski factorisation A = LDLT is to be applied to the solution of

Ax = b, where b = 1

1 . It is found that

L = 1

1

2

1 , D = 2

3

2 .

The next step in the method is to solve Ly = b to get y = 1

1

2

. Form the

upper triangular system of equations needed to complete the solution.

[4 marks]

(d) Solve these equations. [2 marks]

(e) What is meant by the order of convergence of an iterative process? [1 mark]

(f ) State the Newton-Raphson formula for solving f(x) = 0 for scalar x. What is

the order of convergence of this method? [2 marks]

(g) This method is used to solve f(x) = x2 4 = 0 using IEEE Double Precision

with a certain starting value x0. It is found that the third iterate x3 ' 2.0006,

and x4 ' 2.00000009. Very roughly, how many signifificant decimal digits of

accuracy would you expect in x5? Explain your answer. [4 marks]

8CST.2001.4.9

10 Computer Graphics and Image Processing

(a) Describe an algorithm to draw a straight line using only integer arithmetic.

You may assume that the line is in the fifirst octant, that the line starts and

ends at integer co-ordinates, and that the function setpixel(x, y) turns on the

pixel at location (x, y). [8 marks]

(b) Describe Douglas and Pucker's algorithm for removing superflfluous points from

a line chain. [10 marks]

(c) Under what circumstances would it be sensible to employ Douglas and Pucker's

algorithm? [2 marks]

(a) Using complex exponentials, prove the following trigonometric identity, which describes the multiplicative modulation of one cosine wave by another as being simply the sum of a different pair of cosine waves: cos(ax) cos(bx) = 1 2 cos((a + b)x) + 1 2 cos((a b)x) [3 marks] (b) The function sinc(x) = sin(x) x for x 6= 0 as plotted here plays an important role in the Sampling Theorem. By considering its Fourier transform, show that this function is unchanged in form after convolution with itself, and show that it even remains unchanged in form after convolution with any higher frequency sinc function sinc(ax) for a > 1, but that if 0 < a < 1, then the result is instead that lower frequency sinc function sinc(ax). Figure The sinc function sinx x 0 0.2 0.4 0.6 0.8 1 1.2 -W 0 W Figure Aliasing eect example [5 marks] (c) Let V be an inner product space spanned by an orthonormal system of vectors {e1, e2, . . . , en} so that i 6= j the inner product hei , ej i = 0, but every ei is a unit vector so that hei , eii = 1. We wish to represent a data set consisting of vectors u span{e1, e2, . . . , en} in this space as a linear combination of the orthonormal vectors: u = Xn i=1 aiei . Derive how the coefficients ai can be determined for any vector u, and comment on the computational advantage of representing the data in an orthonormal system. [7 marks] (d) Show how a generating (or "mother") wavelet (x) can spawn a family of "daughter" wavelets jk(x) by simple shifting and scaling operations, and explain the advantages of representing continuous functions in terms of such a family of self-similar dilates and translates of a mother wavelet. [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 marks]

The following code uses a Scanner object to read a text file called dogYears.txt. Notice that each line of this file contains a dog's name followed by an age. Ihe program then outputs this data to the console. The output looks like this: Tippy 2 Rex i Desdemona 5 1. Your task is to use the Scanner methods that will initialize the variables name1, name2, name3, age1, age2, age3 so that the execution of the three println statements below will result in the desired output. String name1, name2, name3; int age1, age2, age3; Scanner inputData = new Scanner(new FileReader("dog Years.txt"); //code here System.out.printin(name1+ "+age1); System.out.println(name2+" "+age2); System.out.println(name3+" "+age3); 2. Complete the Java program below so that it reads the text file dogNames.bxt and writes

its contents to another text file, "doggoneData.txt" The file dogNames.txt contains just the single word "Tippy", so the output in "doggoneData.bxt" should simply look like this: Tippy Your task is to complete the code below by using the variable name along with the appropriate Scanner and PrintWriter methods. String name; Scanner inputfile = new Scanner(new FileReade("dog Names.txt"); PrintWriter outputFile = new PrintWriter('doggoneData. bxt"); /code here inputFile.close0; outputFile.close0; 3. Complete the Java program below so that it reads the text file dogYears.txt and writes its contents to another text file "doggoneData.bxt". Here the input file gives ages in dog years, and the output file reports these figures in human years (so Tippy is 2 in the input file, and 14- 7 times older - in the output file.)