Computing Theory
COSC 1107/1105
Assignment 2: Computability
Assessment Type Individual assignment. Submit online via Canvas → As-
signments → Assignment 2. Marks awarded for meeting re-
quirements as closely as possible. Clarifications/updates may
be made via announcements/relevant discussion forums.
Due Date Tuesday 19th October 2021, 11:59pm (note new date)
Marks 100 worth 25% of your final assessment
1 Overview
This assignment requires you to demonstrate your knowledge of some key issues in com-
putability, and of non-determinism. There is also a part which deals with the Platypus
game.
2 Assessment details
1. Passwords (8 marks)
The Dwarves of the Lonely Mountain have a lot of time on their hands, and are
very worried about information about the size of their horde of gold getting around
various parts of Middle-Earth, as they fear invasion by all and sundry if any hint
of their true wealth were to be revealed. Accordingly King Durin XIII decrees
that all records are to be written only in encrypted Khuzdul, in which there are
32 distinct characters. The encryption process is based on a Khuzdul keyword of
n characters in length. Durin’s advisors inform him that an evil wizard with an
army of orc-slaves at his disposal (or a few of the lesser intelligent hobbits :-)) could
exhaustively search through all possible keywords at a rate of 100,000 keywords per
day. Durin decrees that the royal Khuzdul keyword must be secure ”until the end
of the age”.
(a) Determine an appropriate interpretation of Durin’s statement ”until the end
of the age”. In other words, define what you think this means and hence what
the maximum such time would be. (2 marks)
(b) Given your previous answer, how long should the royal keyword be? Explain
your answer. (2 marks)
(c) Sillibo, a passing hobbit, happens to point out to Durin that not all of the
32 Khuzdul characters are equally likely to be used, as there are 18 that are
only used very rarely. Hence an outsider might concentrate on the other 14
and make significant progress. Assuming Sillibo is correct, calculate how long
it would take an evil wizard to discover a Khuzdul keyword of length given in
your previous answer, assuming that at most 14 different Khuzdul characters
are used in the keyword. (2 marks)
(d) Sillibo also tells Durin, who is very impressed with the hobbit’s knowledge,
that the royal keyword should not include a Khuzdul translation of the name
”Arkenpebble” (a famous jewel revered by all dwarves), as this is very well
known to all in Middle-Earth, and hence will certainly be known to any evil
wizard. Given the translation of Arkenpebble into Khuzdul uses 6 characters,
evaluate Sillibo’s claim. In other words, is the royal keyword still as secure as
Durin would like if it does contain the translation of Arkenpebble and the evil
wizard correctly guesses this. (2 marks)
2. Computability (14 marks)
The generalised 3-player Platypus game is defined as follows. Let M1, M2, and M3
be Turing machines, which share the same tape. The tape is initially blank. The
initial configuration of the three machines is as shown below.
As in the Platypus game, each machine takes turns to move (but there is no scoring
involved).
(a) Show that the halting problem for the 3-player generalised Platypus game is
undecidable. You may use any reduction you like. (6 marks)
(b) Suppose the 3-player generalised Platypus game is played on a Turing machine
with a finite tape (making the halting problem decidable), and that this prob-
lem has been shown to be NP-complete. Given your above reduction from
some problem A to the 3-player generalised Platypus game, can this reduction
be used to conclude that A is NP-complete? Why or why not? Explain your
answer. (2 marks)
(c) Consider the two Turing machines M1 and M2 below.
M1: Whatever the input is, M1 overwrites each character in the input, resulting
in a totally blank tape. M1 then terminates.
M2: Given an input string w, M2 outputs another Turing machine Mw which
will take a blank tape as input, print w on the tape and then terminate.
Explain how these two Turing machines are used in at least three reductions
of the Halting problem to other undecidable problems. (6 marks)
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3. Nondeterminism (8 marks)
Consider the incomplete NFA M0 below, whose alphabet is {0, 1}.
Use M0 to create three more NFAs M1, M2 and M3 according to the constraints
below. Explain in one or two English sentences how you constructed each NFA.
ˆ Each of M1, M2 and M3 must contain at least 10 transitions (potentially but
not necessarily including λ-transitions) and must be an NFA but not a DFA.
Specifically there must be at least one combination of state and input (either
0 or 1) for which there are at least two possible states. Put another way,
removing all λ-transitions must not result in a DFA.
ˆ Use JFLAP to transform M1, M2 and M3 into equivalent DFAs.
ˆ The sizes of the DFA resulting from the determinising algorithm must be as
below. Note that the JFLAP implementation of this often omits an “error”
state, i.e. it may be necessary to add an extra state to the result from JFLAP
in order to account for this. The size constraints below assume a fully deter-
ministic DFA; one way to check for this is that if the DFA has k states, there
must be exactly 2k transitions (one for each of 0 and 1 in each state).
(a) The size of the DFA corresponding to M1 is 2. (3 marks)
(b) The size of the DFA corresponding to M2 is 5 (2 marks)
(c) The size of the DFA corresponding to M3 is at least 9 (this may be harder
than you think!) (3 marks)
4. Pumping Lemma (20 marks)
(a) There are three errors in the statement of the Pumping Lemma for regular
languages below. Find all three, and state how to correct them. Explain each
of your corrections. (4 marks)
Let L be a regular language. Then ∃n ≥ 1 such that for some w ∈ L such that
|w| ≥ n, ∀x, y, z such that w = xyz where
i. |xy| < n+ 1
ii. y 6= λ
iii. xyiz ∈ L for all i ≥ 1
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(b) One fine day, as soon as lockdown was over, Elladan and Elrohir talk an
afternoon walk in the gardens of Lothlorien. While contemplating the meaning
of existence underneath the massive ancient trees, they come across ten pieces
of parchment, which appear to have been hastily ripped up during the recent
War of the Ring, on which are written some strange runes that they cannot
read. Desperate to find out what these mean, they send the fragments to
Imladris for analysis, where the wisest still in Middle Earth would be found.
After several anxious weeks, a passing eagle delivers a package to them. Inside
they find a letter from Elrond himself, explaining that despite their best efforts,
they were unable to fully translate the runes, and so there are some places
where there are several possible translations of particular phrases. Elladan
and Elrohir therefore have two tasks. The first one is to arrange the fragments
in the most likely order. Then they have to select from each of the possible
translations so that the overall message makes sense.
The fragments are below, with the parts where there are alternative transla-
tions labelled with capital letters, like this:
(Translated text) A (Translated text)
A1: Alternative 1
A2: Alternative 2
A3: Alternative 3
1: Let w be A
A1: 0n2n1210n2n. A2: 2n1212n. A3: 0n1210n. A4: 1n1211n.
2: and so xyiz is B
B1: 2n0n+j1212n0n B2: 2n+j1212n B3: 0n+j1210n B4: 1n+j1211n
3: Then ∃n ≥ 1 such that C and D,
C1: for all w ∈ L and |w| ≥ n C2: for some w ∈ L and |w| ≥ n C3: for all
w ∈ L and |w| ≤ n C4: for some w ∈ L and |w| ≤ n
D1: ∃x, y, z such that w = xyz, |xy| ≤ n D2: ∀x, y, z such that w = xyz, |xy| ≤
n D3: ∃x, y, z such that w = xyz, |xy| ≥ n D4: ∀x, y, z such that w =
xyz, |xy| ≥ n
4: Finrod’s Tale: A proof that the language L =E is F .
E1: {w121|w ∈ {0, 1, 2}∗} E2: {121w|w ∈ {0, 1, 2}∗} E3: {w000w|w ∈
{0, 1, 2}∗} E4: {w121w|w ∈ {0, 1, 2}∗}
F1: regular F2: context-free F3: not regular F4: empty
5: As the Pumping Lemma requires xyiz ∈ L, this is a G
G1: problem G2: paradox G3: contradiction G4: tautology
6: and so we have shown that our assumption is false, i.e. that L is H.
H1: regular H2: context-free H3: not regular H4: empty
7: Assume L is I.
I1: regular I2: context-free I3: not regular I4: empty
8: Now consider J
J1: i = 0 J2: i = 1 J3: i = 2 J4: i = 3
9: so we have w ∈ L, K and |xy| ≤ n, and so we have L for some 1 ≤ j ≤ n.
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K1: |w| ≤ n K2: |w| < n K3: |w| ≥ n K4: |w| > n
L1: y = 0j L2: y = 1j L3: y = (02)j L4: y = (12)j
10: y 6= λ and M
M1: xyiz 6∈ L for all i ≥ 0 M2: xyiz 6∈ L for some i ≥ 0 M3: xyiz ∈ L for all
i ≥ 0 M4: xyiz ∈ L for some i ≥ 0
You do not need to write the proof out in full. Just indicate the
fragment order and your choice for each of the letters in a sequence like the
one below.
9K2 8J3 4E4 ...
(6 marks)
(c) Let L be any regular language and let M be a DFA for it with n states. Explain
how you can use the Pumping Lemma to show that L is infinite iff there is a
string w ∈ L such that n ≤ |w| ≤ 2n− 1. (4 marks)
(d) Let L be any regular language over {a, b, c}. Show how the Pumping Lemma
can be used to demonstrate that in order to determine whether or not L is
empty, we need only test at most (3n − 1)/2 strings. (2 marks)
(e) The Pumping Lemma for context-free languages is below.
Let L be a context-free language. Then there is an n ≥ 1 such that for any
string w ∈ L with |w| ≥ n there exists strings x, y, z, u, v such that w = xyzuv
and
i. |yzu| ≤ n
ii. |y|+ |u| > 0
iii. xyizuiv ∈ L for all i ≥ 0
Use this to show that the language L = {ai2+i | i ≥ 0} is not context-free by
filling in the gaps below. (4 marks)
Proof: Assume . So the Pumping Lemma applies, and so for
any string w ∈ L with |w| ≥ n there exist strings x, y, z, u, v such that w =
xyzuv and
i. |yzu| ≤ n
ii. |y|+ |u| > 0
iii. xyizuiv ∈ L for all i ≥ 0
Let . So w ∈ L and |w| ≥ n, and w = xyzuv, |yzu| ≤ n, |y| + |u| > 0
and xyizuiv ∈ L for all i ≥ 0. Now as , this means |yzu| = |y|+ |z|+
|u| ≤ n and so |y|+ |u| ≤ n.
Let i = 0 and consider |xy0zu0v| = |xzv| = −|y| − |u| = n2 + n −
(|y| + |u|) ≥ n2 + n − n > = (n − 1)2 + (n − 1). So n2 + n =
> |xy0zu0v| > (n− 1)2 + (n− 1), and so xy0zu0v 6∈ L. This is a contradiction,
and so L is not context-free.
5. Intractable problems (10 marks)
Intractable problems are decidable problems, but for which the best known solution
is exponential (or worse). Describe two intractable problems and their practical
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application. You should write one introductory paragraph on intractable problems,
and two further paragraphs, one on each problem, and a reason that you selected
each one. Some suggestions will be given in class and on Canvas.
Please note that I am not at all interested in what you can find on Google or
Wikipedia or anything like that. What I really want to see is some evidence of
you thinking about intractable problems and what effect these have, such as the
relationship between the vertex cover problem and sensor networks.
Perhaps an interesting way to address this is to consider what differences it would
make if your intractable problem could be solved efficiently. For example, if we
could solve say the Travelling Salesperson problem in O(nn) time, what would be
possible that is impossible now?
Another possibility is to consider how intractability can be a useful thing, such
as keeping information secure in cryptosystems or in related applications such as
blockchain.
Whatever problems you choose, please avoid the temptation to ’cut, copy and edit’;
as soon as you do that, you have done something wrong. I would far prefer to read
your own words and your own perspective on these problems.
6. Universality (24 marks)
In a nutshell, you are expected to revise and extend your work on this topic in
Assignment 1.
In Assignment 1, you investigated one of the following three topics, or came up
with your own related topic or creative story.
ˆ Two-dimensional Turing machines
ˆ Small universal Turing machines
ˆ Notable universal Turing machines
For this assignment, you are expected to either continue your investiga-
tion from Assignment 1 on the same topic in more depth, or to make a
different choice. In other words, you can either continue with your choice from
Assignment 1, or make a different one now. Whatever your decision, you are ex-
pected to write about 1800-2000 words (9 or 10 paragraphs) overall. This
should include a revised version of your Assignment 1 submission, so that if you
continue with the same choice as in Assignment 1, this is will be an extended form
of that work. If you make a different choice, that is fine, but you should include your
(potentially revised) Assignment 1 submission as part of this submission. So you
have two different choices for the two assignments, you are expected to write about
the same length on each; if you have the same choice for each, you should write
about 1800-2000 words overall. Either way, the submission for Assignment 2
will involve around 1000 words over and above what you submitted for
Assignment 1.
As in Assignment 1, you may also propose an alternative topic, or write a creative
story involving a Turing machine of some kind. You can do this even if you did not
choose either of these in Assignment 1. However, for any alternative topic or
creative story, please seek approval from the lecturer before commencing
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work on it. This is to make sure that what you are doing is appropriate for this
assignment — I would hate to see an outcome where you do a lot of work, only for
it not to count because it does not address the intended content.
One alternative topic of particular interest is quantum computing; if anyone is
interested in pursuing this, you are strongly encouraged to do so (but as above,
please do consult me first). Another possibility is zero-knowledge proofs, but
again please consult me before doing this.
A further point is that you can present your information in other ways
that a formal report if you wish . Some suggestions are below. Please keep in
mind that you still need to discuss the technical content; the point is to find a way
that assists you with this, rather than being a blockage for you.
ˆ Pick a side in the debate about the 2007 universal TM competition
ˆ Langton’s Ant vs Paterson’s worm
ˆ ‘Cellular automata are better than Turing machines’
ˆ Write a children’s story, movie scene, poem, . . .
ˆ 2D TMs as a game, map, drawing a picture, annotating photos, . . .
ˆ Implementation of some aspects (be careful of rabbit holes!)
ˆ Experiment with Java implementation of 2D Universal TM
ˆ Langton’s ant with ‘boundaries’ (see an example here)
You will be marked according to the rubric below.
Points Description Details
18-24 Exemplary You have explored your chosen topic well.
Your report is clear and well-written,
and has the appropriate length.
This is interesting and informative.
12-17 Accomplished You have explored your chosen topic reasonably well.
Your report is generally good,
but can be improved with some more attention to detail,
particularly concerning the amount of technical detail.
6-11 Developing Your report needs some further work,
either to increase the level of content or
to improve the choice of material and its presentation.
0-5 Beginning Your report does not show evidence
of sufficient investigation, and is lacking in detail.
7. The Platypus game. (16 marks)
We have previously talked about running as large a tournament as possible with the
Platypus game. The way we will do this in this assignment is for each of your to run
a tournament of 2,500 machines. From these, you will report your top 10 machines
(see below for details). These 10 will then be part of a knock-out tournament to
determine the overall winner.
Before answering the questions below, do the following.
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ˆ Get your allocation of 2,500 machines. These will in a OneDrive directory
that will be shared with you. Once you have the link, look for a file in the
folder with your student number, i.e. if your student number is 7654321, look
for the file 7654321.pl. Store this somewhere that suits you with a name like
machines.pl for convenience.
ˆ Open a new SWI-Prolog shell and consult both platypus.pl and machines.pl
(or whatever you called it). Then run the command below. This will classify
your list of machines into three categories, and will generate one file each cate-
gory named none.pl, reachable.pl and unreachable.pl, as per the descrip-
tion below. This will be in the directory specified by the results directory
predicate in platypus.pl; you can change this to something more convenient
for you if you like.
?- classify machines.
None The machine has no platypus state in the fourth row of columns 1-6
Reachable It is possible reach the platypus state from the kangaroo state
Unreachable It is not possible reach the platypus state from the kangaroo state
The point of this analysis is to work out whether it is possible for the machine
to terminate the game or not. An example of each class of machine is below.
In the first case, the machine cannot terminate the game, as it is not possible
for it to go from the start state (kangaroo) to the platypus state as there is
no transition that will move the machine into the platypus state.
Y G Y G Y G Y
K K E E W W P
g y y g y y g
Emu Emu Wombat Kangaroo Wombat Emu Kangaroo
w gg gg w w w gg
The second and third cases occur when there is indeed such a platypus state.
When it is possible to move from the kangaroo state to the platypus state
(depending on the cells on the tape of course), then the machine is classified
as reachable, as in the machine below. This is because it is possible to move
from the kangaroo state to the emu state (column 1 or 2), from the emu state
to the wombat state (column 3) and from the wombat state to the platypus
state (column 6).
Y G Y G Y G Y
K K E E W W P
g y y g y y g
Emu Emu Wombat Kangaroo Wombat Platypus Kangaroo
w gg gg w w w gg
It is also possible that even if such a platypus state exists, it is not possible
to move into that state from the kangaroo state. In this case, the machine is
classified as unreachable, as in the machine below. In this machine we can get
from the kangaroo state to the wombat state and vice-versa, but we can never
get to the emu state or platypus state from either of these.
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Y G Y G Y G Y
K K E E W W P
g y y g y y g
Wombat Wombat Platypus Kangaroo Wombat Kangaroo Kangaroo
w gg gg w w w gg
Intuitively, machines classified as reachable are in some sense “genuine” with
the others being “imposters”. Accordingly, having separate tournaments for
each seems only fair.
ˆ Run a tournament for each of the following classes of machines
– All 2,500 machines
– Only machines classified as reachable
– Only machines classified none or unreachable (ie all those not classified as
reachable).
To do this, you will first have to prepare a file containing only the machines
as above (the above classify machines command will more or less do this
for you). You will also need to use the command below, which will run the
tournament with all of the options below.
?- my tournament([competition]).
Variation Description
Tree 5 points for whenever either tree is reached
Green 2 points rather than 1 for changing green to yellow
Animal 1 point every time a change of animal occurs
Tiebreaker A random starting configuration is chosen with game length is 200
(a) For each of your tournaments, report the overall time taken, the top 10 ma-
chines (by ’football’ ranking), the overall number of wins and draws, and the
number of winless machines. How many machines were classified as none,
reachable and unreachable respectively? Report your results in a table as be-
low. (2
marks)
Class Number Percentage
Reachable
Unreachable
None
(b) Re-run your tournament of all machines, but this time you are to include 10
extra machines of your own choice. These should be different from any ma-
chines in your list already, but otherwise you are free to choose them however
you like.
You can use platypus.pl from Canvas (link here) to assist with this if you
wish. This will check whether your 10 added machines are ’legal’, and whether
or not they were already part of your allocation. To do this, consult platypus.pl
and machines.pl as above, and a third file extra.pl. Then run the command
?- check new.
This will then output whether your added machines are legal, and whether
they are already part of your allocation or not (and if they are, which ones are
already in their allocation).
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You should report where each of these 10 machines finish in the ’football’
ranking, and your reasons for choosing each of them. (2 marks)
(c) Were you surprised how your chosen 10 machines performed? What can you
conclude from this about high performing Platypus machines? If you had to
choose one machine to represent you in a tournament, what machine would it
be? Briefly explain your decision. (4 marks)
(d) In the previous assignment, your calculated the largest Platypus tournament
you can play on your machine in 4 hours, ie 4 × 60 × 60 = 14, 400 seconds.
This is of course for the ’standard’ 2-player game. 3-player and 4-player tour-
naments will of course take longer. Calculate the largest 3- and 4-player tour-
naments you can play on your machine in 4 hours. You may assume that a 3-
or 4-player match takes the same time as a 2-player one. You may also find
the following table useful (see the notes on 3- and 4-player tournaments for
how these are derived). (2 marks)
Players Matches required
2 n(n+ 1)/2
3 n(n+ 1)(n+ 2)/6
4 n(n+ 1)(n+ 2)(n+ 3)/24
(e) If the Platypus tournament were to be run again, what alterations would you
recommend? Some ideas are below; you can add others as you see fit. You
should have at least three suggestions. (6 marks)
ˆ Once students are allocated their machines, they play a tournament amongst
these to find the best 10. These 10 then take on the best 10 from other
students. This could include the possibility of students choosing their 10
machines from their allocation or from the set of machines which are not
allocated to any student.
ˆ The initial tape and scoring processes are changed from tournament to
tournament. These are announced in advance, and allow choices to be
made for which machines will be used.
ˆ Non-player machines can be added. These are not competitors, but may
change the tape in ways that influence the game. Such machines would
not be allowed to terminate the game (presumably by allowing them tran-
sitions for a green cell with a platypus).
ˆ When a player has a platypus on a green cell, the game does not halt,
but is “rebooted”, i.e. the tape reverts to its initial state, the player which
’halted’ the game gets a bonus or a penalty, and the machines involved
are changed in some way. This change could be swapping rows 3, 4 and
5 of column 1 and 2 (kangaroo), and the same for columns 3 and 4 (emu)
and 5 and 6 (wombat). There could be a maximum of say 5 reboots per
game.
ˆ (insert your idea here!)
3 Submission
You should submit a PDF file, and all tournament.csv files from your tournaments. Do
not use a zip file. Just add attach all relevant files to the submission on Canvas. No
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other file formats will be accepted.
In addition, for Questions 4b and 4e, you must fill in your answers on the Canvas
Quiz here. As noted in announcements, this will assist with marking assignments more
quickly.
4 Marking guidelines
Your assessment will be marked according to the criteria below.
ˆ Completeness and accuracy of your answers to the first five questions.
ˆ Completion of your section of the Platypus game tournament.
ˆ Quality of your comments on the Platypus game tournament.
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