Computing Theory
COSC 1107/1105
Assignment 1: Fundamentals
Assessment Type Individual assignment. Submit online via Canvas → As-
signments → Assignment 1. Marks awarded for meeting re-
quirements as closely as possible. Clarifications/updates may
be made via announcements/relevant discussion forums.
Due Date Week 6, Sunday 28th August 2022, 11:59pm
Marks 130 worth 15% of your final assessment
1 Overview
This assignment is intended both for introducing you to some basic concepts that we
will use in various ways later in the course, and to provide some early feedback on your
progress. The are six key concepts that we will return to again and again, which are formal
languages, regular expressions, grammars, finite state automata, pushdown automata and
Turing machines. A common thread in all of these is nondeterminism. This will come
up in various contexts, as you will see. Much of this assignment is concerned with these
concepts, to ensure that you are well-versed in these fundamentals. There is another part
which deals with the Platypus game.
2 Assessment details
A Note on Notation of Regular Expressions: Unfortunately there isn’t a uniformly
accepted standard notation for regular expressions. Given we are using JFLAP, our
notation should be as consistent as practicable with that, but that also means some
things get quite cumbersome. The two main issues are the specification of alternatives,
and how to abbreviate some obvious patterns like letters and numbers.
So in this assignment the following syntactic rules will be used.
ˆ ’+’ will be used to denote both alternatives (as in (1 + 2)∗) and also to denote one
or more applications of Kleene star (as in a+, meaning the language {an|n ≥ 1}).
You must take its application within the context in which it is applied (so you will
need to use your brains!).
ˆ Ranges such as all letters or all digits will be represented as [a−z] meaning (a+b+
c+d+e+f+g+h+ i+j+k+ l+m+n+o+p+q+r+s+ t+u+v+w+x+y+z).
Hopefully the reason for using ranges is now obvious!
1. Regular expressions and languages (17 marks)
The game of Buckitch has a history that goes back centuries, including being played
at the legendary school Warthogs. Matches at Warthogs began shortly after the
first recorded match in 1317, and have been held regularly since between the four
Warthogs houses of Liongate, Crowfoot, Serpentine and Dragonbreath. Records
of all Buckitch matches at Warthogs since 1317 have been kept in a handwritten
archive. To save precious parchment and ink, the records of each match were kept
as a string, using one character for each house (l, c, s, and d respectively) in the
match, and including the date, winner and scores. Such strings were written as
follows.
D1D2M1M2Y1Y2Y3Y4H1H2WS1S2S3S4S5S6
where
ˆ D1D2 are two digits of the day of the date
ˆ M1M2 are two digits of the month of the date
ˆ Y1Y2Y3Y4 are four digits of the year of the date
ˆ H1 and H2 are the characters representing the two houses involved
ˆ W is the character for the winning house
ˆ S1S2S3 is the three-digit winning house’s score
ˆ S4S5S6 is the three-digit losing house’s score
Note that D1 can only be 0, 1, 2 or 3, M1 can only be 0 or 1 and Y1 can only be 1
or 2. Note also that the score is always a multiple of 10, and can be assumed to be
no more than 990.
Buckitch matches are never drawn or incomplete. If no result is possible on the
given day, the winner is determined by random choice. This means it is possible
for the winning and losing score to be the same.
Scores were written one after the other on the parchment as one enormous string. In
order to analyse the history of Buckitch, it is necessary to write regular expressions
to identify specific matches of interest somewhere in this string.
(a) Give a regular expression for the following cases.
i. Any match in the year 1737 won by Dragonbreath. (1 mark)
ii. Any match on the 1st of April in which either score was 540. (1 mark)
iii. Any match involving Liongate after 1999. (2 marks)
iv. Any sequence of wins for Crowfoot. (3 marks)
v. Any sequence of losses in August for Serpentine before 2000. (4 marks)
(b) Is it possible to give a regular expression for the following cases? Explain
either why it is possible or why it is not.
i. Any match in which the scores are equal. (2 marks)
ii. The longest losing sequence for each house. (2 marks)
iii. Any match played on a date which is a palindrome. (2 marks)
2
2. Grammars (18 marks)
(a) Consider the G1 grammar below.
S → aaSb | Ab
A→ cAc | B
B → ddB | λ
i. Give a derivation of a string in L(G1) which contains at least 4 occurrences
of b. (2 marks)
ii. Give a derivation of length at least 8 of a string in L(G1). (2 marks)
iii. Is λ ∈ L(G1)? Explain your answer. (1 marks)
iv. Express L(G1) in set notation. (3 marks)
v. Let G2 be the grammar obtained from G1 by adding the rule S → AbS
to G1. This makes G2 as below. What is L(G2)? How does it differ from
L(G1)? (3 marks)
S → aaSb | Ab | AbS
A→ cAc | B
B → ddB | λ
(b) Warthogs School provides its students with identifiers of the form N+I+H
where
ˆ N is a 5-digit number over {1, 3, 5, 7, 9}
ˆ I is a string over {a, b, e, f} of length at least four in which the number of
a’s and f ’s is the same
ˆ H is a string of length at least two consisting solely of one of the characters
{l, c, s, d}
For example, 13579+baaefef+llll and 53399+fafabbbe+ccc are valid identi-
fiers, whereas 13579+baefef+llll and 53399+fafabbbe+csl are not.
i. Give a grammar whose language contains all valid identifiers, and only
such identifiers. (2 marks)
ii. Give the derivation in your grammar for the two valid identifiers above.
(2 marks)
iii. Warthogs School now decrees that N can be at least 5 digits long or more,
but must contain at least one 7. Give a grammar for this new version of
identifiers. (3 marks)
3
3. Finite State Automata (18 marks)
Consider the finite state automatonM1 below. You can download this from Canvas
here.
(a) Give three examples of strings in L(M1) of length at least 6. There must be
at least one example for which execution finishes in the state q6, and similarly
for q9 and q12. (3 marks)
(b) Explain whyM1 is nondeterministic. Do not include any ’missing’ cases in your
explanation (i.e. cases in which there is no transition for a given combination
of state and input). (3 marks)
(c) What is L(M1)? Explain your answer. (4 marks)
(d) Is it possible to create an equivalent machine M2 with fewer transitions or
states than M1? Explain your answer. (4 marks)
(e) Consider the machine M3 which is derived from M1 as follows. Is L(M3) the
same as L(M1)? Explain your answer. (4 marks)
ˆ Delete state q2 (and transitions δ(q0, λ) = q2 and δ(q2, 0) = q10).
ˆ Delete the transition δ(q0, λ) = q3.
ˆ Delete the transition δ(q0, λ) = q1.
ˆ Add the transition δ(q0, 0) = q3.
ˆ Delete the transition δ(q0, 0) = q1.
4
4. Pushdown Automata (12 marks)
Consider the PDA M1 below. You can download this from Canvas here.
(a) Show the execution of the strings dabba, bbbbdcb , cacadcbc and abcdddcbadd
in M1. (4 marks)
(b) What is L(M1)? (use set notation). Explain your answer. (4 marks)
(c) Construct a PDA M2 such that L(M2) = {aibicjdj | i, j ≥ 0}. (2 marks)
(d) Explain why there is no PDA for the language {aibjcidj | i, j ≥ 0} (2 marks)
5
5. Turing machines (20 marks)
Consider the Turing machine M1 below. You can download this from Canvas here.
(a) What is L(M1)? Explain your answer. Show some examples of strings accepted
and rejected to justify your answer. You must show at least one accepted string
of length at least 6, and at least one rejected string of length at least 6. (4
marks)
(b) Consider the Turing machine M2 which is obtained from M1 by deleting the
transition below fromM1. What is L(M2)? How does this compare to L(M1)?
Explain your answer. (3 marks)
State Input Output Direction New State
q6 c c L q6
(c) Recall the game of Buckitch from Question 1 and its method of recording
scores as strings. Give a Turing machine M3 which accepts a string recording
a single match if any one of the following conditions holds. (6 marks)
ˆ Changing one digit in one of the scores will make them the same. For
example 130 and 180 satisfy this; 270 and 480 do not, and neither do 310
and 140.
ˆ The scores were a substring of the date.
ˆ The match was won by Dragonbreath either on the 17th day of a month
or a year ending in 17 but not both.
(d) Is M4 deterministic? Explain your answer. (2 marks)
6
(e) Explain how you could use M4 as the basis for a Turing machine M5 which
would output which of the above three conditions was satisfied by a given
string. You do not need to explicitly construct M5, but describe how it would
be done. (2 marks)
(f) 17th July in years which end in 17 (such as 1917 or 2017) are very special
dates in Buckitch history known as Seventeens. On such days a marathon
tournament has been played at Warthogs since 1317, with all four houses
playing as many matches as they can in that 24-hour period from midnight
to midnight. The results of each match are recorded as above. On July 18th
2017, Serpentine made the outrageous claim to have won more matches in
Seventeens than the other three houses combined, but produced no evidence
whatsoever to back up this claim. Accordingly, the Head of Warthogs, Pumbaa
the Magnificent, has asked you to design a Turing machine which, given as
input a string containing the entire records of Buckitch at Warthogs, will
output the number of matches played and won by each house on Seventeens.
Explain how you would design such a Turing machineM5. There is no need to
explicitly construct this machine. You may use any variety of Turing machine
that you wish, with particular regard to making the machine as simple as
possible to understand (as it will certainly be keenly scrutinised by the Head
of Serpentine, Scarface Shape). (3 marks)
6. Universality (20 marks)
Universal Turing machines are often discussed, but are rarely explicitly defined.
This question requires you to report on one of the following topics. You should
choose the one that you find most interesting. Some pointers and resources for each
topic will be made available on Canvas.
This will form part 1 of an investigation; you will be able to build on
and extend on this topic (or explore a different one if you wish) in
Assignment 2.
ˆ Two-dimensional Turing machines
ˆ Small universal Turing machines
ˆ Notable universal Turine machines
Some more detail on each of these is below. Whichever topic you choose, you are
expected to write around 800-1000 words (approximately 4 or 5 paragraphs). You
are also encouraged to consider the use of the Adobe Creative Cloud
suite, to which all RMIT students have access. You can find the details about the
Adobe Creative Cloud at this link.
You may also propose an alternative topic of your choice if you wish. In particular,
anyone wishing to write a creative story involving a Turing machine of some
kind is strongly encouraged to do so. However, for any alternative topic or
creative story, please seek approval from the lecturer before commencing
work on it.
Two-dimensional Turing machines
7
Turing machines were conceived in a less visual era. Whilst in principle the re-
striction to a one-dimensional tape does not reduce the scope of programs that can
be written, there is a modern reason why a two-dimensional version is much more
interesting: the predominance of the computer monitor as an output device. In
particular, in the modern computing environment, there is every reason to consider
abstract computations which use images as basic symbols, rather than numerals or
similar strings (which, for all their mathematical properties, are visually rather pro-
saic). There are several varieties of two-dimensional such machines that have some
rather curious properties, such as Langton’s ant, Turmites and Paterson’s worms.
Small universal Turing machines Universal Turing machines are often rather
large. In 2007, a competition was held to determine whether or not a given 2-state
3-symbol machine was universal or not. The question was settled in the affirmative,
with the winner of a US$25,000 prize being a 20-year-old undergraduate. The quest
for small universal machines continues, as there is some issue about the precise
definition of universality used in the competition. You can write a report about the
competition itself, or on aspects of the quest for small universal Turing machines.
Perhaps pick a side in the debate (i.e. was the winning machine actually universal?
Should the definition of universality be changed as a result?) and argue for it. Or
argue for both sides and come to your own conclusion.
Notable universal Turing machines It is remarkably difficult to find ’good’
explicit examples of Turing machines. Some well-known examples include Turing’s
original machine, and machines built in particular ways by Minsky, Colmerauer
and Wolfram. For example, Colmerauer built his machine as a means of teaching
assembly language programming (!!), and intended it to be programmable. Minsky,
on the other hand, derived his machine from principles of tag systems, and while
it is certainly universal, it is harder to imagine programming it. There is also a
relatively recent universal machine in two dimensions constructed by Dershowitz
and Dowek, for which a Java implementation is available via Github.
7. The Platypus game For this task you will need to be familiar with the
Platypus game. (25 marks)
You have been allocated a number of machines, based on your student number.
(a) Play a tournament amongst your entire list of machines. There is SWI-Prolog
code available for you to do this in Canvas. The main thing you need to do is
to use your particular list of machines for the tournament. You should report
your results as follows.
i. Report the top 10 machines performance, ranked in ’football’ order, ie by
number of wins, and if the wins are equal, by the ratio of points score for
to points scored against. You should include both the tournament.csv
file generated by the software in your submission, as well as include a table
in your PDF file with the top 10 according to this ranking. (2 marks)
ii. Examine your top 10 machines. Are they any key similarities or differences
between them? (2 marks)
iii. How does your top 10 change if the ranking is based on overall points for,
rather than as above? (2 marks)
iv. Were there any machines without a win? (1 mark)
8
v. Suggest at least one way in which the game rules can be changed which
would alter the outcome of the tournament. Keep in mind that each
tournament typically involves thousands of games, so any such change
must not require input from the user during the game. (2 marks)
(b) Time how long it takes your tournament to run. Record that time along with
the basics of the machine on which it was run. For example, “My tournament
involved 42,132 matches which took 156.5 seconds on a Windows 10 desktop
with an Intel i7 processor and 16GB of RAM.” (2 marks)
(c) A tournament of this form involving n teams requires n(n + 1)/2 matches.1
Use your time above to calculate the average time it takes for a match on your
machine, and use this value to calculate long it would take to run a tournament
for 100, 1,000, 10,000, 100,000, and 1,000,000 matches. Present your results
in the form of a calculation and a table of the form below. (3 marks)
n Matches Estimated time
100
1,000
10,000
100,000
1,000,000
(d) Calculate the largest Platypus tournament you can play on your machine in
4 hours, ie 4 × 60 × 60 = 14, 400 seconds. Use the value calculated above for
how long it takes you to play a match. (2 marks)
(e) The Platypus game is entirely deterministic, and hence a little boring as en-
tertainment. Suggest at least two ways in which the game can be extended to
increase player involvement in the game. (4 marks)
(f) As discussed in class, some variations of the Platypus game seem appropriate.
Your tasks is to rerun your tournament with your allocated machines, with
different parameters, and to compare the results. You should include at least
the variations below (and more if you wish). The code to do all this will be
provided.
Variation Description
Standard No changes; this is the original version
Tree 5 points for whenever either tree is reached
Green 2 points rather than 1 for changing green to yellow
Short Maximum game length of 50 rather than 100
Long Maximum game length of 200 rather than 100
Tiebreaker A random starting configuration is chosen with game length is 200
For each of the variations above, you should report the following results.
ˆ Time taken
ˆ Top 10 machines
ˆ Number of wins
ˆ Number of draws
ˆ Number of winless machines
1This will be explained in class.
9
Report your results in a table like this.
Standard Tree Green Short Long Tiebreaker
Time
Top 10
Wins
Draws
Winless machines
You should identify your top ten machines by the overall number, ie in the
range 1 to 268,435,456. (5 marks)
3 Submission
You should submit two files. One should be the tournament.csv generated when you ran
your tournament. The other should be a PDF containing your answers to the questions.
No other file formats will be accepted.
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 Quality of your report on your chosen topic for Question 6.
ˆ Completion of your section of the Platypus game tournament.
ˆ Quality of your comments on the Platypus game tournament.
5 Rubric
Your assignment will be graded in accordance with the rubric in Canvas.