computing theory assignment

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Computing Theory COSC

Computing Theory COSC 1107/1105

Assignment 1: Fundamentals

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.

   ’+’ 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 a n n   1 ). You must take its application within the context in which it is applied (so you will need to use your brains!).

   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!

• Regular expressions and languages (15 marks)

The game of Buckscratch has a history that goes back centuries, including being played at the legendary school Pigspimple. Matches at Pigspimple began shortly after the first recorded match in 1177, and have been held regularly since between the four Pigspimple houses of Echidna, Goanna, Possum and Wallaby. Records of all Buckscratch matches at Warthogs since 1177 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 ( e, g, p, , and w respectively) in the match, and including the date, winner and scores. Such strings were written as follows.

D 1 D 2 M 1 M 2 Y 1 Y 2 Y 3 Y 4 H 1 H 2 SESc 1    −   Sc 2

    D 1 D 2 are two digits of the day of the date

    M 1 M 2 are two digits of the month of the date

    Y 1 Y 2 Y 3 Y 4 are four digits of the year of the date

    H 1 and H 2 are the characters representing the two houses involved

    S is the sequence of scores in the match

    E is either the character q or the character t (indicating what caused the match to end) 1

    S 1 and S 2 are the total score for each team in the match (separated by the character −)

Note that D 1 can only be 0 , 1 , 2 or 3, M 1 can only be 0 or 1 and Y 1 can only be 1 or 2.

The sequence of scores is in the format of the character representing one house, followed by a sequence of consecutive scores for that house. Each score can either be 1, 2 or 3 (referred to as a “single”, “double” and “triple” respectively. For example, a sequence of scores for a match between Echidna ( e ) and Wallaby ( w ) could be e 131111 w 22 e 13123 w 2111111.

When recording H 1 and H 2 , scribes obeyed the rule that these would be strictly in alphabetical order. Hence H 1 could only be one of { e, g, p } and H 2 could only be one of { g, p, w }.

Scores were written one after the other on the parchment as one enormous string. In order to analyse the history of Buckscratch, it is necessary to write regular expressions to identify specific matches of interest somewhere in this string. Scribes were attentive and diligent, but inevitably it has been observed that sometime there were occasionally errors made by scribes. This means that any regular expressions for such purposes must be precise, and cannot assume that any given entry is error- free.

1 The rules of Buckscratch are unclear and subject to debate, but it seems that there were at least two ways in which matches came to an end.

• Any match taking place on July 7th involving Echnida or Goanna. (1 mark)
• Any match on or after January 1st 1900 in which the end event was q . (1 marks)
• Any ”syntactically correct” record of a match. This means the recorded entry must have M 1 being either 0 or 1, the houses must be one of the four above, and scores must consist of digits. Anything not ”syntactically correct” must be an obvious error, such as the date 496618888 or the score sequence e 1 e 1 e 1 d 2 d 2.                                                                                         (3 marks)
• Any match on a day in August involving Goanna in which the opposition scored at least three triples.                                                                                                (2 marks)
• Any match in which either of the scores is a palindrome. (2 marks)
• The longest sequence of consecutive matches for each house.    (2 marks)
• A sequence of matches in which no triples were scored by either side. (2 marks)
• Determining which house was the first to reach 100 victories.   (2 marks)
• Grammars (15 marks)

Consider the grammar below.

S → SHN | λ N → D | DN

D → 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9

H → e | g | p | w

• Is λ ∈ L ( G )?  Explain your answer (1 mark)
• Express L ( G ) in set notation. Explain why your answer is correct. (4 marks)
• Sneaky Wobbler, the legendary housemaster of Goanna house at Pigspim- ple, claims that this grammar correctly specifies the sequence of scores in a Buckscratch match (i.e. the sequence labelled S above). Explain why Sneaky’s claim is not correct.                                                                                                      (3 marks)
• Give a (correct) grammar for specifying the two houses and the possible se- quences of scores (i.e. the string labelled H 1 H 2 S above). Explain how you derived your grammar and why it is correct.                                                                                                      (4 marks).
• Finite  State  Automata (18 marks)

Consider the finite state automaton M 1 below. You can download this from Canvas here.

• Explain why M 1 is non-deterministic. Give at least four examples of states for which there is more than one successor state for some input in your explana- tion. (3
• Is it possible to create an equivalent machine M 2 with fewer transitions or states than M 1 ?  Explain your answer.  Note that by equivalent we mean L ( M 1 )   =   L ( M 2 ).                                                                                                         (4 marks)
• Consider the machine M 3 which is derived from M 1 as follows. Is M 3 equivalent to M 1 ? Explain your answer.                                                                                                      (4 marks)

   Delete all transitions from state q 5 .

   Add the transition δ ( q 5 , λ ) = q 1 .

   Delete all transitions from state q 10 .

   Delete the transition  δ ( q 15 , λ ) = q 11 .

   Delete the transition  δ ( q 19 , λ ) = q 15 .

   Add the transition δ ( q 19 , 3) = q 12 .

• Pushdown Automata (15 marks)

Consider the pushdown automaton M 4 below. You can download this from Canvas here.

Give a PDA M 5 such that L ( M 5 ) =    a i b j c k b l a m      i + j < l + m, i, j, k, l, m                                                                     0  . Explain why your machine is correct.                                                                                                    (4 marks)

• Turing machines (12 marks)

Consider the Turing machine M 6 below. You can download this from Canvas here.

• Consider the Turing machine M 7 obtained from M 6 by deleting the transitions δ ( q 1 , c ) = ( c, R, q 1 ) and δ ( q 1 , Y ) = ( Y, R, q 1 ). Is M 7 equivalent to M 6 ? Explain your answer.                                                                                                      (4 marks)
• Consider now the Turing machine M 8 obtained from M 6 adding the transition

δ ( q 3 , a ) = ( a, R, q 3 ). Is M 8 equivalent to M 6 ? Explain your answer. (4 marks)

• Get Creative! (20 marks)

This will form part 1 of an investigation or creative artefact; you will be able to build on and extend on this topic (or explore a different one if you wish) in Assignment 2.

You are strongly encouraged to use 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.

There are many aspects of Computing Theory that allow you to show your in- vestigative or creative skills. This could be in the form of an investigation into a particular topic (Universal Turing machines, Langton’s Ant, Paterson’s Worms, two-dimensional Turing machines, …) or by coming up with a creative artefact (story, movie,  VR experience, images, …) which takes as its subject a topic rele- vant to Computing Theory. Investigations will typically generate empirical data, often by using existing software (with some possible modification of your own if need be). One such investigation would be to use Colmerauer’s universal Turing machine (or any other explicitly defined universal Turing machine). Another would be to generate your own experimental data on well-known concepts such as Lang- ton’s Ant or Patersons’s Worms. If creative practice is more to your liking, then coming up with a creative story involving a Turing machine of some kind, or a topic of similar relevance to Computing Theory (such as the Enigma machine, or Chomsky’s work on grammars). You could also present this in animated or video form, or as text.

You may also propose an alternative topic of your choice if you wish. The idea is to allow you some ’space’ to develop your own understanding of a particular topic of interest. But please do seek the approval of the lecturer for any alternative topic.

Some more details on particular topics are below.

Two-dimensional Turing machines

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.

• The Platypus game For this task you will need to be familiar with the Platypus  game.                                                                                                              (30 marks)

You have been allocated a number of machines, based on your student number.

• 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)
• Examine your top 10 machines. Are they any key similarities or differences between them? In your analysis, what makes these the top performers? (2 marks)
• How does your top 10 change if the ranking is based on overall points for, rather than as above? (2 marks)
• Examine your bottom 10 machines. Were there any machines without a win? What is the difference between these machines and the top perform- ers? (2
• 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)
• A tournament of this form involving n teams requires n ( n + 1) / 2 matches. 2 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)

Calculate the largest Platypus tournament you can play on your machine in 3 hours, ie 3    60     60 = 10 , 800 seconds. Use the value calculated above for how long it takes you to play a match.                                                                                                        (2 marks)

• 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.

   Time taken

   Top 10 machines

   Number of wins

   Number of draws

   Number of winless machines Report your results in a table like this.

You should identify your top ten machines by the overall number, ie in the range 1 to 268,435,456.                                                                                              (5 marks)

• One suggestion from previous semesters is to have a “Battle Royale”, in which each student in the class nominates a particular machine, and a single game is played in which all machines compete at once. The object is to see which machine can avoid termination for the longest time (making this a bit like “The Hungar Games” if you are familiar with it). In other words, the objec- tive of each machine is not to score the most points, but to avoid termination. To make this fair, we would need to require each machine to contain a platy- pus somewhere in the fourth row, and that this state be reachable from the kangaroo state (so that it is possible for the machine to terminate).

Which of your machines, if any, would you choose as your representative in this event? Would you prefer to use some other machine? Explain your answer. (4 marks)

3         Submission

You should submit the following.

   A PDF file containing your answers to the questions.

   All .csv files generated by running your tournaments.

No other file formats will be accepted.

4         Rubric

Your assignment will be graded in accordance with the rubric in Canvas (which will be available shortly).

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One of the most popular courses for students interested in learning computer programming and networking is Computer Science Engineering (CSE). Computer architecture and organization, data base management systems, distributed computing systems, data structures and algorithms design, computer networks, operating systems, cloud computing, and software testing are all topics covered in computer science engineering. Students who want to pursue a profession in computer science frequently enrol in these classes. Obviously, people find it difficult to finish and achieve decent scores when faced with a massive computer science homework. Students spend sleepless hours finishing projects and homework, yet they still fall short of their goal of being in the top 5% of their class. Online Assignment Expert has a team of computing theory assignment help professionals that can assist you in comprehending complicated topics and delivering high-quality assignments and homework at a reasonable cost.

Students are expected to demonstrate their understanding of topics and check their conclusions' integrity by examining the data. This could be more difficult and demand a lot of effort, resulting in fines if the tasks aren't completed on time. We have experts who could provide with the Computer science assignment help that is most reliable and good in quality, it is tailored to provide insights into current learning methodologies and the latest topics' latest inputs. If you want assistance with computer science coursework, please contact our professionals. We are the industry leaders in providing assistance with computer assignments.

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Computer engineering combines electronics engineering and computer science ideas. Computer hardware engineering & computer software engineering are two main categories. Circuit design, Programming, microprocessors, microcontrollers, and computer networks are all examples of software and hardware components of computing. Different approaches will be used by the computer to integrate the system's components into other machines.

Computer automation, architecture, programming languages (C, C++, Java, .Net, etc.), data structures, computer graphics, multimedia, operating systems, software testing & quality are all topics covered in computer science engineering.

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Computer science is a vast subject with several sub-fields. To produce reliable assignments, we have specialists that have obtained their PhDs from reputable colleges throughout the world. Our computer science assignment help professionals are experts in the following areas of computer science engineering and can assist you with any of them. Computer Science Assignment: computer science assignment assistance is provided by us with some of the most prominent computer science engineering subjects that are now trending and these are:

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Computing Theory:

Theoretical computing is a branch of computer science that deals with tackling issues in an efficient manner by employing some model of computation and solving them with an algorithm. It is a theory that encompasses the whole field of computer science. It demonstrates how various algorithms may be used to address computer science challenges. So, it's essentially a research project that deduces mathematical logic implantations. This course provides a mathematical explanation of computer algorithms as well as information on how they function. Furthermore, the course is accountable for explaining the analysis of any algorithm utilising mathematical logics. It is a large academic area that has entailed a lot of study in the past.

A mathematical abstraction is used by Computer scientists of computers called a computation model to perform a thorough computing theory. There are several models in use, but the Turing machine is the most often used. The Turing machine is studied by computer scientists as it is easy to define, used to prove findings and can be examined, and shows what are the most powerful "reasonable" model of computing. It is known to everyone that an unlimited memory capacity is hold by the machine of Turing and each decidable issue answered via a Turing computer will use just a limited amount of memory always.

Every issue that a Turing machine can solve (decide) may theoretically be resolved (decided) via a computer with a limited memory capacity. Because of its enormous size, it is separated into three different branches: computational complexity theory, automata theory, and computation theory.

• Automata:  This theory focuses on the abstract machines (to be very precise, abstract mathematical machines) along with the problems regarding computation which can be resolved by the machines. The term "automata" refers to these abstract machineries. Automata is derived from the Greek term automaton, which means " something capable of acting independently." Because automata are frequently classified, automata theory is strongly tied to formal language theory according to the types of formal languages they can understand. A finite representation of an infinite collection of formal languages can be an automaton.
• Theory of computation:  This largely addresses the topic of how computer-solvable a problem is. Because it is a sample of a complex problem that is both difficult to answer & simple to pose with a computer named Turing, the notion that a Turing machine cannot solve the halting issue is one among the most popular fundamental leads to theory of computability. The halting issue result is the basis for most of theory of computability.

It is closely connected to recursion theory, a school of mathematical logic that eradicates the hurdles of investigating Computing models that may be reduced to a Turing model. Recursion theory is referred to as computability theory by several mathematicians and computer theorists who research it.

• Theory of computational complexity:  Complexity theory posits that a computer can not only solve a problem, but also that it can do so efficiently. There are two important factors to consider: space complexity and time complexity, that refer to the number of steps necessary to do a computation and the amount of memory required. To determine how much space and time a specific strategy needs, the amount of time or space required to solve a problem is proportional to the magnitude of the input problem, according to computer experts.

Importance of Computing Theory:

Theoretical computing is a basic subject in the computer science discipline. Any student graduating in this field should be familiar with this subject since it will assist them in addressing computational problems quickly. Algorithms are at the heart of computer science, thus knowing their mathematical behaviour and understanding how they work inside is essential for solving any difficult issue. Students will be able to tie theoretical computer science knowledge to some mathematical foundation after taking these courses, resulting in a solid basic backbone for computer science graduates. The goal of the course is to discover how an algorithm is designed to work and to mathematically verify it by evaluating various issues that are solved using such algorithms. Computer scientists who work in this subject will benefit greatly from this course. Different models of computing are being researched; for example, the most important study is being conducted on the Turing Machine, which has attracted many experts from across the world. As a result, maintaining this course in the curriculum is a requirement of the computer science field.

Why Students Seek Computing Theory Assignment Help?

Understanding the theory of computing is a challenging endeavour since it combines a great deal of mathematics with computer technology. As a result, it combines two disciplines to build theoretical foundations. In this course, things are discussed at a very basic level, requiring a thorough understanding of all of the essential principles. Students must have understanding of grammar, may, and more machines in order to solve difficulties in automata theory or Turing machines, which are themselves quite difficult to comprehend.

Strong programming skills is required to implement the notions of the theory of computing. To tackle the difficulties, much of the coding is done in the C programming language. It is necessary to master the fundamental ideas first as it includes the working of mathematical algorithms. Many students find themselves in a tight spot and then lookout for reliable computer engineering assignment help .

Online Assignment Expert assists you by simplifying your learning and tackling complicated computing issues in the most straightforward and straightforward manner possible. You may contact our big pool of experts at any time of the year through us and post your assignment difficulty, inquiry, or project statement.

We assist you by providing the finest and most efficient assistance to your problem within the time constraints that you specify. We offer online services to connect with students all around the world along with one-on-one tutoring sessions. Our services are available to our customers 365 days a year, seven days a week and 24 hours a day. Pupils can contact us for assistance with any challenging problems in computing theory.

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Computer Science > Computer Science and Game Theory

Title: facility assignment with fair cost sharing: equilibrium and mechanism design.

Abstract: In the one-dimensional facility assignment problem, m facilities and n agents are positioned along the real line. Each agent will be assigned to a single facility to receive service. Each facility incurs a building cost, which is shared equally among the agents utilizing it. Additionally, each agent independently bears a connection cost to access a facility. Thus, an agent's cost is the sum of the connection cost and her portion of the building cost. The social cost is the total cost of all agents. Notably, the optimal assignment that minimizes the social cost can be found in polynomial time. In this paper, we study the problem from two game-theoretical settings regarding the strategy space of agents and the rule the assignment. In both settings, agents act strategically to minimize their individual costs. In our first setting, the strategy space of agents is the set of facilities, granting agents the freedom to select any facility. Consequently, the self-formed assignment can exhibit instability, as agents may deviate to other facilities. We focus on the computation of an equilibrium assignment, where no agent has an incentive to unilaterally change her choice. We show that we can compute a pure Nash equilibrium in polynomial time. In our second setting, agents report their positions to a mechanism for assignment to facilities. The strategy space of agents becomes the set of all positions. Our interest lies in strategyproof mechanisms. It is essential to note that the preference induced by the agents' cost function is more complex as it depends on how other agents are assigned. We establish a strong lower bound against all strategyproof and anonymous mechanisms: none can achieve a bounded social cost approximation ratio. Nonetheless, we identify a class of non-trivial strategyproof mechanisms for any n and m that is unanimous and anonymous.

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