At the end of the course, you will be familiar with the basic computational models of Finite State and Pushdown Automata and with the classes of grammars that generate the languages recognized by these abstract computational devices. We shall also see that there are problems for which no algorithmic solution will ever exist, and realizing the existence of such inherently unsolvable problems has been one of the main achievements of 20th century mathematics. Roughly speaking, the subject matter of the theory of computability is the precise definition of the informal concept of algorithm, and the characterization of those problems that can (or, perhaps more intriguingly, cannot) conceivably be solved by means of algorithms. In the process of developing such a theory, we shall touch upon topics such as:
As an alternative, excellent description of this course I strongly recommend the preface of Computers Ltd.: What They Really Can't Do (Oxford University Press) by David Harel. This lovely little book by one of Computer Science's best expositors is highly recommended reading. (See the review of this book in Plus magazine.)
As supplementary reading on the science of computing as a whole, I also strongly recommend the books:
Note! You are not required to buy these two books. I'll suggest supplementary reading from these sources if and when I believe it may aid your general understanding of the topics that we cover in the course.
Note: The above exercises are not really
part of this
course. How many you attempt to solve depends only upon
your level of
confidence with the mathematics that will be used as
the course
progresses.
Our working language for the course
will be
English.
Below, I am attempting to give you an a priori description of parts of
the plan of the course. You are invited to take it with a large pinch of salt. The topic of each actual lecture
may vary, depending on how fast the course progresses, and on how
receptive you are to the topics of the course.
In general, I shall give you more exercises than you might conceivably
be able to solve during one exercise class. The exercises marked with
a star are those that I consider more important for your
understanding. All the exercises will be "doable", and working them
out will greatly increase your understanding of the topics covered in
the course. The best advice I can give you is to work them all out by
yourselves, and to make sure you understand the solutions if the other
members of your group (or me) gave you the solutions on a golden
plate. Above all, don't give up if you cannot find the key to the
solutions right away. Problem solving is often a matter of mental
stamina as much as creativity.
Above all, try to be critical of your solutions to the exercises. You
may not need (some of) the material covered in this course in your
future careers, but having an analytical and critical attitude will
always help you.
For further advice on how to learn this material (and, in fact, the
material in any course) I strongly recommend that you look at the
slides for the talk Psychologists'
tips on how and how not to learn by Wilfrid Hodges. In
particular, try to reflect upon the hints he gives, and ask yourselves
how much you practice what he preaches. You might also wish to read How
to Read Mathematics by Shai Simonson and
Fernando Gouveau --- a collection of useful, down-to-earth tips on how
to read, and learn from, mathematical texts
Information on the exam is now
available.
This page will be actively modified over the autumn term
2004, and is currently undergoing heavy restructuring. You are invited
to check it regularly during the autumn term. The page is dormant in
the spring term.
Prerequisites
The course assumes knowledge of basic
discrete mathematics, such as
the one that you have been using in
the course Matematik 2B
and in Algorithms and
Data
Structures. Those of you who would like to refresh their
memory,
or who are not confident about their understanding of the
basic
mathematics used in this course, are strongly encouraged to
read,
e.g., Chapter 0 of Sipser's book and/or Chapter 1 of the
book
Elements of the Theory of Computation (2nd edition) by
Lewis
and Papadimitriou. As
practice makes perfect, I also recommend that you work out some of
the
exercises to be found in those references. Try, for example,
exercises
0.2-0.5, 0.7, 0.10-0.11 in Sipser's book and/or exercises
1.1.1-1.1.4,
1.5.1-1.5.2 in the book by Lewis and
Papadimitriou.Course Material and Lecture Plan
The course will consist
of a series of 15 lectures, which will usually take
place at 10:15am
on Mondays and Fridays during term time. The
location of the
lectures will be room A4-106. Look at
the semester calendar
for updated information on the schedule for this (and all other)
courses. In particular, there will be at least two
days on which we'll have lectures both in the morning and in the
afternoon. Details will be available from this web page and on the
semester calendar.
[A
poetic proof of the pumping lemma for regular languages
courtesy of Harry
Mairson]
[For the curious minds amongst yourselves: If you
wish to read about Paul
Erdös, look at The
Erdös Number Project. There is also N is a
Number, a wonderful documentary film about Erdös by
George Paul Csicsery.]
Exercise
Classes and Advice on Modus Operandi
As usual, each lecture will have an associated exercise class. The
exercises mentioned on the web page for a lecture should be solved
during the exercise class that precedes the following lecture. For
example, the exercises listed on the web page for lecture 1 should be
solved before lecture 2. There will be no exercise class preceding the
first lecture.
Exam
The exam will be individual and oral. Each student will be examined
for at most 20 minutes on one of the topics listed in the exam
prospectus that will be distributed during the last lecture. The topic
will be selected randomly by each student. The student will be
informed of his/her mark (on the standard 13-scale) after the oral
exam. The student can choose to hold the exam in either Danish or
English.