This course answers two main questions:
What are the fundamental limits of computation?
What makes some problems easy, and others hard?
The first question is the study of the area of Computability theory. Most of its questions are solved, which is what makes this subject fun. We are concerned with such extremal, almost philosophical questions. What even is computation? What even is a computer? Are all problems solvable? We will explore several models of computation and explore their relative power, and weaknesses.
The second question is the study of Complexity theory. Most of its questions are unsolved. This subject does not have a happy ending (and perhaps won’t, in our lifetimes) but this contrast is what makes it interesting. We may not know how to solve certain questions, but ironically, we know a lot about how hard these questions are.
I like to think of this course as a finale to your CS degree. It is simultaneously the most important and least important course you will take. It is the least important as it doesn’t develop any single technical skill. It is the most important, as it develops your ability to conceptualize and theorize. This is the course where you will learn why computer science gets to be called a science. It puts the rest of your degree into context.
This course has a lot of pre-reqs, some of which I would disagree should be a requirement. All you really need is good proof skills, like those found CS2050. If you think you might be rusty, please refresh chapter zero of the Sipser book.
The book for the course is Introduction to the Theory of Computation by Michael Sipser. It is an excellent textbook, can’t count how many times I’ve read it. The notes and lectures for the course are the authoritative reference, but it is expected you follow along with Sipser’s book. Later on, I may reference the Arora-Barak and Li-Vitanyi books.
In order to accomodate students who wish to take this course fully remote, I have decided to make all assignments take home. The tradeoff here is that the difficulty will increase. You will have three exams. They are open note and open book, but not open internet. You will also have ten problem sets. Your tentative exam dates are:
Jun 05 Exam 1
Jul 01 Exam 2
Jul 31 Exam 3
This is subject to change as I realize what takes more or less time.
| Day | Title |
|---|---|
| May 12 01A | Introduction |
| May 12 01B | Deterministic Finite Automata |
| May 14 02A | Nondeterminism |
| May 14 02B | Powerset Construction |
| May 19 03A | Regular Expressions |
| May 19 03B | The Pumping Lemma |
| May 21 04A | Context Free Grammars |
| May 21 04B | Closure |
| May 28 05A | Syntactic Structures |
| May 28 05B | Chomsky Normal Form |
| Jun 02 06A | Pushdown Automata |
| Jun 02 06B | Each CFG has a PDA |
| Jun 04 07A | Each PDA has a CFG |
| Jun 04 07B | Non Context-Free Languages |
| Jun 09 08A | Turing Machines |
| Jun 09 08B | The Church-Turing Thesis slides, required reading1, required reading 2, required reading 3 |
| Jun 11 09A | Simulation evidence |
| Jun 11 09B | Turing-Completeness |
| Jun 16 10A | Countability |
| Jun 16 10B | Diagonalization |
| Jun 18 11A | Foundations of Mathematics |
| Jun 18 11B | Russell’s Paradox |
| Jun 23 12A | Godel Incompleteness |
| Jun 23 12B | The Halting Problem |
| Jun 25 13A | The Art of Reduction |
| Jun 25 13B | Rice’s Theorem and PCP |
| Jun 30 14A | Tarskian Theory of Truth |
| Jun 30 14B | Kolmogorov Complexity |
| Jul 02 15A | P |
| Jul 02 15B | NP |
| Jul 07 16A | Cook-Levin Theorem |
| Jul 07 16B | Ladner’s Theorem |
| Jul 09 17A | Savitch’s Theorem |
| Jul 09 17B | PSPACE-completeness |
| Jul 14 18A | Hierarchy Theorems |
| Jul 14 18B | The Relativization Barrier |
| Jul 16 19A | Circuit Complexity |
| Jul 16 19B | Circuit Lower Bounds |
| Jul 21 20A | The Polynomial Hierarchy |
| Jul 21 20B | Karp-Lipton Theorem |
Besides the notes we will publish this semester, I recommend you use the following two references:
Submission of any work not your own will result on a zero on the assignment to a report to OSI, which may incur further sanctions.
As a member of the Georgia Tech community, I am committed to creating a learning environment in which all of my students feel safe and included. Because we are individuals with varying needs, I am reliant on your feedback to achieve this goal. To that end, I invite you to enter into dialogue with me about the things I can stop, start, and continue doing to make my classroom an environment in which every student feels valued and can engage actively in our learning community.