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 a computation? What even is a computer? Are all problems solvable? We will explore several models of computation and explore their 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.
Other schools may call a version this course something like "Great Ideas in Computer Science". I like to think of it like, a kind of finale to your CS degree. This is the course where you will learn why computer science gets to be called a science.
This course has a lot of preqs, 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. We will follow the book very closely, and the places we differ from the book will be explicitly mentioned.
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 06 Exam 1
Jun 30 Exam 2
Jul 27 Exam 3
This is subject to change as I realize what takes more or less time. Summer pacing is usually weird. We will split each class into two sections, each one hour, with a ten minute break in between. I will record lectures and post them.
|May 16 01A||Introduction|
|May 16 01B||Deterministic Finite Automata|
|May 18 02A||Nondeterminism|
|May 18 02B||Powerset Construction|
|May 23 03A||Regular Expressions|
|May 23 03B||The Pumping Lemma|
|May 25 04A||Context Free Grammars|
|May 25 04B||Closure|
|May 30 05A||Syntactic Structures|
|May 30 05B||Chomsky Normal Form|
|Jun 01 06A||Pushdown Automata|
|Jun 01 06B||Each CFG has a PDA|
|Jun 06 07A||Each PDA has a CFG|
|Jun 06 07B||Non Context-Free Languages|
|Jun 08 08A||Turing Machines|
|Jun 08 08B||The Church-Turing Thesis|
|Jun 13 09A||Simulation evidence|
|Jun 13 09B||Turing-Completeness|
|Jun 15 10A||Countability|
|Jun 15 10B||Diagonalization|
|Jun 20 11A||A Crisis in Geometry|
|Jun 20 11B||Russell's Paradox|
|Jun 22 12A||Godel Incompleteness|
|Jun 22 12B||The Halting Problem|
|Jun 27 13A||The Art of Reduction|
|Jun 27 13B||Rice's Theorem and PCP|
|Jun 29 14A||Kleene's Recursion Theorem|
|Jun 29 14B||Kolmogorov Complexity|
|Jul 06 15A||P|
|Jul 06 15B||NP|
|Jul 11 16A||Cook-Levin Theorem|
|Jul 11 16B||Ladner's Theorem|
|Jul 13 17A||Savitch's Theorem|
|Jul 13 17B||PSPACE-completeness|
|Jul 18 18A||Hierarchy Theorems|
|Jul 18 18B||The Relativization Barrier|
|Jul 20 19A||Advice|
|Jul 20 19B||Circuits|
|Jul 25 20A||The Polynomial Hierarchy|
|Jul 25 20B||Karp-Lipton Theorem|
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.