MathJax check: If you see a fancy equation here $$NP = \bigcup_k NTIME[n^k]$$ then the math on this page is working. If you don't, then you probably disabled JavaScript or your browser is wack. Sorry.
Introduction
What is computation? Given a definition of a computational model, what problems can we hope to solve in principle with this model? Besides those solvable in principle, what problems can we hope to efficiently solve? This course provides a mathematical introduction to these questions. In many cases we can give completely rigorous answers; in other cases, these questions have become major open problems in both pure and applied mathematics!
By the end of this course, students will be able to classify computational problems given to them, in terms of their computational complexity (Is the problem regular? Not regular? Decidable? Recognizable? Neither? Solvable in P? NP-complete? PSPACE-complete?, etc.) They will also gain a deeper appreciation for some of the fundamental issues in computing that are independent of trends of technology, such as the Church-Turing Thesis and the P versus NP problem. Prerequisites: 6.042 or equivalent mathematical maturity.
Jiatu Li,
jiatuli (at mit dot edu), recitation Friday 1pm-2pm
Jakin Ng, jakinng (at mit dot edu), recitation Friday 11am-12pm
Class: Tuesday and Thursday 2:30-4:00, classroom 37-212
Office hours:
Ryan: Wednesday 11:00am-12:30pm, 32-G678
Jiatu: Tuesday 4pm-5:30pm, Building 32, G5 lounge (next to the 5th floor Gates elevators of Stata)
Jakin: Tuesday 12:30pm-2:00pm, TBD
Rather than emailing questions directly to the teaching staff, please post your questions on Piazza. If you have any problems or feedback for the developers, email team@piazza.com (and feel free to cc rrw).
See lecture 1 for the Piazza access code (or email rrw).
Textbook (optional):
Michael Sipser, Introduction to the Theory of Computation (3rd Edition), Thomson
Note: the 2nd edition of Sipser is also fine for this course, if you can find it cheaper!
Grading: Midterm exam: 25%, Final exam: 35%, Problem Sets: 40%.
Homeworks/psets will come out on Thursdays (except when an exam falls on the next week) and will be due the following Wednesday at 11.59 pm. Late assignments will not be accepted. To account for this, we will drop the lowest homework grade in computing your final grade. If you are unable to submit your homework because of extenuating circumstances such as a family/medical emergency, please have S^3 contact Ryan or the TAs about it (ASAP).
We require that you write your homeworks using a word processor that can generate PDFs. If you can generate a readable typeset PDF of your pset (i.e., in a clean font that is not your handwriting), we are OK with whatever system you use. We strongly recommend that you write your homeworks in LaTeX; we will provide the LaTeX source code for every homework assignment! (Exception: Hand-drawn figures are allowed; you can scan them and merge them in your PDF.) Practically every paper published in computer science and mathematics nowadays is written in LaTeX, so the ability to write in LaTeX is a very useful skill. If you use Windows, MiKTeX is a good choice. If you don't like downloading software but like writing in a web browser, Overleaf is also good. Grading will be done by TAs and/or Graders, via Gradescope.
Most of the work in this course will be mathematical proofs of various cool facts. Your proofs should be as clear as possible. Note, this does not mean your proofs should be long -- in fact, typically, good clear explanations are also short!
We will use pset partners to help you find stuents to collaborate with, on problem sets.
Policy on Collaboration:This is very important; please read carefully. We believe that the best learning often happens when you work with others, and we want to encourage that. But it is equally important that you try to learn on your own! So here are our rules for collaboration on homework:
You must think about all problems on your own and do your very best to find the solutions yourself, before discussing/collaborating with other students. After failing to find the answers yourself, you may discuss problems with other students, and work in groups on the remaining questions.
You must not look for solutions on the web or via LLMs (ChatGPT, Claude, etc). Instead, ask us questions if/when you get stuck.
Cite properly! If you work in a group, include the names of the other people in the group in your written solution. In general, if you do receive a significant idea from someone or somewhere, you must acknowledge that person/book/website in your solution.
You must write up your own solution to every problem; you may not copy answers from another student, or any other source. Also, you may not show your answers to another student. Failing to follow this rule may constitute plagiarism.
Deviating from the above rules may result in a violation of MIT policies. If you have any questions about our policy, please don't hesitate to send a private note to the course staff on Piazza.
How to submit problem sets: via Gradescope, to be announced on Piazza.
Regrade Policy: If you wish to request a regrade on a problem, please do so through Gradescope. Provide a clear description and justification of your request. Our graders will do their best to resolve your requests, but if you have trouble resolving your requests, you may ask that your request be forwarded to the TAs. You may also additionally write directly to us if necessary, and the graders may also forward the regrade requests to the TAs if they are having difficulty resolving the request. Please note that when we are regrading questions, we reserve the right to re-evaluate the entire problem set or exam.
For all pset questions (that do not give away solutions!), please post
to piazza.com/mit/spring2025/61400/home or come visit us at office hours. You can also send private questions (to TAs and instructors only) using Piazza.
02/13 Finish up regexps; Proving languages are not regular
Readings: Sipser 1.4, Sipser Problem 7.40 in 2nd ed (7.25 in 3rd ed) and its solution
Slides: [before class, color pdf][color pdf][grayscale pdf]
02/18 YOUR MONDAY CLASS GOES HERE -- NO 6.1400 TODAY
02/20 DFA Minimization
Readings: Sipser Problem 1.52 in 2nd ed (1.48 in 3rd ed) and its solution
Slides: [before class, color pdf][color pdf]
03/06 Church-Turing Thesis, Decidability and Recognizability
Readings: Sipser 4, 5.1, 5.2
Slides: [before class, color pdf][color pdf]
03/11 Diagonalization, Undecidability, and Reductions
Readings: Sipser 5.3, 6.3
Video: see piazza for a link
Slides: [before class, color pdf][color pdf]
03/20 Oracles and the Recursion Theorem
Readings: Sipser 6.2
Slides: [color pdf]
03/24 and 03/26 SPRING BREAK -- NO CLASS
04/01 "Deep Computability": The Recursion Theorem and the Foundations of Mathematics
Readings: Sipser 6.2, Luca Trevisan's notes on computability and logic
Slides: [color pdf]
04/15 Polynomial-Time Reductions, the Cook-Levin Theorem, and $NP$-completeness
Readings: Sipser 7.4, 7.5, an alternate proof of the Cook-Levin Theorem
Slides: [color pdf]
Midterm: The midterm will be in-class, tentatively scheduled for Tuesday, March 18 in room TBD. For this exam, we allow one single-sided sheet of notes (on the usual 8.5x11 letter paper); otherwise, the exam is closed-book.
IMPORTANT: If you have a conflict with the midterm time, please email the entire course staff (Ryan, Jiatu, Jakin) with the subject line "[6.1400 Midterm Conflict]". We'll schedule a conflict midterm as needed.
Final Exam: Time/Place TBD For this exam, we will allow one double-sided sheet of notes; otherwise, the exam is closed-book.
