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22C:131 Limits of Computation

A sample final exam of problems can be found here.

This will be the course web page where the assignments are posted.

1:05-2:20 MW, 1100 UCC

Instructor: Hantao Zhang
Office: 201B MLH,
Tel: 353 2545
Office hours: MW 3:30-4:30pm, Fr 12:30-1:30pm
Teaching assistant: Nery Chapeton-Lamas
Office: 201N MLH,
Tel: 353 2547
Office hours: Tu., Th, 12:30-2:00pm

Goals and Objectives

This course is a mathematical exploration of the limits of the power of computers. Some of the questions we ask and attempt to answer are the following. Are there problems that cannot be solved on any computer? How does one determine if a given problem can or cannot be computationally solved? If we place bounds on the resources (time and space) available to a computer, then what can be said about which problems can and which problems cannot be solved on a computer? How does the power of a computer change, if it has access to random bits? What happens when we relax the notion of solving a problem to "approximately" solving a problem - does this fundamentally change which problems can and which problems cannot be solved on a computer?

In attempting to answer these questions we will study the following topics:

  1. Computation models: Finite State Automata.
  2. Regular Grammar and Context-free Grammar.
  3. Turing machines. Definitions and examples. Turing-decidable and Turing-recognizable languages.
  4. Enhancements of TMs: multi-tape TMs, non-deterministic TMs. Equivalence of these and the standard TM.
  5. Diagonalization. Acceptance problem is undecidable; Acceptance problem is recognizable; the complement of the Acceptance problem is unrecognizable.
  6. Reductions. Examples of other undecidable languages. Rice's theorem. Post's Correspondence Problem (PCP) is undecidable.
  7. Running time of Turing Machines. The classes P, NP, NP-hard, and NP-complete.
  8. Cook-Levin Theorem, some reductions.
  9. Space complexity, Savitch's Theorem, PSPACE. Quantified boolean formula satisfiability is PSPACE-complete. So is Generalized Geography.
  10. The space heirarchy and the time heirarchy theorems.


Undergraduate algorithms (22C:31) or its equivalent.


Introduction to the Theory of Computation (second edition) by Professor Michael Sipser. The book is available in Union Store and the latest errata can be found at

Homeworks (TEN homeworks, each counts for three percent of final score)

LATE-DUE HOMEWORK ARE NOT ACCEPTED. In general you will be better off turning in what you have on time rather than seeking extra time to complete your work. There will be no make-up exams in general and exceptions will be rare and only for students whose reasons are included in the University's policy on "Excused Absences from Examinations".

Exams (One midterm and one final exam)

Midterm on Oct 13, 1:05PM (30 percent of final score)

A sample exam problems can be found here

Final exam on Dec 17, 9:45-11:45AM (35 percent of final score)


For the policies on ACADEMIC DISHONESTY and PROCEDURE FOR COMPLAINT, see the Student Academic Handbook, of the Colleage of Liberal Arts and Sciences.


Class Participation (5 percent of final score)

Lecture Notes

You are expected to study all the material in each chapter covered in the class even if that material is not explicitly discussed in class or in the homework.

The lecture notes are a supplement to the course textbooks. They are supposed to help you understand the textbook material better, they are a replacement for neither the textbook nor the lecture itself.

Please only print the lecture notes on the day of the class as it's updating.

Hantao Zhang