CMPSCI 401: Theory of Computation

David Mix Barrington

Spring, 2008

This is the home page for CMPSCI 401. CMPSCI 401 is an advanced undergraduate core course in the theory of computation and will deal with formal language theory (finite automata, regular languages, grammars, and pushdown automata), computability theory, and complexity theory.

Instructor Contact Info: David Mix Barrington, 210 CMPSCI building, 545-4329, office hours M 1:30-2:30, Tue 11-12, Wed 2:30-3:30, Thu 1-2. I generally answer my email fairly reliably.

TA Contact Info: Vimal Mathew, vml@cs.umass.edu, 5-1596. Office hours in LGRT 220, Tuesdays 2-3 pm.

The course is primarily intended for undergraduates in computer science and related majors such as mathematics or computer engineering. CMPSCI 311 (theory of algorithms) is the primary prerequisite, though this may be negotiable for students with a strong mathematics background. The mathematical techniques taught in CMPSCI 250 (or similar courses like MATH 300 or MATH 455) will be used heavily. No programming will be assigned, but familiarity with programming at the level of at least CMPSCI 187 (data structures) will sometimes be assumed. This is probably the mathematically most difficult course in the CMPSCI undergraduate curriculum. It is a semi-elective course -- it fills requirements for any CMPSCI major and many MATH majors, but should be taken only by students whose mathematical ability and/or motivation is average or above relative to CMPSCI majors.

The textbook for the course is Introduction to the Theory of Computation by Michael Sipser, second edition. This is a very good book: I will be following it very closely for my lectures, and it is a very good long-term reference, but it is rather expensive. The first edition is available more cheaply, and the main text of the two is virtually identical, but the second edition will be the source of the problems I assign and it has a large number of solved exercises. I have ordered the book through the Jeffrey Amherst College Store in downtown Amherst.

The course will meet for three lecture meetings a week, MWF 11:15-12:05 in LGRT 204. There is no formal attendance requirement but there may be occasional graded in-class activities.

There is also an scheduled honors section for the course called CMPSCI H07, a one-credit seminar intended for undergraduates in Commonwealth College (but open to non-CC students as well if there is room). I will lead a seminar on some topics related to the main course, such its connections to formal logic and abstract algebra. This seminar meets Fridays 3:30-4:30 p.m. in room A311 LGRC.

• Course Requirements and Grading
• Homework Assignment Directory (with HW#6)
• Exam Directory (with real final and solution)
• Questions and Answers on Homework (with some on HW#4)
• Syllabus

Announcements (22 May 2008):

(22 May) I have graded the finals and emailed each of you your individual grades. I was disappointed with the final, in particular that so many of you didn't absorb the central definition of Question 7 (reference corrected 16 May 2009) . I changed the scale for the final to A = 95, C = 60. The 25 scores, in descending order, were 109, 106, 104, 98, 97, 92, 90, 88, 86, 84, 82, 81, 78, 78, 71, 63, 56, 54, 53, 51, 49, 48, 46, 30, 28. The final grade distribution for the course was A 8 (four of them unofficial A+), A- 2, B+ 5, B 0, B- 0, C+ 3, C 2, C- 1, D+ 2, D 1.
• (16 May) The final exam and its solutions are now up. I hope to have grades ready on Monday or Tuesday -- I will email individual reports to each of you. Good luck with your other exams!
• (16 May) The final meeting of the honors section will be Monday 19 May, 1:30-3:30 p.m., in room 303 in the Computer Science Building.
• (14 May) Paper copies of the HW#6 solutions are available in the CMPSCI main office, in the grey metal cabinet to the left of the reception desk. I will be in a faculty meeting all morning but generally around this afternoon.
• (9 May) I have just posted the practice final exam. Solutions will follow, probably Sunday night.
• (9 May) A correction to the syllabus -- the "plus" material referred to for the last lecture is largely contained in section 10.5 of Sipser. Our brief tour of circuit complexity will cover both sections 9.3 and 10.5.
• (28 April) I've posted the final homework assignment , which is due on the last day of classes (Tuesday 13 May) by 4:00 p.m. either to me directly or to the CMPSCI main office. Solutions for HW#5 will be available in class Wednesday or in my office hours before that.
• (28 April) Two corrections on the syllabus: the final exam is on Thursday 15 May, and we are covering all of Chapter 8.
• (24 April) I've answered two questions on HW#5.
• (13 April) I have graded the exams and set the scale at A = 104 and C = 64. The total range of scores was 120, 120, 117, 111, 108, 107, 106, 105 (all A's or A+'s to here), 100, 97, 96, 90, 83, 71, 69, 66, 65, 63, 61 (the last C), 51, 43, 35, 35, and 29.
• (13 April) I have just posted HW#5, which is due two weeks from tomorrow, 28 April. The last homework, HW#6, will be due at 4:00 p.m. on the last day of classes, Tuesday 15 May.
• (13 April) I have posted the second midterm and its solution. I expect to return graded exams tomorrow, and to post the HW#5 assignment tonight.
• (13 April) Note that we have now been assigned a final exam slot, on Tuesday 15 May 2008 from 1:30-3:30 in room 142, CPMSCI building.
• (7 April) I have just posted solutions to the second practice midterm. On of the problems, 6b, turned out to be much too hard for the exam, but you might still find the solution of interest.
• (7 April) Today in lecture I gave some definitions that I think make it easier to read the two main proofs in section 6.1 of Sipser -- the proof that SELF_TM exists and the proof of Theorem 6.3:
  • The concatenation of a sequence of Turing machines denotes a single machine which operates each of the machines in turn, taking the output of one as input to the next. Thus a machine ABT runs A on its input w to produce some string x, runs B on x to get some string y, then runs T on y to get z.
  • For any string x, the machine P_x erases its input string and outputs x.
  • In the first proof, the machine B is defined so that on input w, it outputs the concatenation of the machine P_w and the string w. If w happens to be a machine, this concatenation will form a machine that first runs P_w and then runs w on the result.
  • In the first proof, R is defined to be the machine P_BB, where B is defined above. (Sipser uses "A" to denote P_B.) We can trace the action of R on any input string w. First P_B erases w and outputs the string B. Then B takes this B as input and outputs P_BB. Since this string is exactly R, R has the property that it outputs its own description.
  • Now we turn to the proof of Theorem 6.3. By "P_x" in this proof, Sipser actually means a machine I will call P'_x. This machine, given any input w, outputs the pair of strings (w.x).
  • In this second proof, B is defined slightly differently as well. Its input must be a pair (w,x), and on this input it outputs a pair (P'_xx, w).
  • The machine T in this second proof takes a pair of strings (y,z) as input and computes the partial function t(y,z).
  • In this proof, R is defined to be the machine P'_BTBT. We can trace the computation of R on a single input R. First P'_BT takes w and outputs (w, BT). Then B takes this pair as input and outputs (P'_BTBT, w). Finally T takes this pair and outputs t(P'_BTBT, w) which is equal to t(R, w) because R is equal to P'_BTBT.
• (7 April) Roberto Olivares, a student in this class, is looking for fellow students with whom to study for the exam, particularly tonight and tomorrow. His email is his first name, at cs.umass.edu (not student.umass.edu).
• (4 April) I just received and answered some thoughtful questions on HW#4. If and when I get more questions I will answer them on this page.
• (4 April) I have just posted a practice exam for the second midterm, which will be next Thursday evening, 7-9 p.m., in ELAB 306.
• (24 March) I have also just posted HW#4, which is due in class on 9 April (two weeks from Wednesday).
• (24 March) As I mentioned in class today, I have to miss class this Friday, 28 March, but we will still have class as grad student Philipp Weis will lecture on section 5.1 of Sipser. The honors class will not meet.
• (24 March) Note that tomorrow is the last day to drop the course with a W grade. Only a few of you need to be thinking about this, as most of you are clearly on track to pass (C or better). But please email me or see me in office hours if you are doing badly -- you should be able to judge your performance from your returned work and the grade scales given below.
• (24 March) Welcome back from break! Vimal has graded HW#3, and solutions and graded papers were handed out in class today. I will use the same scale of A = 90, B = 72, C = 54 for this homework (and for HW#2). There were two 100's on HW#3, exactly half were above 80, and only two were below 50 (along with three people who handed in nothing). As I just clarified on the requirements page, the lowest of the six homeworks will be dropped and the best five will each count for 6% of the final grade.
• (5 Mar) A reminder that undergraduate elections are going on today and tomorrow -- you can choose the student who represents you on the board of trustees of UMass, choose the student president, and vote in a referendum on whether to endorse the negotiations coming out of last fall's "strike".
• (5 Mar) There is a First Friday event this coming Friday, with some recent alumni coming back to talk about jobs in the real world -- 3:30 in room 151 CMPSCI building.
• (5 Mar) The Jeffrey Amherst bookstore will soon return any excess copies of Sipser they have to the publisher, so get your book soon if you still need one from them.
• (4 Mar) I have finished the grading -- if you can't wait for tomorrow's class, you may pick up graded exams in my office hours 11-12 today and if you catch me in my office this afternoon (likely 1:30-4:30). I kept the scale at 105 = A, 88 = B, 70 = C, 53 = D, 35 = F. I was generally pleased with the results -- out of 26 exams there were:
  • 8 A+ (120, 119, 116, 115, 109, 109, 109, 109)
  • 1 A (106)
  • 1 A- (99)
  • 3 B+ (92, 92, 91)
  • 3 B (90, 88, 85)
  • 4 B- (83, 81, 81, 81)
  • 0 C+
  • 2 C (69, 68)
  • 1 C- (62)
  • 1 D+ (60)
  • 1 D (52)
  • 1 F (36)

  The four people below C should talk to me individually.

• (2 Mar) I have just posted solutions for the first midterm. I may or may not have them graded by tomorrow.
• (1 Mar) I have just posted the text for the first midterm. Solutions will follow probably tonight or tomorrow, and I have done about a third of the grading.
• (29 Feb) I'll get the test and solutions for it posted as soon as I can, probably sometime over the weekend. My optimistic goal is to have them graded by Monday. In the meantime, I have posted HW#3, which is due on 14 March, the last lecture before spring break.
• (29 Feb) Honors class today will be in a different room, Prof. Grupen's office. We'll meet outside the normal room, LGRT 311, but be courteous of the presentation that starts there at 3:00.
• (25 Feb) I have just posted solutions to the first practice test. While writing them up I discovered an error in the last question of the test itself, which I have now corrected.
• (25 Feb) I have to skip my office hours tomorrow (Tuesday) because of a dentist's appointment. Vimal will have his office hours 2-3 in LGRT 220, and I expect to be around in general for most of the afternoon.
• (25 Feb) HW#1 has been graded and Vimal has made up paper solutions. I will bring these to class on Wednesday. If you want them earlier, you may be able to catch me tomorrow afternoon. The scale is A = 90, B = 72, C = 54. The mean was around 77, and only a few people were below 70. The three people below 50 should talk to me.
• (25 Feb) Today's lecture finished section 2.2 of Sipser and made a brief start on 2.3 -- I plan to cover all of 2.3 on Wednesday. All of 2.3 will be fair game for Thursday night's midterm, but I will not expect detailed knowledge of the more complicated proofs in Chapter 2.
• (25 Feb) The exam on Thursday is in LGRT 1234, from 7-9 p.m. -- for future reference, the second midterm on 10 April is in ELAB 306.
• (23 Feb) NO CLASS today as the University is closing at noon. I will be in the regular room at the regular time but not present new material -- I'll review whatever anyone wants to discuss. Of course the honors class at 3:30 is completely canceled and I will not be there.
• (22 Feb) As promised, the practice exam for the first midterm is posted. The real exam will be comparable to this in length and difficulty.
• (22 Feb) The weather looks threatening again for tomorrow -- please check the UMass web page or other sources for any official closing, late opening, or early closing.
• (18 Feb) Remember that tomorrow, Tuesday 19 February, is a Monday schedule. Thus we will have class at 11:15 and I will hold office hours 1:30-2:30 as usual for Mondays.
• (18 Feb) I have posted the second homework assignment . I had to decide whether to make it due before the exam, which would give you only a little more than a week for it, or after the exam, which means I can't give out solutions before the exam. I chose the later deadline, but I could be talked out of this.
• (14 Feb) I've received and answered some more good questions in office hours today and by email. I'll be reading email tonight until about midnight, but I can't guarantee that I'll log in tomorrow morning until just before class.
• (12 Feb) Vimal has set his office hour times as Tuesday 2-3 in LGRT 220 (across from the edlab) and he will be there today to talk about HW#1 if needed. I got some good questions in office hours this morning, which I summarized on the Q&A page.
• (12 Feb) The weather forecast for tomorrow is bad -- I will go by the university's decision, so if the university opens before 11:15 we will have class and otherwise we won't. If you are driving in, use your judgment as to whether to risk it if we are open. The lecture, if we have it, will be on section 1.4 but with special attention to the Myhill-Nerode theorem, which Mike presents as the solution to Exercise 1.52 on pages 97 and 98.
• (7 Feb) I have posted the first question and answer on HW#1. I'll add new questions and answer them as I get them.
• (30 Jan) I have posted the first homework assignment, due in class Friday 15 February. I've also posted my office hours, though I will check in class today before finalizing the Thursday time.
• (25 Jan) The bookstore has confirmed by phone that the Sipser books have arrived and are on the shelf. I look forward to meeting you all on Monday!
• (21 Jan) The syllabus and the requirements and grading page are now up.
• (17 Jan) I've ordered the textbook through the Jeffery Amherst College Store in downtown Amherst (through the alley next to Laughing Dog cycle shop). I talked to them today and they do not have the books on the shelf yet -- they will be there by the first day of classes and may in fact be in the building already. I'll confirm here when they are available. As I said above, you might be able to get by with the cheaper first edition but the second is preferred.
• (17 Jan) This area of the main course page will serve as a blog and will be my primary medium to communicate with the students (though I will email everyone for urgent matters). Please check here frequently. Today I'm setting up the basic parts of the course web page.

Last modified 16 May 2009