4005-700-01: Homework, Quizzes, Reading and Slides

Week 1

• Tuesday, Sep 4
  • Covered in class: Course logistics.
  • Covered in class: Section 1.5.
  • Slides for this week: pdf, ppt
  • Next class: Discrete math quiz, September 6, 2-2:25pm. There will be two problems. First, a basic discrete math reading, writing, reasoning question. Second, a proof by mathematical induction (the simplest form of mathematical induction, there will be no structural induction (Section 2.5) on the discrete math quiz). I recommend to use our tutoring center for help.


• Thursday, Sep 6
  • In class: Discrete math quiz (30 min).
  • Covered in class: Sections 2.4, introduction to Section 2.5. In class we said that every string generated by the rules for language L_{0=1} contains the same number of zeros and ones, and we have discussed how to prove this statement by structural induction. I wrote up the proof on the third to last slide (with one case of the analysis left for you to finish). Other examples of structural induction are in Section 2.5 in the book - we will not have much time to get back to it in class (I will briefly go over the proof on the slide) so please read the section, it should be helpful for parts of the homework. Follow the above link for updated slides.
  • Keep in mind our helpful mentoring center if you have any questions about the quiz or any class material.
  • We will skip the proof of the statement that every string with equal number of zeros and ones can be generated by rules 1)-3). We will get back to it later in the quarter.
  • A note for students with disabilities: RIT is committed to providing reasonable accommodations to students with disabilities. If you would like to request accommodations such as special seating, testing modifications, or note taking services due to a disability, please contact the Disability Services Office. It is located in the Eastman Building, Room 2342; the Web site is www.rit.edu/dso. After you receive accommodation approval, it is imperative that you see me during office hours so that we can work out whatever arrangement is necessary.


• Homework 1, due 9/12 6:00pm: (Note: you only have to explain your answer if this is stated in the question.)
  1. Exercise 1.40.
  2. Exercise 1.47. You do not have to give reasons, but you should really think about your answers.
  3. Exercise 1.62. Simplify your answers as much as possible.
  4. Exercise 2.40 (e).
  5. Let Sigma = {a,b}. Give a recursive definition of function f on Sigma* such that for all strings x over Sigma, f(x) is the number of a's in x minus the number of b's in x. (Look at Example 2.24 to see how to recursively define such functions.)
  6. Let L be the language defined in Exercise 2.39 (b). Prove, using structural induction on L, that every string in L contains exactly one a.
  Before you start on the homework, please read the rules on collaboration in the syllabus. Also keep in mind that, as stated in the syllabus, I will stop answering homework questions at 10am the day it is due.

• Tuesday, Sep 11
  • Covered in class: Section 3.1. Read through Section 3.2.
  • Slides: pdf, ppt
  • Read ahead: Section 3.2.


• Thursday, Sep 13
  • Covered in class: solutions of Hw 1.
  • Covered in class: Section 3.2, 3.3, a part of 3.4 (we proved that the language {a^k b^k | k>=0} is not regular, but we haven't discussed the concept of "distinguishability" on strings yet). Slides are updated above.
  • Read ahead: Section 3.5 - operations on regular languages. Think about the last 5 slides.
  • Next class: Quiz on hw1, 9/18, 2-2:20pm.
  • No office hours on Tuesday, 9/18/07.


• Homework 2, due 9/19 6:00pm: Download the pdf. (Note: you only have to explain your answer if this is stated in the question.)

• Slides for this week: pdf, ppt


• Tuesday, Sep 18
  • In class: quiz 2 on material from week 1.
  • Covered in class: Section 3.5 (slides from previous week are now updated)
  • Covered in class: intro to Section 4.1 - think about slides 3 and 4.


• Thursday, Sep 20
  • Discussed in class: solutions of Hw 2.
  • Covered in class: Section 4.1 and a part of Section 4.2 (slides have been updated)
  • Next class we will have Quiz 3. Study material from week 2 and hw 2.


• Homework 3, due 9/26 6:00pm: follow this link

• Slides for this week: pdf, ppt


• Tuesday, Sep 25
  • In class: quiz 3.
  • Covered in class: Section 4.2 and 4.3 (slides have been updated)


• Thursday, Sep 27
  • Covered in class: solutions of problems 4 and 5 on hw 3, the other problems are solved here (thanks to Greg Von Pless we now have the pdf for problem 4 as well)
  • Covered in class: finished Section 4.3, Section 3.4, parts of Sections 5.1 and 5.2 (slides are updated, the color on the slide with the proof of Kleene's Theorem, Part 2, does not show well on the pdf, it is ok in the ppt)
  • Next class: quiz 4, focus on material presented in class in week 3 (and on hw3)


• Homework 4, due 10/3 6:00pm: (Note: you only have to explain your answer if this is stated in the question.)
  1. Use the construction from the proof of Kleene's Theorem to construct an NFA-&Lambda for the following regular expression: (01 + 101*)*(0 + (10)*). It suffices to draw the transition diagram of the final NFA-&Lambda . Apply the construction literally, and do not simplify.
  2. Exercise 4.28(d). It suffices to give the transition diagram of the NFA. Apply the NFA-&Lambda to NFA construction literally, and do not simplify.
  3. Minimize the FA from Exercise 5.16(h). You must apply Algorithm 5.1. Your answer should be of the form of Figure 5.4, i.e., both the table and the FA.
  4. Exercise 5.26. You do not have to explain your answers.
  5. In class, we observed that if M = (Q,&Sigma,q₀,&delta,A) is a FA accepting L, then (Q,&Sigma,q₀,&delta,Q-A) is an FA accepting L'. Does this still work if M is an NFA? If so, prove it. If not, give a counterexample and briefly explain why it is a counterexample.
  6. 1. Give an example of a 2-state NFA M such that the minimal FA for L(M) has four states. For your answer, you should give the transition diagrams of the NFA and the FA, and you should briefly explain why your FA accepts L(M) and why your FA is minimal.
     2. How many equivalence classes does I_L(M) have? Explain your answer.
     3. Does there exist a 2-state NFA M such that the minimal FA for L(M) has five states? If so, give the example at the same level of detail as in part 1. If not, explain why not.

• Tuesday, Oct 2
  • In class: quiz 4.
  • Covered in class: Sections 5.1, 5.2, and 5.3 (slides have been updated)


• Thursday, Oct 4
  • Covered in class: more proofs of nonregularity by the Pumping Lemma (Section 5.3), Section 5.4 (slides have been updated)
  • Covered in class: solutions to Exercise 5.26, distributed solutions to other hw4 problems
  • Next week: the midterm will take place in class on Thursday, October 11. More information can be found here. Recall that there is no homework due week 6 (to give you more time to focus on the midterm preparation).
  • Greg Von Pless (one of our tutors) wrote up the solution to problem 4 on hw3: pdf

• Slides for this week: pdf, ppt


• Tuesday, Oct 9
  • Covered in class: Chapter 6 (slides have been updated)
  • Covered in class: another proof of nonregularity by the pumping lemma.
  • Midterm will take place next class. Remember that you may bring one letter-sized, double-sided cheat-sheet (in your own handwriting).


• Thursday, Oct 11
  • Midterm.


• Homework 5, due 10/17 6:00pm: pdf

• Slides for this week: pdf, ppt


• Tuesday, Oct 16
  • Discussed solutions of the midterm.
  • Covered in class: finished Chapter 6, started Chapter 7 (slides have been updated)


• Thursday, Oct 18
  • Covered in class: finished Chapter 7 (slides have been updated)
  • Solutions of hw5 can be found here
  • Next class: quiz 5. This quiz will test material covered by homework 5.


• Homework 6, due 10/24 6:00pm: follow this link

• Slides for this week: pdf, ppt
• Slides for Thursday: pdf, ppt


• Tuesday, Oct 23
  • In class: quiz 5.
  • Covered in class: Sections 8.1 and 8.2 (slides have been updated)


• Thursday, Oct 25
  • Discussed in class: a solutions of Problem 1 on Quiz 5.
  • Covered in class: finished Section 8.2, Section 8.3, started Chapter 9 (slides have been updated)
  • Solutions of Hw 6 are now available (pdf1 and pdf2). There were several typos in the original version - the current version fixes them (thanks goes to our TA Greg). Let me know if you find any more problems.
  • Several students asked about the DPDA for L = {w in {a,b}* | number of a's in w = 2(number of b's in w)}. As we probably won't have the time to go over this problem in class, I decided to post the solution for everybody's benefit: pdf
  • Next class: quiz 6. The quiz covers material covered by Homework 6.


• Homework 7, due 10/31 6:00pm: pdf

• Slides for this week: pdf, ppt


• Tuesday, Oct 30
  • In class: quiz 6.
  • Covered in class: Chapter 9, except for the nondeterministic TM - think about the last slide before Thursday (slides have been updated)


• Thursday, Nov 1
  • Covered in class: finished Chapter 9 and Closure properties of RE, Rec (slides have been updated)
  • We will most probably not have much time for the Chomsky Hierarchy. Please read the slides, we will briefly discuss it on Tuesday.
  • Solutions of Homework 7 are now available.
  • Next class: quiz 7, our last quiz. It will cover material covered by Homework 7.
  • Information about the final exam is now available.


• Homework 8, due 11/07 6:00pm: (Note: you only have to explain your answer if this is stated in the question)
  1. Exercise 8.5. You do not have to prove your answers; a simple yes or no suffices.
  2. Exercise 8.19 (e). You may use the fact that DCFLs are closed under complement (see page 310/311) and you may use the fact that DCFLs are not closed under intersection.
  3. Exercise 9.35. The encoding can be found in Section 9.6.
  4. The following grammar generates {aⁱ | i is a power of 2 greater than 1}.
     • S -> ACaB
     • Ca -> aaC
     • CB -> DB
     • CB -> E
     • aD -> Da
     • AD -> AC
     • aE -> Ea
     • AE -> Λ
     Give a derivation of aaaaaaaa.
  5. Question 1 from the practice final. Since we haven't yet gone over P and NP in class, you don't have to draw them.
  6. Show that the following decision problem is decidable:

     Given a CFG G, is L(G) not empty?

     Do not give the stupid algorithm from the book (page 312). Instead, think of a way to recursively mark all variables A such that for some string x in Σ*, A =>* x.
  Special hard extra credit question.
  For extra credit, do Exercise 9.52. Your answer to this question should be written on separate pages that are not stapled to the rest of your homework. Credit for this question will only be given for answers that are basically correct. In addition, you will have to explain your answer and how you came up with it to me during office hours. Keep in mind that, as with the rest of the homework, your answer has to be completely your own.

• Slides for this week: pdf, ppt
• Slides for Thursday: pdf, ppt


• Tuesday, Nov 6
  • In class: quiz 7.
  • Covered in class: Chomsky Hierarchy, Decidability (slides have been updated)
  • A note about the slides: my Powerpoint crashed before I got a chance to save the notes from today's class. I copied the notes from the other section and thus some of the names of the programs are different but the essentials are the same. I included one more reduction proof (deciding if a language accepted by a TM is empty) and the last reduction proof I am leaving open as a practice for the final (reduce from the "for a TM T, is L(T) empty?" problem; the proof is a two line program).
  • I created a website with some practice problems for the final exam. As of now there are only construction-of-Turing-machines related practice problems but I might add more later.


• Thursday, Nov 8
  • Covered in class: Post's correspondence problem, Complexity (slides have been updated)
  • Solutions of hw8: pdf
  • Good luck with your finals!