4005-700-02: Homework, Quizzes, Reading and Slides

Week 1

• Monday, Sep 3
  • Covered in class: Course logistics.
  • Covered in class: Section 1.5.
  • Slides for this week: pdf, ppt
  • Next class: Discrete math quiz, September 5, 10-10:25am. 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.


• Wednesday, Sep 5
  • In class: Discrete math quiz (30 min).
  • Covered in class: Sections 2.4 and 2.5, updated slides are uploaded (see link above).
  • In class I said that I will post the proof of the statement on the second to last slide ("Every x with equal number of zeros and ones can be generated by rules 1)-3) defining the language L_{0=1}.") I decided to cover this proof in class at a later time, thus I am not posting the proof at the moment.
  • 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.

• Monday, Sep 10
  • Covered in class: Section 3.1, introduction to Section 3.2.
  • Slides: pdf, ppt
  • Think ahead: how to construct finite automata for the languages described on slide 9?


• Wednesday, Sep 12
  • Covered in class: Sections 3.2, 3.3, a part of 3.4 (a proof that the {a^kb^k | k>=0 } language is not regular), a part of 3.5. Slides above are updated.
  • Think ahead: about the last three slides.
  • 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


• Monday, Sep 17
  • Covered in class: solutions of hw 1.
  • Covered in class: Section 3.5 (slides from previous week have been updated).
  • Covered in class: introduction to Section 4.1 - think about slides 2 and 3.
  • Reminder: Quiz 2 will take place next class, Wednesday, Sept 19, 10-10:20am. Study topics covered in the first week, be sure to understand the solutions of hw 1.


• Wednesday, Sep 19
  • In class: quiz 2.
  • Covered in class: Section 4.1 and introduction to Section 4.2 (slides have been updated, also notice that the proof of the subset construction is a somewhat simplified)


• Homework 3, due 9/26 6:00pm: follow this link
  We have not gone through the formal definition of NFA-lambda in class yet, we will do it on Monday. However, it is intuitive and I encourage you to think about the problem 3 before Monday. All other problems deal with material we have already done in class.

• Slides for Wednesday and next week: pdf, ppt


• Monday, Sep 24
  • Covered in class: solutions of hw2.
  • Covered in class: Section 4.2 and the first part of Section 4.3 (slides from previous week are updated).
  • Next class: Quiz 3 will take place on Wednesday, Sept 26, 10-10:20am. It will be on material covered upto the end of week 2, with emphasis on material covered in hw 2.


• Wednesday, Sep 26
  • In class: quiz 3.
  • Covered in class: Section 4.3 (slides have been updated, I added some color to visualize the proof of Kleene's Thm, Part 2 but for an unknown reason the color does not show well on the pdf, it is fine in the ppt), started Chapter 5.
  • Solutions to questions 1-3 from hw3 are here, we will cover problems 4 and 5 in class on Monday.


• Homework 4, due 10/3 6:00pm: (Note: you only have to explain your answer if this is stated in the question.)
  (Note 2: we have not introduced the relation I_L in class yet, you can find the definition on slide 3. For two strings x,y over Sigma, we say that x I_L y iff L/x=L/y (i.e. if they are indistinguishable with respect to L).)
  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.

• Monday, Oct 1
  • Covered in class: Sections 5.1 and 5.2 (slides have been updated)
  • Covered in class: solutions to Hw3, problems 4 and 5. Solutions to questions 1-3 are here, and now we also have the solution to problem 4 written up (pdf, thanks to Greg Von Pless, one of our tutors)
  • Next class: quiz 4. Focus will be on material covered in week 3 and on hw3.


• Wednesday, Oct 3
  • In class: quiz 4.
  • Covered in class: Sections 5.3 and 5.4 (slides have been updated)
  • Next week: the midterm will take place in class on Wednesday, October 10. 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


• Monday, Oct 8
  • Covered in class: Exercise 5.26 from Hw 4, solutions of other problems on the hw were handed out.
  • Covered in class: Chapter 6 (slides have been updated)
  • Midterm will take place next class. We have voted to start the midterm at 10:10 (and end at 11:50) instead of 10:00 (and end at 11:40). Remember that you may bring one letter-sized, double-sided cheat-sheet (in your own handwriting).


• Wednesday, Oct 10
  • Midterm.


• Homework 5, due 10/17 6:00pm: pdf

• Slides for this week: pdf, ppt


• Monday, Oct 15
  • Discussed solutions of the midterm.
  • Covered in class: finished Chapter 6, started Chapter 7 (slides have been updated)


• Wednesday, Oct 17
  • Covered in class: finished Chapter 7 (slides have been updated)
  • Solutions of hw5 can be found here


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

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


• Monday, Oct 22
  • Covered in class: revisited Section 7.4, covered Section 8.1 and a part of 8.2 (slides have been updated)
  • Next class: quiz 5 on material covered by Hw 5.


• Wednesday, Oct 24
  • In class: quiz 5.
  • Covered in class: finished Section 8.2, (almost completed) 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)}. I decided to post the solution for everybody's benefit: pdf


• Homework 7, due 10/31 6:00pm: pdf (One of the problems asks you to give a Turing machine. Before you start working on this one, it might be easier to give a TM for {a^kb^kc^k | k>0} - we will discuss it in class on Monday. Keep in mind that you are allowed to use any symbols on the tape, not just symbols from Sigma.)

• Slides for this week: pdf, ppt


• Monday, Oct 29
  • Covered in class: finished Chapter 9 (slides have been updated)
  • Discussed in class: parts of solutions of Hw6. Additional typos were discovered in the meantime and we posted an updated version of the solutions - please check the link above.
  • Next class: quiz 6.


• Wednesday, Oct 31
  • In class: quiz 6.
  • Covered in class: Nondeterministic TM, 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 Monday. For one of the homework problems, you are asked to trace a so-called unrestricted grammar. This is a grammar where the rules are of the form (a string of terminals and nonterminals) -> (a string of terminals and nonterminals).
  • Solutions of Homework 7 are now available.
  • 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 Wednesday: pdf, ppt


• Monday, Nov 5
  • Covered in class: Chomsky Hierarchy, Decidability (slides will be updated soon).
  • In class I said that I will not ask you to come up with a reduction on the final. After talking to Prof. Hemaspaandra I have to adjust this statement - I will not ask you to come up with a reduction of the same difficulty level as we discussed in class today. However, for example, you should be able to prove that for two Turing machines T1, T2, it is undecidable to say whether L(T1)=L(T2). Use the fact that it is undecidable to check if, for a given TM T, L(T)=emptyset. (This is the problem from the slide we skipped.)
  • Next class: quiz 7, our last quiz. It will cover material covered by Homework 7.
  • 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.


• Wednesday, Nov 7
  • In class: quiz 7. Search for your graded quizzes and the solutions of the practice final in your mailfolders on Thursday.
  • Covered in class: Post's correspondence problem, Complexity (slides have been updated)
  • Solutions of hw8: pdf
  • Good luck with your finals!