4003-380-03: Homework, 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, 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.


• Thursday, Sep 6
  • 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 we started proving that every string with equal number of zeros and ones can be generated by the rules for language L_{0=1}. Think about the proof and try to finish it. We will dicuss it on Tuesday.
  • 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 4:00pm: (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, introduction to Section 3.2.
  • Slides for week 2 are available: pdf, ppt
  • Think ahead: finite automata for the languages described on slide 10.


• Thursday, Sep 13
  • Covered in class: Section 3.2, 3.3, a part of 3.4 (the proof that L = {a^k b^k | k>=0} is not regular}), a part of 3.5. Slides are updated above.
  • Think ahead: slides with [Section 3.5] in the right top corner (the last five slides).
  • No office hours on Tuesday 9/18/07.


• Homework 2, due 9/19 4:00pm: 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
  • Covered in class: finished Section 3.5 (slides above have been updated).
  • Covered in class: Section 4.1, think about how to define the delta function on slide 6.
  • Next class: we will have a 15 minute quiz on material from the first week (and hw1). The quiz will not be graded and it will not be counted against you. (The main purpose of the quiz is to get everybody on track so that the midterm does not come as a surprise.)


• Thursday, Sep 20
  • In class: practice quiz on material from week 1.
  • Covered in class: Section 4.1 and almost all of Section 4.2. Think about a formal description of the NFA-lambda to NFA construction.


• Homework 3, due 9/26 4:00pm: follow this link

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


• Tuesday, Sep 25
  • Covered in class: solutions of hw2.
  • Covered in class: Section 4.2 and 4.3 (slides have been updated, I tried to include some color to make the proof on the last slide more understandable but my pdf printer did not print the color properly - if you'd like to see the color, check the ppt).
  • Next class: practice quiz on material from week 2 and hw 2.


• Thursday, Sep 27
  • In class: practice quiz 3.
  • Covered in class: solutions of problems 4 and 5 of hw 3, solutions of problems 1-3 can be found here (now we also have the pdf for the proof from problem 4, thanks to Greg Von Pless, one of our tutors)
  • Covered in class: Section 3.4, part of Section 5.1 and part of Section 5.2.


• Homework 4, due 10/3 4:00pm: (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.

• Tuesday, Oct 2
  • Covered in class: finished Sections 5.1, 5.2, started Section 5.3 (slides have been updated)
  • Next class: practice quiz 4, from material covered in week 3 and on hw3.
  • Recall that week 6 is our midterm week. The midterm will take place on Oct 11, in class. It will focus on Chapters 3-5 (review also sections 1.5, 2.4 and 2.5). Additional information about the midterm will be posted on this website by the end of the week. Also recall that there is no homework due week 6.


• Thursday, Oct 4
  • In class: practice quiz 4.
  • Covered in class: Sections 5.3 and 5.4 (slides have been updated)
  • Covered in class: solutions to Exercise 5.26 (except for part j which is false), the solutions to other hw4 problems were distributed in class
  • Next week: the midterm will take place in class on Thursday, October 11. More information can be found here.
  • 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 4:00pm: 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: (almost) finished Chapter 7 (slides have been updated)
  • Solutions of hw5 can be found here


• Homework 6, due 10/24 4:00pm: follow this link

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


• Tuesday, Oct 23
  • Covered in class: finished Chapter 7 (we have not done one of the PDA's on the slides yet - do it on your own as a preparation for the final exam), done Sections 8.1 and 8.2 (slides have been updated)
  • We will have our practice quiz on Thursday. It will cover context-free grammars.


• Thursday, Oct 25
  • In class: practice quiz 5.
  • Covered in class: 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 4:00pm: pdf

• Slides for this week: pdf, ppt


• Tuesday, Oct 30
  • Covered in class: {ss | s in {a,b}*} is not CFL, a proof by the pumping lemma for CFL's.
  • Covered in class: finished Chapter 9 (slides have been updated)
  • Next class: practice quiz on PDA's and TM's.


• Thursday, Nov 1
  • In class: practice quiz 6.
  • Covered in class: Closure properties of RE, Rec; briefly discussed the Chomsky Hierarchy (slides have been updated)
  • We will most probably not have much time to go into more detail with the Chomsky Hierarchy. Please read the slides.
  • Solutions of Homework 7 are now available.
  • Information about the final exam is now available.


• Homework 8, due 11/07 4:00pm: (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.

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


• Tuesday, Nov 6
  • Covered in class: Decidability (slides have been updated)
  • 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!