Problem Set 4: Nonregular Languages

Computer Science 416/685: Computability Theory
Lehman College, City University of New York
Fall 2016

Due Dates:  Oral presentations of material begin on 22 September.
Written descriptions of material presented due one week after presentation.

Topics: Nonregular Languages (Chapter 1).

An integral part of this class is understanding and presenting the problems assigned as homework. Everyone is expected to do all the problems, but we will take turns on who presents the problem solutions to the class (every 2-3 weeks, depending on the number of students in the class). Within a week of presenting a problem solution to the class, you must submit a written description of it, via the Blackboard system. The written solutions will be posted on the Blackboard website for the class, so, they can used by everyone to study for the exams. Since it's hard to write down answers that are concise and are easily readable by all, if you wish to improve a grade on any problem, you may resubmit it for grading.

Undergraduate Problems

All students enrolled should complete the following:

1. Show that the following language is regular:
   {00, 0000, 000000, ...} = {0²ⁿ| n ≥ 1}
2. Use the pumping lemma to show that the following language is not regular:
   {0ⁿ1ⁿ2ⁿ | n ≥ 0}
3. Show that the following language is regular:
   {ε, abc, abcabc, abcabcabc, ...} = {(abc)ⁿ| n ≥ 0}
4. Use the pumping lemma to show that the following language is not regular:
   {www | w ∈ {a,b,c}*}
5. Use the pumping lemma to show that the following language is not regular:
   {0^m1ⁿ | m ≠ n}
6. Is the language:
   {0^m10^m | m ≥ 0} regular or not? Justify your answer.
   

Graduate Problems

Students enrolled for graduate credit, should complete all the undergraduate problems, as well as:

1. Show that the complement of {0ⁿ1ⁿ | n ≥ 0} is not regular
2. Show that the language
   {w | w ∈ {0,1}* is not a palindrome}
   is not regular (a palindrome is a string that reads the same forward and backwards.)