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**Assigned:**
Thu 07 Feb 2008

**Due:**
Thu 14 Feb 2008

- Please review the
grading policy
outlined in the course information page.
- On the
*first page*of each part of your solution write-up, you*must*make explicit which problems are to be graded for "regular credit", which problems are to be graded for "extra credit", and which problems you did not attempt. Please use a table something like the followingProblem 01 02 03 04 05 06 07 08 09 ... Credit RC RC RC EC RC RC NA RC RC ... where "RC" is "regular credit", "EC" is "extra credit", and "NA" is "not applicable" (not attempted). Failure to do so will result in an arbitrary set of problems being graded for regular credit, no problems being graded for extra credit, and a five percent penalty assessment.

- You must also write down with whom you worked on the assignment. If this changes from problem to problem, then you should write down this information separately with each problem.

- For each part of Problem 1 below, you should either prove
that the assertion is true (by appealing to closure properties) or
demonstrate that it is false by providing a counterexample.
- For Problems 2 through 6, you should use the Pumping
Lemma and/or appeal to closure properties. For examples of
complete and correct Pumping Lemma proofs, see Examples 1.73,
1.74, 1.75, 1.76, and 1.77 on pages 80-82 of the Sipser text.
The use of closure properties may also be helpful and may, in fact, yield complete proofs without the need to resort to the Pumping Lemma (as discussed in class).

For examples of proofs for non-regularity via closure properties and the Pumping Lemma, please see the following linked handout.

**Required:** 5 of the following 6 problems

**Points:** 20 pts per problem

- Given any two sets
*A*and*B*, the*set difference**A-B*is constructed by removing the elements in*B*from*A*. For example, if*A*= {1, 2, 3, 4, 5} and*B*= {2, 4, 6, 8}, then*A-B*= {1, 3, 5}. In mathematical terms,Let

*R*be any regular language and let*N*be any non-regular language. (*Note:**R*is*not*the*set*of regular languages; it is a variable which represents any*particular*regular language. Similarly for*N*.)For each of the following assertions, either prove that the assertion is true or demonstrate that it is false by providing a counterexample.

*R-N*is always regular.*N-R*is always non-regular.

- Prove that the following languages are non-regular:
- the language defined in Exercise 1.29 part (b) and
- {
*w*: every prefix of*w*has at least as many 0s as 1s}.

- Prove that the following languages are non-regular:
- the language defined in Problem 1.46 part (c) and
- {
*w*:*w*contains a prime number of 0s and a prime number of 1s}.

- Problems 1.46 part (d) and 1.49 part (b).
- Prove that the following languages are non-regular:
- {0
^{m}1^{n}0^{m+n}:*m*>= 1 and*n*>= 1} and - {0
^{i}1^{j}0^{k}:*j*= max{*i*,*k*}}.

- {0
- Prove that the following languages are non-regular:
- the language defined in Problem 1.35 and
- {
*w*:*w*contains twice as many 0s as 1s}.

jaa@ccs.neu.edu