CSG714 04S: Homework 01

Assigned: Wed 14 Jan 2004
Due: Wed 21 Jan 2004

Instructions

1. Please review the grading policy outlined in the course information page.

2. On the first page 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 following

   Problem	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.

3. 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.

Problems

Required: 7 of the following 9 problems
Points: 14 pts per problem

1. Problem 1.24

   Note: I would strongly suggest that you attempt and solve this problem. The result itself will be useful for many of the problems below, and the ideas needed to prove it are necessary for many of the remaining problems.

2. • Problem 1.25

   • Consider the following language:
     Even₃ = {w | w in {0,1,2}^* and w is an even integer in base 3}.

     Prove that Even₃ is a regular language.

3. Problems 1.26 and 1.27
4. Problem 1.31
5. Probelm 1.32
6. Consider the following language:
   Multiple₅ = {w | w in {0,1}^* and w, treated as a binary number, is a multiple of 5}.

   Prove that Multiple₅ is a regular language.

   Hint: There is nothing special about the number 5; I could just as easily ask the same question for Multiple₆, Multiple₇, etc. Think "mod"...

7. Convert the following finite automata to regular expressions:
   • machine M₁ from Figure 1.4 and
   • machine M₁ from Exercise 1.1.

8. Prove that the regular languages are closed under the following operation:
   HalfPal(A) = {w | ww^R in A}.

   In other words, HalfPal(A) consists of those strings which are the first halves of all even length palindromes in A.

9. Prove that the regular languages are closed under the following operation:
   FirstHalf(A) = {x | there exists y, |x| = |y|, where xy in A}.

   In other words, FirstHalf(A) consists of the first halves of all even length strings in A.

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• Homework 02