CSG714 04S: Homework 06

Assigned: Wed 10 Mar 2004
Due: Wed 17 Mar 2004

General Instructions

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

2. 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 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: 6 of the following 8 problems
Points: 18 pts per problem

1. Problems 5.17, 5.18, and 5.23

2. Prove that the following languages are undecidable without using Rice's Theorem; i.e., you must make a direct reduction from some known undecidable problem.
   • L = {<M> | M is a Turing machine, and L(M) = 0^*1^*}
   • L = {<M> | M is a Turing machine, and M is a decider}

3. Prove that the following language is undecidable.
   {<G> | G is a CFG, and L(G) = (L(G))^R}

   Hint: Review undecidability results for CFLs using PCP.

4. In class, we showed that an algorithm for minimizing finite automata exists. Specifically, we showed that there exists an algorithm which takes as input any DFA D and returns a DFA D₁ where L(D₁) = L(D) and no DFA D₂ exists where L(D₂) = L(D) and D₂ has fewer states than D₁. A variant of this algorithm is given in Problem 7.35.

   Show that no such algorithm for PDAs exists. Specifically, you should formulate a language L which could be decided by such an algorithm and then prove that the language L is undecidable; thus, the algorithm in question cannot exist.

   Hint: We showed in class that that the following language was undecidable.

   {<G> | G is a CFG, and L(G) = {0,1}^*}

5. Repeat the above problem for Turing Machines; i.e., show that an algorithm for minimizing Turing Machines cannot exist.

6. Problems 7.19 and 7.29

7. Problem 7.22

8. Problem 7.37

   Hint: Construct a directed graph with one vertex for each possible literal and directed edges corresponding to constraints. Test for a particular graph property.

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• Homework 05
• Homework 07