CS3800 SP11: Homework 07

Assigned: Sat 03-12-11
Due: Fri 03-18-11

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.

Specific Instructions

1. In Problems 1, 2, and 3, you must explain how one type of machine can simulate one or more other machines. (For example, in part of Problem 1, you must show how a TM can simulate a 2-PDA; in Problems 2, you must show how one TM can simulate two other TMs.) You may describe these simulations at a high level, in English, as we did in class.

2. In Problem 4, you should provide a diagonalization proof, as we did in class and as presented in the text. You must show, by contradiction using diagonlization, that no one-to-one mapping can exist between the infinite sets in question.

Problems

1. Exercise 3.9

   Hint: For part (a), show that 2-PDAs can simulate Turing Machines, and for part (b), show that Turing Machines can simulate 3-PDAs. (For the latter part, you will likely need multi-tape Turing Machines, which can themselves be simulated by ordinary Turing Machines.)

2. Exercise 3.15 (d,e)

   Hint: You may use multi-tape Turing Machines which we have proven to be equivalent in power to ordinary Turing Machines.

3. Exercise 3.15 (b,c)

Hint: You may use multi-tape Turing Machines which we have proven to be equivalent in power to ordinary Turing Machines.

4. Let N be the set of natural numbers; i.e., N = {1, 2, 3, ...}. Prove, by diagonalization, that the cardinality of the power set of N is greater than the cardinality of N; i.e., |2^N| > |N|.

(Hint: Let X be the ordered set {a, d, k, z}. Then one can denote the subset Y = {a, k, z} by the sequence 1011; each "1" indicating that the corresponding element from the original set is present in the subset, and each "0" indicating that the corresponding element is absent.)

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