CS4311: Introduction to Computation Theory hw9

CS 4311 Introduction to Computation Theory
Homework 11: Chapter 5

Due: Wednesday, 11/29/06, beginning of class (Assigned: Wednesday, 11/15/06)

You are required to turn in answers to all the questions. However, only a random subset of the questions will be graded. While discussion with others is permitted and encouraged, the final work should be done individually. You are not allowed to work in groups. The answers, comments, and programs (if any) must be the original work of the author. You are allowed to build on material supplied in the class. If you use any other source than the current class notes and the textbook, specify it clearly.

1. Consider the following language L2.

L2 = {< M,w,q > | M is a TM, q is a state, and M enters state q during the computation on w}.

Use reduction to show that L2 is undecidable.

2. Consider the problem of determining if a Turing machine rejects at least two strings, i.e., rejects two or more strings. Formulate this problem as a language and show that it is undecidable.

3. (Exercise 5.16) Consider the problem of testing whether a two-tape Turing machine ever writes a nonblank symbol on its second tape. Formulate this problem as a language, and show that it is undecidable.

4. Consider the problem of determining whether ε ∈ L(M) where M is a Turing machine. Note that we are not trying to determine whether L(M) is empty, but rather whether M accepts the empty string.

5. Suppose that we prove a language L to be undecidable.