CS4311: Introduction to Computation Theory hw9

CS 4311 Introduction to Computation Theory
Homework 10: Chapters 4 and 5

Due: Wednesday, 11/10/06, beginning of class (Assigned: Friday, 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. (Exercise 4.4) Let AεCFG = {<G> | G is a CFG that generates ε}. Show that AεCFG is decidable.

2. Consider the problem of determining if a DFA accepts any string of length 3. Formulate this problem as a language and show that it is decidable.

3. Consider the following language L1.

L1 = {< M,w > | M is a TM and input w causes M to move its head to the blank portion of the tape}.

Show that L1 is decidable.

4. 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.

5. 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.