Prof Franklin, Spring 2009
ECS 120, Homework #5 (Covers Sipser 4,5)
Due Tues 26 May, 2009

1. For each of the following sets, explain whether it is countable or uncountable:
a. 0*
b. {0,1}*
c. the powerset of 0*

2. Consider the language L = {<M,w> | M is a TM and M accepts w in at most 2^|w| steps}.
a Show that L is a decidable language.
b. If the proof strategy for Theorem 4.11 were used to try to show that L was undecidable, explain where the proof would break down.

3. Consider the language L = {<M,w> | M is a TM and M accepts w in more than 2^|w| steps}.
a. Is L is a Turing-recognizable language? Explain why or why not.
b. Is the complement of L a Turing-recognizable language? Explain why or why not.

4. This question is about the Post Correspondence Problem.
a. Construct the Modified Post Correspondence Problem instance for TM
M_2 (from Chapt 3, p 144) with input w = 0.  

b. Solve the MPCP instance from part (a), and show that the solution
corresponds to an accepting computation of M_2.

c. Demonstrate that the MPCP instance from part (a) does not work
as a PCP instance, i.e., has solutions that do not correspond
to accepting computations of M_2.

d. Convert the MPCP instance from part (a) into a PCP instance.