Limited TM

We've looked at ways to try to increase the power of Turing Machines. Let's limit one.

Define Γ = Σ = { 1, blank }. Does this have the power of a standard Turing Machine?

Concurrent TM

Concurrency is all the rage in modern programming languages. If we add to a standard TM a second, independently controlled read/write head, controlled by a separate DFA, does that increase the power of the TM?

Note: We were not able, during recitation, to show how to simulate this with a standard TM. I have since figured out how to do this, assuming that the two read/write heads are "synchronized" in the Java sense, so that the two heads aren't operating on the same square at the same time.

I think this is a good exercise, so I won't give away my solution. :-)

Let ℕ be the set of natural numbers, size(set) be the size of a set, and ℵ₀ be size(ℕ).

If S, T, and U are denumerable sets, size(S) = size(T) = ℵ₀,

1. What is size(S - T)?
   
   
2. What is size(S ∪ T)?
   
   
3. What is size(S ∩ T)?
   
   
4. What is size(S × T)?
   
   
5. What is size(S × T × U)?