MHB Can Turing Machines Enumerate Union and Cartesian Product of Constructible Sets?

  • Thread starter Thread starter mathmari
  • Start date Start date
AI Thread Summary
The discussion revolves around the properties of constructible enumerable sets, specifically sets $S_1$ and $S_2$. It is established that if both sets are constructible enumerable, they can be enumerated by a Turing machine. The participants confirm that the union of these sets, $S_1 \cup S_2$, is also constructible enumerable by constructing a Turing machine that alternates enumeration between elements of $S_1$ and $S_2$. Additionally, the conversation addresses how to demonstrate that the Cartesian product $S_1 \times S_2$ is constructible enumerable. It is suggested to use a pairing function, similar to the proof that $\mathbb{N} \times \mathbb{N}$ is countable, or to leverage the property that recursively enumerable sets are semidecidable and can be accepted by a Turing machine.
mathmari
Gold Member
MHB
Messages
4,984
Reaction score
7
Hey! :o

We have that the sets $S_1$ and $S_2$ are constructible enumerable, that means that there is Turing machine that enumerates them, right?

To show that the set $S_1 \cup S_2$ is also constructible enumerable, we construct a Tuiring machine that enumerates alternately one element of $S_1$ and one of $S_2$.
Is this correct?

How can we show that the cartesian product $S_1 \times S_2$ is also constructible enumerable?
 
Technology news on Phys.org
mathmari said:
We have that the sets $S_1$ and $S_2$ are constructible enumerable, that means that there is Turing machine that enumerates them, right?
I am not familiar with this term. Did you mean "recursively enumerable"?

mathmari said:
To show that the set $S_1 \cup S_2$ is also constructible enumerable, we construct a Tuiring machine that enumerates alternately one element of $S_1$ and one of $S_2$.
Is this correct?
Yes.

mathmari said:
How can we show that the cartesian product $S_1 \times S_2$ is also constructible enumerable?
You can use a pairing function from the proof that $\Bbb N\times\Bbb N$ is countable. Or you can use the fact that r.e. sets are semidecidable, i.e., accepted by some TM.
 
Thread 'Is this public key encryption?'
I've tried to intuit public key encryption but never quite managed. But this seems to wrap it up in a bow. This seems to be a very elegant way of transmitting a message publicly that only the sender and receiver can decipher. Is this how PKE works? No, it cant be. In the above case, the requester knows the target's "secret" key - because they have his ID, and therefore knows his birthdate.
Thread 'Project Documentation'
Trying to package up a small bank account manager project that I have been tempering on for a while. One that is certainly worth something to me. Although I have created methods to whip up quick documents with all fields and properties. I would like something better to reference in order to express the mechanical functions. It is unclear to me about any standardized format for code documentation that exists. I have tried object orientated diagrams with shapes to try and express the...
Back
Top