If I use the shell to generate a very long random number, I'd expect the output to belong to me.
Secondly, no, it doesn't depend on who run the AI. Because running the AI isn't called creating the result. The one who run it didn't provide creative input. How is it different from telling someone else "go create"? You didn't create the result either.
Another point - copyright by definition is given as an incentive to create. I don't see much need for an incentive to press a button and do nothing after that.
My understanding is that's not true, see: https://en.wikipedia.org/wiki/Illegal_number. I agree it's stupid but not up to me.
> Also, consider a generative algorithm, that creates every possible combination. Do you expect to potentially own everything that wasn't yet created? Because at some point, that algorithm will get to it.
Well yes, given infinite time, you could write a program to generate every possible sentence. If I saved those sentences on an infinitely large hard drive, yes, I would expect to own the rights to each one.
Note, that doesn't mean I'd be able to sue anyone else who ever speaks. The color of bits  matters.
IMO, the alternative of assigning copyright to the algorithm's creator gets really messy. If an author uses GPT-2 for writing inspiration, and at some point, they copy out a GPT-2 paragraph verbatim, does the author not own that paragraph? Same question for music which incorporates AI-generated samples (or just randomly generated samples).
No, you should not expect it. Because the process wasn't creative. And there is no incentive needed for it either, because it's automated. Check again the definition of copyright, and why it's given in the first place. That should be the way to analyze, whether it's applicable or not.