I need some help to demonstrate that this set below is decidable, semi-decidable, or undecidable. Here's the set:
H = {p| |Images(fp)| >= 10}
explanation: an element p is on H if the function computed by the progam p has at least 10 images.
I think I have to make a reduction to another language and show that the language reduced is semi-decidable but I'm not capable to do it.
2026-03-28 02:02:20.1774663340
Demonstrate that a language is semi-decidable
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Why bother to make a reduction? If there is such a decider, just make a program $M$ that takes input $x$, tests whether $H( t \mapsto x(x)(t) )$, and if yes then outputs a program that has less than $10$ images, and if no then outputs a program that has at least $10$ images. You can then show that $M(M)$ halts and produces a program that is both in and out of $H$, contradictory. This method generalizes to any non-trivial output behaviour (number of images is just one instance). This shows that $H$ is not decidable.
After that, figure out whether $H$ can be accepted by some program or not. If so, then you're done.