Non-computable vs computable functions

So I am currently working on assignment 3, and was able to draft out my proofs for question 1 - 4 pretty easily, but I found myself stuck for the longest time on coming up with a proof for question 5.

And through reviewing the course notes and lecture slides, I was able to find a clue:

Clue 1. If f reduces to g, then f can be implemented using g and perhaps some other functions, like how halt(f, i) can be created using initialized(g, v) and f_prime(x).

So I began by trying to re-create halt with the implementation of true_that, which I was able to do with the help of another function. But yet, for the longest time, I still couldn't link this piece with an idea for a conclusive solution. And then I realized (because I'm really slow otherwise this shouldn't have taken a whole day):

Clue 2. Then if f is non-computable, then that means g is also non-computable. (Like halt, which we've already proven in class to be non-computable -and we already know what f ang g are in this case, hint hint).


It took me an entire day from when I first started working on the assignment to reach this epiphany, so I decided to take the time and share it with the world (obviously I can't say exactly HOW I did it, but this should definitely set whoever just happens to be desperately looking for help on the right track. And if you're desperate enough to feel the need to browse through 150 slogs, then props to you, and you're welcome (hopefully).)


Now I'm going to go back to pasting all my notations back onto MS word clipboard, so I can write my proofs properly. Why can't I just save them permanently, ugh.


I guess I'm just having a harder time than I thought grasping the idea of a well-defined, yet non-computable function, because it's so abstract and very difficult to visualize. But I do find the idea of an almost paradoxical  function fascinating, and will probably review this topic in finer detail for the exam.

Good luck to everyone, hope you'll have an easier time than I do with the last one.


Δ~~~~Δ
ξ ･ェ･ ξ ξ　~　ξ ξ　　 ξ ξ　　　“~～~～〇 ξ　　　　　　 ξ ξ　ξ　ξ~～~ξ　ξ 　ξ_ξξ_ξ　ξ_ξξ_ξ 　　ヽ(´･ω･)ﾉ 　　　 |　 / 　　　 UU"

P.S: I follow this japanese emoji twitter, so if you think I'm ever going to run out of these, you are wrong.