#decidability

Number 195 in an ongoing series celebrating remarkable women in science, technology, engineering, and mathematics.

Trust Me I’M A Doctor…. Working towards a Healthy Prescriptive Consent According to Decision Theory by David J John O Regan in Biomedical Journal of Scientific & Technical Research (BJSTR) https://biomedres.us/fulltexts/BJSTR.MS.ID.000393.php

Note, it’s not that no one ever cared to implement such an algorithm, or that it would just be an unreasonably inefficient brute-force-ish method. The interesting thing is, such an algorithm cannot exist—it is doomed to (sometimes) output wrong results or not give results at all.

Recursive Functions and Metamathematics: Problems of Completeness and Decidability, Gödel’s Theorems (Synthese Library) Recursive Functions and Metamathematics: Problems of Completeness and Decidability, Gödel’s Theorems (Synthese Library) by Roman Murawski…

that is so wrong as to be totally disgusting”

For imagine we have a procedure called P that for specified input permits you to see whether specified source code, with all of its faults, defines a routine that eventually halts.   You feed in your program, with suitable data, and P gets to work, and a little while later (in finite compute time) correctly infers whether infinite looping behavior occurs.   If there will be no looping, then P prints out ‘Good.’ That means work on this input will halt, as it should. But if it detects an unstoppable loop, then P reports ‘Bad!’ --- which means you’re in the soup.   Well, the truth is that P cannot possibly be, because if you wrote it and gave it to me, I could use it to set up a logical bind that would shatter your reason and scramble your mind.   Here’s the trick that I’ll use -- and it’s simple to do. I’ll define a procedure, which I will call Q, that will use P’s predictions of halting success to stir up a terrible logical mess.   For a specified program, say A, one supplies, the first step of this program called Q I devise is to find out from P what’s the right thing to say of the looping behavior of A run on A.   If P’s answer is ‘Bad!’, Q will suddenly stop. But otherwise, Q will go back to the top, and start off again, looping endlessly back, till the universe dies and turns frozen and black.   And this program called Q wouldn’t stay on the shelf; I would ask it to forecast its run on itself. When it reads its own source code, just what will it do? What’s the looping behavior of Q run on Q?   If P warns of infinite loops, Q will quit; yet P is supposed to speak truly of it! And if Q’s going to quit, then P should say ‘Good.’ Which makes Q start to loop! (P denied that it would.)   No matter how P might perform, Q will scoop it: Q uses P’s output to make P look stupid. Whatever P says, it cannot predict Q: P is right when it’s wrong, and is false when it’s true!   I’ve created a paradox, neat as can be --- and simply by using your putative P. When you posited P you stepped into a snare; Your assumption has led you right into my lair.   So where can this argument possibly go? I don’t have to tell you; I’m sure you must know. A reductio: There cannot possibly be a procedure that acts like the mythical P.   You can never find general mechanical means for predicting the acts of computing machines; it’s something that cannot be done. So we users must find our own bugs. Our computers are losers!