It’s incredibly disconcerting for so many brilliant thinkers to accept and repeat the circular logic about the paradoxical anti-halt machine “g” — if you make some g which contains f, then I can make f such that it detects this and halts. If you make some g which invokes f, then I can make f which detects this and halts. By definition of the problem, “f” is the outer main function and paradoxical “g” is trapped in the closure of f which would mean f can control the process, not g. The whole basis for both Gödel Incompleteness Theorems and the Halting Problem is based on this idea we can make some paradox machine that does the opposite of whatever we think, without continuing to consider if such recursive properties might be detectable.
A more difficult case would be the machines which loop forever and never hit a prior state, tape tuple, but even in this case, I truly believe the difference is ultimately decidable because an n-state Turing machine which does eventually halt after many steps would necessarily do so by some form of countdown or count up to a limit. This periodicity would presumably be apparent in the frequency domain of the state transition matrix. Also, we might try to connect the (state, tape) tuples (which are enumerable) to the complex plane, and then we could look them up on the Mandelbrot set to see if they converge or diverge. Perhaps we could pick some natural threshold of closeness to the edge of the set where we would categorize machines as being more difficult to analyze. Seems like it would be a natural connection…
(disclaimer, I’m a programmer, not a computational complexity research scientist in this area really, so I am likely wrong, I just think we ought to be wayyyy more skeptical about the bold claims of Gödel and Turing!)
It’s incredibly disconcerting for so many brilliant thinkers to accept and repeat the circular logic about the paradoxical anti-halt machine “g” — if you make some g which contains f, then I can make f such that it detects this and halts. If you make some g which invokes f, then I can make f which detects this and halts. By definition of the problem, “f” is the outer main function and paradoxical “g” is trapped in the closure of f which would mean f can control the process, not g. The whole basis for both Gödel Incompleteness Theorems and the Halting Problem is based on this idea we can make some paradox machine that does the opposite of whatever we think, without continuing to consider if such recursive properties might be detectable.
A more difficult case would be the machines which loop forever and never hit a prior state, tape tuple, but even in this case, I truly believe the difference is ultimately decidable because an n-state Turing machine which does eventually halt after many steps would necessarily do so by some form of countdown or count up to a limit. This periodicity would presumably be apparent in the frequency domain of the state transition matrix. Also, we might try to connect the (state, tape) tuples (which are enumerable) to the complex plane, and then we could look them up on the Mandelbrot set to see if they converge or diverge. Perhaps we could pick some natural threshold of closeness to the edge of the set where we would categorize machines as being more difficult to analyze. Seems like it would be a natural connection…
(disclaimer, I’m a programmer, not a computational complexity research scientist in this area really, so I am likely wrong, I just think we ought to be wayyyy more skeptical about the bold claims of Gödel and Turing!)
WDYT?