I have read that problems resemblings collatz have been shown to be undecidable. Conway proved that apparantly but Im not sure if the proof was constructive.
So I wonder : What is the simplest collatz like problem that is undecidable ?
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It is problematic to compare the difficulty of undecidable problems. – Peter Dec 20 '13 at 12:45
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Simple refers to the collatz like problem. That is in terms of complexity and length. For instance Collatz has two operations for even or for odd. So Collatz has 2 'if' conditions. 255 'if' conditions is less simple than 10 'if' conditions , assuming they are both linear operators. Cleary the difficulty of impossible things ( deciding the undecidable ) cannot be compared. I suppose it was a joke. – mick Dec 20 '13 at 23:25
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Hm, how to be sure that a given statement can't be stated in a "simpler" way? – Stefan Mesken Jun 19 '14 at 01:17
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@user155124 I would think that a qualitatively "pretty simple" answer would probably suffice, if "simplest" can't be quantified. – Alexander Gruber Jun 19 '14 at 03:09
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See also What is the “nearest” problem to the Collatz conjecture that has been successfully resolved?. – lhf Jun 19 '14 at 03:31
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I don't know about simplest, but this answer by Gerry Myerson cites two papers that prove that some natural generalizations of the Collatz problem are undecidable:
Conway, "Unpredictable iterations", 1972
Kurtz and Simon, "The Undecidability of the Generalized Collatz Problem", 2007
There is also a more recent, quite short paper:
- Lehtonen, Two undecidable variants of Collatz’s problems, 2008
the PDF seems freely available
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If i get the second paper correctly , they claim ( accepted ?? ) that the iterations $f(3n) = 2n , f(3n -1) = 4n -1 , f(3n-2) = 4n-2 $ are undecidable.
Is this well accepted ? Im no expert in turing machines. And is this sequence studied somewhere ?
– mick Jul 05 '24 at 18:48 -
... by the heuristic here https://math.stackexchange.com/questions/4665870/collatz-variant-7-x-1 we expect the sequence to go to infinity for most imputs. – mick Jul 05 '24 at 18:57