I am interested in the "nearest" (and "most complex") problem to the Collatz conjecture that has been successfully solved (which Erdos famously said "mathematics is not yet ripe for such problems"). It has been proven that a class of "Collatz-like" problems is undecidable. However, problems that are vaguely similar such as Hofstadter's MIU game (resolved, but admittedly more of a toy problem) are indeed decidable or have been solved.

Related questions

Collatz Conjecture & Grammars / Automata

up vote 22 down vote accepted

An extended comment:

Collatz-like sequences can be computed by small Turing machines having few symbols and states. In "Small Turing machines and generalized busy beaver competition" by P. Michel (2004), there is a nice table that positions Collatz-like problems between decidable TMs (for which the halting problem is decidable) and Universal TMs.

enter image description here

There are TMs that compute Collatz-like sequences for which the decidability is still an open problem: $TM(5,2)$, $TM(3,3)$ and $TM(2,4)$ (where $TM(k,l)$ is the set of Turing Machine with $k$ states and $l$ symbols). I don't know if the results have been inproved.

From the comclusion of the paper:

... The present Collatz-like line is already on its lowest possible level, with the possible exception of $TM(4,2)$, but we conjecture that all machines in this set can be proved to be decidable...

See also "The complexity of small universal Turing machines: a survey" by D. Woods and T. Neary (2007).

Another example of Collatz-like problem for which decidability is an open problem is the Post's tag system: $\mu = 2, v=3,0\rightarrow 00, 1 \rightarrow 1101$; for a recent analysis see "On the boundaries of solvability and unsolvability in tag systems. Theoretical and Experimental Results" by L. De Mol (2009).

  • 8
    to complement the answer: Conway showed that there are Collatz-like sequences which are undecidable ams.org/mathscinet-getitem?mr=392904. i.e. a collatz-like sequence can itself simulate a universal turing machine. – Sasho Nikolov Jun 2 '12 at 16:23
  • thx! the mitchell survey/results are very cool! fyi clarification in the table, a "T" in a cell indicates a TM(k,l) has been shown to exist that is equivalent to the collatz conjecture. the perspective also suggests that the collatz conjecture is not merely an isolated theoretical curiosity but possibly a surface phenomenon of something deeper in computability theory. ps also very interested if any once-open "collatz like problems" have ever been solved...? – vzn Jun 2 '12 at 16:23

Consider the function $T: \mathbb N \rightarrow \mathbb N$, where $T(n)=n/2$ when $n$ is even and $T(n)=n+1$ when $n$ is odd. Then it is known that for any $n \in \mathbb N$, there exists a $k \in \mathbb N$ such that $T^{(k)}(n)=1$.

If instead of $T(n)=n+1$ when $n$ is odd, we had defined $T(n)=3n+1$ when $n$ is odd, we would have the Collatz Conjecture, so I think this is the nearest problem to the Collatz Conjecture that has ever been resolved.

  • 2
    I don't think this satisfies the "most complex" part of the question, as a motivated grade school student can identify the key idea behind the proof of your statement with a bit of thought. – Yonatan N Jul 14 '15 at 22:17
  • But if it is more complex and still resolved, it won't resemble the Collatz Conjecture anymore. Furthermore, the title of his question indicates that he gives priority to "nearest" over "most complex". – Craig Feinstein Jul 15 '15 at 14:25

Your Answer

 
discard

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.