Has the 3-tag system investigated by Emil Post $(0\to00, 1\to1101)$ been solved? Is there a decision algorithm to determine which starting strings terminate, which end up in a cycle, and which (if any) grow without bound? 


Also, what cycle structures are there? Setting $a=$ '00' and $b=$ '1101', the only cycles I know of begin with $ab, b^2 a^2$, combinations of these, and $a^2 b^3 (a^3 b^3)^n$. Are there any more? 



Is Post's tag system solved?
Here are the two irreducible repeating patterns that Liesbeth de Mol discovered, together with a third high-period irreducible repeating pattern discovered by Rich Schroeppel:


* $b^3 a^5 b^5$ (period 40);
* $a b^2 a b^3 a^3 b^3 a^2 b^2 a^4 b^2$ (period 66);
* $a b^3 a b a b a^2 b^2 a b^9 a^2$ (period 282);


This was discovered [by 2004](https://www.experimentalmath.info/workshop2004/schroeppel-talk.pdf), and as far as I know no other irreducible repeating patterns are known yet.


