Talk:Collatz-like

Definition

FWIW, I think the current definition is a bit too restrictive for the general idea of Collatz-like. Notice Bigfoot where each subfunction can depend upon all the inputs and the Tetration Machine example where the function don't need to be linear/polynomials.

I think of Collatz-like as any situation where the the function is defined piecewise based on parity of the inputs.

That said, I think the linear case is one of the most important and should definitely be documented well! Sligocki (talk) 14:15, 30 May 2025 (UTC)

Thanks Shawn. If you want the definition to be inclusive to TMs like Bigfoot and TetraMach, perhaps we could define an $n$-ary CLF $f({\vec {x}}):\omega ^{n}\hookrightarrow \omega ^{n}$ to be of the form

$$f({\vec {x}})={\begin{cases}(\rho _{1}({\vec {x}}),\dots ,\rho _{n}({\vec {x}}))&{\text{proj}}_{1}({\vec {x}})\equiv 0\mod z\\(\rho _{n+1}({\vec {x}}),\dots ,\rho _{2n}({\vec {x}}))&{\text{proj}}_{1}({\vec {x}})\equiv 1\mod z\\\quad \quad \quad \quad \quad \vdots &\quad \quad \quad \quad \quad \vdots \\(\rho _{nz-n+1}({\vec {x}}),\dots ,\rho _{nz}({\vec {x}}))&{\text{proj}}_{1}({\vec {x}})\equiv -1\mod z,\end{cases}}$$

where $z\geq 2$ is a positive integer, ${\text{proj}}_{1}$ is the first projection function, and $\rho _{1},\dots \rho _{nz}$ are (again not necessarily distinct) $n$-ary general recursive functions. I've shortened the input $n$-tuple $(x_{1},\dots ,x_{n})$ to ${\vec {x}}$ to save space. This includes the current def., of course; a subsection is to be written highlighting Conway's generalized Collatz mappings which are equivalent to the current definition's linear case anyhow.

This is notationally heavy. It can probably be simplified by sacrificing space lol. The fact that general recursive functions are defined only in the domain of nonnegative integers $\omega$ doesn't seem to pose any grave issues. The antihydra problem of whether $A^{\circ x}(8,0)=(y,-1)$ for some $x,y$ can be easily reworded with as $A^{\circ x}(8,1)=(y,0)$ instead.

This new definition would not include functions which choose their subfunctions based on the modularity of multiple input values ... have we encountered one such beast yet? :) -- ISquillante (talk) 16:40, 30 May 2025 (UTC)

Thanks for all your attention here! I'm always excited to see someone working to improve the wiki. I do think that is closer to my idea of what a Collatz-like function is, but (as you say) notationally it is quite heavy. Also I wonder if what we're defining at this point is just equivalent to general recursive functions (I assume that any Collatz-like function is a general recursive functions). I think there is a challenge here between making a definition that is maximally general (to not miss any examples), but also useful. I think my intuition is to allow Collatz-like to be described intuitively (a function which is piecewise defined based on remainders), but then perhaps subsections or other articles focussing on specific rigorously defined types of Collatz-like function (say linear ones on one variable which come up a ton!). I think that strikes the balance that allow Bigfoot and tetrational machines to be Collatz-like (which is how many of us see them), while noting a common simpler definition that can actually be used to prove things. What do you think? Sligocki (talk) 17:05, 30 May 2025 (UTC)

I also think this aligns nicely with cosmo's suggestion of talking about Conway's "Generalised Collatz Maps" which I think are equivalent to the linear Collatz-like functions of one variable I'm talking about. Sligocki (talk) 17:09, 30 May 2025 (UTC)

Also, note, someone created a Consistent Collatz page a while ago that might fit in with this idea, but I'm not really sure how to unite them ... Sligocki (talk) 14:17, 30 May 2025 (UTC)