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 $f(\overrightarrow{x}):{\omega }^n\hookrightarrow {\omega }^n$ to be of the form

$$f(\overrightarrow{x})=\{\begin{array}{ll}({\rho }_1(\overrightarrow{x}),\dots ,{\rho }_n(\overrightarrow{x}))& {\text{proj}}_1(\overrightarrow{x})\equiv 0modz\\ ({\rho }_{n+1}(\overrightarrow{x}),\dots ,{\rho }_{2n}(\overrightarrow{x}))& {\text{proj}}_1(\overrightarrow{x})\equiv 1modz\\ ⋮& ⋮\\ ({\rho }_{nz-n+1}(\overrightarrow{x}),\dots ,{\rho }_{nz}(\overrightarrow{x}))& {\text{proj}}_1(\overrightarrow{x})\equiv -1modz,\end{array}$$

where $z\ge 2$ 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 $\overrightarrow{x}$ 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)

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)