Talk:Collatz-like - Revision history

Sligocki: /* Definition */

2025-05-30T17:09:49Z

Definition

← Older revision		Revision as of 17:09, 30 May 2025
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	::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? [[User:Sligocki\|Sligocki]] ([[User talk:Sligocki\|talk]]) 17:05, 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? [[User:Sligocki\|Sligocki]] ([[User talk: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. [[User:Sligocki\|Sligocki]] ([[User talk: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 ... [[User:Sligocki\|Sligocki]] ([[User talk:Sligocki\|talk]]) 14:17, 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 ... [[User:Sligocki\|Sligocki]] ([[User talk:Sligocki\|talk]]) 14:17, 30 May 2025 (UTC)

Sligocki: /* Definition */

2025-05-30T17:05:52Z

Definition

← Older revision		Revision as of 17:05, 30 May 2025
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	: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 <math>\omega</math> doesn't seem to pose any grave issues. The antihydra problem of whether <math>A^{\circ x}(8,0)=(y,-1)</math> for some <math>x,y</math> can be easily reworded with as <math>A^{\circ x}(8,1)=(y,0)</math> instead.		: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 <math>\omega</math> doesn't seem to pose any grave issues. The antihydra problem of whether <math>A^{\circ x}(8,0)=(y,-1)</math> for some <math>x,y</math> can be easily reworded with as <math>A^{\circ x}(8,1)=(y,0)</math> 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? :) [[User:ISquillante\|-- ISquillante]] ([[User talk:ISquillante\|talk]]) 16:40, 30 May 2025 (UTC)		: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? :) [[User:ISquillante\|-- ISquillante]] ([[User talk: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? [[User:Sligocki\|Sligocki]] ([[User talk:Sligocki\|talk]]) 17:05, 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 ... [[User:Sligocki\|Sligocki]] ([[User talk:Sligocki\|talk]]) 14:17, 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 ... [[User:Sligocki\|Sligocki]] ([[User talk:Sligocki\|talk]]) 14:17, 30 May 2025 (UTC)

ISquillante: /* Definition */

2025-05-30T16:41:36Z

Definition

← Older revision		Revision as of 16:41, 30 May 2025
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	That said, I think the linear case is one of the most important and should definitely be documented well! [[User:Sligocki\|Sligocki]] ([[User talk:Sligocki\|talk]]) 14:15, 30 May 2025 (UTC)		That said, I think the linear case is one of the most important and should definitely be documented well! [[User:Sligocki\|Sligocki]] ([[User talk: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 <math>n</math>-ary CLF <math>f(\vec{x}):\omega^n\hookrightarrow\omega^n</math> to be of the form
			<math display="block">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} </math>
			:where <math>z\ge2</math> is a positive integer, <math>\text{proj}_1</math> is the first projection function, and <math>\rho_1,\dots\rho_{nz}</math> are (again not necessarily distinct) <math>n</math>-ary ''general recursive functions''. I've shortened the input <math>n</math>-tuple <math>(x_1,\dots,x_n)</math> to <math>\vec{x}</math> 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 <math>\omega</math> doesn't seem to pose any grave issues. The antihydra problem of whether <math>A^{\circ x}(8,0)=(y,-1)</math> for some <math>x,y</math> can be easily reworded with as <math>A^{\circ x}(8,1)=(y,0)</math> 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? :) [[User:ISquillante\|-- ISquillante]] ([[User talk: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 ... [[User:Sligocki\|Sligocki]] ([[User talk:Sligocki\|talk]]) 14:17, 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 ... [[User:Sligocki\|Sligocki]] ([[User talk:Sligocki\|talk]]) 14:17, 30 May 2025 (UTC)

Sligocki: /* Definition */

2025-05-30T14:17:02Z

Definition

← Older revision		Revision as of 14:17, 30 May 2025
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	That said, I think the linear case is one of the most important and should definitely be documented well! [[User:Sligocki\|Sligocki]] ([[User talk:Sligocki\|talk]]) 14:15, 30 May 2025 (UTC)		That said, I think the linear case is one of the most important and should definitely be documented well! [[User:Sligocki\|Sligocki]] ([[User talk:Sligocki\|talk]]) 14:15, 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 ... [[User:Sligocki\|Sligocki]] ([[User talk:Sligocki\|talk]]) 14:17, 30 May 2025 (UTC)

Sligocki: /* Definition */ new section

2025-05-30T14:15:06Z

Definition: new section

New page

== 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! [[User:Sligocki|Sligocki]] ([[User talk:Sligocki|talk]]) 14:15, 30 May 2025 (UTC)

← Older revision		Revision as of 17:09, 30 May 2025
Line 20:		Line 20:

	::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? [[User:Sligocki\|Sligocki]] ([[User talk:Sligocki\|talk]]) 17:05, 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? [[User:Sligocki\|Sligocki]] ([[User talk: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. [[User:Sligocki\|Sligocki]] ([[User talk: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 ... [[User:Sligocki\|Sligocki]] ([[User talk:Sligocki\|talk]]) 14:17, 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 ... [[User:Sligocki\|Sligocki]] ([[User talk:Sligocki\|talk]]) 14:17, 30 May 2025 (UTC)