New variant of outcomes.png

Stream: Equational

[Harald Husum]

In https://github.com/teorth/equational_theories/issues/424, I've been experimenting with modifying our outcomes.png in different ways. Just wanted to share a new variant. This time the equations are sorted by the number of equivalence classes they imply, instead of the raw number of equational implications. As before, the equations are grouped by equivalence class.

Probably no revolutionary new patterns to gleam from this, but I find the picture nice to look at.

outcomes.png

[Terence Tao]

Can you remind me what the color scheme represents? Also, is there some way to incorporate duality to make the image more symmetric? That would be aesthetically pleasing, if nothing else.

[Harald Husum]

The color scheme is the same as for the original outcomes.png (the one on our dashboard).

• Muted blue and red are indirect implicitly proven implications and anti-implications, respectively.
• Light blue and red are direct explicitly proven implications and anti-implications.

[Harald Husum]

Terence Tao said:

Also, is there some way to incorporate duality to make the image more symmetric? That would be aesthetically pleasing, if nothing else.

I haven't yet entirely understood what two equations being dual entails, other than that there is some sort of symmetry between them. Is there a primer for this written down somewhere?

[Jared green]

if i interpreted it correctly, its taking each occurance of the operation and flipping it, placing the inputs on opposite sides, once each.

[Harald Husum]

The picture above led me to be fascinated by Equation 895.

It's the fifth equivalence group when you follow the diagonal up from the bottom right. Since it is sorted towards the bottom it means it's a relatively strong law, with lots of implications. Still, there's only 6 equations in the equivalence class, which breaks with the pattern of other strong laws, which seem to amass big equivalence classes. I'm guessing this is also just an artifact of the limitation of the project to a maximum of 4 operations per equation, but it still caught my eye.

[Harald Husum]

Jared green said:

if i interpreted it correctly, its taking each occurance of the operation and flipping it, placing the inputs on opposite sides, once each.

Do you know what this implies in terms of the relationship between the implications and anti-implications of an equation and its dual?

[Jared green]

the fully dual implication has the same truth value as the initial implication

[Jared green]

that is, with a dual antecedent and a dual consequent

[Harald Husum]

I take it that means the structure of implications and anti-implications originating from dual equations should look the same, only with potentially different equations involved?

I'd need to think about how that could be used in the picture...

[Vlad Tsyrklevich]

Perhaps the middle is self-duals with the other duals coming out on it's left/right? EDIT: That doesn't really make sense, after all how do you organize them in the center.

[Michael Bucko]

I also made another variant (optimized with numpy and Kahn’s algorithm), but don't consider it useful enough yet.
newimage.jpg

[Michael Bucko]

Vectorized the elements of the outcome json and turned it into a network. So close to being symmetrical, yet so asymmetrical.
newimage.jpg