Outlier hunting

Stream: Equational

[Douglas McNeil]

In the spirit of https://github.com/teorth/equational_theories/issues/278, I figured I'd make some pictures to see if anything leapt out.. but looking at the current dataset, rather than the finite magmas, because 178981952 is big. (Interestingly enough, o1 had trouble understanding the right isomorphism check to use, but o1-preview was able to sort it out.)

So here are some heatmaps with num_vars (number of variables, from 1 to 6) and var_count (the sum of the multiplicities, e.g. x = x ◇ (y ◇ x) has num_vars 2 and var_count = sum({"x": 3, "y": 1}.values()) = 4. The colours are the fractions of True implications (1=green, 0=red).

image.png

The num_vars = (1,1) and (1,2) slices (so x and x,y) look fun. num_vars=(1,2) in particular is very mischievous: I had to put the annotations on to confirm it, but there are slight differences in var_count=(3,5), (3,6), and (4,6): they're almost, but not quite, zero.

These thirteen implications are a little unusual, as they're an exception to what would otherwise be ruled out by the variable number heuristic.

           i             eq_i     j                       eq_j  num_vars_i  num_vars_j  var_count_i  var_count_j
1525552    3        x = x ◇ x   326        x ◇ y = x ◇ (y ◇ y)           1           2            3            5
1755558    3        x = x ◇ x   375        x ◇ y = (x ◇ x) ◇ y           1           2            3            5
15574694   3        x = x ◇ x  3319  x ◇ y = x ◇ (y ◇ (y ◇ y))           1           2            3            6
16527576   3        x = x ◇ x  3522  x ◇ y = x ◇ ((y ◇ y) ◇ y)           1           2            3            6
17433518   3        x = x ◇ x  3715  x ◇ y = (x ◇ x) ◇ (y ◇ y)           1           2            3            6
17466376   3        x = x ◇ x  3722  x ◇ y = (x ◇ y) ◇ (x ◇ y)           1           2            3            6
18372318   3        x = x ◇ x  3915  x ◇ y = (x ◇ (x ◇ x)) ◇ y           1           2            3            6
19325200   3        x = x ◇ x  4118  x ◇ y = ((x ◇ x) ◇ x) ◇ y           1           2            3            6
20977488   3        x = x ◇ x  4470  x ◇ (y ◇ y) = (x ◇ x) ◇ y           1           2            3            6
15574699   8  x = x ◇ (x ◇ x)  3319  x ◇ y = x ◇ (y ◇ (y ◇ y))           1           2            4            6
18372323   8  x = x ◇ (x ◇ x)  3915  x ◇ y = (x ◇ (x ◇ x)) ◇ y           1           2            4            6
16527596  23  x = (x ◇ x) ◇ x  3522  x ◇ y = x ◇ ((y ◇ y) ◇ y)           1           2            4            6
19325220  23  x = (x ◇ x) ◇ x  4118  x ◇ y = ((x ◇ x) ◇ x) ◇ y           1           2            4            6

BTW, thanks to the infrastructure people for making the datasets so easy for external amateurs to play with!

[Terence Tao]

Thanks for this! It's good to see some actual data-driven visualization to support the theoretical heuristics.

So I see that one-variable equations almost never imply equations in two or more variables (which makes sense, they can only really see close to the "diagonal" of the magma in some sense), whereas equations with four or more variables tend to imply a huge chunk of the other laws (probably because they are so overdetermined that they only admit the trivial magma in many cases.

The exceptional implications you pointed out all involve longer laws that collapse to the identity after applying the hypothesis one or two times to shorten the equation. I guess it might actually be possible to prove that this is the only way in which a one-variable law could imply a higher-variable law.