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groups=:[: +/\ 2 </\ 0 , *
ooze=: [ >. [ +&* [ * [: ; groups@[ <@(* * 2 < >./)/. +
percolate=: ooze/\.@|.^:2^:_@(* (1 + # {. 1:))
trial=: percolate@([ >: ]?@$0:)
simulate=: %@[ * [: +/ (2 e. {:)@trial&15 15"0@#
Explanation:
groups uses a sum fold on successive prefixes to identify regions of contiguous bits.
2 0 2 0 1 0 1 0 0 1 1 1 1 1 0
2 0 2 0 1 0 1 0 0 1 1 1 1 1 0
* 2 0 2 0 1 0 1 0 0 1 1 1 1 1 0
1 0 1 0 1 0 1 0 0 1 1 1 1 1 0
(0, *) 2 0 2 0 1 0 1 0 0 1 1 1 1 1 0
0 1 0 1 0 1 0 1 0 0 1 1 1 1 1 0
(2 </\ 0, *) 2 0 2 0 1 0 1 0 0 1 1 1 1 1 0
1 0 1 0 1 0 1 0 0 1 0 0 0 0 0
([: +/\ 2 </\ 0, *) 2 0 2 0 1 0 1 0 0 1 1 1 1 1 0
1 1 2 2 3 3 4 4 4 5 5 5 5 5 5
ooze propagates 2s from its right argument to adjacent groups of non-zero locations in its left argument.
2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 + 0 0 1 1 1 1 0 2 0 2 0 1 1 1 1
2 0 3 1 2 1 1 2 0 3 1 2 2 2 1
2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 (groups@[ <@(* * 2 < >./)/. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 1
┌───┬───┬───┬─────┬───────────┐
│0 0│1 1│0 0│0 0 0│1 1 1 1 1 1│
└───┴───┴───┴─────┴───────────┘
2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 ([: ; groups@[ <@(* * 2 < >./)/. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 1
0 0 1 1 0 0 0 0 0 1 1 1 1 1 1
2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 ([ * [: ; groups@[ <@(* * 2 < >./)/. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 1
0 0 2 0 0 0 0 0 0 1 1 1 1 1 0
2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 ([ +&* [ * [: ; groups@[ <@(* * 2 < >./)/. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 1
1 0 2 0 1 0 1 0 0 2 2 2 2 2 0
2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 ([ >. [ +&* [ * [: ; groups@[ <@(* * 2 < >./)/. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 1
2 0 2 0 1 0 1 0 0 2 2 2 2 2 0
Note that u/. uses the key in the left argument to gather corresponding values from the right argument (which are then processed by u).
2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 (groups@[ </. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 1
┌───┬───┬───┬─────┬───────────┐
│2 0│3 1│2 1│1 2 0│3 1 2 2 2 1│
└───┴───┴───┴─────┴───────────┘
2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 (groups@[ <@(>./)/. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 1
┌─┬─┬─┬─┬─┐
│2│3│2│2│3│
└─┴─┴─┴─┴─┘
2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 (groups@[ <@(2 < >./)/. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 1
┌─┬─┬─┬─┬─┐
│0│1│0│0│1│
└─┴─┴─┴─┴─┘
2 0 2 0 1 0 1 0 0 1 1 1 1 1 0 (groups@[ <@(* * 2 < >./)/. +) 0 0 1 1 1 1 0 2 0 2 0 1 1 1 1
┌───┬───┬───┬─────┬───────────┐
│0 0│1 1│0 0│0 0 0│1 1 1 1 1 1│
└───┴───┴───┴─────┴───────────┘
We use +&* because 1 +&* 2 is 2, but also 2 +&* 2 is 2:
+&* table 0 1 2
┌───┬─────┐
│+&*│0 1 2│
├───┼─────┤
│0 │0 1 1│
│1 │1 2 2│
│2 │1 2 2│
└───┴─────┘
percolate changes 1s in the first row to 2s, then flips over that result and performs an ooze right fold on suffixes, and then does that again (which restores the original orientation and propagates 2s in all directions). And it repeats this until all changes have been propagated.
A
1 1 1 0 0 0 0 0 1 1 1 1 0 1 0
1 1 1 1 0 1 0 0 1 1 0 0 0 1 0
0 1 1 0 0 0 1 1 0 0 1 1 0 1 0
0 1 1 1 0 1 1 1 0 1 1 0 0 1 0
1 0 1 1 0 1 1 1 1 0 1 1 0 1 1
0 0 1 0 0 1 1 0 1 0 1 1 1 0 1
0 1 1 0 1 0 1 0 0 1 0 1 1 0 0
1 0 1 0 1 1 0 0 0 0 1 0 1 0 0
1 1 1 0 1 0 0 0 0 0 0 0 0 1 1
1 1 0 1 0 1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 0 0 1 0 1 0 0 0 1
1 0 1 1 1 1 1 0 1 0 0 1 1 0 0
1 1 1 1 0 1 1 1 1 1 0 0 1 0 1
1 1 1 1 1 0 0 1 1 1 1 0 0 1 0
1 1 1 0 1 1 0 0 1 0 1 0 0 1 1
(1 + # {. 1:) A
2 1 1 1 1 1 1 1 1 1 1 1 1 1 1
(* (1 + # {. 1:)) A
2 2 2 0 0 0 0 0 2 2 2 2 0 2 0
1 1 1 1 0 1 0 0 1 1 0 0 0 1 0
0 1 1 0 0 0 1 1 0 0 1 1 0 1 0
0 1 1 1 0 1 1 1 0 1 1 0 0 1 0
1 0 1 1 0 1 1 1 1 0 1 1 0 1 1
0 0 1 0 0 1 1 0 1 0 1 1 1 0 1
0 1 1 0 1 0 1 0 0 1 0 1 1 0 0
1 0 1 0 1 1 0 0 0 0 1 0 1 0 0
1 1 1 0 1 0 0 0 0 0 0 0 0 1 1
1 1 0 1 0 1 1 0 1 1 1 1 1 1 1
1 1 0 1 1 1 0 0 1 0 1 0 0 0 1
1 0 1 1 1 1 1 0 1 0 0 1 1 0 0
1 1 1 1 0 1 1 1 1 1 0 0 1 0 1
1 1 1 1 1 0 0 1 1 1 1 0 0 1 0
1 1 1 0 1 1 0 0 1 0 1 0 0 1 1
|. (* (1 + # {. 1:)) A
1 1 1 0 1 1 0 0 1 0 1 0 0 1 1
1 1 1 1 1 0 0 1 1 1 1 0 0 1 0
1 1 1 1 0 1 1 1 1 1 0 0 1 0 1
1 0 1 1 1 1 1 0 1 0 0 1 1 0 0
1 1 0 1 1 1 0 0 1 0 1 0 0 0 1
1 1 0 1 0 1 1 0 1 1 1 1 1 1 1
1 1 1 0 1 0 0 0 0 0 0 0 0 1 1
1 0 1 0 1 1 0 0 0 0 1 0 1 0 0
0 1 1 0 1 0 1 0 0 1 0 1 1 0 0
0 0 1 0 0 1 1 0 1 0 1 1 1 0 1
1 0 1 1 0 1 1 1 1 0 1 1 0 1 1
0 1 1 1 0 1 1 1 0 1 1 0 0 1 0
0 1 1 0 0 0 1 1 0 0 1 1 0 1 0
1 1 1 1 0 1 0 0 1 1 0 0 0 1 0
2 2 2 0 0 0 0 0 2 2 2 2 0 2 0
ooze/\. |. (* (1 + # {. 1:)) A
2 2 2 0 2 2 0 0 1 0 1 0 0 1 1
2 2 2 2 2 0 0 1 1 1 1 0 0 1 0
2 2 2 2 0 1 1 1 1 1 0 0 1 0 1
2 0 1 1 1 1 1 0 1 0 0 1 1 0 0
2 2 0 1 1 1 0 0 1 0 1 0 0 0 1
2 2 0 1 0 1 1 0 1 1 1 1 1 1 1
2 2 2 0 1 0 0 0 0 0 0 0 0 1 1
1 0 2 0 1 1 0 0 0 0 1 0 1 0 0
0 2 2 0 1 0 1 0 0 1 0 1 1 0 0
0 0 2 0 0 1 1 0 1 0 1 1 1 0 2
1 0 2 2 0 1 1 1 1 0 1 1 0 2 2
0 2 2 2 0 1 1 1 0 1 1 0 0 2 0
0 2 2 0 0 0 1 1 0 0 1 1 0 2 0
2 2 2 2 0 1 0 0 2 2 0 0 0 2 0
2 2 2 0 0 0 0 0 2 2 2 2 0 2 0
ooze/\.@|.^:2 (* (1 + # {. 1:)) A
2 2 2 0 0 0 0 0 2 2 2 2 0 2 0
2 2 2 2 0 1 0 0 2 2 0 0 0 2 0
0 2 2 0 0 0 1 1 0 0 1 1 0 2 0
0 2 2 2 0 1 1 1 0 1 1 0 0 2 0
1 0 2 2 0 1 1 1 1 0 1 1 0 2 2
0 0 2 0 0 1 1 0 1 0 1 1 1 0 2
0 2 2 0 1 0 1 0 0 1 0 1 1 0 0
2 0 2 0 1 1 0 0 0 0 1 0 1 0 0
2 2 2 0 1 0 0 0 0 0 0 0 0 1 1
2 2 0 2 0 2 2 0 1 1 1 1 1 1 1
2 2 0 2 2 2 0 0 1 0 1 0 0 0 1
2 0 2 2 2 2 2 0 1 0 0 1 1 0 0
2 2 2 2 0 1 1 1 1 1 0 0 1 0 1
2 2 2 2 2 0 0 1 1 1 1 0 0 1 0
2 2 2 0 2 2 0 0 1 0 1 0 0 1 1
ooze/\.@|.^:2^:2 (* (1 + # {. 1:)) A
2 2 2 0 0 0 0 0 2 2 2 2 0 2 0
2 2 2 2 0 1 0 0 2 2 0 0 0 2 0
0 2 2 0 0 0 1 1 0 0 1 1 0 2 0
0 2 2 2 0 1 1 1 0 1 1 0 0 2 0
1 0 2 2 0 1 1 1 1 0 1 1 0 2 2
0 0 2 0 0 1 1 0 1 0 1 1 1 0 2
0 2 2 0 1 0 1 0 0 1 0 1 1 0 0
2 0 2 0 1 1 0 0 0 0 1 0 1 0 0
2 2 2 0 1 0 0 0 0 0 0 0 0 2 2
2 2 0 2 0 2 2 0 2 2 2 2 2 2 2
2 2 0 2 2 2 0 0 2 0 1 0 0 0 1
2 0 2 2 2 2 2 0 2 0 0 1 1 0 0
2 2 2 2 0 2 2 2 2 2 0 0 1 0 1
2 2 2 2 2 0 0 2 2 2 2 0 0 1 0
2 2 2 0 2 2 0 0 2 0 2 0 0 1 1
ooze/\.@|.^:2^:3 (* (1 + # {. 1:)) A
2 2 2 0 0 0 0 0 2 2 2 2 0 2 0
2 2 2 2 0 1 0 0 2 2 0 0 0 2 0
0 2 2 0 0 0 1 1 0 0 1 1 0 2 0
0 2 2 2 0 1 1 1 0 1 1 0 0 2 0
1 0 2 2 0 1 1 1 1 0 1 1 0 2 2
0 0 2 0 0 1 1 0 1 0 1 1 1 0 2
0 2 2 0 1 0 1 0 0 1 0 1 1 0 0
2 0 2 0 1 1 0 0 0 0 1 0 1 0 0
2 2 2 0 1 0 0 0 0 0 0 0 0 2 2
2 2 0 2 0 2 2 0 2 2 2 2 2 2 2
2 2 0 2 2 2 0 0 2 0 2 0 0 0 2
2 0 2 2 2 2 2 0 2 0 0 1 1 0 0
2 2 2 2 0 2 2 2 2 2 0 0 1 0 1
2 2 2 2 2 0 0 2 2 2 2 0 0 1 0
2 2 2 0 2 2 0 0 2 0 2 0 0 1 1
ooze/\.@|.^:2^:_ (* (1 + # {. 1:)) A
2 2 2 0 0 0 0 0 2 2 2 2 0 2 0
2 2 2 2 0 1 0 0 2 2 0 0 0 2 0
0 2 2 0 0 0 1 1 0 0 1 1 0 2 0
0 2 2 2 0 1 1 1 0 1 1 0 0 2 0
1 0 2 2 0 1 1 1 1 0 1 1 0 2 2
0 0 2 0 0 1 1 0 1 0 1 1 1 0 2
0 2 2 0 1 0 1 0 0 1 0 1 1 0 0
2 0 2 0 1 1 0 0 0 0 1 0 1 0 0
2 2 2 0 1 0 0 0 0 0 0 0 0 2 2
2 2 0 2 0 2 2 0 2 2 2 2 2 2 2
2 2 0 2 2 2 0 0 2 0 2 0 0 0 2
2 0 2 2 2 2 2 0 2 0 0 1 1 0 0
2 2 2 2 0 2 2 2 2 2 0 0 1 0 1
2 2 2 2 2 0 0 2 2 2 2 0 0 1 0
2 2 2 0 2 2 0 0 2 0 2 0 0 1 1
(^:_ repeats an operation until it reaches a [[wp:Fixed_point_(mathematics)|fixed point]]. In other words, it's basically a while loop.)
trial generates an arbitrary connection matrix and runs percolate on it. You'll typically get a different result from each trial.
0.6 0: 15 15
0
0.6 (] ?@$ 0:) 15 15
0.251168 0.233985 0.14541 0.219665 0.854985 0.0104645 0.449358 0.397573 0.0221767 0.701012 0.5692 0.320853 0.9029 0.0698325 0.439166
0.919973 0.31569 0.733443 0.397144 0.314414 0.0626569 0.125752 0.222472 0.859704 0.517682 0.249954 0.420026 0.974397 0.353249 0.199624
0.0632196 0.729989 0.0384723 0.962052 0.583483 0.794639 0.101726 0.987826 0.0711966 0.0199436 0.692024 0.281396 0.23614 0.445332 0.669898
0.817156 0.896721 0.128014 0.499834 0.86062 0.544871 0.296865 0.762644 0.583945 0.325163 0.701976 0.863764 0.631463 0.590224 0.0233733
0.577801 0.607615 0.893612 0.372065 0.692356 0.385318 0.948463 0.179848 0.482163 0.40125 0.690535 0.48328 0.936104 0.813984 0.319394
0.311636 0.27107 0.565229 0.152493 0.990904 0.856595 0.0755649 0.511731 0.636254 0.0199192 0.0299945 0.465337 0.472827 0.978434 0.207082
0.343524 0.388673 0.169093 0.133254 0.520914 0.736237 0.663096 0.757243 0.901397 0.367862 0.94831 0.45684 0.0569871 0.0211516 0.833437
0.745867 0.62734 0.790395 0.909088 0.359723 0.265804 0.176521 0.34355 0.0484308 0.775815 0.593603 0.023714 0.359704 0.621523 0.567126
0.866712 0.572275 0.679043 0.27958 0.649579 0.91871 0.229361 0.495236 0.646061 0.980648 0.563149 0.461033 0.630641 0.115773 0.460841
0.00359967 0.198523 0.509623 0.995623 0.395574 0.130878 0.458647 0.304251 0.552992 0.491661 0.0503197 0.523992 0.554198 0.818936 0.392742
0.696368 0.3988 0.931908 0.776482 0.23695 0.686757 0.799919 0.471909 0.441598 0.145376 0.470236 0.728451 0.567937 0.26908 0.564895
0.826714 0.202945 0.673436 0.655122 0.61951 0.0459954 0.777457 0.968675 0.418669 0.862087 0.836782 0.274477 0.483467 0.345968 0.530893
0.294519 0.286444 0.522912 0.455377 0.0480425 0.789532 0.22362 0.116726 0.563241 0.107526 0.000663275 0.36509 0.140703 0.694587 0.533685
0.155445 0.320069 0.102258 0.844235 0.981544 0.306235 0.627662 0.638738 0.806254 0.753405 0.391762 0.201705 0.07424 0.707749 0.373488
0.58236 0.384308 0.636697 0.577945 0.953101 0.527706 0.992441 0.242658 0.514722 0.269795 0.621798 0.528643 0.0910917 0.072488 0.746536
0.6 ([ >: ] ?@$ 0:) 15 15
1 1 0 1 1 0 1 1 1 1 1 1 1 0 1
0 1 1 1 1 1 1 0 0 1 0 0 1 1 1
0 1 1 1 1 1 0 1 1 1 0 0 0 0 0
1 0 1 0 1 0 1 1 0 0 0 1 0 1 1
0 1 0 0 1 1 1 0 1 0 0 0 1 1 1
1 0 0 1 0 1 0 1 1 1 0 1 0 1 1
0 1 0 1 1 1 0 1 0 1 1 1 1 1 1
1 1 1 0 1 0 1 1 0 1 0 1 1 1 0
0 0 1 1 1 1 1 1 1 1 1 1 0 1 1
1 1 1 1 0 1 1 1 1 1 1 1 1 1 0
1 1 0 0 1 0 1 1 0 1 0 1 1 1 1
1 1 1 0 1 0 1 1 1 1 1 1 1 0 1
0 1 1 0 1 1 1 1 0 1 1 1 0 0 0
1 1 0 1 1 1 1 1 1 1 0 1 1 0 1
0 1 1 1 1 0 1 0 0 1 1 0 0 1 0
0.6 ([ >: ] ?@$ 0:) 15 15
0 1 0 1 0 0 1 1 0 1 1 1 1 1 1
0 1 1 1 0 1 1 1 1 1 1 0 0 1 0
1 1 1 0 0 1 0 1 1 0 1 0 1 0 0
1 0 1 0 1 1 1 0 0 1 0 1 1 1 0
1 1 1 1 0 0 0 1 1 0 0 1 0 1 1
0 1 1 1 0 1 1 1 0 1 0 1 1 0 1
1 0 1 0 0 1 1 0 1 1 1 1 1 0 1
1 1 1 0 1 1 1 0 1 0 0 1 0 1 0
1 1 1 0 1 0 1 1 1 1 1 1 0 1 1
1 0 1 1 1 0 1 0 1 1 1 1 1 0 1
1 0 1 0 1 1 1 1 1 1 1 1 1 0 1
1 0 1 0 1 0 1 1 1 1 0 1 0 0 1
1 1 0 1 0 1 1 1 1 1 1 0 1 1 0
0 1 1 1 0 0 1 1 0 1 1 0 0 0 0
1 1 0 1 0 1 1 1 1 0 0 1 1 1 0
0.6 percolate@([ >: ] ?@$ 0:) 15 15
0 0 0 0 2 2 2 2 2 0 2 0 0 0 2
0 1 0 2 0 2 2 0 2 0 2 2 0 0 0
1 0 2 2 2 2 2 2 2 0 0 2 0 0 1
0 2 2 0 2 0 2 0 2 0 2 2 2 0 1
2 2 2 2 0 2 2 2 2 0 2 2 0 0 1
2 2 2 0 1 0 2 2 0 0 2 2 2 2 0
0 0 2 2 0 0 2 2 2 0 2 2 0 2 2
0 2 2 0 0 2 2 2 2 2 0 2 2 0 2
0 2 0 0 0 2 0 0 0 0 0 2 2 0 0
0 2 0 2 2 2 2 0 2 2 2 2 0 0 0
0 2 2 2 2 2 0 0 2 2 2 0 1 0 2
2 0 2 0 0 0 2 2 0 0 0 0 0 0 2
2 2 2 2 2 2 2 2 2 2 2 2 0 2 2
0 0 2 2 2 2 0 2 2 0 0 2 2 2 0
0 1 0 0 2 0 2 2 0 0 1 0 2 0 1
simulate runs a sequence of 15 by 15 trials and averages them. Since each trial is different, each run of simulate is different.
40 (%@[ * [: +/ (2 e. {:)@trial&15 15"0@#) 0.5
0.1
40 ((2 e. {:)@trial&15 15"0@#) 0.5
0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0
40 ((2 e. {:)@trial&15 15"0@#) 0.5
0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0
40 ([: +/ (2 e. {:)@trial&15 15"0@#) 0.5
6
40 ([: +/ (2 e. {:)@trial&15 15"0@#) 0.5
4
40 (%@[ * [: +/ (2 e. {:)@trial&15 15"0@#) 0.5
0.1
40 (%@[ * [: +/ (2 e. {:)@trial&15 15"0@#) 0.5
0.125
Note that the alternative implementation is different. (For example, there, percolate generates an arbitrary 15 by 15 connection matrix, and finds connections on that - rather like trial in this implementation.)