Recovering the rule behind a Lenia body

1One snapshot of Orbium was enough to work out its rule

Orbium’s rule has three dials. The radius RThe kernel radius: how far, in cells, a cell’s neighbourhood reaches. Orbium’s native value is 13. sets how far each cell looks, the growth centre mThe growth centre: the neighbourhood density at which a cell grows fastest. Orbium’s native value is 0.15. sets the neighbourhood sum at which a cell grows fastest, and the growth width sThe growth width: how far the neighbourhood density can stray from m before growth turns into decay. Orbium’s native value is 0.017. sets how far that sum can stray before growth turns into fading. Orbium needs R = 13, m = 0.150 and s = 0.017. We gave a search the update law, the ring-shaped neighbourhood, the growth function and Orbium’s starting cells, hid the three dials, and it returned R = 12.94, m = 0.1526 and s = 0.0180, off by 0.5, 1.7 and 5.9 percent.

The search guessed settings in rounds of 96, grew each guess from the same starting cells for 600 steps, scored how closely the grown field matched Orbium’s at step 600, and guessed again around the best. The score is the field matchOne minus half the summed absolute difference between the candidate’s mass field and the reference, both scaled to unit mass and aligned at their centres of mass, minus small penalties for differences in total mass and in the number of occupied cells. A perfect match scores 1., which centres both fields on their mass and counts how much of that mass overlaps, so 1 is a perfect copy. The search started from R = 10, m = 0.12 and s = 0.024, a rule that breaks the seed into rings, and ran 28 broad rounds over R from 9 to 17, m from 0.1 to 0.2 and s from 0.01 to 0.03, then 12 narrow rounds around the winner, 3,840 grown guesses in all. The broad rounds already put the radius and the growth centre close. The narrow rounds raised the match from 0.878 to 0.969, mostly by pulling the width in from 0.0194.

The creature the recovered rule grows is a working Orbium. It keeps its shape through step 1,200, twice as long as the search ever looked, and it glides like the original: between steps 100 and 1,200 it covers 716 cells against Orbium’s 706, both in straight lines on the same heading, at about 0.64 cells per step, a distance of nearly four widths of their 192-cell world.

More of the creature’s history made the answer worse. A second search scored on snapshots at steps 100, 300 and 600, with the same two-stage budget, returned R = 12.80, m = 0.1435 and s = 0.0185, each farther from Orbium’s value than the single-snapshot answer, although its creature also kept its shape through step 1,200. This is one creature and one comparison, so it shows only that here the two earlier snapshots gave the search nothing it could use.

When you know every part of the rule but three numbers and where the creature began, one picture of the grown creature is enough to recover the rule, and the rule you recover keeps the creature going past the moment you checked. A creature we invent comes with none of that handed over.

2A creature’s size and outline decide about half of how it travels

Flow Lenia changes one thing about Lenia that matters here: the total amount of stuff on the grid never changes, so instead of cells filling and emptying, mass streams from cell to cell along currents that the mass itself sets up [2][2]E. Plantec, G. Hamon, M. Etcheverry, P.-Y. Oudeyer, C. Moulin-Frier, and B. W.-C. Chan, “Flow-Lenia: Towards open-ended evolution in cellular automata through mass conservation and parameter localization.” arXiv preprint 2212.07906, 2022. doi: 10.48550/arXiv.2212.07906.. Its creatures glide, spin, pulse and tear apart. If you knew only what one of them looks like, how big it is, how stretched out and whether it is in pieces, how well could you guess how far and how fast it goes? About halfway. Across 8,155 creatures, eight measurements of their shape account for a little over half of the differences in how far they travel and how fast (53 percent of each), and for even more of how much their insides churn from step to step (74 percent).

The creatures come from one screen of 8,192 random rules in which two interacting channels, which you can picture as two inks, act on each other through ten neighbourhood shapes [3][3]B. W.-C. Chan, “Lenia and expanded universe,” in Proceedings of the 2020 conference on artificial life (ALIFE 2020), MIT Press, 2020, pp. 221–229. doi: 10.1162/isal_a_00297.. We measured six behaviours for each creature: speed, path length, displacement, centre velocity, temporal variance (how much the field changes from step to step) and energy. Most of their variation lies along two directions. The first, locomotionThe first principal axis of the six behaviour measurements, dominated by path length, displacement and centre velocity. It holds 62 percent of the behavioural variation., is mostly how far a creature goes, and the second, activityThe second principal axis of the six behaviour measurements, dominated by how much the field changes over time (its temporal variance) and its energy. It holds 28 percent of the behavioural variation., is mostly how much it churns; together they hold 90 percent of the variation.

A straight-line fit on the eight shape measurementsRadius of gyration, occupancy, mean mass, the number of separate pieces, the largest piece’s share of mass and its elongation, and two moment measures of density and elongation, each standardized across the creatures. explains 61 percent of the variation in activity and 55 percent in locomotion (R² 0.610 and 0.552), against 16 percent for the third direction, how winding a creature’s path is. Size and elongation do most of the work: each on its own explains about a third of the variation in path length, and all eight together explain 45 percent of displacement. Across all 112 screens of the same harvest, four kinds of rule with 8,192 random rules each, the medians are somewhat lower in the same order (45 percent for path length, 40 for speed, 36 for displacement and 73 for temporal variance). These are in-sampleThe linear fit is scored on the same creatures it was fitted to. With eight predictors and thousands of creatures the optimism is small, but a fit on held-out bodies is the stronger test. fits, scored on the creatures they were fitted to.

Draw a creature’s size and outline and you have already decided about half of how it will travel. The other half, where any unusual gait would have to come from, is invisible in the outline, so a search that only checks the picture has no reason to find it.

3Look-alike creatures grown by different rules still travel and heal differently

Half is an average over thousands of creatures that look nothing alike, so it leaves the sharper question open: if two creatures look the same, do they behave the same? Mostly not. We collected families of look-alikes grown by different rules, and in six of seven families the look-alikes differ in how they travel and how they heal after a cut, by more than one rule’s own creatures differ from each other. They still behave more alike than creatures grown by random rules nearby.

To build a family we started from one Flow Lenia creature and its rule, nudged the rule’s numbers in many directions, and kept only the nudged rules whose creature looks like the original as closely as a tiny random nudge would leave it. From each family we kept the 48 shape-matesRules that grow the starting rule’s shape to within the smallest shape change a tiny random change of that rule produces. Each starting rule has 48 of them, chosen as far apart from each other as the search allowed. whose rules lie farthest apart. These rules are simpler than the screen above, with one ink and three neighbourhood shapes. Every mate was grown from the same eight starting blobs for 1,600 steps, once left alone and once with its centre cut out at step 800, and we measured how far and how fast it went, how much energy its field held and how far its body drifted from the uncut copy after the cut. One rule grown from different blobs already gives slightly different creatures, so the yardstick is that rule’s own scatterHow much one rule’s behaviour varies when the same rule is grown from different starting fields, measured over 64 fields and scaled to the eight fields each shape-mate was grown from. over 64 blobs.

The 48 mates of rule 2300 differ in behaviour by 5.2 times that scatter, and the other families with a clear answer lie between 1.6 and 5.2 times. The exception is rule 1643, where tiny nudges of the rule barely change the creature at all; its mates differ by less than one rule varies on its own. An eighth family, around rule 3029, yielded too few look-alikes to measure. For comparison each mate was paired with a rule nudged the same distance in a random direction with no requirement on its look, and in every behaviour of every family the mates spread less than these neighbours, between 0.06 and 0.97 times as much (median 0.43). Looking alike narrows behaviour without fixing it.

Where the differences come from depends on the behaviour. Energy follows the body. Look-alikes are never perfect copies, and at five of the seven families twelve shape descriptors measured on four of the blobs predict 55 to 91 percent of the mates’ energy differences on the other four. Speed and healing do not follow the body. The speeds of rule 2300’s mates repeat from one blob to the next as reliably as their energies, yet no description of the body we have, its outline or its interior, predicts more than a quarter of them. Around rule 2221, how far each mate drifts after the cut repeats almost exactly across blobs (a correlation of 0.96 between two halves of them), nothing in the body predicts any of it, and the direction the mate’s rule was nudged in predicts 64 percent.

Over whole kinds of rule, looking alike says even less about the rule. Among a quarter of a million random rules of each of four kinds, two- and three-ink rules with ten to twenty neighbourhood shapes, the creatures that look most alike come from rules almost exactly as far apart as two rules picked at random (0.997 to 0.999 of the random distance), about one percent of the closeness the sample allows.

Two rules can grow the same-looking creature and make it travel or heal differently, and for some of those differences the body shows nothing while the rule’s numbers predict them. A picture of the creature leaves that part of the rule unwritten.

4A more intricate body is no better at healing a cut

Cut a hole in the middle of a creature and watch whether it comes back. It seems natural to guess that a more elaborate body, with more structure left to rebuild from, would recover better, and across 160 Flow Lenia creatures it does not: what little link there is points the other way.

The 160 are three-ink rules with fifteen neighbourhood shapes, 40 creatures that hold together and 120 grown by random rules, all from the same starting patch. At step 400 we cleared a 40 by 40 square at the centre of each, and by step 1,000 we measured how far it had ended up from an uncut twin in how many cells it occupies, how spread out it is and how much of its mass sits in its largest piece, which we call form driftThe mean relative difference between the cut creature and an undamaged copy in occupancy, radius of gyration and the largest piece’s share of mass, over the steps from the cut at 400 to step 1,000. Zero means the cut left no trace on these measures.. We scored how intricate each body is by how badly its picture compresses, since a smooth blob compresses well and a body full of filaments and holes does not, and call that complexityHow poorly the body’s mass map compresses: the compressed size of the image relative to its raw size, averaged over four downsampling scales. Intricate bodies score high, smooth or empty ones low..

More intricate bodies drifted slightly further from their uncut twins (r = 0.16 across the 160). The creatures that hold together were more intricate than the random ones and drifted further after the cut (0.34 against 0.25 on average), and only 31 of the 160 landed in the intricate, well-healed corner.

An elaborate body is no more likely to heal. If you want a creature that survives a cut, you have to ask for that directly, by cutting it during the test that judges it.

5A snapshot leaves out where a creature starts and how it heals

Orbium’s rule came back from one snapshot because we handed over where it started and every part of its rule but three numbers, and a creature we invent arrives with neither. Its outline would settle about half of how it travels. The rest, along with how it heals, depends on which of the many rules that grow that outline the search happens to find, and how elaborate the creature looks gives no hint about the healing.

Our bet is that a search over the starting patch and the rule together can grow a creature it has seen in only one snapshot, and keep it. The test hides both the rule and the starting cells of a known creature, searches both against one snapshot of the grown body, then grows the fitted rule from new starting cells and cuts the result. The bet fails if the fitted rules reach the snapshot only from the one start they were found with, or if their creature drifts further after the cut than the original does, because then a snapshot pins down a picture and leaves the creature that should keep it unspecified.

If the bet holds, asking for a creature means handing over a picture of it and the damage it has to survive, and letting the search work out where it begins.

6A body is what its rulebook keeps rebuilding

We ask whether different kinds of systems share principles of organization: how a body takes shape, how it holds that shape and what it does when part of it is taken away. Lenia is a model organism for this because it comes with its rulebook, and this report runs the question backwards, from a body to the rulebook that grows it. For Orbium, with the seed and all but three numbers of the rule handed over, one frame of the mature body was enough, and the rulebook it gave back grows a body that keeps its shape and keeps gliding long after the frame it was read from. A body, here, is something the rulebook keeps rebuilding, and one picture of it says enough about that rebuilding to recover the rule.

The Flow Lenia creatures show how much more has to be said once nothing is handed over. Shape fixes about half of how a body travels and leaves the rest to hidden variation that a picture does not show. Look-alikes grown by different rulebooks behave more alike than bodies grown by random neighbouring rules, which suggests that shape and behaviour read some of the same combinations of the rule, but when we cut them and ask whether each body keeps its future, bodies that looked the same before the cut still answer differently. How intricate a body looks says nothing about that answer, so a request for a body that survives damage has to put the damage into the test that scores it.

7Methods

Four kinds of rule run these experiments, and each section uses one.

All Flow Lenia runs use the corrected engine, LeniaCLI build 060e90a71ec1 (A.1). The six behaviour measurements are mean speed, path length, displacement and centre velocity, each log-transformed, and the field’s temporal variance and energy, all standardized before the principal axes are taken; the screen’s creatures are those that passed its persistence filters. Each shape fit is an ordinary least-squares regression of one behaviour or axis on the eight standardized shape measurements, and the size and elongation groups are fitted the same way on their own subsets. The look-alike energy, speed and drift predictions are ridge regressions across the 48 mates scored on held-out fields, from the twelve shape descriptors, from a heading-free description of the body’s interior, and from both. The family-wide rule distances are measured after sorting kernels and also minimized over every kernel and channel relabelling, on 128-cell worlds grown once for 1,500 steps.