Lenia
Lenia - Mathematical Life Forms
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We report a new model of artificial life called Lenia (from Latin lenis "smooth"), a two-dimensional cellular automaton with continuous space-time-state and generalized local rule. Computer simulations show that Lenia supports a great diversity of complex autonomous patterns or "lifeforms" bearing resemblance to real-world microscopic organisms. More than 400 species in 18 families have been identified, many discovered via interactive evolutionary computation. We present basic observations of the model regarding the properties of space-time and basic settings. We provide a board survey of the lifeforms, categorize them into a hierarchical taxonomy, and map their distribution in the parameter hyperspace. We describe their morphological structures and behavioral dynamics, propose possible mechanisms of their self-propulsion, self-organization and plasticity. Finally, we discuss how the study of Lenia would be related to biology, artificial life, and artificial intelligence.
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Comprehensive list of resources on the topic of digital morphogenesis (the creation of form through code). Includes links to major articles, code repos, creative projects, books, software, and more.
Among the long-term goals of artificial life are to simulate existing biological life and to create new life forms using artificial systems. These are expressed in the fourteen open problems in artificial life [1], in which number three is of particular interest here:
Determine whether fundamentally novel living organizations can exist.
There have been numerous efforts in creating and studying novel mathematical models that are capable of simulating complex life-like dynamics. Examples include particle systems like Swarm Chemistry [2], Primordial Particle Systems (PPS) [3]; reaction-diffusion systems like the U-Skate World [4]; cellular automata like the Game of Life (GoL) [5], elementary cellular automata (ECA) [6]; evolutionary systems like virtual creatures [7], soft robots [8, 9]. These models have a common theme — let there be countless modules or particles and (often localized) interactions among them, a complex system with interesting autonomous patterns will emerge, just like how life emerged on Earth 4.28 billion years ago [10].
Life can be defined as the capabilities of self-organizing (morphogenesis), self-regulating (homeostasis), self-directing (motility), self-replicating (reproduction), entropy reduction (metabolism), growth (development), response to stimuli (sensitivity), response to environment (adaptability), and evolving through mutation and selection (evolvability) (e.g., [11, 12, 13, 14]). Models of artificial life are able to reproduce some of these capabilities with various levels of fidelity. Lenia, the subject of this paper, is able to achieve many of them, except self-replication that is yet to be discovered.
Lenia also captures a few subjective characteristics of life, like vividness, fuzziness, aesthetic appeal, and the great diversity and subtle variety in patterns that a biologist would have the urge to collect and catalogue them. If there is some truth in the biophilia hypothesis [15] that humans are innately attracted to nature, it may not be too far-fetched to suggest that these subjective experiences are not merely feelings but among the essences of life as we know it.
The diverse life in Lenia, although being interesting, should not be treated as a mere curiosity. As shown in this paper, this form of artificial life might be deeply linked to biological life in surprising ways, like metamerism / multicellularity, symmetry, plasticity, and common issues like the species problem. They warrant scientific investigations.
Due to similarities between life on Earth and Lenia, we borrow terminologies and concepts from biology, like taxonomy (corresponds to categorization), binomial nomenclature (naming), ecology (parameter space), morphology (structures), behavior (dynamics), physiology (mechanisms), and allometry (statistics). We also borrow space-time (grid and time-step) and fundamental laws (local rule) from physics. With a few caveats, these borrowings are useful in providing more intuitive characterization of the system, and may facilitate discussions on how Lenia or similar models could give answers to life [16], the universe [17], and everything.
A cellular automaton
(CA, plural: cellular automata) is a mathematical model where a grid of sites, each having a particular state at a moment, are being updated repeatedly according to each site’s neighboring sites and a local rule. Since its conception by John von Neumann and Stanislaw Ulam
[18, 19], various CAs have been invented and investigated, the most famous being Stephen Wolfram’s one-dimensional elementary cellular automata (ECA) [6, 17] and John H. Conway’s two-dimensional Game of Life (GoL) [5, 20].GoL is the starting point of where Lenia came from. GoL consists of a two-dimensional square grid, 2-state sites, 8-site neighborhood, and a totalistic update rule with survival/birth intervals {2, 3} and {3}. It produces a whole universe of interesting patterns [21] ranging from simple “still lifes”, “oscillators” and “spaceships”, to complex constructs like pattern emitters, self-replicators, logic gates, and even fully operational computers thanks to its Turing completeness [22].
Several aspects of GoL can be generalized via the following path:
discrete (unary extended fractional) continuous
A unary property (e.g. dead-or-alive state) can be extended into a range (multi-state), which is equivalent to a fractional property by being normalized into the unit range, and becomes continuous by further splitting the range into infinitesimals (real number state). The local rule can be generalized from the basic ECA/GoL style (totalistic sum, survival/birth intervals, replacing update) to smooth parameterized operations (weighted sum, growth mapping, incremental update).
By comparing various CAs including ECA, GoL, continuous GoL [23], Larger-than-Life (LtL) [24, 25, 26], RealLife [27], SmoothLife^{1}^{1}1SmoothLife and Lenia, being independent developments, exhibit striking resemblance in model and generated patterns. This can be considered an instance of “convergent evolution” in generalizing GoL. [28], Primordia^{2}^{2}2Primordia is a precursor to Lenia, written in JavaScript/HTML by the author circa 2005. It had multi-states and extended survival/birth intervals, with automatic pattern detection., and Lenia, we observe the evolution of generalization with increasing genericity and continuity (Table 1, Figure 1). This suggests that Lenia is currently at the latest stage of generalizing GoL, although there may be room for further generalizations.
We describe the methods of constructing and studying Lenia, including its mathematical definition, in silico simulation, strategies of evolving new lifeforms, and how to perform observational and statistical analysis.
Mathematically, a CA is defined by a 5-tuple^{4}^{4}4Conventionally a CA is defined by a 4-tuple , here the timeline is added to indicate its variability. , where is the -dimensional lattice or grid, is the timeline, is the state set, is the neighborhood (of the origin), is the local rule.
Define as the configuration or pattern (i.e. collection of states over the whole grid) at time . is the state of site , and is the state collection over the site’s neighborhood . The global rule is such that . Starting from an initial configuration , the grid is updated according to the global rule for each time-step , leading to the following time-evolution:
(1) |
After repeated updates (or generations):
(2) |
Take GoL as an example, , where is the two-dimensional discrete grid; is the discrete timeline; is the unary state set; is the Moore neighborhood (Chebyshev norm) including the site itself and its 8 neighbors (Figure 2(a)).
The totalistic neighborhood sum of site x is:
(3) |
Every site is updated synchronously according to the local rule:
(4) |
Discrete Lenia (DL) generalizes GoL by extending and then normalizing the space-time-state dimensions. DL is used for computer simulation and analysis, and with normalization, patterns from different dimensions can be compared. Continuous Lenia (CL) is hypothesized to exist by taking the dimensions of DL to their continuum limits.
The state set is extended to with maximum . The neighborhood is extended to a discrete ball (Euclidean norm) of range that (Figure 2(b)).
To normalize, define or redefine as the space resolution, time resolution, and state resolution, and their reciprocals as the site distance, time step (fractional), and state precision, respectively. The space-time-state dimensions are scaled by the reciprocals, so that
(5) |
and the neighborhood is normalized to become a discrete unit ball . (Figure 2(c)) As the resolutions approach infinity and the differences become infinitesimals , it is conjectured that the space-time-state dimensions will approach their continuum limits, i.e. the Euclidean space, the real timeline, and real number states of the unit interval
(6) |
and the neighborhood will approach the continuous unit ball . (Figure 2(d))
However, there is a cardinality leap between the discrete dimensions in DL and the continuous dimensions in CL. The existence of the space continuum limit was proved mathematically in [27], and our computer simulations provide empirical evidence for continuum limit of space and time (see “Physics” section). Further rigorous proofs are needed.
To apply Lenia’s local rule to every site at time , the potential distribution is calculated by convolution of its neighborhood with a kernel :
in DL | |||||
in CL | (7) |
Feeding the potential into the growth mapping yields the growth distribution
(8) |
Every state is updated by adding a small fraction of the growth and clipped back to the unit interval , and the time is now . (In CL, the time step is replaced by infinitesimal )
(9) |
where is the clip function.
The kernel is constructed by kernel core which determines the inner “texture” of the kernel, kernel shell which determines its “skeleton”, and normalization which makes sure . The convolution with kernel (i.e. weighted sum) is a generalization of the totalistic sum in GoL.
The kernel core is any unimodal function satisfying and usually . By taking polar distance as argument, it creates a uniform ring around the site (Figure 3(a, d)). The function can be exponential, polynomial, trigonometric, trapezoidal, or rectangular, etc. (Figure 3(a-c)); the exponential kernel core is used throughout this paper.
(10) |
The kernel shell
takes a vector parameter
(kernel peaks) of size (the rank) and copies the kernel core into concentric rings of equal thickness with peak heights (Figure 3(b, e)).(11) |
Finally, the kernel is normalized:
(12) |
where in DL, or in CL.
Notes on parameter :
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A vector of rank is equivalent to an extended vector with trailing zeros while space resolution is scaled by factor , e.g. with scaled by 3.
A vector where is equivalent to a scaled vector where while the kernel is unchanged due to normalization, e.g. .
Consequently, all possible of rank as a -dimensional hypercube can be projected onto its -dimensional hypersurfaces where . (see Figure 10 for )
The growth mapping is any unimodal, nonmonotonic function with parameters (growth center and growth width) satisfying , and roughly corresponds to the width of its peak (cf. in [23]). The function can be exponential, polynomial, trigonometric, trapezoidal, or rectangular, etc. (Figure 3(f-g)); the exponential growth mapping is used throughout this paper.
(13) |
The growth mapping is a generalization of the survival/birth intervals in GoL, where the positive, near zeros, and negative values correspond to the birth, survival, and death intervals, respectively.
GoL can be considered a special case of discrete Lenia with , using a variant of the rectangular kernel core:
(14) |
and the rectangular growth mapping with .
In summary, discrete and continuous Lenia are defined as:
(15) | ||||
(16) |
The associated dimensions are: space-time-state resolutions , differences , infinitesimals . The associated parameters are: growth center , growth width , kernel peaks of rank . The mutable core functions are: kernel core , growth mapping .
Discrete Lenia (DL) can be implemented with the pseudocode below, assuming an array programming language is used (e.g. Python with NumPy, MATLAB, Wolfram).
Interactive programs have been written in JavaScript / HTML5, Python, and MATLAB to provide user interface for new species discovery (Figure 4(a-b)). Non-interactive program has been written in C#.NET for automatic traverse through the parameter space using a flood fill algorithm (breath-first or depth-first search), providing species distribution, statistical data and occasionally new species.
State precision can be implicitly implemented as the precision of floating-point numbers. For values in the unit interval , the precision ranges from to (about to ) using 32-bit single-precision, or from to (about to ) using 64-bit double-precision [33]. That means using double precision.
Discrete convolution can be calculated as the sum of element-wise products:
(17) |
or alternatively, using discrete Fourier transform (DFT) according to the convolution theorem:
(18) |
Efficient calculation can be achieved using fast Fourier transform (FFT) [34], pre-calculation of the kernel’s FFT , and parallel computing like GPU acceleration. The DFT/FFT approach automatically produces a periodic boundary condition.
Symbol @ indicates the data type of two-dimensional matrix of floating-point numbers.
function pre_calculate_kernel(beta, dx) @radius = get_polar_radius_matrix(SIZE_X, SIZE_Y) * dx @Br = size(beta) * @radius @kernel_shell = beta[floor(@Br)] * kernel_core(@Br % 1) @kernel = @kernel_shell / sum(@kernel_shell) @kernel_FFT = FFT_2D(@kernel) return @kernel, @kernel_FFT end function run_automaton(@world, @kernel, @kernel_FFT, mu, sigma, dt) if size(@world) is small @potential = elementwise_convolution(@kernel, @world) else @world_FFT = FFT_2D(@world) @potential_FFT = elementwise_multiply(@kernel_FFT, @world_FFT) @potential = FFT_shift(real_part(inverse_FFT_2D(@potential_FFT))) end @growth = growth_mapping(@potential, mu, sigma) @new_world = clip(@world + dt * @growth, 0, 1) return @new_world, @growth, @potential end function simulation() R, T, mu, sigma, beta = get_parameters() dx = 1/R; dt = 1/T; time = 0 @kernel, @kernel_FFT = pre_calculate_kernel(beta, dx) @world = get_initial_configuration(SIZE_X, SIZE_Y) repeat @world, @growth, @potential = run_automaton(@world, @kernel, @kernel_FFT, mu, sigma, dt) time = time + dt display(@world, @potential, @growth) end end
For implementations requiring an interactive user interface, one or more of the following components are recommended:
[noitemsep]
Controls for starting and stopping CA simulation
Panels for displaying different stages of CA calculation
Controls for changing parameters and space-time-state resolutions
Controls for randomizing, transforming and editing the current configuration
Controls for saving, loading, and copy-and-pasting configurations
Clickable list for loading predefined patterns
Utilities for capturing the display output (e.g. static or animated image, movie clip)
Controls for customizing the layout (e.g. grid size, color map)
Controls for auto-centering, auto-rotating and temporal sampling
Panels or overlays for displaying real-time statistical analysis
A pattern can be stored for publication and sharing using a data exchange format (e.g. JSON, XML) that includes the run-length encoding (RLE) of the two-dimensional array At and its associated parameters , or alternatively, using a plaintext format (e.g. CSV) for further analysis or manipulation in numeric software.
A long list of interesting patterns can be saved in a JSON/XML list for program retrieval. To save storage space, patterns can be stored with space resolution as small as possible (usually ) thanks to Lenia’s scale invariance (see “Physics” section) and re-enlarged upon retrieval.
Most of computer simulations, experiments, statistical analysis, image and video capturing for this paper were done using the following environments and settings:
[noitemsep]
Hardware: Apple MacBook Pro (OS X Yosemite), Lenovo ThinkPad X280 (Microsoft Windows 10 Pro)
Software: Python 3.7.0, MathWorks MATLAB Home R2017b, Google Chrome browser, Microsoft Excel 2016
State precision: double precision
Kernel core and growth mapping: exponential
A self-organizing, autonomous pattern in Lenia is called a lifeform, and a kind of similar lifeforms is called a species. Up to the moment, more than 400 species have been discovered. Interactive evolutionary computation (IEC) [35]
is the major force behind the generation, identification and selection of new species. In evolutionary computation (EC), the fitness function is usually well known and can be readily calculated. However, in the case of Lenia, due to the non-trivial task of pattern recognition, as well as aesthetic factors, evolution of new species often requires human interaction.
Interactive computer programs provide user interface and utilities for human users to carry out mutation and selection operators manually. Mutation operators include parameter tweaking and configuration manipulation. Selection operators include observation via different views for fitness estimation (Figure
3(c-f)) and storage of promising patterns. Selection criteria include survival, long-term stability, aesthetic appeal, and novelty.Listed below are a few evolutionary strategies learnt from experimenting and practicing.
Initial configurations with random patches of non-zero sites were generated and put into simulation using interactive program. This is repeated using different random distributions and different parameters. Given enough time, naturally occurring lifeforms would emerge from the primordial soup, for example Orbium, Scutium, Paraptera, and radial symmetric patterns. (Figure 5(a))
Using an existing lifeform, parameters were changed progressively or abruptly, forcing the lifeform to die out (explode or evaporate) or survive by changing slightly or morphing into another species. Any undiscovered species with novel structure or behavior were recorded. (Figure 5(b))
Transient patterns captured during random generation could also be stabilized into new species in this way.
Long-chain lifeforms (e.g. Pterifera) could first be elongated by temporary increasing the growth rate (decrease or increase ), then stabilized into new species by reversing growth. Shortening could be done in the opposite manner.
Starting from an existing lifeform, automatic program was used to traverse the parameter space (i.e. continuous parameter tweaking). All survived patterns were recorded, among them new species were occasionally found. Currently, automated exploration is ineffective without the aid of artificial intelligence (e.g. pattern recognition), and has only been used for simple conditions (rank 1, mutation by parameter tweaking, selection by survival). (Figure 5(c))
Patterns were edited or manipulated (e.g. enlarging, shrinking, mirroring, single-side flipping, recombining) using our interactive program or other numeric software, and then parameter tweaked in attempt to stabilize into new species. (Figure 5(d))
By using computer simulation and visualization and taking advantage of human’s innate ability of spatial and temporal pattern recognition, the physical appearances and movements of known species were being observed, documented and categorized, as reported in the “Morphology” and “Behavior” sections. Using automatic traverse program, the distributions of selected species in the parameter space were charted, as reported in the “Ecology” section. A set of criteria, based on the observed similarities and differences among the known species, were devised to categorize them into a hierarchical taxonomy, as reported in the “Taxonomy” section.
Statistical methods were used to analyze lifeforms to compensate limitations in human observation regarding subtle variations and long-term trends. A number of statistical measures were calculated over the configuration (i.e. mass distribution) and the positive-growth distribution
[noitemsep]
Mass is the sum of states,
Volume is the number of positive states,
Density is the density of states,
Growth is the sum of positive growth,
Centroid is the center of states,
Growth center is the center of positive growth,
Growth-centroid distance is the distance between the two centers,
Linear speed is the linear moving rate of the centroid,
Angular speed is the angular moving rate of the centroid,
Mass asymmetry is the mass difference across the directional vector,
where
Angular mass is the second moment of mass from the centroid,
Gyradius is the root-mean-square of site distances from the centroid,
Note: SI units in microscopic scale were borrowed as units of measure, e.g. “mm” for length, “rad” for angle, “s” for time, “mg” for states (cf. “lu” and “tu” in [4]).
Based on multivariate time-series of statistical measures over a predefined time period, the following “meta-measures” were calculated:
[noitemsep]
Summary statistics (mean, median, standard deviation, minimum, maximum, quartiles)
Quasi-period (estimated using e.g. autocorrelation, periodogram)
Degree of chaos (e.g. Lyapunov exponent, attractor dimension)
Probability of survival
The following charts were plotted using various parameters, measures and meta-measures:
Over 1.2 billion measures were collected using automatic traverse program and analyzed using numeric software like Microsoft Excel. Results are presented in the “Physiology” section.
Constant motions like translation, rotation and oscillation render visual observation difficult. It is desirable to separate the spatial and temporal aspects of a moving pattern so as to directly assess the static form and estimate the motion frequencies (or quasi-periods).
Linear motion can be removed by auto-centering, to display the pattern centered at its centroid .
Using temporal sampling, the simulation is displayed one frame per time-steps. When any rotation is perceived as near stationary, the rotation frequency is approximately the display frequency . Calculate the sampled angular speed where n is the number of radial symmetric axes. Angular motion can be removed by auto-rotation, to display the pattern rotated by .
With the non-translating, non-rotating pattern, any global or local oscillation frequency can be determined as again using temporal sampling.
Results of the study of Lenia will be outlined in various sections: Physics, Taxonomy, Ecology, Morphology, Behavior, Physiology, and Case Study.
We present general results regarding the effects of basic CA settings, akin to physics where one studies how the space-time fabric and fundamental laws influence matter and energy.
For sufficiently fine space resolution (), patterns in Lenia are minimally affected by spatial similarity transformations including shift, rotation, reflection and scaling (Figure 6(d-g)). Shift invariance exists in all homogenous CAs; reflection invariance is enabled by symmetries in neighborhood and local rule; scaleinvariance is enabled by large neighborhoods (as in LtL [25]); rotation invariance is enabled by circular neighborhoods and totalistic or polar local rules (as in SmoothLife [28] and Lenia). Our empirical data of near constant metrics of Orbium over various space resolutions further supports scale invariance in Lenia (Figure 7(a-b)).
The local rule of discrete Lenia (DL) can be considered the Euler method for solving the local rule
of continuous Lenia (CL) rewritten as ordinary differential equation (ODE):
(19) | ||||
(20) |
The Euler method should better approximate the ODE as step size diminishes, similarly DL should approach its continuum limit CL as decreases. This is supported by empirical data of asymptotic metrics of Orbium over increasing time resolutions (Figure 7(c-d)) towards an imaginable “true Orbium” (Figure 6(i)).
Choices of kernel core and growth mapping (the core functions or “fundamental laws”) alters the “textures” of a pattern but not its overall structure and dynamics (Figure 6(b-c)). Smoother core functions (e.g. exponential) produce smoother patterns, rougher ones (e.g. rectangular) produce rougher patterns. This plasticity suggests that similar lifeforms should exist in SmoothLife which resembles Lenia with rectangular core functions, as supported by similar creatures found in both CAs (Figure 1(f-g)).
We present the classification of Lenia lifeforms into a hierarchical taxonomy, a process comparable to the biological classification of Terrestrial life [38].
The most famous moving pattern in GoL is the diagonally-moving “glider” (Figure 1(a)). It was not until LtL [26] that scalable digital creatures were discovered including the glider analogue “bugs with stomach”, and SmoothLife [28] was the first to produce an omni-directional bug called the “smooth glider” (Figure 1(f)), which was rediscovered in Lenia as Scutium plus variants (Figure 1(e-g) left). We propose the phylogeny of the glider:
Glider Bug with stomach Smooth glider Scutium and variants
Phylogenies of other creatures are possible, like the “wobbly glider” and Pyroscutium (Figure 1(f-g) right).
Principally there are infinitely many types of lifeforms in Lenia, but a range of visually and statistically similar lifeforms were grouped into a species, defined such that one instance can be morphed smoothly into another by continuously adjusting parameters or other settings. Species were further grouped into higher taxonomic ranks — genera, families, orders, classes — with decreasing similarity and increasing generality, finally subsumed into phylum Lenia, kingdom Automata, domain Simulata, and the root Artificialia. Potentially other kinds of artificial life can be incorporated into this Artificialia tree (see Appendix A).
Below are the current definitions of the taxonomic ranks.
[noitemsep]
A genus is a group of species with the same global morphology and behavior but differ locally, occupy adjacent niches, and have discontinuity in statistical trends. Abrupt but reversible transformation among member species is possible.
A subfamily (or series) is a series of genera with increasing number of “units” or “vacuoles”, occupy parallel niches of similar shapes.
A family is a collection of subfamilies with the same architecture or body plan, composed of the same set of components arranged in similar ways.
An order is a rough grouping of families with similar architectures and statistical qualities, e.g. speed.
A class is a high-level grouping of how lifeforms influenced by the arrangement of kernel.
Appendix A gives a proposed taxonomic tree of Lenia according to the above taxonomical ranks, and a proposed “tree of artificial life” in a boarder context. Appendix B gives the details of classes and families.
Much like real-world biology, the taxonomy of Lenia is tentative and is subject to revisions or redefinitions when more data is available.
Following Theo Jansen for naming artificial life using biological nomenclature (Animaris spp.) [39], each Lenia species was given a binomial name that describes its geometric shape (genus name) and behavior (species name) to facilitate analysis and communication. Alphanumeric code was given in the form “BGUs” with initials of genus or family name (G) and species name (s), number of units (U), and rank (B).
Suffix “-ium” in genus names is reminiscent of a bacterium or chemical elements, while suffixes “-inae” (subfamily), “-idae” (family), and “-iformes” (order) were borrowed from actual animal taxa. Numeric prefix ^{5}^{5}5Prefixes used are: Di-, Tri-, Tetra-, Penta-, Hexa-, Hepta-, Octa-, Nona-, Deca-, Undeca-, Dodeca-, Trideca-, etc. in genus names indicates the number of units, similar to organic compounds and elements (IUPAC names)
We describe the parameter space of Lenia (“geography”) and the distribution of lifeforms (“ecology”).
The four classes of CA rules [40, 17] corresponds to the four landscapes in the Lenia parameter space (Figure 9):
[noitemsep]
Class 1 (homogenous “desert”) produces no global or local pattern but a homogeneous (empty) state
Class 2 (cyclic “savannah”) produces regional, periodic immobile patterns (e.g. Circidae)
Class 3 (chaotic “forest”) produces chaotic, aperiodic global filament network (“vegetation”)
Class 4 (complex “river”) generates localized complex structures (lifeforms)
In the -dimensional -- parameter hyperspace, a lifeform only exists in a continuous parameter range called its niche. Each combination of parameters is called a locus (plural: loci).
For a given , a - map is created by plotting the niches of selected lifeforms on a vs. chart. Maps of rank-1 species have been extensively charted and were used in taxonomical analysis (Figure 9).
A -cube is created by marking the existence (or the size of - niche) of a lifeform at every locus. As noted in “Definition” section, a -dimensional hypercube can be reduced to its -dimensional hypersurfaces, perfect for visualization in the three-dimensional case (Figure 10).
We present the study of structural characteristics, or “morphology”, of Lenia lifeforms. See Figure 8 and Appendix B for the family codes (O, S, P, etc.).
The structures of Lenia lifeforms can be summarized into the following types of architecture:
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Segmented architecture is the serial combination of a few basic components. It is prevalent in class Exokernel (O, S, P, H; also: Ct, U, K). Details in “Anatomy” and “Metamerism” below.
Radial architecture is the radial arrangement of repeating units. It is the defining architecture of Radiiformes in class Endokernel (D, R, B, L, F; also: C, V). Details in “Symmetry” below.
Swarm architecture is the versatile and volatile structure formed by a cluster of granular masses. They are not confined to a particular geometry or locomotive tendency, in extreme case formless and directionless. Common in class Mesokernel (E, G; also: Q, A).
The following is an inventory of components found in segmented species of class Exokernel. Their internal structures or “anatomy” are described (Figure 11(a-c, f)).
[noitemsep]
The orb (disk) is a circular disk divided into two halves by a central stalk, has a front mouth and a rear tail (plus two side tails). Can be found as singular Orbium or compound Orbidae.
The scutum (shield, plural: scuta) is a circular disk with a thick, dense shield at the front, sometimes with an opening (valve). Can be found as singular Scutium or compound Scutidae.
The wing or pteron (plural: ptera) comes in two flavors:
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The orboid wing (disk-like wing) is a distorted orb with a pivot sac, has a budding mechanism to create a new sac in each cycle, which becomes mature and finally dissolved at the tail. Can be found as singular Gyrorbium or in concave Scutiformes.
The scutoid wing (shield-like wing) is a distorted scutum with portions of the shield missing. Can be found as singular Gyropteron or in convex Scutiformes.
The vacuole (sac) is a circular disk found between two wings in long-chain Scutiformes, with thick walls between vacuoles and wings.
It is likely that many of the components are interrelated, e.g. the orboid wing related to the orb as suggested by smooth transition between Paraptera cavus and Parorbium; the scutoid wing related to the scutum as suggested by similarity between Paraptera arcus and Tetrascutium.
In segmented architecture, multiple components can be combined serially through fusion or adhesion, in a fashion comparable to metamerism in biology (or multicellularity if we consider the components as “cells”) (Figure 11(f-g, i)).
[noitemsep]
Fusion of multiple orbs (shared halves) forms long-chain Synorbinae; fusion of two orboid wings (shared pivot sac) forms linear Synptera or rotating Helicium.
Adhesion of multiple orbs forms long-chain Parorbinae; of multiple scuta forms Megaloscutinae; of two wings plus vacuoles forms Vacuopterinae or Helicidae.
Long-chain species exhibit different degrees of convexity, especially obvious in longer chains (Figure 11(i)). When ordered by convexity: Scutidae convex Pteridae (arcus subgenus) linear Orbidae concave Pteridae (cavus subgenus) (Figure 8 column 1). Sinusoidal Pteridae (sinus subgenus) have hybrid convexity.
Higher-rank segmented Ctenidae, Uridae, Kronidae also exhibit linear metamerism and convexity with more complicated set of components.
Structural symmetry is a prominent characteristic of Lenia life, including the following types:
[noitemsep]
Bilateral symmetry (dihedral group D1) is present in segmented architecture (O, S, P, Ct, U, K), also weakly bilateral in some with swarm architecture (G, Q, E).
Radial symmetry (dihedral group Dn) is rotational plus reflectional symmetry, caused by bilateral repeating units in radial architecture (R, L, F), also weakly radial in some with swarm architecture (E).
Rotational symmetry (cyclic group Cn) is rotational without reflectional symmetry, caused by asymmetric repeating units in radial architecture (D, R, L) (Figure 11(h)).
Spherical symmetry (orthogonal group O(2)) is a special case of radial symmetry (C).
Secondary symmetries:
[noitemsep]
Spiral symmetry is secondary rotational symmetry derived from twisted bilaterals (H, V).
Biradial symmetry is secondary bilateral symmetry derived from radials (B, R).
Deformed bilateral symmetry is bilateral with heavy asymmetry (e.g. in O, S, Q).
Asymmetry is present in amorphous species (A).
Asymmetry also plays a significant role in shaping the lifeforms and guiding their movements, causing various degrees of angular movements (detailed in “Physiology” section).
Asymmetry is usually intrinsic in a species, as demonstrated by experiments where a slightly asymmetric form (e.g. Paraptera pedes, Echinium limus) was mirrored into perfect symmetry and remained metastable, but slowly restored to its natural asymmetric form after the slightest perturbation (e.g. rotate by 1°).
Many detailed local patterns arise in higher-rank species owing to their complex kernels (Figure 11(d-e, j-l)):
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Decoration is the addition of tiny ornaments (e.g. dots, circles, crosses), prevalent in class Endokernel.
Serration is a ripple-like sinusoidal boundary or pattern, common in class Exokernel and Mesokernel.
Caudation is a tail-like structure behind a long-chain lifeform (e.g. P, K, U), akin to “tag-along” in GoL.
Liquefaction is the degradation of an otherwise regular structure into a chaotic “liquified” tail.
We present the study of dynamical behaviors of Lenia lifeforms, or “ethology”, in analogy to the study of animal behaviors in biology.
Pattern movements in GoL include stationary (fixed, oscillation), directional (orthogonal, diagonal, rarely oblique), and indefinite growth (linear, sawtooth, quadratic) [21]. SmoothLife added omnidirectional movement to the list [28]. Lenia supports a qualitatively different repertoire of behavioral dynamics, which can be described in two levels: global locomotion and local gaits.
The overall movement of lifeforms can be summarized into modes of locomotion (Figure 12(a-c, e)):
[noitemsep]
Stationarity (S) means the pattern stays still with negligible directional movement or rotation.
Rotation (R) is the angular movement around the centroid which has minimal movement.
Translocation (T) is the linear movement in certain direction.
Gyration (G) is the angular movement around a non-centroid center, basically a combination of translocation and rotation.
In formula,
(21) |
where is the quasi-period, is a shift by distance due to linear speed , is a rotation (around the centroid) by angle due to angular speed .
Stationarity: | |
---|---|
Rotation: | |
Translocation: | |
Gyration: |
The local details of movements are identified as different gaits (Figure 12(e-i)):
[noitemsep]
Fixation (_{F}) means negligible or no fluctuation during locomotion.
Oscillation (_{O}) is the periodic fluctuation during locomotion
Alternation (_{A}) is the global oscillation plus out-of-phase local oscillations (see “Physiology” section).
Deviation (_{D}) is a small departure from the regular locomotion, e.g. slightly curved linear movement, slight movements in the rotating or gyrating center.
Chaoticity (_{C}) is the chaotic, aperiodic movements during any mode of locomotion.
Any gait or gait combination can be applied to any locomotive mode (e.g. chaotic deviated alternating translocation) and is represented by the combined code (e.g. T_{CDA}). See Table 2 for possible combinations.
Spontaneous metamorphosis is a highly chaotic behavior in Lenia, where a “shapeshifting” species frequently switch among different morphological-behavioral templates
, forming a continuous-time Markov chain. Each template often resembles an existing species. The set of possible templates and the transition probabilities matrix are determined by the species and parameter values (Figure
12(j)).An extreme form of spontaneous metamorphosis is exhibited by the Amoebidae, where the structure and locomotive patterns are no longer recognizable, while a bounded size can still be maintained.
These stochastic or amoeboid behaviors denied the previous assumption that morphologies and behaviors are fixed qualities in a species, but are actually probabilistic (albeit the probability usually close to one).
Besides the limited and controlled behaviors mentioned above where the total mass remains bounded and non-zero, there are unlimited behaviors with indefinite growth or shrinkage.
[noitemsep]
Explosion is the uncontrolled indefinite growth where the mass quickly expands in all directions, forming a chaotic “vegetation” (cf. class 3 CA).
Evaporation is the uncontrolled indefinite shrinkage where insufficient growth causes the pattern to vanish eventually (cf. class 1 CA).
Elongation is the controllable indefinite growth where a long-chain lifeform keeps lengthening along tangential directions.
Contraction is the controllable indefinite shrinkage where a long-chain lifeform keeps shortening along tangential directions until vanished.
Elongation and contraction are complementary behaviors utilized to evolve long-chain species of desired lengths. Microscopically, vacuoles in the long-chain are being constantly created or absorbed via binary fission or fusion.
Growth rates were estimated by time-series of mass. Linear and circular elongation show linear growth; spiral elongation (in Helicidae) and explosion show quadratic growth. Same for negative growth rates in case of contraction and evaporation.
Using the interactive program as “particle collider”, we investigated the reactions among individual lifeforms especially Orbidae acting as physical or chemical particles. These particles often exhibit elasticity and resilience during collision, engage in inelastic (sticky) collision, and seem to exert a kind of weak “attractive force” when two particles are nearby or a “repulsive force” when getting too close.
Collision of two Orbium particles with different incident angles and starting positions would result in one of the followings:
[noitemsep]
Deflection, two Orbium disperse in different angles.
Reflection, one Orbium retains its course while one Orbium bounces back in opposite direction.
Fusion, two Orbium fuse together into one Synorbium (Figure 12(k)).
Absorption, two Orbium form an unstable intermediate and then only one Orbium survives.
Annihilation, the resultant mass evaporates.
Detonation, the resultant mass explodes into infinite growing vegetation.
Starting from a composite Orbidae, the following outcomes are possible:
[noitemsep]
Fission, one Synorbinae breaks down into multiple smaller Synorbinae or Orbium.
Parallelism, multiple Orbium travel in parallel with “forces” subtly balanced, forming a Parorbinae.
The exact mechanisms, or “physiology”, of morphogenesis (self-organization) and homeostasis (self-regulation) in Lenia are not well understood. Here we will present a few observations and speculations.
A striking result in analyzing Lenia is the correlations between structural symmetries/asymmetries (“Morphology” section) and behavioral dynamics (“Behavior” section).
At a global scale, the locomotive modes correspond to the types of overall symmetry.
[noitemsep]
Stationarity is associated with radial symmetry (including spherical symmetry).
Rotation is associated with rotational symmetry (including secondary spiral symmetry).
Translocation is associated with bilateral symmetry (including secondary biradial symmetry).
Gyration is associated with deformed bilateral symmetry.
At a local scale, the locomotive gaits correspond to the dynamical developments of asymmetry.
[noitemsep]
Fixation implies that any asymmetry remains static.
Oscillation implies that the asymmetry changes over time.
Alternation implies that the asymmetries between units are dynamically out of phase.
Deviation implies that the net balance of asymmetries is unevenly distributed among units.
Chaoticity implies a stochastic component in the development of asymmetry.
These correlations (Table 2) hold in most conditions, except e.g. a bilateral species that is stationary.
A closer look in the symmetry-behavior correlations suggests the mechanisms of how motions arise.
[noitemsep]
A bilateral species has reflectional symmetry along the lateral (left-right) axis, but heavy asymmetry along the longitudinal (rostro-caudal) axis that may be the cause of directional movement along this axis.
A deformed bilateral species has the lateral reflectional symmetry broken, that introduce an angular component to its linear motion.
A radial species has bilateral repeating units arranged radially, all directional vectors along the radii (either pulling in to or pushing out from the centroid) cancel out, thus overall remain stationary.
A rotational species has asymmetric repeating units with lateral reflectional symmetry broken, that initiates an angular rotation around the centroid.
On top of these global movements, the dynamical qualities of asymmetry — static/dynamical, in-phase/out-of-phase, balanced/unbalanced, regular/stochastic — lead to the dynamical qualities of locomotion (i.e. gaits).
Based on these reasonings, we propose the following stability-motility hypothesis in Lenia (potentially applicable to real-world physiology or evolutionary biology):
Symmetry provides stability; asymmetry provides motility.
Distribution of asymmetry determines locomotive mode; its development determines gait.
The alternation gait, that is global oscillation plus out-of-phase local oscillations, demonstrates phenomena like long-range synchronization and internal clockwork, suggests the existence of self-organization and internal communication among components.
Alternating translocation (T_{A}) in a simple bilateral species, where the two halves are in opposite phases, leads to spatiotemporal reflectional (i.e. glide) symmetry at half-cycle, in addition to the full oscillation
(22) | ||||
(23) |
where is the quasi-period, is a shift, is a flip (along the lateral axis). (Figure 12(g))
In alternating long-chain species, where the two wings are oscillating out-of-phase but the main chain remains static, demonstrates long-range synchronization that faraway local structures are able to synchronize with each other. ^{6}^{6}6We performed experiments to show that alternation is self-recovering, meaning that it is actively maintained by the species and is not coincidental.
Alternating gyration (G_{A}) is a special case only found in Vagorbium (a variant of Gyrorbium) where it gyrates to the opposite direction every second cycle, resulting in a zig-zag trajectory (Figure 12(d)).
Alternating stationarity (S_{A}) occurs in stationary radial lifeforms (with repeating units), leads to spatiotemporal reflectional or rotational symmetry at half-cycle, giving an optical illusion of “inverting” motions (Figure 12(a)).
(24) | ||||
(25) |
where is a rotation (around the centroid).
Alternating rotation (R_{A}) is an intricate phenomenon found in rotational species, especially family Dentidae. Consider a Dentidae species with repeating units, two adjacent units are separated spatially by angle and temporally by a phase difference of cycle (Figure 13).
After cycle, the pattern recreates itself with rotation due to angular speed , plus an extra rotational symmetry of m units (angle ) due to alternation, which leads to spatiotemporal rotational symmetry at cycle
(26) | ||||
(27) |
This giving an optical illusion that local features (e.g. a hole) are being transferred from one unit to another inside the rotating species (Figure 13 outer arrows).
The values of and depend on the species, seem to follow no obvious trend, except that (adjacent phase transfer) in even-sided species and (unit-skipping) in odd-sided species (Table 3).
Besides direct observation, patterns in Lenia can be studied though statistical measurement and analysis, akin to “allometry” or “biostatistics” in biology.
For example, various behaviors can be inferred by the average (mean or median), variability (standard deviation or interquartile length) or phase space trajectory of various statistical measures (Table 4).
A few general trends can be deduced from measure charts, for example, linear speed is found to be roughly inverse proportional to density. From the linear speed vs. mass chart (Figure 14), genera form strata according to linear speed (OPSHC), and species form clusters according to mass, while cluster shapes may convey further information, e.g. twin clusters indicate a separation in convexity.
In the previous sections, we outlined the general characterizations of Lenia life from various perspectives. Here we combine these aspects and provide a focused study on one representative genus — Paraptera — as a concrete example of qualitative and quantitative analysis.
From a wider perspective, Paraptera is closely related to two other genera Parorbium and Tetrascutium, altogether they comprise the rank-1 unit-4 group.
In the - map (Figure 15), their niches comprise the Parorbium-Paraptera-Tetrascutium complex. The narrow bridge between Parorbium and Paraptera indicates that continuous transformation is possible, while the correspondence between the small tip on Paraptera (species P. arcus valvatus) and Tetrascutium suggests a remote relationship (as hinted by their similar morphology).
Allometric methods were employed to further segregate the complex (Figure 16, Table 4). A few potential species with distinct traits (e.g. jumping or walking) were isolated in measure charts. After verified in computer simulations that they exhibit unique morphology or behavior, new species names were assigned (Table 5).
In genus Paraptera, a cross section at was further studied, where five species exist in (Figure 15 red dotted line, Table 6).
To assess their behavioral traits, five number summaries of statistical measures (mass and mass asymmetry ) were plotted against , together with snapshot phase space trajectories at a few loci (Figure 17, also see Figure 12(e-i)).
At higher values, Paraptera arcus saliens (P4as) has high variability and near zero , corresponding to perfect bilateral symmetry and jumping behavior (locus a). P. cavus pedes (P4cp) has high variability, corresponding to alternating asymmetry and walking behavior (locus d).
P4as and P4cp coexist over . Just outside of this niche overlap, the two species slowly transform into each other, as shown by the spiral phase space trajectories (loci b, c). Slow transformations also occur between P4cp and P4sp (locus f).
Irregularity and chaos arise at lower values. P. sinus pedes (P4sp) has non-zero , corresponding to asymmetric morphology and deflected movement (locus g). P. sinus pedes furiosus (P4spf) has chaotic phase space trajectory, corresponding to deformed morphology and chaotic movement (locus h).
At the edge of chaoticity, P. sinus pedes rupturus (P4spr) has even higher and rugged variability, often encounters episodes of acute deformation but eventually recovers (locus i). Outside the lower boundary, the pattern fails to recover and finally disintegrates.
Standard CAs like GoL and ECA consider only the nearest sites as neighborhood, while more recent variants like LtL, SmoothLife and Lenia have extended neighborhoods, and are able to control over the “granularity” of space. The latter ones are still technically discrete, but are likely approximating a continuous class of models called Euclidean automata (EA) [27]. We call them geometric cellular automata (GCA).
GCAs and standard CAs are fundamentally different, with a number of contrasting qualities between their generated patterns (Table 7). In standard CAs, a large number of interesting patterns are concentrated in specific rules like GoL, while GCAs patterns are scattered over the parameter space. Also, the distinction of “digital” vs. “analog” goes beyond a metaphor, in that standard CAs like GoL and ECA rule 110 are actually capable of (digital) universal computation [22, 41], while whether a certain kind of “analog computer” is possible in GCAs remains to be seen.
As GCAs being approximants of EAs, these contrasting qualities may well exist between EAs and CAs.
Here we deep dive into the very nature of Lenia regarding the unpredictability, fuzziness, quasi-periodicity, resilience and lifelikeness of its generated patterns, at times using GoL for contrast.
GoL patterns are either persistent, which are guaranteed to follow the same dynamics every time, or temporary, which will eventually stabilize as persistent patterns or vanish. Lenia patterns, on the other hand, have various types of persistence:
[noitemsep]
Transient patterns that only last for a short time.
Quasi-stable patterns that are able to sustain for a few to hundreds of cycles.
Stable patterns that survive as long as simulations went, seemingly everlasting.
Metastable patterns that are stable, but transform into other patterns after slight perturbations.
Chaotic patterns that “walk a thin line” between chaos and self-destruction.
Markovian patterns that shapeshift among templates, each has its own degree of persistence.
Given a static pattern, it is unpredictable whether it belongs to which persistent level unless we put it into simulation for a considerable (potentially infinite) amount of time, a situation akin to the halting problem and the undecidability in class 4 CAs [42]. This uncertainty results in the vague niche boundaries of many Lenia species (especially at lower values with much chaos).
Even at persistent level 3, in contrast to the “glider” in GoL that will move diagonally forever, we can never be 100% sure that a seemingly stable Orbium will not eventually die out.
No two patterns in Lenia are the same. There are various levels of fuzziness and subtle varieties. Within a species, slightly differences in parameters, rule settings, or initial configurations would result in slightly different patterns (see Figure 6). Even during a pattern’s lifetime, no two cycles are the same.
Consider the phase space trajectories of recurrent patterns (Figure 18), evidently every trajectory corresponds to an attractor (or a strange attractor if chaotic). Yet, behind a group of similar patterns, there seems to be another kind of “attractor” that draws them into a common morphological-behavioral template.
In philosophy, essentialism proposes that every entity in the world can be identified by a set of intrinsic features or an “essence”, be it an ideal form (Plato’s idealism) or a natural kind (Aristotle’s hylomorphism). So in Lenia, is there something like “Orbium-ness” in all instances and occurrences of Orbium? Could this be identified or utilized objectively and quantitatively?
The notions of an “attractor” or “essence” and the tolerance of fuzziness around such central tendency is what makes the definition of a Lenia “species” possible.
Unlike GoL where a recurrent pattern returns to the exact same pattern after an exact period of time, a recurrent pattern in Lenia returns to similar patterns after slightly irregular periods or quasi-periods
, which are probably normally distributed around an average. Lenia has various types of periodicity:
[noitemsep]
Aperiodicity, transient non-recurrent patterns.
Quasi-periodicity, quasi-stable, stable or metastable patterns.
Chaotic periodicity, chaotic patterns, with wide-spread period distribution.
Markovian periodicity, Markovian patterns, each template has its own quasi-period.
Essentially, in discrete Lenia, there are finite, albeit astronomically large, number of possible configurations . Given enough time, an initial pattern would eventually return to the exact same pattern, an argument not unlike Nietzsche’s “eternal recurrence”. But in Lenia, there would be numerous approximate recurrences between two exact recurrences, while in continuous Lenia, exact recurrence may even be impossible.
Given the fuzziness and irregularity in Lenia patterns, they are surprisingly resilient and exhibit phenotypic plasticity. By adjusting their structures and dynamics elastically, they are able to absorb deformations and transformations, adapt to environmental changes (parameters and rule settings), or react to head-to-head collisions, and continue to survive.
A speculative mechanism for the plasticity (also self-organization and self-regulation in general) is postulated as the kernel resonance hypothesis (Figure 19). In the potential distribution, a network of potential peaks can be observed. Possibly, each peak was formed by the overlapping or “resonance” of kernel rings casted by mass lumps from various locations, while the collection of peaks in turn determine the location of the mass lumps. In this way, the mass lumps (i.e. peaks in the mass distribution) influence each other reciprocally and may self-organize into local structures, providing the basis of morphogenesis.
The kernel resonance would be dynamic over time, and may even be self-regulating due to the mass-potential-mass feedback loop, providing the basis of homeostasis. Plasticity may stem from static buffering and dynamical flexibility provided by such feedback loop.
Besides the superficial resemblance, Lenia life may have deeper connections with biological life.
Both Lenia and Earth life exhibit structural symmetry and similar symmetry-locomotion relationships (Figure 20(b-c)).
Radial symmetry is universal in Lenia order Radiiformes. In biological life, radial symmetry is exhibited in microscopic protists (diatoms, radiolarians) and primitive animals historically grouped as Radiata (jellyfish, corals, comb jellies, echinoderm adults). These radiates are sessile, floating or slow-moving, similarly, Lenia radiates are usually stationary or rotating with little linear movement.
Bilateral symmetry is ubiquitous among Lenia families. In biological life, the group Bilateria (vertebrates, arthropods, mollusks, various “worm” phyla) with the same symmetry are the most successful branch of animals since their proliferation and rapid diversification near the Cambrian explosion 542 million years ago [43]. These bilaterians are optimized for efficient directed locomotion, while similarly, Lenia bilaterians engage in fast linear movements.
The parameter space of Lenia, earlier visualized as a geographical landscape (“Ecology” section), can also be thought of as an adaptive landscape. Species niches correspond to fitness peaks in the adaptive landscape, indicate successful adaptation of those structural-dynamical templates to ranges of parametric environmental settings.
Among all body plans (correspond to Earth animal phyla or Lenia families), some may be considered more adaptive as indicated by higher biodiversity, wider ecological distribution, and perhaps greater complexity. In Earth life, the champions are the insects (in terms of biodiversity), the nematodes (in terms of ecosystem breadth and individual count), and the mammals (producing intelligent species like cetaceans and primates). In Lenia, family Pterifera is the most successful in class Exokernel with its high diversity (number of species), wide distribution (total niche area), and high complexity.
The parallels between two systems regarding adaptive landscapes and quantitative assessment of adaptability may provide insights in evolutionary biology and evolutionary computation.
One common difficulty encountered in the studies of both Earth and Lenia life is the precise definition of a “species”, or the species problem. In evolutionary biology, several species concepts have been proposed [46]:
[noitemsep]
Morphological species, based on phenotypic differentiation [47]
Phenetic species, based on numerical clustering (cf. phenetics) [48]
Genetic species, based on genotypic clustering [49]
Biological species, based on reproductive isolation [50]
Ecological species, based on niche isolation [53]
Similar concepts are used in combination for species identification in Lenia, including morphological (similar morphology and behavior), phenetic (statistical cluster) and ecological (niche cluster). However, species concepts face problems in various situations, for example, in Lenia’s case, species aggregates or convergent evolution, and in Earth’s case, niche complex or shapeshifting lifeforms. It remains an open question whether clustering into species and grouping into higher taxa can be carried out objectively and systematically.
Here are a few open questions we hope to answer:
[noitemsep]
What are the enabling factors and mechanisms of how self-organization, self-regulation, self-direction, adaptability, etc. emerge in Lenia?
How do interesting phenomena like symmetry, metamerism, spontaneous metamorphosis, particle collision, alternation, etc. arise in Lenia?
How is Lenia life related to biological life and other forms of artificial life?
Can Lenia life be grouped into species and higher taxa objectively and systematically?
Does continuous Lenia exist as the continuum limit of discrete Lenia? If so, do corresponding “ideal” lifeforms exist in this Euclidean Automaton?
Is Lenia Tuning-complete and capable of universal computation?
Is Lenia capable of open-ended evolution that generates unlimited novelty and complexity?
Do self-replicating or pattern-emitting lifeforms exist in Lenia?
Do lifeforms exist in other variants of Lenia (e.g. 3D)?
To answer these questions, the following approaches of future works are suggested.
For the sheer joy of discovering new species, and for further understanding Lenia and artificial life, we need better capabilities of species discovery and identification.
Automatic and accurate species identification could be achieved via computer vision and pattern recognition using machine learning or deep learning techniques, e.g. by feeding the grid to a convolutional neural networks (CNNs) or feeding the measures time-series to a recurrent neural networks (RNNs).
Interactive evolutionary computation (IEC) currently used for new species discovery could be advanced to allow crowdsourcing, web or mobile applications with intuitive interface would allow online users to simulate, mutate, select and share interesting patterns (cf. Picbreeder [54], Ganbreeder [55]
). Web performance and functionality could be improved using WebAssembly, OpenGL, TensorFlow.js, etc.
Alternatively, evolutionary computation (EC) and similar methodologies could be used for automatic and efficient exploration of the search space, as has been successfully used for evolving new body parts or body plans [39, 9, 56]. Patterns could be represented in genetic (indirect) encoding using Compositional Pattern-Producing Network (CPPN) [57] or Bezier splines [58]
, which are then evolved using genetic algorithms like NeuroEvolution of Augmenting Topologies (NEAT)
[59]. Novelty-driven and curiosity-driven algorithms are promising approaches [60, 61, 62].Grid traversal of the parameter space (depth-first or breath-first search) is still useful in collecting statistical data, but it needs more reliable algorithms, especially for high-rank metamorphosis-prone species.
All species data collected from automation or crowdsourcing could be stored in a central database for further analysis. Using well-established techniques in related scientific disciplines, the data could be used for dynamical systems analysis (e.g. quasi-period distribution, Lyapunov exponents, transition probabilities matrix), shape analysis (computational anatomy, statistical shape analysis), time-series analysis (cf. in astronomy [63]
), and automatic classification (unsupervised or semi-supervised learning).
We could also explore variants and further generalizations of Lenia, for example, higher-dimensional space (especially 3D) [64, 65, 66]; different kinds of grids (e.g. hexagonal, Penrose tiling, irregular mesh) [67, 68, 69]; different structures of kernel (e.g. non-concentric rings); other updating rules (e.g. asynchronous, heterogeneous, stochastic) [70, 71, 72].
It has been demonstrated that Lenia shows a few signs of a living system:
[noitemsep]
Self-organization, patterns develop well-defined structures
Self-regulation, patterns maintain dynamical equilibria via oscillation etc.
Self-direction, patterns move consistently through space
Adaptability, patterns adapt to changes via plasticity
Evolvability, patterns evolve via manual operations and potentially genetic algorithms
We should seek whether these are merely superficial resemblances with biological life or are indicators of deeper connections. In the latter case, Lenia could contribute to the endeavors of artificial life in attempting to “understand the essential general properties of living systems by synthesizing life-like behavior in software” [73], or could even add to the debate about the definitions of life as discussed in astrobiology and virology [74, 75]. In the former case, Lenia can still be regarded as a “mental exercise” on how to study a complex system using various methodologies.
Lenia could also be served as a “machine exercise” to provide a substrate or testbed for parallel computing, artificial life and artificial intelligence. The heavy demand in matrix calculation and pattern recognition could act as a benchmark for machine learning algorithms and hardware acceleration; the huge search space of patterns, possibly in higher dimensions, could act as a playground for evolutionary algorithms in the quest of algorithmizing and ultimately understanding open-ended evolution.
[76][noitemsep]
Showcase video of Lenia (produced using Python program) at https://vimeo.com/277328815
Source code of Lenia at http://github.com/Chakazul/Lenia
Source code of Primordia at http://github.com/Chakazul/Primordia
I would like to thank David Ha, Sam Kriegman, Tim Hutton, Kyrre Glette, and Pierre-Yves Oudeyer for valuable discussions, suggestions and insights.
Probability and Phase Transition
, pp. 49–67, Dordrecht: Springer, 1994.W. Kahan, “Lecture notes on the status of ieee standard 754 for binary floating-point arithmetic.”
http://http.cs.berkeley.edu/~wkahan/ieee754status/ieee754.ps, 1997-10-01. [Online].D. Ha, “Reinforcement learning for improving agent design.” arXiv preprint, 2018.
A. Baranes and P. Y. Oudeyer, “Active learning of inverse models with intrinsically motivated goal exploration in robots,”
Robotics and Autonomous Systems, vol. 61, no. 1, pp. 49–73, 2013.The notion of “life”, here interpreted as self-organizing autonomous entities in a broader sense, may include biological life, artificial life, and other possibilities like extraterrestrial life:
Vitae
Tree Biota 4.5cm Biological life
Tree Artificialia 4.5cm Artificial life
Tree Xenobiota (?) 4.5cm Extraterrestrial life (undiscovered)
The biological tree of life, except the uncertain situation of viruses [75], is widely accepted as:
Biota
Superdomain Acytota
Domain Vira (?)
Superdomain Cytota
Domain Bacteria, Archaea
Domain Eukaryota
Kingdom Protista, Plantae, Fungi, Animalia
We propose the following tree of artificial life, based on lifeforms from Lenia and other models:
Artificialia
Domain Synthetica 4.5cm “Wet” biochemical synthetic life
Domain Mechanica 4.5cm “Hard” mechanical or robotic life, e.g. [39]
Domain Simulata 4.5cm “Soft” computer simulated life
Kingdom Sims 4.5cm Evolved virtual creatures, e.g. [7, 8, 9]
Kingdom Greges 4.5cm Particle swarm solitons, e.g. [77, 78, 30, 3]
Kingdom Turing 4.5cm Reaction-diffusion solitons, e.g. [29, 4, 66]
Kingdom Automata 4.5cm Cellular automata patterns
Phylum Discreta 4.5cm Non-scalable, e.g. [20, 21]
Phylum Lenia 4.5cm Scalable, e.g. [25, 28]
The current taxonomy of Lenia, according to the definitions of taxonomical ranks (“Taxonomy” section), is:
Phylum Lenia
Subphylum Stereolenia 4.5cm Three-dimensional (undiscovered)
Subphylum Planolenia 4.5cm Two-dimensional
Class Exokernel
Order Orbiformes
Family Orbidae
Order Scutiformes
Family Scutidae, Pterifera, Helicidae, Circidae
Class Mesokernel
Order Echiniformes
Family Echinidae, Geminidae, Ctenidae, Uridae
Class Endokernel
Order Kroniformes
Family Kronidae, Quadridae, Volvidae
Order Radiiformes
Family Dentidae, Radiidae, Bullidae, Lapillidae, Folidae
Order Amoebiformes
Family Amoebidae
The followings are the proposed three Lenia classes (Figure 10).
[noitemsep]
Class Exokernel are lifeforms that have strong outer kernels rings.
Class Mesokernel are lifeforms that have kernel rings with similar peak heights.
Class Endokernel are lifeforms that have strong inner kernels rings.
Class Exokernel
Order Orbiformes
“disk bugs” are composed of halved circular disks (orb), including the singular Orbium and Gyrorbium, long-chain
textsftextsfParorbinae and Synorbinae series. Orbidae are special in Lenia being the fastest moving, rank-1 only, and free from the upper state bound ().
Order Scutiformes
“shield bugs” are composed of thick circular disks (scutum), including singular Scutium, diploid Pyroscutim, and long-chain Megaloscutinae series.
“winged bugs” are composed of wings (pteron) and sacs (vacuole), including singular Gyropteron, diploid Synptera and Paraptera, and long-chain Vacuopterinae series.
“helix bugs” are rotating versions of Pterifera, including long-chain Perissohelicinae and Artiohelicinae series with odd and even number of vacuoles.
“circle bugs” are single or multiple concentric rings.
Class Mesokernel
Order Echiniformes
“spiny bugs” is a “waste-bin taxon” with thorny or wavy species.
“twin bugs” have compartments with heavy lamination, including versatile twin-compartment Diplogemininae and long-chain Polygermininae series.
“comb bugs” resemble Pterifera but have parallel narrow stripes between wings.
“tail bugs” are large lifeforms with tails of various lengths.
Class Endokernel
Order Kroniformes
“crown bugs” are complex versions of Scutidae and Pterifera, including singular Kronium and deviations like Argentokronium and Aurokronium, and long-chain Vacuokroninae series.
“square bugs” includes versatile Quadrium composed of square grid of masses.
“twist bugs” are possibly complex versions of Helicidae.
Order Radiiformes
“tooth bugs” are rotating species with gear-like units.
“radial bugs” are regular or star polygons, including triangular Trigonium, cross-like Crucium, star-like Asterium, and snowflake-like Nivium.
“bubble bugs” are bilateral species consist of an inner ring and an outer bubbled layer.
“gem bugs” are small rings distributed around a circle.
“petal bugs” are stationary, thick, radial species with petal-like units.
Order Amoebiformes
“amoeba bugs” are volatile species without well-defined shape or behavior.