Lenia is a family of cellular automata created by Bert Wang-Chak Chan.[1][2][3] It is intended to be a continuous generalization of Conway's Game of Life, with continuous states, space and time. As a consequence of its continuous, high-resolution domain, the complex autonomous patterns ("lifeforms" or "spaceships") generated in Lenia are described as differing from those appearing in other cellular automata, being "geometric, metameric, fuzzy, resilient, adaptive, and rule-generic".[1]

A sample autonomous pattern from Lenia.
An animation showing the movement of a glider in Lenia.

Lenia won the 2018 Virtual Creatures Contest at the Genetic and Evolutionary Computation Conference in Kyoto,[4] an honorable mention for the ALIFE Art Award at ALIFE 2018 in Tokyo,[5] and Outstanding Publication of 2019 by the International Society for Artificial Life (ISAL).[6]

Rules

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Iterative updates

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Let   be the lattice or grid containing a set of states  . Like many cellular automata, Lenia is updated iteratively; each output state is a pure function of the previous state, such that

 

where   is the initial state and   is the global rule, representing the application of the local rule over every site  . Thus  .

If the simulation is advanced by   at each timestep, then the time resolution  .

State sets

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Let   with maximum  . This is the state set of the automaton and characterizes the possible states that may be found at each site. Larger   correspond to higher state resolutions in the simulation. Many cellular automata use the lowest possible state resolution, i.e.  . Lenia allows for much higher resolutions. Note that the actual value at each site is not in   but rather an integer multiple of  ; therefore we have   for all  . For example, given  ,  .

Neighborhoods

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A 9-square Moore neighborhood like those used in Game of Life.
 
The "ball" neighborhoods used by Lenia.

Mathematically, neighborhoods like those in Game of Life may be represented using a set of position vectors in  . For the classic Moore neighborhood used by Game of Life, for instance,  ; i.e. a square of size 3 centered on every site.

In Lenia's case, the neighborhood is instead a ball of radius   centered on a site,  , which may include the original site itself.

Note that the neighborhood vectors are not the absolute position of the elements, but rather a set of relative positions (deltas) with respect to any given site.

Local rule

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There are discrete and continuous variants of Lenia. Let   be a vector in   within   representing the position of a given site, and   be the set of sites neighboring  . Both variations comprise two stages:

  1. Using a convolution kernel   to compute the potential distribution  .
  2. Using a growth mapping   to compute the final growth distribution  .

Once   is computed, it is scaled by the chosen time resolution   and added to the original state value: Here, the clip function is defined by   .

The local rules are defined as follows for discrete and continuous Lenia:

 

Kernel generation

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The kernel shell, kernel skeleton, and growth mappings for Lenia.

There are many ways to generate the convolution kernel  . The final kernel is the composition of a kernel shell   and a kernel skeleton  .

For the kernel shell  , Chan gives several functions that are defined radially. Kernel shell functions are unimodal and subject to the constraint   (and typically   as well). Example kernel functions include:

 

Here,   is the indicator function.

Once the kernel shell has been defined, the kernel skeleton   is used to expand it and compute the actual values of the kernel by transforming the shell into a series of concentric rings. The height of each ring is controlled by a kernel peak vector  , where   is the rank of the parameter vector. Then the kernel skeleton   is defined as

 

The final kernel   is therefore

 

such that   is normalized to have an element sum of   and   (for conservation of mass).   in the discrete case, and   in the continuous case.

Growth mappings

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The growth mapping  , which is analogous to an activation function, may be any function that is unimodal, nonmonotonic, and accepts parameters  . Examples include

 

where   is a potential value drawn from  .

Game of Life

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The Game of Life may be regarded as a special case of discrete Lenia with  . In this case, the kernel would be rectangular, with the function and the growth rule also rectangular, with  .

Patterns

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Some of the wide variety of "species" in Lenia.

By varying the convolutional kernel, the growth mapping and the initial condition, over 400 "species" of "life" have been discovered in Lenia, displaying "self-organization, self-repair, bilateral and radial symmetries, locomotive dynamics, and sometimes chaotic nature".[7] Chan has created a taxonomy for these patterns.[1]

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Cellular automata as a convolutional neural network.[8]

Other works have noted the strong similarity between cellular automata update rules and convolutions. Indeed, these works have focused on reproducing cellular automata using simplified convolutional neural networks. Mordvintsev et al. investigated the emergence of self-repairing pattern generation.[9] Gilpin found that any cellular automaton could be represented as a convolutional neural network, and trained neural networks to reproduce existing cellular automata[8]

In this light, cellular automata may be seen as a special case of recurrent convolutional neural networks. Lenia's update rule may also be seen as a single-layer convolution (the "potential field"  ) with an activation function (the "growth mapping"  ). However, Lenia uses far larger, fixed, kernels and is not trained via gradient descent.

See also

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References

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  1. ^ a b c Chan, Bert Wang-Chak (2019-10-15). "Lenia: Biology of Artificial Life". Complex Systems. 28 (3): 251–286. arXiv:1812.05433. doi:10.25088/ComplexSystems.28.3.251.
  2. ^ "Lenia". chakazul.github.io. Retrieved 2021-10-12.
  3. ^ Roberts, Siobhan (2020-12-28). "The Lasting Lessons of John Conway's Game of Life". The New York Times. ISSN 0362-4331. Retrieved 2021-10-13.
  4. ^ "The virtual creatures competition". virtualcreatures.github.io. Retrieved 2021-10-12.
  5. ^ "ALife Art Award 2018". ALIFE Art Award 2018. Retrieved 2021-10-12.
  6. ^ "2020 ISAL Awards: Winners".
  7. ^ "Lenia". chakazul.github.io. Retrieved 2021-10-13.
  8. ^ a b Gilpin, William (2019-09-04). "Cellular automata as convolutional neural networks". Physical Review E. 100 (3): 032402. arXiv:1809.02942. doi:10.1103/PhysRevE.100.032402. ISSN 2470-0045.
  9. ^ Mordvintsev, Alexander; Randazzo, Ettore; Niklasson, Eyvind; Levin, Michael (2020-02-11). "Growing Neural Cellular Automata". Distill. 5 (2): e23. doi:10.23915/distill.00023. ISSN 2476-0757.