Lenia

Let $\backslash mathcal\{L\}$ be the lattice or grid containing a set of states $S^\backslash mathcal\{L\}$. Like many cellular automata, Lenia is updated iteratively; each output state is a pure function of the previous state, such that

$$\backslash Phi(A^0)\; =\; A^\{\backslash Delta\; t\},\; \backslash Phi(A^\{\backslash Delta\; t\})\; =\; A^\{2\backslash Delta\; t\},\; \backslash ldots,\; \backslash Phi(A^t)\; =\; A^\{t\; +\; \backslash Delta\; t\},\backslash ldots$$

where $A^0$ is the initial state and $\backslash Phi\; :\; S^\backslash mathcal\{L\}\; \backslash rightarrow\; S^\backslash mathcal\{L\}$ is the global rule, representing the application of the local rule over every site $\backslash mathbf\{x\}\backslash in\backslash cal\{L\}$. Thus $\backslash Phi^N(A^t)\; =\; A^\{t\; +\; N\backslash Delta\; t\}$.

If the simulation is advanced by $\backslash Delta\; t$ at each timestep, then the time resolution $T\; =\; \backslash frac\{1\}\{\backslash Delta\; t\}$.

Let $S\; =\; \backslash \{0,\; 1,\; \backslash ldots,\; P-1,\; P\backslash \}$ with maximum $P\; \backslash in\; \backslash Z$. This is the state set of the automaton and characterizes the possible states that may be found at each site. Larger $P$ correspond to higher state resolutions in the simulation. Many cellular automata use the lowest possible state resolution, i.e. $P\; =\; 1$. Lenia allows for much higher resolutions. Note that the actual value at each site is not in $[0,P]$ but rather an integer multiple of $\backslash Delta\; p\; =\; \backslash frac\{1\}\{P\}$; therefore we have $A^t(\backslash mathbf\{x\})\; \backslash in\; [0,\; 1]$ for all $\backslash mathbf\{x\}\; \backslash in\; \backslash mathcal\{L\}$. For example, given $P\; =\; 4$, $\backslash mathbf\{A\}^t(\backslash mathbf\{x\})\; \backslash in\; \backslash \{0,\; 0.25,\; 0.5,\; 0.75,\; 1\backslash \}$.

File:Moore neighborhood.svg

File:Lenia neighborhood.png

Mathematically, neighborhoods like those in Game of Life may be represented using a set of position vectors in $\backslash R^2$. For the classic Moore neighborhood used by Game of Life, for instance, $\backslash mathcal\{N\}\; =\; \backslash \{-1,\; 0,\; 1\backslash \}^2$; i.e. a square of size 3 centered on every site.

In Lenia's case, the neighborhood is instead a ball of radius $R$ centered on a site, $\backslash mathcal\{N\}\; =\; \backslash \{\backslash mathbf\{x\}\; \backslash in\; \backslash mathcal\{L\}\; :\; \backslash lVert\; \backslash mathbf\{x\}\; \backslash rVert\_2\; \backslash leq\; R\backslash \}$, 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.

There are discrete and continuous variants of Lenia. Let $\backslash mathbf\{x\}$ be a vector in $\backslash R^2$ within $\backslash mathcal\{L\}$ representing the position of a given site, and $\backslash mathcal\{N\}$ be the set of sites neighboring $\backslash mathbf\{x\}$. Both variations comprise two stages:

1. Using a convolution kernel $\backslash mathbf\{K\}\; :\; \backslash mathcal\{N\}\; \backslash rightarrow\; S$ to compute the potential distribution $\backslash mathbf\{U\}^t(\backslash mathbf\{x\})=\backslash mathbf\{K\}\; *\; \backslash mathbf\{A\}^t(\backslash mathbf\{x\})$.
2. Using a growth mapping $G\; :\; [0,\; 1]\; \backslash rightarrow\; [-1,\; 1]$ to compute the final growth distribution $\backslash mathbf\{G\}^t(\backslash mathbf\{x\})=G(\backslash mathbf\{U\}^t(\backslash mathbf\{x\}))$.

Once $\backslash mathbf\{G\}^t$ is computed, it is scaled by the chosen time resolution $\backslash Delta\; t$ and added to the original state value:$$\backslash mathbf\{A\}^\{t+\backslash Delta\; t\}(\backslash mathbf\{x\})\; =\; \backslash text\{clip\}(\backslash mathbf\{A\}^\{t\}\; +\; \backslash Delta\; t\; \backslash ;\backslash mathbf\{G\}^t(\backslash mathbf\{x\}),\backslash ;\; 0,\backslash ;\; 1)$$Here, the clip function is defined by $\backslash operatorname\{clip\}(u,a,b):=\backslash min(\backslash max(u,a),b)$ .

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

$$\backslash begin\{align\}$$

\mathbf{U}^t(\mathbf{x}) &= \begin{cases}

\sum_{\mathbf{n} \in \mathcal{N}} \mathbf{K(n)}\mathbf{A}^t(\mathbf{x}+\mathbf{n})\Delta x^2, & \text{discrete Lenia} \\

\int_{\mathbf{n} \in \mathcal{N}} \mathbf{K(n)}\mathbf{A}^t(\mathbf{x}+\mathbf{n})dx^2, & \text{continuous Lenia}

\end{cases} \\

\mathbf{G}^t(\mathbf{x}) &= G(\mathbf{U}^t(\mathbf{x})) \\

\mathbf{A}^{t+\Delta t}(\mathbf{x}) &= \text{clip}(\mathbf{A}^t(\mathbf{x}) + \Delta t\;\mathbf{G}^t(\mathbf{x}),\; 0,\; 1)

\end{align}

File:Screenshot from 2021-10-12 18-26-15.png

There are many ways to generate the convolution kernel $\backslash mathbf\{K\}$. The final kernel is the composition of a kernel shell $K\_C$ and a kernel skeleton $K\_S$.

For the kernel shell $K\_C$, Chan gives several functions that are defined radially. Kernel shell functions are unimodal and subject to the constraint $K\_C(0)\; =\; K\_C(1)\; =\; 0$ (and typically $K\_C\backslash left(\backslash frac\{1\}\{2\}\backslash right)\; =\; 1$ as well). Example kernel functions include:

$$K\_C(r)\; =\; \backslash begin\{cases\}$$

\exp\left(\alpha - \frac{\alpha}{4r(1-r)}\right), & \text{exponential}, \alpha=4 \\

(4r(1-r))^\alpha, & \text{polynomial}, \alpha=4 \\

\mathbf{1}_{\left[\frac{1}{4},\frac{3}{4}\right]}(r), & \text{rectangular} \\

\ldots, & \text{etc.}

\end{cases}

Here, $\backslash mathbf\{1\}\_A(r)$ is the indicator function.

Once the kernel shell has been defined, the kernel skeleton $K\_S$ 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 $\backslash beta\; =\; (\backslash beta\_1,\; \backslash beta\_2,\; \backslash ldots,\; \backslash beta\_B)\; \backslash in\; [0,1]^B$, where $B$ is the rank of the parameter vector. Then the kernel skeleton $K\_S$ is defined as

$$K\_S(r;\backslash beta)=\backslash beta\_\{\backslash lfloor\; Br\; \backslash rfloor\}\; K\_C(Br\; \backslash text\{\; mod\; \}\; 1)$$

The final kernel $\backslash mathbf\{K\}(\backslash mathbf\{n\})$ is therefore

$$\backslash mathbf\{K\}(\backslash mathbf\{n\})\; =\; \backslash frac\{K\_S(\backslash lVert\; \backslash mathbf\{n\}\; \backslash rVert\_2)\}$$

such that $\backslash mathbf\{K\}$ is normalized to have an element sum of $1$ and $\backslash mathbf\{K\}\; *\; \backslash mathbf\{A\}\; \backslash in\; [0,\; 1]$ (for conservation of mass). $|K\_S|\; =\; \backslash textstyle\; \backslash sum\_\{\backslash mathcal\{N\}\}\; \backslash displaystyle\; K\_S\; \backslash ,\; \backslash Delta\; x^2$ in the discrete case, and $\backslash int\_\{N\}\; K\_S\; \backslash ,dx^2$ in the continuous case.

The growth mapping $G\; :\; [0,\; 1]\; \backslash rightarrow\; [-1,1]$, which is analogous to an activation function, may be any function that is unimodal, nonmonotonic, and accepts parameters $\backslash mu,\backslash sigma\; \backslash in\; \backslash R$. Examples include

$$G(u;\backslash mu,\backslash sigma)\; =\; \backslash begin\{cases\}$$

2\exp\left(-\frac{(u-\mu)^2}{2\sigma^2}\right)-1, & \text{exponential} \\

2\cdot\mathbf{1}_{[\mu\pm3\sigma]}(u)\left(1-\frac{(u-\mu)^2}{9\sigma^2}\right)^\alpha-1, & \text{polynomial}, \alpha=4 \\

2\cdot\mathbf{1}_{[\mu\pm\sigma]}(u)-1, & \text{rectangular} \\

\ldots, & \text{etc.}

\end{cases}

where $u$ is a potential value drawn from $\backslash mathbf\{U\}^t$.

The Game of Life may be regarded as a special case of discrete Lenia with $R\; =\; T\; =\; P\; =\; 1$. In this case, the kernel would be rectangular, with the function$$K\_C(r)\; =\; \backslash mathbf\{1\}\_\{\backslash left[\backslash frac\{1\}\{4\},\backslash frac\{3\}\{4\}\backslash right]\}(r)\; +\; \backslash frac\{1\}\{2\}\backslash mathbf\{1\}\_\{\backslash left[0,\backslash frac\{1\}\{4\}\backslash right)\}(r)$$and the growth rule also rectangular, with $\backslash mu\; =\; 0.35,\; \backslash sigma\; =\; 0.07$.

File:Lenia species.png

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".{{Cite web|title=Lenia|url=https://chakazul.github.io/lenia.html|access-date=2021-10-13|website=chakazul.github.io}} Chan has created a taxonomy for these patterns.

File:Cellular automata and convnets.png

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.{{Cite journal|last=Mordvintsev|first=Alexander|last2=Randazzo|first2=Ettore|last3=Niklasson|first3=Eyvind|last4=Levin|first4=Michael|date=2020-02-11|title=Growing Neural Cellular Automata|url=https://distill.pub/2020/growing-ca|journal=Distill|language=en|volume=5|issue=2|pages=e23|doi=10.23915/distill.00023|issn=2476-0757|doi-access=free}} Gilpin found that any cellular automaton could be represented as a convolutional neural network, and trained neural networks to reproduce existing cellular automata

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" $\backslash mathbf\{K\}$) with an activation function (the "growth mapping" $G$). However, Lenia uses far larger, fixed, kernels and is not trained via gradient descent.

• Conway's Game of Life
• Cellular automaton
• Self-replication
• Pattern formation
• Morphogenesis

• [https://github.com/Chakazul/Lenia The Github repository for Lenia]
• [https://chakazul.github.io/lenia.html Chan's website for Lenia]
• [https://www.youtube.com/watch?v=G5P8eu6gUSo&ab_channel=stanfordonline An invited seminar at Stanford given by Chan]

{{reflist}}

{{Conway's Game of Life}}

Category:Cellular automaton rules