User:Chakazul/CA

Lenia
$${\partial \over \partial t} \mathbf a^t(\vec x) = \delta \bigg( {1 \over \kappa_N} \cdot \Big[ \beta_{\lfloor kr \rfloor} \gamma ( kr \bmod 1 ) \Big] \vec x, \vec y \in \mathbb R^2, t \in \mathbb R, \mathbf a(\cdot) \in \mathbb \mathbb R$$
 * \mathbf a^t(\vec x); \mu,\sigma \bigg) \quad \quad

Level of Generalization
ECA: 1D discrete space, discrete time, discrete state, square range-1 neighborhood, discrete non-totalistic + lookup transition rule

GoL: 2D discrete space, discrete time, discrete state, square range-1 neighborhood, discrete totalistic + interval transition rule

LtL: 2D discrete space, discrete time, discrete state, square long range neighborhood, discrete totalistic + interval transition rule

RL: continuous space, discrete time, discrete state, square long range neighborhood, discrete totalistic + interval transition rule

SL: 2D continuous space, discrete time, continuous state, circular long range neighborhood + circular cell, smooth totalistic + interval transition rule

CSA: 2D continuous space, discrete time, continuous state, circular long range neighborhood, discrete totalistic neighborhood rule, continuous transition rule

Lenia: 2D continuous space, continuous time, continuous state, circular unit-range neighborhood, continuous radial + differential transition rule

Summary
$$\mathcal A_\text{GoL} = (\mathcal L, \mathcal T, S, N, \phi) = \Big( \mathbb Z^2, \ \mathbb Z, \ \{0,1\}, \ \{ \vec x \mid \lVert \vec x \rVert_\infty \leq 1 \}, \ \phi: \mathbf a^t|_{\vec x + N} \longmapsto \lambda_{s_0, b_0, b_1, s_1}(\mathbf n * \mathbf a^t(\vec x), \mathbf a^t(\vec x)) \Big)$$

$$\mathcal A_\text{Lenia} = (\mathcal L, \mathcal T, S, N, \phi) = \Big( \mathbb R^2, \ \mathbb R, \ [0,1], \ \{ \vec x \mid \lVert \vec x \rVert_2 \leq 1 \}, \ \phi: \mathbf a^t|_{\vec x + N} \longmapsto \delta_{\mu,\sigma} (\mathbf n_{\beta,\gamma} * \mathbf a^t(\vec x)) \Big)$$

$${\partial \over \partial t} \mathbf a^t = \delta ( \mathbf n * \mathbf a^t ) \quad \quad \mathbf n ( \vec y ) = { \kappa (|\vec y|) \over \kappa_N } \quad \quad \kappa_N = \int_N \mathbf \kappa(|\vec y|) \ \mathrm d |\vec y| \quad \quad \kappa (r) = \gamma ( kr \bmod 1 ) \cdot \beta_{\lfloor kr \rfloor}$$ ... normalized kernel n

$${\partial \over \partial t} \mathbf a^t = \delta \Big( {\mathbf n * \mathbf a^t \over \mathbf n * \mathbf 1_N} \Big) \quad \quad \mathbf n ( \vec y ) = \kappa ( |\vec y| ) \quad \quad \kappa (r) = \gamma ( kr \bmod 1 ) \cdot \beta_{\lfloor kr \rfloor}$$ ... normalized potential N

where \gamma(1-r) = \gamma(r)$$ is the unimodal kernel core function, e.g. $$\gamma(r) = (4r(1-r))^\alpha, \alpha = 4$$
 * $$\lambda : [0,1]^2 \longrightarrow S$$ is the interval transition function
 * $$\delta : [0,1] \longrightarrow [-1,1]$$ is the unimodal differential transition function, e.g. $$\textstyle \delta(n) = 2 \Big( \max(0, 1 - {|n-\mu|^2 \over 9\sigma^2}) \Big)^\lambda - 1, \lambda = 4 $$
 * $$\mu \in [0,1]$$ is the central of growth region
 * $$\sigma \in [0,1]$$ is the width of growth region
 * $$\kappa : [0,1] \longrightarrow [0,1]$$ is the multimodal kernel shell function, generates periodic kernel using triangular wave or sawtooth wave
 * $$\textstyle \gamma : [0,1] \longrightarrow [0,1], \gamma(0) = 0, \gamma({1 \over 2}) = 1,
 * $$k \in \mathbb N^+$$ is the number of peaks
 * $$\beta \in \mathbb [0,1]^k, \max \beta = 1$$ are the peaks

Variations

 * $$S = [0, \infty)$$
 * $$\eta \in \mathbb [0,1]^{k+1}, \eta_{k+1} = 0$$ are the valleys

$$\kappa (r) = \gamma ( kr \bmod 1 ) \cdot ( \beta_{\lfloor kr \rfloor} - \eta_{\lfloor kr+0.5 \rfloor} ) + \eta_{\lfloor kr+0.5 \rfloor}$$
 * $$\gamma' : [0,1] \longrightarrow [0,1], \gamma'(0) = 0, \gamma'(1) = 1$$ is the monotonic kernel core function, e.g. $$\gamma'(r) = (r(2-r))^\alpha, \alpha = 4$$

$$\kappa (r) = \gamma' \Big( \Big| (2kr+1) \bmod 2 - 1 \Big| \Big) \Big( \beta_{\lfloor kr \rfloor} - \eta_{\lfloor kr+0.5 \rfloor} \Big) + \eta_{\lfloor kr+0.5 \rfloor}$$

Notations

 * $$N$$ uppercase = set
 * $$\mathcal N$$ script uppercase = set of sets
 * $$\mathbb N$$ blackboard uppercase = numeric set
 * $$\mathbf n$$ bold lowercase = map
 * $$n$$ lowercase = scalar
 * $$\vec n$$ lowercase with arrow = vector
 * $$\phi$$ lowercase Greek = function / parameters
 * $$\Phi$$ uppercase Greek = set of functions / set of tuples

Topics
Complex Life Forms in Continuous Space-Time-Value Cellular Automata -- Lenia
 * Introduction
 * About Cellular Automata
 * A Brief History
 * Game of Life, Life-like, Larger than Life, RealLife, SmoothLife, 3D, Hexagonal
 * Level of Generalization
 * Discrete to Continuous (Space, Value, Time)
 * Rule Generalization (Neighborhood Rule, Transition Mapping)
 * Basic Definition
 * Spacetime
 * Value and Configuration
 * Neighborhood
 * Normalization
 * Convolution Kernel, Kernel Shell and Kernel Core
 * Transition Mapping
 * Local and Global Transition Rule
 * Parameter Space
 * Methods
 * Computer Program
 * Fast Fourier Transform
 * Online Webpage (LeniaLab)
 * Offline Program (LeniaPetri)
 * Experimentation
 * Random Search
 * Parameter Tuning
 * Parameter Space Exploration
 * Configuration Editing
 * Analysis
 * Charting
 * Results
 * Types of Biomass
 * Class 2/3: Deserts and Jungles
 * Class 4: Rivers
 * Global Structures a.k.a. Vegetation
 * Local Structures a.k.a. Life Forms
 * Taxonomy of Life Forms (OSC PH GQ KV RADF)
 * Orbidae, Scutidae, Circidae
 * Pterifera, Helicidae
 * Geminidae, Quadridae
 * Kronidae, Volvidae
 * Radidae, Astridae, Dentidae, Floridae
 * Diversity
 * Transition Parameter Niches — μ-σ Plane
 * Kernel Parameter Niches — β Cube
 * Anatomy
 * Global Anatomy
 * Multicellularity
 * Symmetry
 * Convexity
 * Body Parts
 * Local Anatomy
 * Ornamentation
 * Serration
 * Physiology
 * Persistence
 * Quasi-periodicity and Chaos
 * Shift, Rotation and Scale Invariance
 * Kernel Core Invariance
 * Spacetime Continuum Limits
 * Mass and Speed
 * Behavior
 * Global Behaviors
 * Stationarity, Reflection
 * Translation, Deflection, Vagrancy, Expension
 * Rotation, Gyration, Undulation, Vacillation
 * Metamorphosis
 * Local Behaviors
 * Solidification, Oscillation, Alternation, Valvation
 * Ambiguation, Liquefaction
 * Transcient Behaviors
 * Elastic Collision
 * Chain Increment and Decrement
 * Statistics
 * Statistical Features
 * Image Features
 * Feature Analysis
 * Discussion
 * Analogue in Physics and Biology
 * Potential and Field
 * Partical Collision
 * Resonance
 * Prospects
 * Other Variations (Unbounded Value, Other Kernels)
 * Automatic Life Form Detection
 * Automatic Analysis
 * Implications
 * Unsupervised Learning
 * General Purpose AI