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Existence of Life in Lenia

Craig Calcaterra, Axel Boldt

Abstract

Lenia is a continuous generalization of Conway's Game of Life. Bert Wang-Chak Chan has discovered and published many seemingly organic dynamics in his Lenia simulations since 2019. These simulations follow the Euler curve algorithm starting from function space initial conditions. The Picard-Lindelöf Theorem for the existence of integral curves to Lipschitz vector fields on Banach spaces fails to guarantee solutions, because the vector field associated with the integro-differential equation defining Lenia is discontinuous. However, we demonstrate the dynamic Chan is using to generate simulations is actually an arc field and not the traditional Euler method for the vector field derived from the integro-differential equation. Using arc field theory we prove the Euler curves converge to a unique flow which solves the original integro-differential equation. Extensions are explored and the modeling of entropy is discussed. Keywords: arc fields; discontinuous vector fields; integro-differential equations; entropy models

Existence of Life in Lenia

Abstract

Lenia is a continuous generalization of Conway's Game of Life. Bert Wang-Chak Chan has discovered and published many seemingly organic dynamics in his Lenia simulations since 2019. These simulations follow the Euler curve algorithm starting from function space initial conditions. The Picard-Lindelöf Theorem for the existence of integral curves to Lipschitz vector fields on Banach spaces fails to guarantee solutions, because the vector field associated with the integro-differential equation defining Lenia is discontinuous. However, we demonstrate the dynamic Chan is using to generate simulations is actually an arc field and not the traditional Euler method for the vector field derived from the integro-differential equation. Using arc field theory we prove the Euler curves converge to a unique flow which solves the original integro-differential equation. Extensions are explored and the modeling of entropy is discussed. Keywords: arc fields; discontinuous vector fields; integro-differential equations; entropy models
Paper Structure (11 sections, 8 theorems, 79 equations)

This paper contains 11 sections, 8 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Lenia: continuous version of GoL
  3. Integro-differential equation
  4. Lenia arc field
  5. Arc field theory
  6. Clip function properties
  7. Existence and Uniqueness of Solutions to the Lenia Arc Field
  8. Solution to the integro-differential equation
  9. Basic Properties
  10. Comparison with Asymptotic Lenia
  11. Extensions

Key Result

Proposition 1

The Lenia arc field $\left( LineDefLeniaArcField\right)$ on $M:=\mathcal{B}\left( \mathbb{R}^{n},\left[ 0,1\right] \right)$ with supremum metric $d_{\infty}$ has speed globally bounded by $\max\left| G\right|$.

Theorems & Definitions (19)

  • Example 1
  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Theorem 1
  • Example 2
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 9 more