Hopeful windows and fractals in cellular automata and combinatorial games

TL;DR

This paper explores the connection between cellular automata and combinatorial games, introducing new game classes and analyzing fractal patterns and convergence behaviors within these systems.

Contribution

It extends triangle placing games to include cellular automata updates and investigates fractal structures and convergence in specific CA subclasses.

Findings

Identification of cellular automata rules related to game outcomes

Analysis of fractal patterns in CA evolution

Insights into partial convergence behaviors

Abstract

This paper studies 2-player impartial combinatorial games, where the outcomes correspond to updates of cellular automata (CA) which generalize Wolfram's elementary rule 60 and rule 110 (Cook 2004). The games extend the class of \emph{triangle placing games} (Larsson 2013) where at each stage of the game the previous player has the option to block certain hopeful moves of the next player. We also study fractals and partial convergence in a subclass of the CA.