Freezing, Bounded-Change and Convergent Cellular Automata

08/19/2019
by   Nicolas Ollinger, et al.
0

This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension , and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-Kurka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension 1, but also dimension 1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension.

READ FULL TEXT

Authors

page 4

10/11/2002

On the Cell-based Complexity of Recognition of Bounded Configurations by Finite Dynamic Cellular Automata

This paper studies complexity of recognition of classes of bounded confi...
04/20/2018

Universality in Freezing Cellular Automata

Cellular Automata have been used since their introduction as a discrete ...
05/02/2021

Fixed Point Constructions in Tilings and Cellular Automata

The fixed point construction is a method for designing tile sets and cel...
02/21/2022

Generating Hard Problems of Cellular Automata

We propose two hard problems in cellular automata. In particular the pro...
11/25/2012

Visualization and clustering by 3D cellular automata: Application to unstructured data

Given the limited performance of 2D cellular automata in terms of space ...
05/18/2022

The Structure of Configurations in One-Dimensional Majority Cellular Automata: From Cell Stability to Configuration Periodicity

We study the dynamics of (synchronous) one-dimensional cellular automata...
12/05/2018

The Spread of Voting Attitudes in Social Networks

The Shapley-Shubik power index is a measure of each voters power in the ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.