Freezing, Bounded-Change and Convergent Cellular Automata

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...

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Published in:Discrete Mathematics and Theoretical Computer Science Vol. 24, no. 1; no. Automata, Logic and Semantics; pp. 1 - 37
Main Authors: Ollinger, Nicolas, Theyssier, Guillaume
Format: Journal Article
Language:English
Published: Nancy DMTCS 01.01.2022
Discrete Mathematics & Theoretical Computer Science
Subjects:
[info.info-cc]computer science [cs]/computational complexity [cs.cc]
[info.info-dm]computer science [cs]/discrete mathematics [cs.dm]
[math.math-ds]mathematics [math]/dynamical systems [math.ds]
Cellular automata
complexity
computability
Computational Complexity
Computer Science
Convergence (Mathematics)
convergent cellular automata
Discrete Mathematics
Dynamical Systems
Freezing
freezing cellular automata
Mathematical research
Mathematics
France
computability
complexity
convergent cellular automata
freezing cellular automata
ISSN:1365-8050, 1462-7264, 1365-8050
Online Access:Get full text
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Summary: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-K\r{u}rka, 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.
Bibliography:SourceType-Scholarly Journals-1
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ISSN:1365-8050
1462-7264
1365-8050
DOI:10.46298/dmtcs.5734