# Cellular automata over generalized Cayley graphs

@article{Arrighi2017CellularAO,
title={Cellular automata over generalized Cayley graphs},
author={Pablo Arrighi and Simon Martiel and Vincent Nesme},
journal={Mathematical Structures in Computer Science},
year={2017},
volume={28},
pages={340 - 383}
}
• Published 29 May 2017
• Mathematics, Computer Science
• Mathematical Structures in Computer Science
It is well-known that cellular automata can be characterized as the set of translation-invariant continuous functions over a compact metric space; this point of view makes it easy to extend their definition from grids to Cayley graphs. Cayley graphs have a number of useful features: the ability to graphically represent finitely generated group elements and their relations; to name all vertices relative to an origin; and the fact that they have a well-defined notion of translation. We propose a…
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## References

SHOWING 1-10 OF 53 REFERENCES
Causal graph dynamics
• Mathematics, Computer Science
Inf. Comput.
• 2013
The theory of cellular automata is extended to arbitrary, time-varying graphs and the intuitive idea of a labelled graph which evolves in time is formalised - but under the natural constraint that information can only ever be transmitted at a bounded speed, with respect to the distance given by the graph.
Hyperbolic Recognition by Graph Automata
• Computer Science
ICALP
• 2002
Graph automata were first introduced by P. Rosenstiehl under the name of intelligent graphs, surely because a network of finite automata is able to know some properties about its own structure, and this hypothesis seems to be absolutely essential for the modelisation of the physical reality.
Graph-Rewriting Automata as a Natural Extension of Cellular Automata
• Computer Science
• 2009
This work considers three different constructions of rule sets to show that various network evolution is possible: hand- coding, evolutionary generation, and exhaustive search.
Reversible quantum cellular automata
• Computer Science
• 2004
The main structure theorem asserts that any quantum cellular automaton is structurally reversible, i.e., that it can be obtained by applying two blockwise unitary operations in a generalized Margolus partitioning scheme.
One-Dimensional Quantum Cellular Automata over Finite, Unbounded Configurations
• Mathematics
LATA
• 2008
It is shown that QCA always admit a two-layered block representation, and hence the inverse QCA is again a QCA, a striking result since the property does not hold for classical one-dimensional cellular automata as defined over such finite configurations.
An Intrinsically Universal Family of Causal Graph Dynamics
• Computer Science
MCU
• 2015
This work presents an intrinsically universal family of Causal Graph Dynamics, and gives insight on why it seems impossible to improve this result to the existence of a unique intrinsically universal instance.
Cellular Automata and Groups
• Mathematics
Encyclopedia of Complexity and Systems Science
• 2009
Theorem: Finitely Generated Amenable Groups + Local Embeddability and Sofic Groups + Uniform Structures + Complements of Functional Analysis + Ultrafilters = 6.
Relational Growth Grammars - A Graph Rewriting Approach to Dynamical Systems with a Dynamical Structure
• Computer Science
UPP
• 2004
Relational growth grammars (RGG) are a graph rewriting formalism which extends the notations and semantics of Lindenmayer systems and which allows the specification of dynamical processes on