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}
}
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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9 Citations
Background Citations
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9 Citations

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References

SHOWING 1-10 OF 53 REFERENCES
Simulations between Cellular Automata on Cayley Graphs
Causal graph dynamics
TLDR
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
TLDR
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
TLDR
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
TLDR
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.
Graph automata: natural expression of self-reproduction
One-Dimensional Quantum Cellular Automata over Finite, Unbounded Configurations
TLDR
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
TLDR
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
TLDR
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
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…
...
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