# 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… Expand

#### Topics from this paper

#### 9 Citations

Causal dynamics of discrete manifolds

- Mathematics, Computer Science
- NCMA
- 2018

This work formalizes, and proves theorems about, the intuitive idea of a discrete manifold which evolves in time, subject to two natural constraints: the evolution does not propagate information too fast; and it acts everywhere the same. Expand

Reversible causal graph dynamics: invertibility, block representation, vertex-preservation

- Computer Science
- Natural Computing
- 2019

Causal Graph Dynamics extends Cellular Automata to arbitrary time-varying graphs of bounded degree by proving that the inverse of a causal graph dynamics is a causalGraph dynamics, and that these reversible graph dynamics can be represented as finite-depth circuits of local reversible gates. Expand

Dynamical Triangulation Induced by Quantum Walk

- Physics, Computer Science
- Symmetry
- 2020

Numerical simulations show that the number of triangles and the local curvature grow as $\alpha$ and $\beta$ parametrize the way geometry changes upon the local density of the walker, and that, in the long run, flatness emerges. Expand

Reversibility vs local creation/destruction

- Mathematics, Computer Science
- ArXiv
- 2018

This paper obtains reversible local node creation/destruction—in three relaxed settings, whose equivalence the authors prove for robustness, both by theoretical computer science considerations and theoretical physics concerns. Expand

Accretive Computation of Global Transformations of Graphs

- Computer Science
- ArXiv
- 2021

An algorithm is presented which computes online the global transformation of a finite graph in an accretive manner and a local criterion is given for a rule system to extend to a graph global transformation. Expand

D C ] 1 7 M ar 2 02 1 Accretive Computation of Global Transformations of Graphs

- 2021

The framework of global transformations aims at describing synchronous rewriting systems on a given data structure. In this work we focus on the data structure of graphs. Global transformations of… Expand

Accretive Computation of Global Transformations

- Computer Science, Mathematics
- 2021

This work focuses on data structures that can be captured as presheaves and study the computational aspects of fully synchronous rule systems on a given data structure to obtain an online algorithm. Expand

A Category Theoretic Interpretation of Gandy's Principles for Mechanisms

- Computer Science, Philosophy
- DCM/ITRS
- 2018

This work gives category-theoretic axioms describing locally deterministic updates to finite objects describing what properties such a category should have and shows that every updating functor satisfying the authors' conditions is computable. Expand

Lindenmayer Systems and Global Transformations

- Computer Science
- UCNC
- 2019

Global transformations, a category-based formalism for capturing computing models which are simultaneously local, synchronous and deterministic, are introduced through the perspective of… Expand

#### References

SHOWING 1-10 OF 53 REFERENCES

Simulations Between Cellular Automata on Cayley Graphs

- Mathematics, Computer Science
- LATIN
- 1995

It is shown that cellular automata on any planar, modular graph are equivalent to cellular automaton on the grid Z2, and interpreted in terms of planar parallel machines. Expand

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. Expand

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. Expand

Graph-Rewriting Automata as a Natural Extension of Cellular Automata

- Mathematics
- 2009

We introduce a framework called graph-rewriting automata to model evo- lution processes of networks. It is a natural extension of cellular automata in the sense that a fixed lattice space of cellular… Expand

Reversible quantum cellular automata

- Physics, Mathematics
- 2004

We define quantum cellular automata as infinite quantum lattice systems with discrete time dynamics, such that the time step commutes with lattice translations and has strictly finite propagation… Expand

Graph automata: natural expression of self-reproduction

- Mathematics
- 2002

Abstract A variety of models of self-reproduction process have been proposed since von Neumann initiated this field with his self-reproducing automata. Almost all of them are described within the… Expand

One-Dimensional Quantum Cellular Automata over Finite, Unbounded Configurations

- Mathematics, Computer Science
- 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. Expand

An Intrinsically Universal Family of Causal Graph Dynamics

- Computer Science, Mathematics
- 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. Expand

Cellular Automata and Groups

- Mathematics, Computer Science
- 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. Expand

Relational Growth Grammars - A Graph Rewriting Approach to Dynamical Systems with a Dynamical Structure

- Mathematics, 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… Expand