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Cellular Automata are discrete-time dynamical systems on a spatially extended discrete space which provide paradigmatic examples of nonlinear phenomena. Their stochastic generalizations, i.e., Probabilistic Cellular Automata (PCA), are…

Statistical Mechanics · Physics 2015-06-16 Emilio N. M. Cirillo , P. -Y. Louis , W. M. Ruszel , C. Spitoni

This paper presents solutions to Density Classification Task (DCT) using a variant of Cellular Automata (CA) called Programmable Cellular Automata (PCA). The translation property as well as the density preserving property of fundamental CA…

Cellular Automata and Lattice Gases · Physics 2009-02-17 Sudhakar Sahoo , Pabitra Pal Choudhury , Amita Pal , Birendra Kumar Nayak

We study self-similarity in one-dimensional probabilistic cellular automata (PCA) using the renormalization technique. We introduce a general framework for algebraic construction of renormalization groups (RG) on cellular automata and apply…

Statistical Mechanics · Physics 2011-08-22 Erik Edlund , Martin Nilsson Jacobi

Probabilistic Cellular Automata (PCA) are simple models used to study dynamical phase transitions. There exist mean field approximations to PCA that can be shown to exhibit a phase transition. We introduce a model interpolating between a…

Mathematical Physics · Physics 2015-06-17 Jean Bricmont , Hanne Van Den Bosch

We concern with a special class of binary cellular automata, i.e., the so-called particle cellular automata (PCA) in the present paper. We first propose max-plus expressions to PCA of 4 neighbors. Then, by utilizing basic operations of the…

Exactly Solvable and Integrable Systems · Physics 2015-05-20 Daisuke Takahashi , Junta Matsukidaira , Hiroaki Hara , Bao-Feng Feng

This article presents a new characterization of controllability and regional controllability of Deterministic Cellular Automata (CA for short). It focuses on analyzing these problems within the framework of control theory, which have been…

Dynamical Systems · Mathematics 2025-01-07 Sara Dridi

There have been several non-axiomatic approaches taken to define Quantum Cellular Automata (QCA). Partitioned QCA (PQCA) are the most canonical of these non-axiomatic definitions. In this work we show that any QCA can be put into the form…

Quantum Physics · Physics 2010-10-13 Pablo Arrighi , Jonathan Grattage

The paper proposes a simple formalism for dealing with deterministic, non-deterministic and stochastic cellular automata in a unifying and composable manner. Armed with this formalism, we extend the notion of intrinsic simulation between…

Formal Languages and Automata Theory · Computer Science 2012-08-15 Pablo Arrighi , Nicolas Schabanel , Guillaume Theyssier

In this article we study a class of shift-invariant and positive rate probabilistic cellular automata (PCA) on rooted d-regular trees $\mathbb{T}^d$. In a first result we extend the results of [10] on trees, namely we prove that to every…

Cellular Automata and Lattice Gases · Physics 2019-02-01 Bruno Kimura , Wioletta Ruszel , Cristian Spitoni

Probabilistic cellular automata are prototypes of non equilibrium critical phenomena. This class of models includes among others the directed percolation problem (Domany Kinzel model) and the dynamical Ising model. The critical properties…

Statistical Mechanics · Physics 2008-02-03 Franco Bagnoli , Paolo Palmerini , Raul Rechtman

We study locally interacting processes in discrete time, often called probabilistic cellular automata, indexed by locally finite graphs. For infinite regular trees and certain generalized Galton-Watson trees, we show that the marginal…

Probability · Mathematics 2025-10-28 Daniel Lacker , Kavita Ramanan , Ruoyu Wu

Algorithms developed to solve many-body quantum problems, like tensor networks, can turn into powerful quantum-inspired tools to tackle problems in the classical domain. In this work, we focus on matrix product operators, a prominent…

Statistical Mechanics · Physics 2024-11-27 Heitor P. Casagrande , Bo Xing , William J. Munro , Chu Guo , Dario Poletti

The zig-zag process is a piecewise deterministic Markov process in position and velocity space. The process can be designed to have an arbitrary Gibbs type marginal probability density for its position coordinate, which makes it suitable…

Probability · Mathematics 2019-12-24 Joris Bierkens , Pierre Nyquist , Mikola C. Schlottke

Classical Cellular Automata (CCAs) are a powerful computational framework for modeling global spatio-temporal dynamics with local interactions. While CCAs have been applied across numerous scientific fields, identifying the local rule that…

Systems and Control · Electrical Eng. & Systems 2025-07-01 Faizal Hafiz , Amelia Kunze , Enrico Formenti , Davide La Torre

Probabilistic timed automata are classical timed automata extended with discrete probability distributions over edges. We introduce clock-dependent probabilistic timed automata, a variant of probabilistic timed automata in which transition…

Logic in Computer Science · Computer Science 2017-07-17 Jeremy Sproston

Cellular Automata(CA) is a discrete computing model which provides simple, flexible and efficient platform for simulating complicated systems and performing complex computation based on the neighborhoods information. CA consists of two…

Distributed, Parallel, and Cluster Computing · Computer Science 2011-12-12 Debasis Das , Rajiv Misra

We consider the problem of metastability in a probabilistic cellular automaton (PCA) with a parallel updating rule which is reversible with respect to a Gibbs measure. The dynamical rules contain two parameters $\beta$ and $h$ which…

Statistical Mechanics · Physics 2009-10-31 Stephen Bigelis , Emilio N. M. Cirillo , Joel L. Lebowitz , Eugene R. Speer

We study lattice spin systems and analyze the evolution of Gaussian concentration bounds (GCB) under the action of probabilistic cellular automata (PCA), which serve as discrete-time analogues of Markovian spin-flip dynamics. We establish…

Probability · Mathematics 2025-07-09 Jean-René Chazottes , Frank Redig , Edgardo Ugalde

Cellular automata (CAs) are fully-discrete dynamical models that have received much attention due to the fact that their relatively simple setup can nonetheless express highly complex phenomena. Despite the model's theoretical maturity and…

Cellular Automata and Lattice Gases · Physics 2025-07-10 Michiel Rollier , Kallil M. C. Zielinski , Aisling J. Daly , Odemir M. Bruno , Jan M. Baetens

We propose using Probabilistic Cellular Automata (PCA) to address inverse problems with the Bayesian approach. In particular, we use PCA to sample from an approximation of the posterior distribution. The peculiar feature of PCA is their…

Computation · Statistics 2026-03-24 Danilo Costarelli , Michele Piconi , Alessio Troiani