Stochastic cellular automaton

Last updated

A stochastic cellular automaton (SCA), also known as a probabilistic cellular automaton (PCA), is a type of computational model. It consists of a grid of cells, where each cell has a particular state (e.g., "on" or "off"). The states of all cells evolve in discrete time steps according to a set of rules.

Contents

Unlike a standard cellular automaton where the rules are deterministic (fixed), the rules in a stochastic cellular automaton are probabilistic. This means a cell's next state is determined by chance, according to a set of probabilities that depend on the states of neighboring cells. [1]

Despite the simple, local, and random nature of the rules, these models can produce complex global patterns through processes like emergence and self-organization. They are used to model a wide variety of real-world phenomena where randomness is a factor, such as the spread of forest fires, the dynamics of disease epidemics, or the simulation of ferromagnetism in physics (see Ising model).

As a mathematical object, a stochastic cellular automaton is a discrete-time random dynamical system. It is often analyzed within the frameworks of interacting particle systems and Markov chains, where it may be called a system of locally interacting Markov chains. [2] [3] See [4] for a more detailed introduction.

Formal definition

From the perspective of probability theory, a stochastic cellular automaton is a discrete-time Markov process. The configuration of all cells at a given time is a state in a product space . Here, is a graph representing the grid of cells (e.g., ), and each is the finite set of possible states for the cell (e.g., ).

The transition probability, which defines the dynamics, has a product form:

where is the next configuration and is a probability distribution on .

Locality is a key requirement, meaning the probability of a cell changing its state depends only on the states of its neighbors. This is expressed as , where is a finite neighborhood of cell and are the states of the cells in that neighborhood. See [5] for a more detailed introduction from this point of view.

Examples of stochastic cellular automaton

Majority cellular automaton

There is a version of the majority cellular automaton with probabilistic updating rules. See the Toom's rule.

Relation to lattice random fields

PCA may be used to simulate the Ising model of ferromagnetism in statistical mechanics. [1] Some categories of models were studied from a statistical mechanics point of view.

Cellular Potts model

There is a strong connection [6] between probabilistic cellular automata and the cellular Potts model in particular when it is implemented in parallel.

Non Markovian generalization

The Galves–Löcherbach model is an example of a generalized PCA with a non Markovian aspect.

References

  1. 1 2 Vichniac, G. (1984), "Simulating physics with cellular automata", Physica D, 10 (1–2): 96–115, Bibcode:1984PhyD...10...96V, doi:10.1016/0167-2789(84)90253-7 .
  2. Toom, A. L. (1978), Locally Interacting Systems and their Application in Biology: Proceedings of the School-Seminar on Markov Interaction Processes in Biology, held in Pushchino, March 1976, Lecture Notes in Mathematics, vol. 653, Springer-Verlag, Berlin-New York, ISBN   978-3-540-08450-1, MR   0479791
  3. R. L. Dobrushin; V. I. Kri︠u︡kov; A. L. Toom (1978). Stochastic Cellular Systems: Ergodicity, Memory, Morphogenesis. Manchester University Press. ISBN   9780719022067.
  4. Fernandez, R.; Louis, P.-Y.; Nardi, F. R. (2018). "Chapter 1: Overview: PCA Models and Issues". In Louis, P.-Y.; Nardi, F. R. (eds.). Probabilistic Cellular Automata. Springer. doi:10.1007/978-3-319-65558-1_1. ISBN   9783319655581. S2CID   64938352.
  5. P.-Y. Louis PhD
  6. Boas, Sonja E. M.; Jiang, Yi; Merks, Roeland M. H.; Prokopiou, Sotiris A.; Rens, Elisabeth G. (2018). "Chapter 18: Cellular Potts Model: Applications to Vasculogenesis and Angiogenesis". In Louis, P.-Y.; Nardi, F. R. (eds.). Probabilistic Cellular Automata. Springer. doi:10.1007/978-3-319-65558-1_18. hdl:1887/69811. ISBN   9783319655581.

Further reading