Toom's rule
[1][2] It is a modification of the 2-dimensional majority vote rule and can have more robust memory when considered as a thermal physical system in statistical field theory.
[1] Toom's rule is a cellular automaton that acts on a 2-dimensional square lattice.
[1] The general version of Toom's rule is probabilistic and can be stated as: The deterministic version can be recovered by setting p=q=0.
Toom's rule is an example of a probabilistic cellular automata (see Stochastic cellular automaton), defined on the lattice
the system will tend to favor one spin over the other, which can be interpreted as the effect of a magnetic field.
[2] When p=q=0, there is a steady state with a staircase-like boundary (interface) between +1 and -1 spins.
Additional models have been built to study the interface dynamics as a 1D spin configuration (on
as the rate (probability) that each +1 exchanges places with the first -1 spin on its right.
[1] The correlation functions, partition function, Markov Matrices, and their eigenvalues can be computed for this finite Toom rule.
and large n < L, the density function follows the same inverse square law as above:
Toom's rule is a dynamical variant of the Ising model.
There are many dynamical rules for the Ising model where the steady state is Gibbsian.
For this reason, the 2D Ising model can be seen as a memory storing one bit of information in the ground state.
This memory is robust in the sense that if errors cause some spins to flip, returning to the ground state will preserve the stored information.
These errors occur due to thermal noise in the system.
Therefore it is said this memory is robust in the presence of thermal noise.
However, if there is a local magnetic field which favors one ground state over the other, then the Ising model is no longer a reliable memory since there is only one ground state.
The 2-dimensional majority vote cellular automaton (CA) is analogous to the Ising model.
The majority vote CA evolves each site in the lattice by taking the spin value of current site plus that of the 4 neighboring sites and makes this spin +1 in the next time step if the sum is positive and -1 if the sum is negative.
These are states such that the spins on the lattice do not change when acted upon by the CA.
Therefore the majority vote CA can be used to store information.
We can define terms analogous to thermal noise and magnetic field as T=p+q and h=(p-q)/(p+q) respectively.
Similarly to the Ising model the majority vote CA can reliably store information for small values of T. Unlike the Ising mode, if T is small enough, this is even true for arbitrary values of h.[4][5] Toom's model can be applied to fault-tolerant computation.
