Toom's rule
Toom's rule is a 2-dimensional cellular automata model created by Andrei Toom in 1978 (see for an English translation). This model is both more robust and simpler than the 2-dimensional majority vote rule (see for more details).
Toom's rule is a cellular automata that acts on a 2-dimensional square lattice. At each site in this lattice is a spin with the value +1 or -1. At time the bits are initialized to some value. At each discrete time step the lattice evolves according to Toom's rule. This rule is applied at each site simultaneously.
A deterministic version of Toom's rule can be stated simply as:
At each site in the lattice if the spin of the current (center) site plus the neighboring spin to the North plus the neighboring spin to the East is greater than 0, then the current spin has the spin +1 in the next time step. Toom's rule is sometimes called the NEC rule since it involves the North, East, and Center sites. If this sum is less than 0, then the current spin has spin -1 in the next time step. As there are 3 spins the sum can never equal 0.
Toom's rule, however, is a probabilistic rule and can be stated as:
(1) Apply the deterministic version of Toom's rule.
Toom's rule is a case of probabilistic cellular automata (see the article ).
The 2-dimensional ferromagnetic Ising model in the absence of local magnetic fields has two ground states. One with all spins in the lattice having +1 (spin up) and the other with all spins in the lattice having -1 (spin down). For this reason, the 2D Ising model can be seen as a memory storing one bit of information in the ground state.
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