Block cellular automaton
A block cellular automaton or partitioning cellular automaton is a special kind of cellular automaton in which the lattice of cells is divided into non-overlapping blocks (with different partitions at different time steps) and the transition rule is applied to a whole block at a time rather than a single cell. Block cellular automata are useful for simulations of physical quantities, because it is straightforward to choose transition rules that obey physical constraints such as reversibility and conservation laws.
A block cellular automaton consists of the following components:
In each time step, the transition rule is applied simultaneously and synchronously to all of the tiles in the partition. Then, the partition is shifted and the same operation is repeated in the next time step, and so forth. In this way, as with any cellular automaton, the pattern of cell states changes over time to perform some nontrivial computation or simulation.
The simplest partitioning scheme is probably the Margolus neighborhood, named after Norman Margolus, who first studied block cellular automata using this neighborhood structure. In the Margolus neighborhood, the lattice is divided into 2-cell blocks (or 2 × 2 squares in two dimensions, or 2 × 2 × 2 cubes in three dimensions, etc.) which are shifted by one cell (along each dimension) on alternate timesteps.
A closely related technique due to K. Morita and M. Harao consists in partitioning each cell into a finite number of parts, each part being devoted to some neighbor. The evolution proceeds by exchanging the corresponding parts between neighbors and then applying on each cell a purely local transformation depending only on the state of the cell (and not on the states of its neighbors). With such a construction scheme, the cellular automaton is guaranteed to be reversible if the local transformation is itself a bijection. This technique may be viewed as a block cellular automaton on a finer lattice of cells, formed by the parts of each larger cell; the blocks of this finer lattice alternate between the sets of parts within a single large cell and the sets of parts in neighboring cells that share parts with each other.
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