Cyclic cellular automaton
Griffeath's cyclic cellular automaton, whose random levels organize into spiral waves.
- Name:
cyclic_cellular_automaton - Field:
chemistry/kinetics - Formalism: lattice dynamics
- References: Griffeath 1988; Fisch, Gravner, and Griffeath 1991; Dilão 2023
Factory
@register_model(
field='chemistry/kinetics',
tags=('lattice', 'excitable media', 'spirals', 'cellular automaton'),
references=('Griffeath 1988', 'Fisch, Gravner, and Griffeath 1991', 'Dilão 2023'),
defaults={'until': 200, 'dt': 1},
)
def cyclic_cellular_automaton(
shape: Sequence[int] = (64, 64),
*,
n: int = 14,
threshold: int = 1,
random_state: int | np.random.Generator | None = 0,
) -> System:
"""Return Griffeath's cyclic cellular automaton, whose random levels organize into spiral waves.
Args:
shape:
The number of rows and columns.
n:
The number of levels.
threshold:
The number of von Neumann neighbours at the following level that advance a cell.
random_state:
The seed or generator of the initial levels.
Returns:
system:
The model, with the levels `cells.level`.
Examples:
>>> from skcomplex.models.chemistry.kinetics import cyclic_cellular_automaton
>>> cyclic_cellular_automaton((10, 10), n=8).elements[0].values['n']
8
"""
levels = np.random.default_rng(random_state).integers(0, n, size=tuple(shape)).astype(float)
space = LatticeSpace(tuple(shape))
return System(
elements=[
LatticeElement(
'cells',
element=CyclicCellElement,
space=space,
values={'level': levels, 'n': n, 'threshold': threshold},
),
],
interactions=[
LocalRuleInteraction(
'advance',
target='cells',
state='level',
rule=_advance_cyclically,
neighborhood=VonNeumannNeighborhood(center=True),
),
],
)
Components
class CyclicCellElement(Element):
"""Cell of Griffeath's cyclic cellular automaton, with `n` levels that follow each other in a cycle.
A cell moves from level `k` to level `k + 1` modulo `n` when at least `threshold` of its neighbours hold that
level, by the local rule of `cyclic_cellular_automaton`, so every level eats its predecessor and random initial
levels organize into spiral waves.
Args:
label:
The label of the cell or the lattice group.
visual:
How the element is drawn, or `None` for `default_visual_`.
level:
The initial level, from 0 to `n - 1`.
n:
The number of levels.
threshold:
The number of neighbours at the following level that advance a cell.
Examples:
>>> from skcomplex.models.chemistry.kinetics import CyclicCellElement
>>> CyclicCellElement('cell').n
14.0
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = None,
*,
level: Annotated[float, StateSpec(domain=NN)] = 0,
n: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 14.0,
threshold: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 1.0,
) -> None:
super().__init__(label, visual=visual)
self.level = level
self.n = n
self.threshold = threshold
class LocalRuleInteraction(Interaction):
"""Local rule of a cellular automaton, which sets the next value of every cell from its ordered neighbourhood.
The rule is a table of the next value of every configuration of the neighbourhood, `TableSpec`, a rule of the sum
of the neighbourhood, `TotalisticSpec`, a rule of the cell and the sum of its neighbours, `OuterTotalisticSpec`, an
object whose `table()` returns a `TableSpec`, or a function. A function receives one array per neighbour of every
read state, in neighbourhood order, with one value per cell, the uniform array `random` when the rule is stochastic,
and the parameters of the bound group it names, and returns the next values of the states, one array per state.
Args:
label:
The label of the interaction, unique among the parts of its system.
target:
The path of the group on the cells of a lattice, the nodes of a network, or a Cayley graph.
state:
The name of the state the rule sets, or a tuple of the names of the states it sets together.
rule:
The table, the totalistic rule, or the function.
neighborhood:
The ordered neighbourhood of every cell, such as `VonNeumannNeighborhood(center=True)`.
reads:
The names of the states whose neighbourhoods a function reads, or `None` for the states it sets.
schedule:
`'synchronous'`, `'random-sequential'`, `'random-site'`, `'checkerboard'`, or `'colored'`.
stochastic:
Whether a function receives the uniform random array `random`.
Raises:
ValidationError:
The state or the read states are not names, or the neighbourhood is not a neighbourhood.
Examples:
>>> from skcomplex.core.interactions import LocalRuleInteraction
>>> from skcomplex.core.spaces import VonNeumannNeighborhood
>>> from skcomplex.specs import TotalisticSpec
>>> parity = TotalisticSpec({1: 1, 3: 1})
>>> around = VonNeumannNeighborhood(n_dim=1, center=True)
>>> rule = LocalRuleInteraction('parity', target='cells', state='cell', rule=parity, neighborhood=around)
>>> rule.check().schedule
'synchronous'
"""
def __init__(
self,
label: str,
*,
target: Annotated[str, RoleSpec()],
state: str | tuple[str, ...],
rule: object,
neighborhood: BaseNeighborhood,
reads: Sequence[str] | None = None,
schedule: str = 'synchronous',
stochastic: bool = False,
) -> None:
super().__init__(label)
self.target = target
self.state = state
self.rule = rule
self.neighborhood = neighborhood
self.reads = reads
self.schedule = schedule
self.stochastic = stochastic
def _get_names(self) -> tuple[str, ...]:
"""Return the names of the states the rule sets."""
return (self.state,) if isinstance(self.state, str) else tuple(self.state)
def _check(self) -> None:
"""Raise an error for states that are not names, or a neighbourhood that is not one.
Raises:
ValidationError:
The state or the read states are not names, or the neighbourhood is not a neighbourhood.
"""
super()._check()
names = [*self._get_names(), *(self.reads or ())]
if not names or not all(isinstance(name, str) and name.isidentifier() for name in names):
error_msg = f'The local rule {self.label!r} sets {self.state!r} and reads {self.reads!r}. Give state names.'
raise ValidationError(error_msg)
if not isinstance(self.neighborhood, BaseNeighborhood):
error_msg = f'The neighbourhood of the local rule {self.label!r} is {self.neighborhood!r}. Give one.'
raise ValidationError(error_msg)
def _get_laws(self, namespace: SimpleNamespace) -> list[BaseLaw]:
"""Return the local rule law of the states of the target.
Args:
namespace:
The namespace of the symbols the laws can use, with `namespace.target`.
Returns:
laws:
The local rule law.
"""
targets = tuple(getattr(self._get_member(namespace, 'target'), name) for name in self._get_names())
reads = (
None
if self.reads is None
else tuple(getattr(self._get_member(namespace, 'target'), name) for name in self.reads)
)
return [
LocalRuleLaw(
targets[0] if isinstance(self.state, str) else targets,
self.neighborhood,
self.rule,
reads=reads,
schedule=self.schedule,
stochastic=self.stochastic,
label=self.label,
),
]