Diffusion automaton
A cellular automaton of diffusion, c' = c + r Σ (c_neighbour - c), started from a square of ones.
- Name:
diffusion_automaton - Field:
computer_science/automata - Formalism: lattice dynamics
- References: Downey 2018
Factory
@register_model(
field='computer_science/automata',
tags=('lattice', 'diffusion', 'conservation'),
references=('Downey 2018',),
defaults={'until': 100, 'dt': 1},
)
def diffusion_automaton(shape: Sequence[int] = (50, 50), *, r: float = 0.1, width: int = 10) -> System:
"""Return a cellular automaton of diffusion, `c' = c + r Σ (c_neighbour - c)`, started from a square of ones.
The total content is conserved on the torus, and the square spreads into a Gaussian bump.
Args:
shape:
The number of rows and columns of the torus.
r:
The share of the difference with a neighbour exchanged in a step.
width:
The side of the central square of cells that start with content 1.
Returns:
system:
The model, with the contents `cells.level`.
Examples:
>>> from skcomplex.models.computer_science.automata import diffusion_automaton
>>> float(diffusion_automaton((20, 20), width=4).elements[0].values['level'].sum())
16.0
"""
level = np.zeros(tuple(shape))
rows, columns = (shape[0] - width) // 2, (shape[1] - width) // 2
level[rows : rows + width, columns : columns + width] = 1.0
return System(
elements=[
LatticeElement(
'cells',
element=DiffusionCellElement,
space=LatticeSpace(tuple(shape)),
values={'level': level, 'r': r},
),
],
interactions=[
LocalRuleInteraction(
'diffusion',
target='cells',
state='level',
rule=_diffuse,
neighborhood=VonNeumannNeighborhood(center=True),
),
],
)
Components
class DiffusionCellElement(Element):
"""Cell of a lattice with a content and the share of the difference it exchanges with each neighbour in a step.
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 content.
r:
The share of the difference with a neighbour exchanged in a step, below `1 / (2 d)` in `d` dimensions.
Examples:
>>> from skcomplex.models.computer_science.automata import DiffusionCellElement
>>> DiffusionCellElement('cell', r=0.2).r
0.2
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = None,
*,
level: Annotated[float, StateSpec(domain=RealSet.closed_open(0, oo))] = 0.0,
r: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.1,
) -> None:
super().__init__(label, visual=visual)
self.level = level
self.r = r
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,
),
]