Lamplighter
The parity automaton on a ball of the lamplighter group, a linear rule over the field of two elements.
- Name:
lamplighter_automaton - Field:
computer_science/automata - Formalism: lattice dynamics
- References: Ceccherini-Silberstein and Coornaert 2023; Grigorchuk and Zuk 2001
Factory
@register_model(
field='computer_science/automata',
tags=('group', 'cellular automaton', 'linear', 'lamplighter'),
references=('Ceccherini-Silberstein and Coornaert 2023', 'Grigorchuk and Zuk 2001'),
defaults={'until': 20, 'dt': 1},
observables=_density_observables,
)
def lamplighter_automaton(
*,
radius: int = 7,
density: float = 0.1,
random_state: int | np.random.Generator | None = 0,
) -> System:
"""Return the parity automaton on a ball of the lamplighter group, a linear rule over the field of two elements.
Every cell takes the sum modulo 2 of itself and the cells at the generators `a` and `b`, so a few ones spread into
nested, self-similar patterns over the Diestel-Leader graph of the group.
Args:
radius:
The radius of the ball.
density:
The probability that a cell starts at 1.
random_state:
The seed or generator of the initial cells.
Returns:
system:
The model, with the cells `cells.alive`.
Examples:
>>> from skcomplex.models.computer_science.automata import lamplighter_automaton
>>> lamplighter_automaton(radius=3).elements[0].check().prepare().size_
39
"""
space = CayleySpace(lamplighter(), ball=radius)
cells = _random_cells(space.check().prepare().size_, density, random_state)
parity = TotalisticSpec({total: total % 2 for total in range(len(LAMPLIGHTER_MEMORY.elements) + 1)})
return System(
elements=[LatticeElement('cells', element=BinaryCellElement, space=space, values={'alive': cells})],
interactions=[
LocalRuleInteraction('parity', target='cells', state='alive', rule=parity, neighborhood=LAMPLIGHTER_MEMORY),
],
)
Components
class BinaryCellElement(Element):
"""Cell of an automaton whose state `alive` is 0 or 1, changed by the local rule of an interaction.
Args:
label:
The label of the cell or the lattice group.
visual:
How the element is drawn, or `None` for `default_visual_`.
alive:
The initial value, 0 or 1.
Examples:
>>> from skcomplex.models.computer_science.automata import BinaryCellElement
>>> BinaryCellElement('cell', alive=1).alive
1
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = None,
*,
alive: Annotated[float, StateSpec(domain=FiniteEnumeratedSet([0, 1]))] = 0,
) -> None:
super().__init__(label, visual=visual)
self.alive = alive
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,
),
]