Fredkin's self-replicating parity automaton
Fredkin's parity automaton, whose local law copies any initial pattern into translated replicas.
- Name:
fredkin_replicator - Field:
computer_science/automata - Formalism: lattice dynamics
- References: Fredkin 1990; Dilão 1993; Dilão 2023
Select this system
from skcomplex.systems import create_simulation, create_system, describe
system = create_system('fredkin_replicator')
entry = describe('fredkin_replicator')
trajectory = create_simulation('fredkin_replicator').prepare().run().trajectory_
Factory
@register_system(
field='computer_science/automata',
tags=('lattice', 'replicator', 'linear', 'parity'),
references=('Fredkin 1990', 'Dilão 1993', 'Dilão 2023'),
defaults={'until': 32, 'dt': 1},
)
def fredkin_replicator(
shape: Sequence[int] = (128, 128),
*,
initial: npt.ArrayLike | None = None,
radius: float = 4.0,
) -> System:
"""Return Fredkin's parity automaton, whose local law copies any initial pattern into translated replicas.
Every cell takes the sum modulo 2 of its four nearest neighbours, `x(i, j, n + 1) = x(i - 1, j, n) +
x(i + 1, j, n) + x(i, j - 1, n) + x(i, j + 1, n) (mod 2)`. The law is linear over the integers modulo 2, and
raising it to the power `2^k` gives the sum of the pattern translated by `2^k` cells in each of the four
directions, so at the times `2^k` the initial pattern reappears four times, far enough apart not to overlap once
`2^k` exceeds its width, and at other times the copies multiply in more complex arrangements. By default the
pattern is a circumference, as in Dilão's example.
Args:
shape:
The number of rows and columns of the torus.
initial:
The initial cells, an array of the shape, or `None` for a circumference of the radius in the middle.
radius:
The radius of the default circumference.
Returns:
system:
The system, with the cells `cells.alive`.
Examples:
>>> from skcomplex.systems.computer_science.automata import fredkin_replicator
>>> int(fredkin_replicator((32, 32)).elements[0].values['alive'].sum())
32
"""
if initial is None:
rows, columns = np.indices(tuple(shape))
distance = np.hypot(rows - shape[0] // 2, columns - shape[1] // 2)
values = (np.abs(distance - radius) < HALF_CELL).astype(float)
else:
values = np.asarray(initial, dtype=float)
space = LatticeSpace(tuple(shape))
return System(
elements=[LatticeElement('cells', element=BinaryCellElement, space=space, values={'alive': values})],
interactions=[
LocalRuleInteraction(
'rule',
target='cells',
state='alive',
rule=life((1, 3), (1, 3)),
neighborhood=VonNeumannNeighborhood(),
),
],
)
Components
Elements:
BinaryCellElement, fromElement
Interactions:
LocalRuleInteraction
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.systems.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. A
state of several components per cell, such as a truncated power series of `PowerSeriesSpec`, reaches a function as
one row of components per cell. A `StateTranslationNeighborhood` reads a group space at the product of every site
and the group element that the first read state holds there.
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.components.interactions import LocalRuleInteraction
>>> from skcomplex.core.components.spaces import VonNeumannNeighborhood
>>> from skcomplex.core.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.schedule
'synchronous'
"""
law_classes: ClassVar[tuple[type[BaseLaw], ...]] = (LocalRuleLaw,)
def __init__(
self,
label: str,
*,
target: Annotated[str, RoleSpec(elements=(ElementKind.ANY,), states=('state', 'reads'))],
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,
),
]