Muller's pre-injective non-surjective automaton on F2
Muller's automaton on the free group F2, pre-injective but not surjective.
- Name:
muller_automaton - Field:
computer_science/automata - Formalism: lattice dynamics
- References: Ceccherini-Silberstein and Coornaert 2023
Select this system
from skcomplex.systems import create_simulation, create_system, describe
system = create_system('muller_automaton')
entry = describe('muller_automaton')
trajectory = create_simulation('muller_automaton').prepare().run().trajectory_
Factory
@register_system(
field='computer_science/automata',
tags=('group', 'cellular automaton', 'free group', 'Garden of Eden'),
references=('Ceccherini-Silberstein and Coornaert 2023',),
defaults={'until': 10, 'dt': 1},
observables=_pair_observables,
)
def muller_automaton(
*,
radius: int = 5,
modulus: int = 2,
initial: npt.ArrayLike | None = None,
random_state: int | np.random.Generator | None = 0,
) -> System:
"""Return Muller's automaton on the free group F2, pre-injective but not surjective.
The alphabet is `A = H x H` for the cyclic group `H` of order `modulus`, and every cell becomes
`p1(x(ga)) + p2(x(gb)) + p1(x(ga^-1)) + p2(x(gb^-1))`, where `p1(h, k) = (h, 0)` and `p2(h, k) = (k, 0)`
(Ceccherini-Silberstein and Coornaert 2023, Section 5.11). The image lies in `H x {0}`, so the automaton is not
surjective, yet two configurations that differ on a finite set have different images, so it is pre-injective, which
the Garden of Eden theorem forbids on amenable groups. For a prime modulus it is the linear automaton over `K^2` of
Section 8.10. Cells on the boundary of the ball read 0 outside it.
Args:
radius:
The radius of the ball of F2.
modulus:
The order of `H`, at least 2.
initial:
The initial pairs, an array of shape `(sites, 2)`, or `None` for random ones.
random_state:
The seed or generator of the random initial pairs.
Returns:
system:
The system, with the coordinates `cells.first` and `cells.second`.
Raises:
ParameterError:
The modulus is below 2.
Examples:
>>> from skcomplex.systems.computer_science.automata import muller_automaton
>>> muller_automaton(radius=2).elements[0].prepare().size_
17
"""
return _pair_system(_muller, radius=radius, modulus=modulus, initial=initial, random_state=random_state)
Components
Elements:
PairCellElement, fromElement
Interactions:
LocalRuleInteraction
class PairCellElement(Element):
"""Cell of an automaton over the product `H x H` of the integers modulo `n` with itself, or `K^2` for a prime `n`.
Args:
label:
The label of the cell or the lattice group.
visual:
How the element is drawn, or `None` for `default_visual_`.
first:
The initial first coordinate, an integer from 0 to `modulus - 1`.
second:
The initial second coordinate, an integer from 0 to `modulus - 1`.
modulus:
The order `n` of the cyclic group `H`, the size of the field `K` when prime.
Examples:
>>> from skcomplex.systems.computer_science.automata import PairCellElement
>>> PairCellElement('cell', first=1).modulus
2
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = None,
*,
first: Annotated[float, StateSpec(domain=ZZ)] = 0,
second: Annotated[float, StateSpec(domain=ZZ)] = 0,
modulus: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 2,
) -> None:
super().__init__(label, visual=visual)
self.first = first
self.second = second
self.modulus = modulus
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,
),
]