Surjective non-pre-injective linear automaton on F2
A linear automaton over K^2 on the free group F2, surjective but not pre-injective.
- Name:
surjective_linear_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('surjective_linear_automaton')
entry = describe('surjective_linear_automaton')
trajectory = create_simulation('surjective_linear_automaton').prepare().run().trajectory_
Factory
@register_system(
field='computer_science/automata',
tags=('group', 'cellular automaton', 'free group', 'linear', 'Garden of Eden'),
references=('Ceccherini-Silberstein and Coornaert 2023',),
defaults={'until': 10, 'dt': 1},
observables=_pair_observables,
)
def surjective_linear_automaton(
*,
radius: int = 5,
modulus: int = 2,
initial: npt.ArrayLike | None = None,
random_state: int | np.random.Generator | None = 0,
) -> System:
"""Return a linear automaton over `K^2` on the free group F2, surjective but not pre-injective.
Every cell becomes `q1(x(ga)) + q1(x(ga^-1)) + q2(x(gb)) + q2(x(gb^-1))`, where `q1(h, k) = (h, 0)` and
`q2(h, k) = (0, h)`, over the field of a prime number `modulus` of elements (Ceccherini-Silberstein and Coornaert
2023, Section 8.11). The configuration `(0, 1)` at the identity and 0 elsewhere maps to 0, so the automaton is not
pre-injective, while a preimage of every configuration is built by induction on the word length, so it is
surjective, which the Garden of Eden theorem forbids on amenable groups. Cells on the boundary of the ball read 0
outside it.
Args:
radius:
The radius of the ball of F2.
modulus:
The prime size of the field `K`.
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 surjective_linear_automaton
>>> surjective_linear_automaton(radius=2).elements[0].prepare().size_
17
"""
return _pair_system(_surjective_linear, 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,
),
]