Group-valued difference automaton
The automaton τ(x)(n) = x(n)^-1 x(n + 1) whose alphabet is a finite group, on a ring.
- Name:
group_difference_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('group_difference_automaton')
entry = describe('group_difference_automaton')
trajectory = create_simulation('group_difference_automaton').prepare().run().trajectory_
Factory
@register_system(
field='computer_science/automata',
tags=('lattice', 'group alphabet', 'surjective', 'symmetric group'),
references=('Ceccherini-Silberstein and Coornaert 2023',),
defaults={'until': 30, 'dt': 1},
)
def group_difference_automaton(
size: int = 60,
*,
group: PermutationGroup_generic | None = None,
initial: npt.ArrayLike | None = None,
random_state: int | np.random.Generator | None = 0,
) -> System:
"""Return the automaton `τ(x)(n) = x(n)^-1 x(n + 1)` whose alphabet is a finite group, on a ring.
Over the integers, with the generator `s0 = 1` of infinite order and a nontrivial group alphabet, the automaton is
surjective but not injective (Ceccherini-Silberstein and Coornaert 2023, Exercise 3.24): every constant
configuration maps to the identity. The ring is the finite quotient of order `size`, where the product of the image
around the ring is the identity. Cells hold group elements by their number in the Cayley table of Sage, the
identity first.
Args:
size:
The number of cells of the ring.
group:
The finite group, a Sage permutation group, or `None` for the symmetric group of degree 3.
initial:
The initial numbers of the elements, or `None` for random ones.
random_state:
The seed or generator of the random initial elements.
Returns:
system:
The system, with the group elements `cells.value`.
Examples:
>>> from skcomplex.systems.computer_science.automata import group_difference_automaton
>>> group_difference_automaton(10).elements[0].prepare().size_
10
"""
alphabet = SymmetricGroup(3) if group is None else group
product = np.asarray(alphabet.cayley_table().table(), dtype=np.intp)
inverse = np.argmin(product, axis=1).astype(np.intp)
if initial is None:
values = np.random.default_rng(random_state).integers(0, product.shape[0], size).astype(float)
else:
values = np.asarray(initial, dtype=float)
rule = functools.partial(_group_difference, product=product, inverse=inverse)
return System(
elements=[
LatticeElement('cells', element=GroupCellElement, space=LatticeSpace(size), values={'value': values}),
],
interactions=[
LocalRuleInteraction('difference', target='cells', state='value', rule=rule, neighborhood=FORWARD),
],
)
Components
Elements:
GroupCellElement, fromElement
Interactions:
LocalRuleInteraction
class GroupCellElement(Element):
"""Cell of an automaton whose alphabet is a group, holding a group element by its number.
The number is the site of the element in a group space, or its position in the list of a finite group, and a
negative number names no element, the fill value read outside a ball.
Args:
label:
The label of the cell or the lattice group.
visual:
How the element is drawn, or `None` for `default_visual_`.
value:
The number of the initial group element, 0 for the identity.
Examples:
>>> from skcomplex.systems.computer_science.automata import GroupCellElement
>>> GroupCellElement('cell', value=2).value
2
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = None,
*,
value: Annotated[float, StateSpec(domain=ZZ)] = 0,
) -> None:
super().__init__(label, visual=visual)
self.value = value
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,
),
]