Subtraction automaton over the naturals
The automaton τ(x)(n) = max(0, x(n) - x(n + 1)) over the naturals, whose image is not closed.
- Name:
subtraction_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('subtraction_automaton')
entry = describe('subtraction_automaton')
trajectory = create_simulation('subtraction_automaton').prepare().run().trajectory_
Factory
@register_system(
field='computer_science/automata',
tags=('lattice', 'infinite alphabet', 'naturals', 'non-closed image'),
references=('Ceccherini-Silberstein and Coornaert 2023',),
defaults={'until': 10, 'dt': 1},
)
def subtraction_automaton(size: int = 41, *, height: int = 10, initial: npt.ArrayLike | None = None) -> System:
"""Return the automaton `τ(x)(n) = max(0, x(n) - x(n + 1))` over the naturals, whose image is not closed.
The image of the configurations `x_M`, equal to `M` left of the origin and to `max(0, M - n)` from it, is the block
of `M` ones from the origin. These blocks converge to the configuration of ones on the half line, which is no image,
so the image of the automaton over the infinite alphabet `N` is not closed (Ceccherini-Silberstein and Coornaert
2023, Example 3.3.3). The cells sit on a segment with fixed ends that read 0, and start at `x_M` with `M = height`,
its origin a quarter of the way along.
Args:
size:
The number of cells.
height:
The value `M` of the staircase.
initial:
The initial values, or `None` for the staircase.
Returns:
system:
The system, with the cells `cells.value`.
Raises:
ParameterError:
The height is negative.
Examples:
>>> from skcomplex.systems.computer_science.automata import subtraction_automaton
>>> subtraction_automaton(12, height=3).elements[0].values['value'].tolist()
[3.0, 3.0, 3.0, 3.0, 2.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
"""
if height < 0:
error_msg = f'The height of the staircase is a natural number, but it is {height}.'
raise ParameterError(error_msg)
if initial is None:
origin = size // 4
steps = np.arange(size) - origin
values = np.where(steps < 0, height, np.maximum(0, height - steps)).astype(float)
else:
values = np.asarray(initial, dtype=float)
return System(
elements=[
LatticeElement(
'cells',
element=PascalCellElement,
space=LatticeSpace(size, boundary='fixed'),
values={'value': values},
),
],
interactions=[
LocalRuleInteraction('subtraction', target='cells', state='value', rule=_subtract, neighborhood=FORWARD),
],
)
Components
Elements:
PascalCellElement, fromElement
Interactions:
LocalRuleInteraction
class PascalCellElement(Element):
"""Cell of an automaton over the nonnegative integers.
Args:
label:
The label of the cell or the lattice group.
visual:
How the element is drawn, or `None` for `default_visual_`.
value:
The initial value, a nonnegative integer.
Examples:
>>> from skcomplex.systems.computer_science.automata import PascalCellElement
>>> PascalCellElement('cell', value=3).value
3
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = None,
*,
value: Annotated[float, StateSpec(domain=NN)] = 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,
),
]