Even and S-gap subshifts
The shift on the S-gap subshift, the binary sequences whose runs of zeros between ones lie in S.
- Name:
gap_subshift - 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('gap_subshift')
entry = describe('gap_subshift')
trajectory = create_simulation('gap_subshift').prepare().run().trajectory_
Factory
@register_system(
field='computer_science/automata',
tags=('subshift', 'S-gap', 'even shift', 'renewal', 'sofic'),
references=('Ceccherini-Silberstein and Coornaert 2023',),
defaults={'until': 100, 'dt': 1},
observables=_density_observables,
)
def gap_subshift(
size: int = 120,
*,
gaps: Sequence[int] | None = None,
random_state: int | np.random.Generator | None = 0,
) -> System:
"""Return the shift on the S-gap subshift, the binary sequences whose runs of zeros between ones lie in `S`.
With `S` the even numbers it is the even subshift, strictly sofic and not of finite type
(Ceccherini-Silberstein and Coornaert 2023, Exercises 1.85 and 1.92). The cells are a window of a configuration
drawn from the measure of maximal entropy, whose gaps are independent with the probability `λ^-(h + 1)` of the gap
`h`, where `λ` solves `sum_(h in S) λ^-(h + 1) = 1` and `log λ` is the entropy. Every step shifts the window one
cell to the left, and the cell entering on the right is a one with the probability that the gap since the last one
ends there. A gap longer than the window counts as the window. Gaps are taken below ten times the size.
Args:
size:
The number of cells of the window.
gaps:
The allowed numbers of zeros between consecutive ones, the set `S`, or `None` for the even numbers.
random_state:
The seed or generator of the initial window.
Returns:
system:
The system, with the cells `cells.alive`.
Raises:
ParameterError:
No allowed gap lies below ten times the size.
Examples:
>>> from skcomplex.systems.computer_science.automata import gap_subshift
>>> cells = gap_subshift(40).elements[0].values['alive']
>>> runs = [len(run) for run in ''.join(str(int(cell)) for cell in cells).strip('0').split('1') if run]
>>> all(run % 2 == 0 for run in runs)
True
"""
_, probabilities, hazards = _gap_law(_gap_lengths(size, gaps))
cells = _renewal_cells(size, probabilities, random_state)
neighborhood = LatticeNeighborhood(tuple((offset,) for offset in range(1 - size, 2)))
return System(
elements=[
LatticeElement(
'cells',
element=BinaryCellElement,
space=LatticeSpace(size, boundary='fixed', fill=OUTSIDE),
values={'alive': cells},
),
],
interactions=[
LocalRuleInteraction(
'shift',
target='cells',
state='alive',
rule=functools.partial(_gap_shift, hazards=hazards),
neighborhood=neighborhood,
stochastic=True,
),
],
)
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,
),
]