Context-free subshift
A Markov chain on the context-free subshift, ternary rings where every block 0 1^h 2^k 0 has h = k.
- Name:
context_free_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('context_free_subshift')
entry = describe('context_free_subshift')
trajectory = create_simulation('context_free_subshift').prepare().run().trajectory_
Factory
@register_system(
field='computer_science/automata',
tags=('subshift', 'context-free', 'Monte Carlo', 'not sofic'),
references=('Ceccherini-Silberstein and Coornaert 2023',),
defaults={'until': 50, 'dt': 1},
)
def context_free_subshift(
size: int = 60,
*,
initial: npt.ArrayLike | None = None,
random_state: int | np.random.Generator | None = 0,
) -> System:
"""Return a Markov chain on the context-free subshift, ternary rings where every block `0 1^h 2^k 0` has `h = k`.
The subshift is not sofic, since a finite graph cannot match the counts of ones and twos
(Ceccherini-Silberstein and Coornaert 2023, Exercise 6.80). Every step proposes to change every cell of the ring, in
a random order, to one of the two other symbols, and rejects the changes that would leave the subshift, so the
chain samples its admissible rings uniformly. It starts from blocks `0 1^h 2^h`, with `h` drawn from a geometric
law.
Args:
size:
The number of cells of the ring.
initial:
The initial symbols, an admissible ring, or `None` for random balanced blocks.
random_state:
The seed or generator of the initial blocks.
Returns:
system:
The system, with the symbols `cells.symbol`.
Raises:
ParameterError:
The initial ring is not admissible.
Examples:
>>> from skcomplex.systems.computer_science.automata import context_free_subshift
>>> context_free_subshift(12).elements[0].values['symbol'].tolist()
[0.0, 1.0, 1.0, 2.0, 2.0, 0.0, 1.0, 2.0, 0.0, 1.0, 2.0, 0.0]
"""
if initial is None:
generator = np.random.default_rng(random_state)
blocks: list[float] = []
while len(blocks) < size:
height = int(generator.geometric(BALANCED_BLOCK))
blocks += [0.0] + [1.0] * height + [2.0] * height
symbols = np.zeros(size)
cut = max(index for index in range(size + 1) if index == len(blocks) or blocks[index] == 0)
symbols[:cut] = blocks[:cut]
else:
symbols = np.asarray(initial, dtype=float)
if not _is_context_free(symbols):
error_msg = f'The initial ring {symbols.tolist()} holds a block 0 1^h 2^k 0 with h != k.'
raise ParameterError(error_msg)
return System(
elements=[
LatticeElement('cells', element=TernaryCellElement, space=LatticeSpace(size), values={'symbol': symbols}),
],
interactions=[
LocalRuleInteraction(
'moves',
target='cells',
state='symbol',
rule=_context_free_move,
neighborhood=_whole_ring(size),
schedule='random-sequential',
stochastic=True,
),
],
)
Components
Elements:
TernaryCellElement, fromElement
Interactions:
LocalRuleInteraction
class TernaryCellElement(Element):
"""Cell of a subshift over the alphabet `{0, 1, 2}`, such as the context-free subshift.
Args:
label:
The label of the cell or the lattice group.
visual:
How the element is drawn, or `None` for `default_visual_`.
symbol:
The initial symbol, 0, 1, or 2.
Examples:
>>> from skcomplex.systems.computer_science.automata import TernaryCellElement
>>> TernaryCellElement('cell', symbol=2).symbol
2
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = None,
*,
symbol: Annotated[float, StateSpec(domain=FiniteEnumeratedSet([0, 1, 2]))] = 0,
) -> None:
super().__init__(label, visual=visual)
self.symbol = symbol
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,
),
]