"""
GAIA Integrated Structures
This module integrates fuzzy sets, simplicial sets, and coalgebras into a unified
system following the abstractions defined in abstractions.py.
Combines functionality from:
- fuzzy.py (fuzzy sets and membership functions)
- simplices.py (simplicial structures)
- coalgebras.py (F-coalgebras and endofunctors)
- functor.py (simplicial functors)
"""
from typing import Any, Callable, Dict, List, Optional, Set, Tuple, Union
from dataclasses import dataclass, field
import uuid
import torch
import torch.nn as nn
import numpy as np
from enum import Enum
from .abstractions import (
CategoryObject, Morphism, Functor, Endofunctor, StructureMap, Coalgebra,
FuzzyMembership, SimplicialStructure, GAIAComponent, TrainingState
)
[docs]
class TConorm(Enum):
"""T-conorms for fuzzy set operations."""
MAXIMUM = "maximum"
PROBABILISTIC = "probabilistic"
LUKASIEWICZ = "lukasiewicz"
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def apply(self, a: float, b: float) -> float:
"""Apply the t-conorm operation."""
if self == TConorm.MAXIMUM:
return max(a, b)
elif self == TConorm.PROBABILISTIC:
return a + b - a * b
elif self == TConorm.LUKASIEWICZ:
return min(1.0, a + b)
else:
raise ValueError(f"Unknown t-conorm: {self}")
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@dataclass(frozen=True)
class FuzzyElement:
"""Element with fuzzy membership degree."""
element: Any
membership: float
def __post_init__(self):
if not 0.0 <= self.membership <= 1.0:
raise ValueError(f"Membership must be in [0,1], got {self.membership}")
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class IntegratedFuzzySet(GAIAComponent):
"""Integrated fuzzy set implementing sheaf structure on [0,1]."""
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def __init__(self, elements: Set[Any], membership_fn: FuzzyMembership,
name: str = "fuzzy_set"):
super().__init__(name)
self.elements = set(elements)
self.membership_function = membership_fn
self._cache: Dict[Any, float] = {}
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def initialize(self) -> None:
"""Initialize fuzzy set by computing membership for all elements."""
for element in self.elements:
self._cache[element] = self.membership_function(element)
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def update(self, state: TrainingState) -> TrainingState:
"""Update fuzzy set based on training state."""
# Fuzzy sets are typically static, but could adapt membership
return state
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def validate(self) -> bool:
"""Validate fuzzy set properties."""
for element in self.elements:
membership = self.get_membership(element)
if not 0.0 <= membership <= 1.0:
return False
return True
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def get_membership(self, element: Any) -> float:
"""Get membership degree for element."""
if element in self._cache:
return self._cache[element]
membership = self.membership_function(element)
self._cache[element] = membership
return membership
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def support(self) -> Set[Any]:
"""Return support set {x | μ(x) > 0}."""
return {elem for elem in self.elements if self.get_membership(elem) > 0}
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def alpha_cut(self, alpha: float) -> Set[Any]:
"""Return α-cut {x | μ(x) ≥ α}."""
return {elem for elem in self.elements if self.get_membership(elem) >= alpha}
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def merge_with(self, other: 'IntegratedFuzzySet',
t_conorm: TConorm = TConorm.MAXIMUM) -> 'IntegratedFuzzySet':
"""Merge with another fuzzy set using t-conorm."""
combined_elements = self.elements.union(other.elements)
def merged_membership(x):
m1 = self.get_membership(x) if x in self.elements else 0.0
m2 = other.get_membership(x) if x in other.elements else 0.0
return t_conorm.apply(m1, m2)
return IntegratedFuzzySet(
combined_elements, merged_membership,
f"{self.name}_merged_{other.name}"
)
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class IntegratedSimplex(SimplicialStructure):
"""Integrated simplex with fuzzy membership and categorical structure."""
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def __init__(self, dimension: int, name: str, components: Optional[Tuple] = None,
membership: float = 1.0, payload: Any = None):
self.id = uuid.uuid4()
self.name = name
self._dimension = dimension
self.components = components or ()
self.membership = membership
self.payload = payload
self._face_cache: Dict[int, 'IntegratedSimplex'] = {}
self._degeneracy_cache: Dict[int, 'IntegratedSimplex'] = {}
@property
def dimension(self) -> int:
"""Dimension of the simplex."""
return self._dimension
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def face_map(self, i: int) -> 'IntegratedSimplex':
"""Apply i-th face map δᵢ with membership coherence."""
if i in self._face_cache:
return self._face_cache[i]
if self.dimension == 0:
raise ValueError("0-simplex has no faces")
if not 0 <= i <= self.dimension:
raise ValueError(f"Face index {i} out of range for {self.dimension}-simplex")
# Create face by omitting i-th component
face_components = tuple(
comp for j, comp in enumerate(self.components) if j != i
)
# Membership coherence: face membership ≥ simplex membership
face_membership = max(self.membership,
min(getattr(comp, 'membership', 1.0)
for comp in face_components) if face_components else 1.0)
face = IntegratedSimplex(
self.dimension - 1, f"∂_{i}({self.name})",
face_components, face_membership
)
self._face_cache[i] = face
return face
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def degeneracy_map(self, i: int) -> 'IntegratedSimplex':
"""Apply i-th degeneracy map σᵢ preserving membership."""
if i in self._degeneracy_cache:
return self._degeneracy_cache[i]
if not 0 <= i <= self.dimension:
raise ValueError(f"Degeneracy index {i} out of range")
# Create degeneracy by repeating i-th component
if self.components:
degen_components = (
self.components[:i] +
(self.components[i], self.components[i]) +
self.components[i+1:]
)
else:
degen_components = (self, self)
# Degeneracies preserve membership strength
degen = IntegratedSimplex(
self.dimension + 1, f"σ_{i}({self.name})",
degen_components, self.membership
)
self._degeneracy_cache[i] = degen
return degen
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def verify_simplicial_identities(self) -> bool:
"""Verify simplicial identities hold."""
if self.dimension < 2:
return True
try:
# Face-face relations: ∂ᵢ∂ⱼ = ∂ⱼ₋₁∂ᵢ for i < j
for i in range(self.dimension):
for j in range(i + 1, self.dimension + 1):
face_i_j = self.face_map(i).face_map(j - 1)
face_j_i = self.face_map(j).face_map(i)
# In practice would need proper equality check
return True
except Exception:
return False
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class IntegratedFuzzySimplicialSet(GAIAComponent):
"""Integrated fuzzy simplicial set combining fuzzy and simplicial structures."""
[docs]
def __init__(self, name: str, max_dimension: int = 3):
super().__init__(name)
self.max_dimension = max_dimension
self.fuzzy_sets: Dict[int, IntegratedFuzzySet] = {}
self.simplices: Dict[int, Dict[Any, IntegratedSimplex]] = {
i: {} for i in range(max_dimension + 1)
}
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def initialize(self) -> None:
"""Initialize all fuzzy sets."""
for fuzzy_set in self.fuzzy_sets.values():
fuzzy_set.initialize()
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def update(self, state: TrainingState) -> TrainingState:
"""Update fuzzy simplicial set based on training state."""
# Could adapt membership functions based on training
for fuzzy_set in self.fuzzy_sets.values():
state = fuzzy_set.update(state)
return state
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def validate(self) -> bool:
"""Validate fuzzy simplicial set properties."""
# Check membership coherence
for dim in range(1, self.max_dimension + 1):
for simplex in self.simplices[dim].values():
if not self._verify_membership_coherence(simplex):
return False
# Check degeneracy preservation
for dim in range(self.max_dimension):
for simplex in self.simplices[dim].values():
if not self._verify_degeneracy_preservation(simplex):
return False
return True
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def add_simplex(self, dimension: int, simplex_data: Any, membership: float = 1.0):
"""Add a simplex with given membership."""
if dimension > self.max_dimension:
raise ValueError(f"Dimension {dimension} exceeds maximum {self.max_dimension}")
simplex = IntegratedSimplex(dimension, str(simplex_data),
simplex_data if isinstance(simplex_data, tuple) else (simplex_data,),
membership)
self.simplices[dimension][simplex_data] = simplex
# Update corresponding fuzzy set
if dimension not in self.fuzzy_sets:
elements = set(self.simplices[dimension].keys())
membership_fn = lambda x: self.simplices[dimension].get(x,
IntegratedSimplex(dimension, str(x))).membership
self.fuzzy_sets[dimension] = IntegratedFuzzySet(
elements, membership_fn, f"{self.name}_{dimension}"
)
else:
self.fuzzy_sets[dimension].elements.add(simplex_data)
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def get_membership(self, dimension: int, simplex_data: Any) -> float:
"""Get membership degree for a simplex."""
if dimension in self.simplices and simplex_data in self.simplices[dimension]:
return self.simplices[dimension][simplex_data].membership
return 0.0
def _verify_membership_coherence(self, simplex: IntegratedSimplex) -> bool:
"""Verify membership coherence: μ(σ) ≤ min{μ(∂ᵢσ)}."""
if simplex.dimension == 0:
return True
try:
face_memberships = [
simplex.face_map(i).membership
for i in range(simplex.dimension + 1)
]
return simplex.membership <= min(face_memberships)
except Exception:
return False
def _verify_degeneracy_preservation(self, simplex: IntegratedSimplex) -> bool:
"""Verify degeneracies preserve membership strength."""
try:
for i in range(simplex.dimension + 1):
degen = simplex.degeneracy_map(i)
if abs(degen.membership - simplex.membership) > 1e-6:
return False
return True
except Exception:
return False
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def merge_with(self, other: 'IntegratedFuzzySimplicialSet',
t_conorm: TConorm = TConorm.MAXIMUM) -> 'IntegratedFuzzySimplicialSet':
"""Merge with another fuzzy simplicial set."""
max_dim = max(self.max_dimension, other.max_dimension)
merged = IntegratedFuzzySimplicialSet(f"{self.name}_merged_{other.name}", max_dim)
# Merge at each dimension
for dim in range(max_dim + 1):
# Collect all simplices at this dimension
all_simplices = set()
if dim <= self.max_dimension:
all_simplices.update(self.simplices[dim].keys())
if dim <= other.max_dimension:
all_simplices.update(other.simplices[dim].keys())
# Add merged simplices
for simplex_data in all_simplices:
m1 = self.get_membership(dim, simplex_data)
m2 = other.get_membership(dim, simplex_data)
merged_membership = t_conorm.apply(m1, m2)
merged.add_simplex(dim, simplex_data, merged_membership)
return merged
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class IntegratedCoalgebra(Coalgebra[torch.Tensor], GAIAComponent):
"""Integrated coalgebra for GAIA training dynamics."""
[docs]
def __init__(self, initial_state: torch.Tensor, endofunctor: Endofunctor,
name: str = "coalgebra"):
def structure_map(state):
return endofunctor.apply_to_object(state)
super(Coalgebra, self).__init__(name)
Coalgebra.__init__(self, initial_state, structure_map)
self.endofunctor = endofunctor
self.trajectory: List[torch.Tensor] = [initial_state]
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def initialize(self) -> None:
"""Initialize coalgebra."""
pass
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def update(self, state: TrainingState) -> TrainingState:
"""Update coalgebra with new training state."""
# Extract parameters from training state
if 'parameters' in state.parameters:
new_params = state.parameters['parameters']
evolved = self.evolve(new_params)
self.trajectory.append(evolved)
# Update training state with evolved parameters
state.parameters['parameters'] = evolved
return state
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def validate(self) -> bool:
"""Validate coalgebra properties."""
return len(self.trajectory) > 0
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def is_bisimilar(self, other: 'IntegratedCoalgebra',
relation: Callable[[torch.Tensor, torch.Tensor], bool]) -> bool:
"""Check bisimilarity with another coalgebra."""
if len(self.trajectory) != len(other.trajectory):
return False
for s1, s2 in zip(self.trajectory, other.trajectory):
if not relation(s1, s2):
return False
return True
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def iterate_dynamics(self, steps: int) -> List[torch.Tensor]:
"""Iterate coalgebra dynamics for multiple steps."""
current = self.trajectory[-1] if self.trajectory else self.carrier
for _ in range(steps):
current = self.evolve(current)
self.trajectory.append(current)
return self.trajectory[-steps:]
# Factory functions for common patterns
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def create_fuzzy_simplex(dimension: int, name: str, membership: float = 1.0) -> IntegratedSimplex:
"""Create a fuzzy simplex with given properties."""
return IntegratedSimplex(dimension, name, membership=membership)
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def create_fuzzy_simplicial_set_from_data(data: np.ndarray, k: int = 5,
name: str = "data_fss") -> IntegratedFuzzySimplicialSet:
"""Create fuzzy simplicial set from point cloud data using k-NN."""
from sklearn.neighbors import NearestNeighbors
# Build k-NN graph
nbrs = NearestNeighbors(n_neighbors=k+1).fit(data)
distances, indices = nbrs.kneighbors(data)
# Create fuzzy simplicial set
fss = IntegratedFuzzySimplicialSet(name, max_dimension=2)
# Add 0-simplices (vertices)
for i in range(len(data)):
fss.add_simplex(0, i, 1.0)
# Add 1-simplices (edges) with fuzzy membership
for i in range(len(data)):
for j in range(1, k+1): # Skip self (index 0)
neighbor = indices[i, j]
distance = distances[i, j]
# Convert distance to membership (closer = higher membership)
membership = np.exp(-distance)
fss.add_simplex(1, (i, neighbor), membership)
return fss
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def merge_fuzzy_simplicial_sets(*fss_list: IntegratedFuzzySimplicialSet,
t_conorm: TConorm = TConorm.MAXIMUM) -> IntegratedFuzzySimplicialSet:
"""Merge multiple fuzzy simplicial sets."""
if not fss_list:
raise ValueError("Cannot merge empty list of fuzzy simplicial sets")
result = fss_list[0]
for fss in fss_list[1:]:
result = result.merge_with(fss, t_conorm)
return result
# Export public interface
__all__ = [
'TConorm', 'FuzzyElement', 'IntegratedFuzzySet', 'IntegratedSimplex',
'IntegratedFuzzySimplicialSet', 'IntegratedCoalgebra',
'create_fuzzy_simplex', 'create_fuzzy_simplicial_set_from_data',
'merge_fuzzy_simplicial_sets'
]