""" Algebraic Neural Network Implementation This module implements various types of algebraic neural networks that don't require traditional gradient-based training. Instead, they use algebraic structures and operations to process information. Author: Algebraic Neural Network Research """ import numpy as np from typing import List, Callable, Union import math class AlgebraicLayer: """ Base class for algebraic layers that use mathematical operations instead of learned weights. """ def __init__(self, input_size: int, output_size: int, operation: str = "polynomial"): self.input_size = input_size self.output_size = output_size self.operation = operation def forward(self, x: np.ndarray) -> np.ndarray: """Apply algebraic transformation to input.""" raise NotImplementedError("Subclasses must implement forward method") class PolynomialLayer(AlgebraicLayer): """ Layer that applies polynomial transformations without learned weights. Uses fixed polynomial coefficients based on algebraic principles. """ def __init__(self, input_size: int, output_size: int, degree: int = 2): super().__init__(input_size, output_size, "polynomial") self.degree = degree # Generate coefficients using algebraic sequences (e.g., Fibonacci-based) self.coefficients = self._generate_algebraic_coefficients() def _generate_algebraic_coefficients(self) -> np.ndarray: """Generate polynomial coefficients using algebraic sequences.""" # Use golden ratio and algebraic numbers for coefficients phi = (1 + math.sqrt(5)) / 2 # Golden ratio coeffs = [] for i in range(self.output_size): for j in range(self.input_size): # Generate coefficients using algebraic properties coeff = (phi ** (i + 1)) / (j + 1) if j < self.input_size else 1.0 coeffs.append(coeff) return np.array(coeffs).reshape(self.output_size, self.input_size) def forward(self, x: np.ndarray) -> np.ndarray: """Apply polynomial transformation.""" if x.ndim == 1: x = x.reshape(1, -1) result = np.zeros((x.shape[0], self.output_size)) for i in range(self.output_size): # Apply polynomial of specified degree poly_result = np.zeros(x.shape[0]) for degree in range(1, self.degree + 1): poly_term = np.power(x, degree) @ self.coefficients[i] poly_result += poly_term / math.factorial(degree) result[:, i] = poly_result return result class GroupTheoryLayer(AlgebraicLayer): """ Layer based on group theory operations, particularly using cyclic groups and group actions for transformations. """ def __init__(self, input_size: int, output_size: int, group_order: int = 8): super().__init__(input_size, output_size, "group_theory") self.group_order = group_order # Generate group elements (rotations in this case) self.group_elements = self._generate_cyclic_group() def _generate_cyclic_group(self) -> List[np.ndarray]: """Generate cyclic group elements as rotation matrices.""" elements = [] for k in range(self.group_order): angle = 2 * math.pi * k / self.group_order # For 2D case, use rotation matrices if self.input_size == 2: rotation = np.array([ [math.cos(angle), -math.sin(angle)], [math.sin(angle), math.cos(angle)] ]) elements.append(rotation) else: # For higher dimensions, use generalized rotations rotation = np.eye(self.input_size) if self.input_size >= 2: rotation[0, 0] = math.cos(angle) rotation[0, 1] = -math.sin(angle) rotation[1, 0] = math.sin(angle) rotation[1, 1] = math.cos(angle) elements.append(rotation) return elements def forward(self, x: np.ndarray) -> np.ndarray: """Apply group action to input.""" if x.ndim == 1: x = x.reshape(1, -1) results = [] for i, group_element in enumerate(self.group_elements[:self.output_size]): if x.shape[1] == group_element.shape[0]: transformed = x @ group_element.T # Take the norm as a scalar output result = np.linalg.norm(transformed, axis=1) else: # Fallback for size mismatch result = np.sum(x * (i + 1), axis=1) results.append(result) return np.column_stack(results) class GeometricAlgebraLayer(AlgebraicLayer): """ Layer using geometric algebra (Clifford algebra) operations. Implements basic geometric product and algebraic operations. """ def __init__(self, input_size: int, output_size: int): super().__init__(input_size, output_size, "geometric_algebra") # Initialize basis vectors for geometric algebra self.basis_vectors = self._generate_basis_vectors() def _generate_basis_vectors(self) -> List[np.ndarray]: """Generate basis vectors for geometric algebra.""" basis = [] # Scalar basis basis.append(np.ones(self.input_size)) # Vector basis for i in range(self.input_size): e_i = np.zeros(self.input_size) e_i[i] = 1.0 basis.append(e_i) # Bivector basis (for pairs) for i in range(self.input_size): for j in range(i + 1, self.input_size): e_ij = np.zeros(self.input_size) e_ij[i] = 1.0 e_ij[j] = 1.0 basis.append(e_ij) return basis[:self.output_size] def geometric_product(self, a: np.ndarray, b: np.ndarray) -> float: """Compute geometric product of two vectors.""" # Simplified geometric product: dot product + outer product norm dot_prod = np.dot(a, b) # Approximate outer product contribution outer_norm = np.linalg.norm(np.outer(a, b)) return dot_prod + outer_norm def forward(self, x: np.ndarray) -> np.ndarray: """Apply geometric algebra transformations.""" if x.ndim == 1: x = x.reshape(1, -1) results = [] for basis_vector in self.basis_vectors: # Compute geometric product with basis vector result = [] for sample in x: gp = self.geometric_product(sample, basis_vector) result.append(gp) results.append(np.array(result)) return np.column_stack(results) class HaltingOracleLayer(AlgebraicLayer): """ Layer that simulates a halting oracle for specific computational patterns. Uses heuristics and bounded computation to approximate uncomputable halting decisions. """ def __init__(self, input_size: int, output_size: int, max_iterations: int = 1000): super().__init__(input_size, output_size, "halting_oracle") self.max_iterations = max_iterations # Fixed seed for deterministic "non-algorithmic" behavior self.oracle_seed = 42 def _simulate_halting_oracle(self, program_encoding: float) -> float: """Simulate halting oracle decision for encoded program.""" # Use fixed-seed randomness to simulate oracle behavior np.random.seed(int(abs(program_encoding * 1000)) % 2**31) # Simple heuristic: programs with certain patterns are more likely to halt # This is a deterministic approximation of the uncomputable halting problem complexity_factor = abs(program_encoding) % 1.0 if complexity_factor < 0.1: # Very simple programs likely halt return 1.0 elif complexity_factor > 0.9: # Very complex programs might not halt return 0.1 else: # Use bounded computation simulation iterations = int(complexity_factor * self.max_iterations) # Simulate bounded program execution state = program_encoding for i in range(iterations): state = (state * 1.618 + 0.786) % 1.0 # Simple state evolution if abs(state - 0.5) < 0.01: # "Halting condition" return 1.0 return 0.3 # Uncertain/timeout case def forward(self, x: np.ndarray) -> np.ndarray: """Apply halting oracle simulation to inputs.""" if x.ndim == 1: x = x.reshape(1, -1) results = [] for i in range(self.output_size): oracle_results = [] for sample in x: # Encode input as "program" for halting oracle program_encoding = np.sum(sample * (i + 1)) / len(sample) halting_prob = self._simulate_halting_oracle(program_encoding) oracle_results.append(halting_prob) results.append(np.array(oracle_results)) return np.column_stack(results) class KolmogorovComplexityLayer(AlgebraicLayer): """ Layer that approximates Kolmogorov complexity using compression-based heuristics. Provides bounded approximations of the uncomputable Kolmogorov complexity function. """ def __init__(self, input_size: int, output_size: int, precision: int = 8): super().__init__(input_size, output_size, "kolmogorov_complexity") self.precision = precision def _approximate_kolmogorov_complexity(self, data: np.ndarray) -> float: """Approximate Kolmogorov complexity using compressibility heuristics.""" # Convert to discrete representation discretized = np.round(data * (2**self.precision)).astype(int) # Simple compression simulation using pattern detection data_str = ''.join(map(str, discretized)) # Count repetitive patterns (simple compression heuristic) unique_chars = len(set(data_str)) total_chars = len(data_str) # Estimate complexity based on entropy-like measure if total_chars == 0: return 0.0 # Normalized complexity estimate entropy_approx = (unique_chars / total_chars) * np.log2(unique_chars + 1) # Add pattern complexity (longer patterns suggest lower complexity) pattern_factor = 1.0 for pattern_len in range(2, min(5, len(data_str))): patterns = set() for i in range(len(data_str) - pattern_len + 1): patterns.add(data_str[i:i+pattern_len]) if len(patterns) < len(data_str) - pattern_len + 1: pattern_factor *= 0.8 # Reduce complexity for repeated patterns return entropy_approx * pattern_factor def forward(self, x: np.ndarray) -> np.ndarray: """Compute approximate Kolmogorov complexity for inputs.""" if x.ndim == 1: x = x.reshape(1, -1) results = [] for i in range(self.output_size): complexity_results = [] for sample in x: # Different projections for different outputs projection = sample * (i + 1) / (self.output_size) complexity = self._approximate_kolmogorov_complexity(projection) complexity_results.append(complexity) results.append(np.array(complexity_results)) return np.column_stack(results) class BusyBeaverLayer(AlgebraicLayer): """ Layer using Busy Beaver function values and approximations. The Busy Beaver function is uncomputable for general n, but known for small values. """ def __init__(self, input_size: int, output_size: int): super().__init__(input_size, output_size, "busy_beaver") # Known Busy Beaver values: BB(1)=1, BB(2)=4, BB(3)=6, BB(4)=13, BB(5)≥4098 self.known_bb_values = {1: 1, 2: 4, 3: 6, 4: 13, 5: 4098} def _busy_beaver_approximation(self, n: int) -> float: """Get Busy Beaver value or approximation.""" if n <= 0: return 0.0 elif n in self.known_bb_values: return float(self.known_bb_values[n]) elif n <= 5: return float(self.known_bb_values[5]) else: # For n > 5, use exponential approximation # This is a heuristic since BB(n) grows faster than any computable function return self.known_bb_values[5] * (2.0 ** (n - 5)) def forward(self, x: np.ndarray) -> np.ndarray: """Apply Busy Beaver function to transformed inputs.""" if x.ndim == 1: x = x.reshape(1, -1) results = [] for i in range(self.output_size): bb_results = [] for sample in x: # Transform input to discrete parameter for Busy Beaver function # Use log scale to keep values reasonable param = max(1, int(abs(np.sum(sample) * (i + 1)) % 10) + 1) bb_value = self._busy_beaver_approximation(param) # Normalize to prevent overflow normalized_bb = np.log1p(bb_value) bb_results.append(normalized_bb) results.append(np.array(bb_results)) return np.column_stack(results) class NonRecursiveLayer(AlgebraicLayer): """ Layer based on non-recursive sets and functions. Simulates operations on computably enumerable but non-computable sets. """ def __init__(self, input_size: int, output_size: int, enumeration_bound: int = 1000): super().__init__(input_size, output_size, "non_recursive") self.enumeration_bound = enumeration_bound # Simulate a c.e. non-recursive set using a bounded enumeration self.ce_set = self._generate_ce_set() def _generate_ce_set(self) -> set: """Generate a computably enumerable set simulation.""" ce_set = set() # Simulate enumeration of a c.e. set (like the set of Gödel numbers of theorems) for i in range(self.enumeration_bound): # Simple enumeration rule (this is computable, but simulates c.e. behavior) if self._enumeration_rule(i): ce_set.add(i) return ce_set def _enumeration_rule(self, n: int) -> bool: """Rule for enumerating elements (simulates theorem enumeration).""" # Simple mathematical property that creates interesting patterns # Simulates: numbers that can be expressed as sum of two squares for i in range(int(np.sqrt(n)) + 1): remainder = n - i*i if remainder >= 0 and int(np.sqrt(remainder))**2 == remainder: return True return False def _membership_oracle(self, value: float) -> float: """Simulate membership oracle for non-recursive set.""" # Convert continuous value to discrete for set membership discrete_val = int(abs(value * 1000)) % self.enumeration_bound if discrete_val in self.ce_set: return 1.0 else: # For values not yet enumerated, return uncertainty # This simulates the non-recursive nature return 0.5 def forward(self, x: np.ndarray) -> np.ndarray: """Apply non-recursive set operations to inputs.""" if x.ndim == 1: x = x.reshape(1, -1) results = [] for i in range(self.output_size): membership_results = [] for sample in x: # Transform input for membership testing test_value = np.sum(sample * np.arange(len(sample))) * (i + 1) membership = self._membership_oracle(test_value) membership_results.append(membership) results.append(np.array(membership_results)) return np.column_stack(results) class AlgebraicNeuralNetwork: """ Main class for Algebraic Neural Networks that combines different algebraic layers to create a complete network. """ def __init__(self): self.layers = [] def add_layer(self, layer: AlgebraicLayer): """Add an algebraic layer to the network.""" self.layers.append(layer) def forward(self, x: np.ndarray) -> np.ndarray: """Forward pass through all algebraic layers.""" current_input = x for layer in self.layers: current_input = layer.forward(current_input) return current_input def predict(self, x: np.ndarray) -> np.ndarray: """Alias for forward pass.""" return self.forward(x) def create_sample_network() -> AlgebraicNeuralNetwork: """Create a sample algebraic neural network for demonstration.""" network = AlgebraicNeuralNetwork() # Add different types of algebraic layers network.add_layer(PolynomialLayer(4, 6, degree=2)) network.add_layer(GroupTheoryLayer(6, 4, group_order=8)) network.add_layer(GeometricAlgebraLayer(4, 2)) return network def create_uncomputable_network() -> AlgebraicNeuralNetwork: """Create a sample network with uncomputable layers for demonstration.""" network = AlgebraicNeuralNetwork() # Add uncomputable layers network.add_layer(HaltingOracleLayer(4, 5, max_iterations=500)) network.add_layer(KolmogorovComplexityLayer(5, 4, precision=6)) network.add_layer(BusyBeaverLayer(4, 3)) network.add_layer(NonRecursiveLayer(3, 2, enumeration_bound=500)) return network def demo_algebraic_neural_network(): """Demonstrate the algebraic neural network with sample data.""" print("=== Algebraic Neural Network Demo ===\n") # Create sample network network = create_sample_network() # Generate sample input data np.random.seed(42) sample_input = np.random.randn(5, 4) # 5 samples, 4 features each print("Input data shape:", sample_input.shape) print("Input data:\n", sample_input) # Run prediction output = network.predict(sample_input) print("\nOutput data shape:", output.shape) print("Output data:\n", output) # Demonstrate individual layers print("\n=== Individual Layer Demonstrations ===\n") # Polynomial Layer poly_layer = PolynomialLayer(4, 3, degree=2) poly_output = poly_layer.forward(sample_input[0]) print("Polynomial Layer Output:", poly_output) # Group Theory Layer group_layer = GroupTheoryLayer(4, 3, group_order=6) group_output = group_layer.forward(sample_input[0]) print("Group Theory Layer Output:", group_output) # Geometric Algebra Layer geo_layer = GeometricAlgebraLayer(4, 3) geo_output = geo_layer.forward(sample_input[0]) print("Geometric Algebra Layer Output:", geo_output) def demo_uncomputable_neural_network(): """Demonstrate the uncomputable neural network with sample data.""" print("\n=== Uncomputable Neural Network Demo ===\n") # Create uncomputable network network = create_uncomputable_network() # Generate sample input data np.random.seed(42) sample_input = np.random.randn(3, 4) # 3 samples, 4 features each print("Input data shape:", sample_input.shape) print("Input data:\n", sample_input) # Run prediction output = network.predict(sample_input) print("\nOutput data shape:", output.shape) print("Output data:\n", output) # Demonstrate individual uncomputable layers print("\n=== Individual Uncomputable Layer Demonstrations ===\n") # Halting Oracle Layer halting_layer = HaltingOracleLayer(4, 3, max_iterations=100) halting_output = halting_layer.forward(sample_input[0]) print("Halting Oracle Layer Output:", halting_output) # Kolmogorov Complexity Layer kolmogorov_layer = KolmogorovComplexityLayer(4, 3, precision=6) kolmogorov_output = kolmogorov_layer.forward(sample_input[0]) print("Kolmogorov Complexity Layer Output:", kolmogorov_output) # Busy Beaver Layer bb_layer = BusyBeaverLayer(4, 3) bb_output = bb_layer.forward(sample_input[0]) print("Busy Beaver Layer Output:", bb_output) # Non-Recursive Layer nr_layer = NonRecursiveLayer(4, 3, enumeration_bound=100) nr_output = nr_layer.forward(sample_input[0]) print("Non-Recursive Layer Output:", nr_output) if __name__ == "__main__": demo_algebraic_neural_network() demo_uncomputable_neural_network()