algebraic_neural_network.py

"""
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()