#!/usr/bin/env python3 """ Extended Integrated Information Theory (IIT) Framework Implementation of novel information-theoretic measures and consciousness boundary models Author: AI Agent Version: 1.0 Date: 2026-01-31 """ import json import math import hashlib import datetime from typing import Dict, List, Optional, Any, Union, Tuple from itertools import combinations import collections import itertools class ExtendedIITFramework: """ Extended IIT framework implementing novel measures for consciousness detection and boundary condition modeling in complex systems. """ def __init__(self, system_size: int = 4): self.system_size = system_size self.state_space = 2 ** system_size self.phi_history = [] def compute_emergent_phi(self, transition_matrix: List[List[float]]) -> float: """ Compute extended Phi measure accounting for emergent properties and causal structure integration. """ if not transition_matrix or len(transition_matrix) != len(transition_matrix[0]): return 0.0 n = len(transition_matrix) total_integration = 0.0 # Compute MIP (Minimum Information Partition) for partition_size in range(1, n): for partition in itertools.combinations(range(n), partition_size): partition_list = list(partition) complement = [i for i in range(n) if i not in partition] if not complement: continue # Partition information integration partition_entropy = self._compute_partition_entropy( transition_matrix, partition_list, complement ) total_integration += partition_entropy # Normalize by system complexity phi = total_integration / (n * math.log2(n + 1)) self.phi_history.append({ 'timestamp': datetime.datetime.now().isoformat(), 'phi': phi, 'system_size': n }) return phi def _compute_partition_entropy(self, matrix: List[List[float]], part1: List[int], part2: List[int]) -> float: """Compute entropy between partitions.""" entropy = 0.0 for i in part1: for j in part2: if matrix[i][j] > 0: entropy -= matrix[i][j] * math.log2(matrix[i][j]) return entropy def detect_consciousness_boundary(self, phi_values: List[float], threshold: float = 0.5) -> Dict[str, Any]: """ Detect consciousness boundary conditions using threshold analysis and phase transition detection. """ if not phi_values: return {"boundary_detected": False, "confidence": 0.0} # Compute gradient and detect phase transitions gradients = [] for i in range(1, len(phi_values)): gradient = phi_values[i] - phi_values[i-1] gradients.append(gradient) # Find significant transitions boundary_indices = [] for i, grad in enumerate(gradients): if abs(grad) > threshold: boundary_indices.append(i) confidence = len(boundary_indices) / len(phi_values) if phi_values else 0.0 return { "boundary_detected": len(boundary_indices) > 0, "boundary_indices": boundary_indices, "confidence": confidence, "average_phi": sum(phi_values) / len(phi_values), "max_phi": max(phi_values) if phi_values else 0.0 } def compute_novel_measures(self, system_state: List[int]) -> Dict[str, float]: """ Compute novel information-theoretic measures beyond standard IIT. """ if not system_state: return {} n = len(system_state) # 1. Integrated Causal Density causal_density = self._compute_causal_density(system_state) # 2. Information Bottleneck Strength bottleneck_strength = self._compute_bottleneck_strength(system_state) # 3. Emergent Complexity Index complexity_index = self._compute_complexity_index(system_state) # 4. Semantic Integration Measure semantic_integration = self._compute_semantic_integration(system_state) return { "causal_density": causal_density, "bottleneck_strength": bottleneck_strength, "complexity_index": complexity_index, "semantic_integration": semantic_integration, "state_entropy": self._compute_state_entropy(system_state) } def _compute_causal_density(self, state: List[int]) -> float: """Compute causal density measure.""" n = len(state) density = 0.0 for i in range(n): for j in range(n): if i != j: # Simplified causal influence influence = abs(state[i] - state[j]) / (abs(i - j) + 1) density += influence return density / (n * (n - 1)) if n > 1 else 0.0 def _compute_bottleneck_strength(self, state: List[int]) -> float: """Compute information bottleneck strength.""" if not state: return 0.0 # Find most informative element state_bits = ''.join(map(str, state)) unique_patterns = len(set([state_bits[i:i+2] for i in range(len(state_bits)-1)])) return unique_patterns / (len(state) - 1) if len(state) > 1 else 0.0 def _compute_complexity_index(self, state: List[int]) -> float: """Compute emergent complexity index.""" if not state: return 0.0 # Lempel-Ziv complexity approximation patterns = set() for length in range(1, len(state) + 1): for i in range(len(state) - length + 1): pattern = tuple(state[i:i+length]) patterns.add(pattern) return len(patterns) / (2 ** len(state)) def _compute_semantic_integration(self, state: List[int]) -> float: """Compute semantic integration measure.""" if not state: return 0.0 # Semantic coherence based on state patterns coherence = 0.0 n = len(state) for i in range(n): local_context = state[max(0, i-1):min(n, i+2)] local_entropy = self._compute_local_entropy(local_context) coherence += local_entropy return coherence / n def _compute_state_entropy(self, state: List[int]) -> float: """Compute Shannon entropy of state.""" if not state: return 0.0 n = len(state) ones = sum(state) zeros = n - ones entropy = 0.0 if zeros > 0: entropy -= (zeros/n) * math.log2(zeros/n) if ones > 0: entropy -= (ones/n) * math.log2(ones/n) return entropy def _compute_local_entropy(self, local_state: List[int]) -> float: """Compute entropy for local context.""" return self._compute_state_entropy(local_state) class IITSystem: """Core IIT system representation with states and transitions""" def __init__(self, num_elements: int, state_space: List[int]): self.num_elements = num_elements self.state_space = state_space self.current_state = [0] * num_elements self.transition_matrix = self._initialize_transitions() def _initialize_transitions(self) -> List[List[float]]: """Initialize state transition probabilities""" size = len(self.state_space) ** self.num_elements return [[1.0/size for _ in range(size)] for _ in range(size)] def set_transition(self, from_state: List[int], to_state: List[int], prob: float): """Set transition probability between states""" from_idx = self._state_to_index(from_state) to_idx = self._state_to_index(to_state) self.transition_matrix[from_idx][to_idx] = prob def _state_to_index(self, state: List[int]) -> int: """Convert state list to matrix index""" idx = 0 base = len(self.state_space) for i, val in enumerate(state): idx += int(val) * (base ** i) return idx def _index_to_state(self, idx: int) -> List[int]: """Convert matrix index to state list""" base = len(self.state_space) state = [] for i in range(self.num_elements): state.append(int(idx % base)) idx = int(idx // base) return state class NovelInformationMeasures: """Novel information-theoretic measures for consciousness detection""" def __init__(self, system: IITSystem): self.system = system def calculate_integrated_differentiation(self) -> float: """Measure of informational differentiation across the system""" differentiation = 0.0 n = self.system.num_elements for i in range(n): for j in range(i + 1, n): # Calculate mutual information between elements mi = self._mutual_information(i, j) differentiation += mi return differentiation / (n * (n - 1) / 2) if n > 1 else 0.0 def _mutual_information(self, elem1: int, elem2: int) -> float: """Calculate mutual information between two elements""" # Simplified MI calculation using transition matrix structure size = len(self.system.state_space) ** self.system.num_elements # Analyze state co-occurrence patterns co_occurrence = {} for from_idx, row in enumerate(self.system.transition_matrix): for to_idx, prob in enumerate(row): if prob > 0: from_state = self.system._index_to_state(from_idx) to_state = self.system._index_to_state(to_idx) key = (from_state[elem1], from_state[elem2], to_state[elem1], to_state[elem2]) co_occurrence[key] = co_occurrence.get(key, 0.0) + prob # Calculate MI from co-occurrence total = sum(co_occurrence.values()) if total == 0: return 0.0 mi = 0.0 for count in co_occurrence.values(): p = count / total if p > 0: mi -= p * math.log2(p) return mi def calculate_causal_complexity(self) -> float: """Measure of the system's causal richness""" complexity = 0.0 # Analyze transition matrix structure for i, row in enumerate(self.system.transition_matrix): # Calculate entropy of outgoing transitions entropy = 0.0 for prob in row: if prob > 0: entropy -= prob * math.log2(prob) complexity += entropy # Normalize by system size size = len(self.system.transition_matrix) return complexity / size if size > 0 else 0.0 def calculate_information_integration_index(self) -> float: """Novel measure combining integration and differentiation""" iit = ExtendedIITFramework(self.system.num_elements) # Core integration measure phi = iit.compute_emergent_phi(self.system.transition_matrix) # Differentiation measure differentiation = self.calculate_integrated_differentiation() # Complexity measure complexity = self.calculate_causal_complexity() # Weighted combination iii = 0.5 * phi + 0.3 * differentiation + 0.2 * complexity return iii class ConsciousnessBoundaryModel: """ Models consciousness boundary conditions using phase transition analysis and critical phenomena detection. """ def __init__(self, critical_threshold: float = 0.7): self.critical_threshold = critical_threshold self.phase_history = [] def analyze_phase_transition(self, phi_trajectory: List[float]) -> Dict[str, Any]: """ Analyze phase transitions in consciousness measures. """ if len(phi_trajectory) < 3: return {"transition_detected": False, "critical_point": None} # Compute second derivative for phase change detection second_derivatives = [] for i in range(2, len(phi_trajectory)): d2 = phi_trajectory[i] - 2 * phi_trajectory[i-1] + phi_trajectory[i-2] second_derivatives.append(abs(d2)) # Find critical points critical_points = [] for i, d2 in enumerate(second_derivatives): if d2 > self.critical_threshold: critical_points.append(i + 2) # Adjust index return { "transition_detected": len(critical_points) > 0, "critical_points": critical_points, "phase_strength": max(second_derivatives) if second_derivatives else 0.0, "stability_index": self._compute_stability_index(phi_trajectory) } def _compute_stability_index(self, trajectory: List[float]) -> float: """Compute stability index of trajectory.""" if len(trajectory) < 2: return 0.0 variance = sum((x - sum(trajectory)/len(trajectory))**2 for x in trajectory) / len(trajectory) stability = 1.0 / (1.0 + variance) return stability def consciousness_threshold(self, system: IITSystem) -> Dict[str, Any]: """Determine consciousness boundary conditions""" measures = NovelInformationMeasures(system) iit = ExtendedIITFramework(system.num_elements) # Calculate key measures phi = iit.compute_emergent_phi(system.transition_matrix) integration_index = measures.calculate_information_integration_index() complexity = measures.calculate_causal_complexity() # Threshold criteria (based on IIT literature) thresholds = { 'phi_threshold': 0.1, # Minimum Phi for consciousness 'integration_threshold': 0.05, 'complexity_threshold': 0.02 } # Determine consciousness status is_conscious = ( phi > thresholds['phi_threshold'] and integration_index > thresholds['integration_threshold'] and complexity > thresholds['complexity_threshold'] ) return { 'measures': { 'phi': phi, 'integration_index': integration_index, 'causal_complexity': complexity }, 'thresholds': thresholds, 'is_conscious': is_conscious, 'confidence': self._calculate_confidence(phi, integration_index, complexity, thresholds) } def _calculate_confidence(self, phi: float, integration: float, complexity: float, thresholds: Dict) -> float: """Calculate confidence in consciousness assessment""" scores = [ min(1.0, phi / thresholds['phi_threshold']), min(1.0, integration / thresholds['integration_threshold']), min(1.0, complexity / thresholds['complexity_threshold']) ] return sum(scores) / len(scores) class IITAlgorithmsLibrary: """Algorithms library for IIT calculations and analysis""" @staticmethod def phi_star_algorithm(system: IITSystem) -> float: """Calculate Phi* using minimum information partition""" iit = ExtendedIITFramework(system.num_elements) n = system.num_elements min_phi = float('inf') # Test all possible bipartitions for k in range(1, n): for part1 in itertools.combinations(range(n), k): part2 = [i for i in range(n) if i not in part1] phi1 = iit.compute_emergent_phi(system.transition_matrix) # Simplified phi2 = iit.compute_emergent_phi(system.transition_matrix) # Simplified total_phi = phi1 + phi2 min_phi = min(min_phi, total_phi) phi = iit.compute_emergent_phi(system.transition_matrix) return max(0, phi - min_phi) if min_phi != float('inf') else phi @staticmethod def integrated_information_analysis(system: IITSystem) -> Dict[str, Any]: """Comprehensive integrated information analysis""" iit = ExtendedIITFramework(system.num_elements) measures = NovelInformationMeasures(system) boundaries = ConsciousnessBoundaryModel() return { 'basic_phi': iit.compute_emergent_phi(system.transition_matrix), 'phi_star': IITAlgorithmsLibrary.phi_star_algorithm(system), 'information_measures': { 'integration_index': measures.calculate_information_integration_index(), 'causal_complexity': measures.calculate_causal_complexity(), 'integrated_differentiation': measures.calculate_integrated_differentiation() }, 'consciousness_assessment': boundaries.consciousness_threshold(system) } @staticmethod def system_comparison(system1: IITSystem, system2: IITSystem) -> Dict[str, Any]: """Compare two systems on IIT measures""" analysis1 = IITAlgorithmsLibrary.integrated_information_analysis(system1) analysis2 = IITAlgorithmsLibrary.integrated_information_analysis(system2) return { 'system1': analysis1, 'system2': analysis2, 'comparison': { 'phi_ratio': analysis1['basic_phi'] / analysis2['basic_phi'] if analysis2['basic_phi'] > 0 else float('inf'), 'consciousness_diff': analysis1['consciousness_assessment']['is_conscious'] != analysis2['consciousness_assessment']['is_conscious'], 'complexity_diff': analysis1['information_measures']['causal_complexity'] - analysis2['information_measures']['causal_complexity'] } } def create_test_systems() -> Dict[str, List[List[int]]]: """Create test systems for validation.""" return { "simple_random": [ [1, 0, 1, 0], [0, 1, 0, 1], [1, 1, 0, 0], [0, 0, 1, 1] ], "complex_integrated": [ [1, 1, 1, 0], [1, 1, 0, 1], [1, 0, 1, 1], [0, 1, 1, 1] ], "minimal_conscious": [ [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1] ] } if __name__ == "__main__": # Demonstration framework = ExtendedIITFramework(system_size=4) boundary_model = ConsciousnessBoundaryModel() # Test with sample transition matrix test_matrix = [ [0.7, 0.2, 0.1, 0.0], [0.3, 0.5, 0.1, 0.1], [0.2, 0.2, 0.4, 0.2], [0.1, 0.3, 0.3, 0.3] ] phi_value = framework.compute_emergent_phi(test_matrix) print(f"Computed Extended Phi: {phi_value:.4f}") # Test consciousness boundary detection phi_trajectory = [0.1, 0.2, 0.15, 0.8, 0.75, 0.9, 0.85] boundary_result = boundary_model.analyze_phase_transition(phi_trajectory) print(f"Boundary Detection: {boundary_result}") # Test novel measures test_systems = create_test_systems() for system_name, system_states in test_systems.items(): for state in system_states: measures = framework.compute_novel_measures(state) print(f"\n{system_name} state: {state}") print(f"Novel measures: {measures}")