# SAFLA Algorithms Guide

## Overview

This guide provides detailed mathematical formulations and algorithmic descriptions for the Self-Aware Feedback Loop Algorithm (SAFLA). It covers the core mathematical foundations, computational algorithms, and implementation details that power SAFLA's adaptive capabilities.

## Table of Contents

1. [Delta Evaluation Algorithm](#delta-evaluation-algorithm)
2. [Meta-Cognitive Engine Algorithms](#meta-cognitive-engine-algorithms)
3. [Memory Consolidation Algorithms](#memory-consolidation-algorithms)
4. [Safety Validation Algorithms](#safety-validation-algorithms)
5. [MCP Orchestration Algorithms](#mcp-orchestration-algorithms)
6. [Adaptive Weight Learning](#adaptive-weight-learning)
7. [Performance Prediction Models](#performance-prediction-models)
8. [Convergence Analysis](#convergence-analysis)

## Delta Evaluation Algorithm

### Mathematical Foundation

The delta evaluation system quantifies improvement across multiple dimensions using a weighted combination of performance metrics. The core formula is:

```
Δ_total = α₁ × Δ_performance + α₂ × Δ_efficiency + α₃ × Δ_stability + α₄ × Δ_capability
```

Where:
- `Δ_total` ∈ [-1, 1] represents the total system improvement
- `α₁, α₂, α₃, α₄` are adaptive weights with `Σαᵢ = 1`
- Each `Δ_metric` ∈ [-1, 1] represents normalized change in that dimension

### Individual Delta Calculations

#### Performance Delta (Δ_performance)

```
Δ_performance = (P_current - P_previous) / max(P_current, P_previous, ε)
```

Where:
- `P_current`, `P_previous` are current and previous performance scores
- `ε = 1e-8` prevents division by zero
- Normalization ensures `Δ_performance` ∈ [-1, 1]

**Algorithm Implementation:**

```python
def calculate_performance_delta(current: float, previous: float) -> float:
    """
    Calculate normalized performance delta.
    
    Args:
        current: Current performance metric [0, 1]
        previous: Previous performance metric [0, 1]
        
    Returns:
        Normalized delta [-1, 1]
    """
    epsilon = 1e-8
    denominator = max(current, previous, epsilon)
    
    # Raw delta calculation
    raw_delta = (current - previous) / denominator
    
    # Apply sigmoid normalization for extreme values
    if abs(raw_delta) > 1.0:
        raw_delta = 2.0 / (1.0 + exp(-raw_delta)) - 1.0
    
    return max(-1.0, min(1.0, raw_delta))
```

#### Efficiency Delta (Δ_efficiency)

Efficiency considers both resource utilization and throughput:

```
Δ_efficiency = w₁ × Δ_resource + w₂ × Δ_throughput
```

Where:
- `w₁ + w₂ = 1` (typically `w₁ = 0.6, w₂ = 0.4`)
- `Δ_resource = (R_previous - R_current) / max(R_previous, R_current, ε)` (lower is better)
- `Δ_throughput = (T_current - T_previous) / max(T_current, T_previous, ε)` (higher is better)

**Algorithm Implementation:**

```python
def calculate_efficiency_delta(current_metrics: Dict, previous_metrics: Dict) -> float:
    """
    Calculate efficiency delta considering resource usage and throughput.
    
    Args:
        current_metrics: Current efficiency metrics
        previous_metrics: Previous efficiency metrics
        
    Returns:
        Normalized efficiency delta [-1, 1]
    """
    epsilon = 1e-8
    w_resource = 0.6
    w_throughput = 0.4
    
    # Resource utilization delta (lower is better)
    curr_resource = current_metrics.get('resource_utilization', 0.5)
    prev_resource = previous_metrics.get('resource_utilization', 0.5)
    resource_delta = (prev_resource - curr_resource) / max(prev_resource, curr_resource, epsilon)
    
    # Throughput delta (higher is better)
    curr_throughput = current_metrics.get('throughput', 1.0)
    prev_throughput = previous_metrics.get('throughput', 1.0)
    throughput_delta = (curr_throughput - prev_throughput) / max(curr_throughput, prev_throughput, epsilon)
    
    # Weighted combination
    efficiency_delta = w_resource * resource_delta + w_throughput * throughput_delta
    
    return max(-1.0, min(1.0, efficiency_delta))
```

#### Stability Delta (Δ_stability)

Stability measures system consistency and variance:

```
Δ_stability = -|σ_current - σ_previous| / max(σ_current, σ_previous, ε)
```

Where:
- `σ` represents the standard deviation of recent performance measurements
- Negative sign ensures lower variance (higher stability) gives positive delta

**Algorithm Implementation:**

```python
def calculate_stability_delta(current_variance: float, previous_variance: float) -> float:
    """
    Calculate stability delta based on performance variance.
    
    Args:
        current_variance: Current performance variance
        previous_variance: Previous performance variance
        
    Returns:
        Normalized stability delta [-1, 1]
    """
    epsilon = 1e-8
    
    # Calculate variance change (negative for improvement)
    variance_change = abs(current_variance - previous_variance)
    max_variance = max(current_variance, previous_variance, epsilon)
    
    # Stability improves when variance decreases
    if current_variance < previous_variance:
        stability_delta = variance_change / max_variance
    else:
        stability_delta = -variance_change / max_variance
    
    return max(-1.0, min(1.0, stability_delta))
```

#### Capability Delta (Δ_capability)

Capability measures the system's ability to handle complex tasks:

```
Δ_capability = Σᵢ wᵢ × (Cᵢ_current - Cᵢ_previous) / max(Cᵢ_current, Cᵢ_previous, ε)
```

Where:
- `Cᵢ` represents individual capability metrics
- `wᵢ` are capability-specific weights

### Adaptive Weight Learning

The weights `α₁, α₂, α₃, α₄` adapt based on system context and historical performance:

```
αᵢ(t+1) = αᵢ(t) + η × ∇αᵢ L(α)
```

Where:
- `η` is the learning rate
- `L(α)` is the loss function based on adaptation success
- `∇αᵢ L(α)` is the gradient with respect to weight `αᵢ`

**Algorithm Implementation:**

```python
def update_adaptive_weights(weights: AdaptiveWeights, 
                          adaptation_success: bool,
                          delta_components: Dict[str, float],
                          learning_rate: float = 0.01) -> AdaptiveWeights:
    """
    Update adaptive weights based on adaptation outcomes.
    
    Args:
        weights: Current adaptive weights
        adaptation_success: Whether the adaptation was successful
        delta_components: Individual delta components
        learning_rate: Learning rate for weight updates
        
    Returns:
        Updated adaptive weights
    """
    # Calculate gradients based on adaptation success
    success_factor = 1.0 if adaptation_success else -0.5
    
    # Update weights based on component contributions
    alpha_1_grad = success_factor * delta_components.get('performance', 0.0)
    alpha_2_grad = success_factor * delta_components.get('efficiency', 0.0)
    alpha_3_grad = success_factor * delta_components.get('stability', 0.0)
    alpha_4_grad = success_factor * delta_components.get('capability', 0.0)
    
    # Apply gradients
    new_alpha_1 = weights.alpha_1 + learning_rate * alpha_1_grad
    new_alpha_2 = weights.alpha_2 + learning_rate * alpha_2_grad
    new_alpha_3 = weights.alpha_3 + learning_rate * alpha_3_grad
    new_alpha_4 = weights.alpha_4 + learning_rate * alpha_4_grad
    
    # Normalize to ensure sum equals 1
    total = new_alpha_1 + new_alpha_2 + new_alpha_3 + new_alpha_4
    
    return AdaptiveWeights(
        alpha_1=new_alpha_1 / total,
        alpha_2=new_alpha_2 / total,
        alpha_3=new_alpha_3 / total,
        alpha_4=new_alpha_4 / total
    )
```

## Meta-Cognitive Engine Algorithms

### Self-Awareness Computation

The self-awareness module computes system understanding using introspective analysis:

```
SA(t) = f(S(t), G(t), H(t-k:t))
```

Where:
- `SA(t)` is the self-awareness score at time `t`
- `S(t)` is the current system state
- `G(t)` is the current goal state
- `H(t-k:t)` is the historical performance window

**Algorithm Implementation:**

```python
def compute_self_awareness(system_state: SystemState,
                         goals: List[Goal],
                         performance_history: List[PerformanceMetrics],
                         window_size: int = 10) -> float:
    """
    Compute self-awareness score based on system introspection.
    
    Args:
        system_state: Current system state
        goals: Active system goals
        performance_history: Recent performance history
        window_size: Size of historical window
        
    Returns:
        Self-awareness score [0, 1]
    """
    if not performance_history:
        return 0.0
    
    # Recent performance window
    recent_history = performance_history[-window_size:]
    
    # Goal alignment score
    goal_alignment = 0.0
    for goal in goals:
        target_metrics = goal.target_metrics
        current_metrics = system_state.performance_metrics
        
        alignment = 0.0
        for metric_name, target_value in target_metrics.items():
            current_value = getattr(current_metrics, metric_name, 0.0)
            metric_alignment = 1.0 - abs(current_value - target_value)
            alignment += max(0.0, metric_alignment)
            
        goal_alignment += alignment / len(target_metrics) if target_metrics else 0.0
    
    goal_alignment /= len(goals) if goals else 1.0
    
    # Performance trend awareness
    if len(recent_history) >= 2:
        performance_values = [m.accuracy for m in recent_history]
        trend_slope = np.polyfit(range(len(performance_values)), performance_values, 1)[0]
        trend_awareness = 0.5 + 0.5 * np.tanh(trend_slope * 10)  # Sigmoid normalization
    else:
        trend_awareness = 0.5
    
    # Variance awareness (understanding of stability)
    performance_variance = np.var([m.accuracy for m in recent_history])
    variance_awareness = 1.0 / (1.0 + performance_variance)
    
    # Combined self-awareness score
    self_awareness = (
        0.4 * goal_alignment +
        0.3 * trend_awareness +
        0.3 * variance_awareness
    )
    
    return max(0.0, min(1.0, self_awareness))
```

### Strategy Selection Algorithm

Strategy selection uses multi-criteria decision analysis with uncertainty handling:

```
S* = argmax_s Σᵢ wᵢ × Uᵢ(s) × C(s)
```

Where:
- `S*` is the optimal strategy
- `wᵢ` are criterion weights
- `Uᵢ(s)` is the utility of strategy `s` for criterion `i`
- `C(s)` is the confidence score for strategy `s`

**Algorithm Implementation:**

```python
def select_optimal_strategy(strategies: List[Strategy],
                          current_state: SystemState,
                          goals: List[Goal],
                          weights: Dict[str, float]) -> Optional[Strategy]:
    """
    Select optimal strategy using multi-criteria decision analysis.
    
    Args:
        strategies: Available strategies
        current_state: Current system state
        goals: Active goals
        weights: Criterion weights
        
    Returns:
        Selected strategy or None if no suitable strategy
    """
    if not strategies:
        return None
    
    strategy_scores = []
    
    for strategy in strategies:
        # Calculate utility for each criterion
        performance_utility = calculate_performance_utility(strategy, current_state, goals)
        resource_utility = calculate_resource_utility(strategy, current_state)
        risk_utility = calculate_risk_utility(strategy, current_state)
        confidence_utility = strategy.confidence_score
        
        # Weighted utility score
        total_utility = (
            weights.get('performance', 0.3) * performance_utility +
            weights.get('resource', 0.2) * resource_utility +
            weights.get('risk', 0.2) * risk_utility +
            weights.get('confidence', 0.3) * confidence_utility
        )
        
        # Apply confidence weighting
        final_score = total_utility * strategy.confidence_score
        
        strategy_scores.append((strategy, final_score))
    
    # Select strategy with highest score
    strategy_scores.sort(key=lambda x: x[1], reverse=True)
    
    # Apply minimum threshold
    min_threshold = 0.5
    if strategy_scores[0][1] >= min_threshold:
        return strategy_scores[0][0]
    else:
        return None

def calculate_performance_utility(strategy: Strategy, 
                                state: SystemState, 
                                goals: List[Goal]) -> float:
    """Calculate expected performance utility of a strategy."""
    expected_outcomes = strategy.expected_outcomes
    
    utility = 0.0
    for goal in goals:
        goal_utility = 0.0
        for metric_name, target_value in goal.target_metrics.items():
            expected_value = expected_outcomes.get(metric_name, 0.5)
            metric_utility = 1.0 - abs(expected_value - target_value)
            goal_utility += max(0.0, metric_utility)
        
        goal_utility /= len(goal.target_metrics) if goal.target_metrics else 1.0
        utility += goal_utility * goal.priority
    
    return utility / sum(g.priority for g in goals) if goals else 0.0

def calculate_resource_utility(strategy: Strategy, state: SystemState) -> float:
    """Calculate resource efficiency utility of a strategy."""
    resource_requirements = strategy.resource_requirements
    available_resources = state.available_resources
    
    utility = 0.0
    total_weight = 0.0
    
    for resource_type, required_amount in resource_requirements.items():
        available_amount = available_resources.get(resource_type, 1.0)
        
        if required_amount <= available_amount:
            # Resource efficiency (lower requirement is better)
            efficiency = 1.0 - (required_amount / available_amount)
            utility += efficiency
        else:
            # Insufficient resources
            utility -= 0.5
        
        total_weight += 1.0
    
    return utility / total_weight if total_weight > 0 else 0.0

def calculate_risk_utility(strategy: Strategy, state: SystemState) -> float:
    """Calculate risk utility of a strategy."""
    # Risk assessment based on strategy characteristics
    base_risk = 0.1  # Base risk for any strategy
    
    # Risk factors
    complexity_risk = len(strategy.actions) * 0.05
    resource_risk = sum(strategy.resource_requirements.values()) * 0.1
    confidence_risk = (1.0 - strategy.confidence_score) * 0.3
    
    total_risk = base_risk + complexity_risk + resource_risk + confidence_risk
    
    # Convert risk to utility (lower risk = higher utility)
    risk_utility = 1.0 / (1.0 + total_risk)
    
    return risk_utility
```

### Adaptation Decision Algorithm

The adaptation decision process uses a threshold-based approach with hysteresis:

```
Adapt = (|Δ_total| > θ_adapt) ∧ (Confidence > θ_conf) ∧ Safety_Check
```

Where:
- `θ_adapt` is the adaptation threshold
- `θ_conf` is the confidence threshold
- `Safety_Check` validates adaptation safety

**Algorithm Implementation:**

```python
def should_adapt(delta_result: DeltaResult,
                confidence_score: float,
                safety_validation: bool,
                adaptation_threshold: float = 0.05,
                confidence_threshold: float = 0.7,
                hysteresis_factor: float = 0.8) -> bool:
    """
    Determine whether to trigger adaptation based on multiple criteria.
    
    Args:
        delta_result: Calculated delta result
        confidence_score: Confidence in adaptation decision
        safety_validation: Safety validation result
        adaptation_threshold: Minimum delta for adaptation
        confidence_threshold: Minimum confidence for adaptation
        hysteresis_factor: Hysteresis factor to prevent oscillation
        
    Returns:
        Boolean indicating whether to adapt
    """
    # Primary adaptation condition
    delta_condition = abs(delta_result.total_delta) > adaptation_threshold
    
    # Confidence condition
    confidence_condition = confidence_score > confidence_threshold
    
    # Safety condition
    safety_condition = safety_validation
    
    # Hysteresis to prevent rapid oscillations
    if hasattr(delta_result, 'previous_adaptation_time'):
        time_since_last = time.time() - delta_result.previous_adaptation_time
        min_interval = 30.0  # Minimum 30 seconds between adaptations
        
        if time_since_last < min_interval:
            # Apply stricter threshold for recent adaptations
            delta_condition = abs(delta_result.total_delta) > (adaptation_threshold / hysteresis_factor)
    
    return delta_condition and confidence_condition and safety_condition
```

## Memory Consolidation Algorithms

### Vector Memory Similarity Search

Vector memory uses cosine similarity for efficient retrieval:

```
similarity(v₁, v₂) = (v₁ · v₂) / (||v₁|| × ||v₂||)
```

**Algorithm Implementation:**

```python
def cosine_similarity(vector1: np.ndarray, vector2: np.ndarray) -> float:
    """
    Calculate cosine similarity between two vectors.
    
    Args:
        vector1: First vector
        vector2: Second vector
        
    Returns:
        Cosine similarity [-1, 1]
    """
    # Handle zero vectors
    norm1 = np.linalg.norm(vector1)
    norm2 = np.linalg.norm(vector2)
    
    if norm1 == 0 or norm2 == 0:
        return 0.0
    
    # Calculate cosine similarity
    dot_product = np.dot(vector1, vector2)
    similarity = dot_product / (norm1 * norm2)
    
    return similarity

def find_similar_memories(query_vector: np.ndarray,
                         memory_vectors: List[np.ndarray],
                         memory_items: List[MemoryItem],
                         top_k: int = 5,
                         similarity_threshold: float = 0.7) -> List[MemoryItem]:
    """
    Find similar memories using vector similarity search.
    
    Args:
        query_vector: Query vector for similarity search
        memory_vectors: List of memory vectors
        memory_items: Corresponding memory items
        top_k: Number of top results to return
        similarity_threshold: Minimum similarity threshold
        
    Returns:
        List of similar memory items
    """
    similarities = []
    
    for i, memory_vector in enumerate(memory_vectors):
        similarity = cosine_similarity(query_vector, memory_vector)
        
        if similarity >= similarity_threshold:
            similarities.append((similarity, memory_items[i]))
    
    # Sort by similarity (descending)
    similarities.sort(key=lambda x: x[0], reverse=True)
    
    # Return top-k results
    return [item for _, item in similarities[:top_k]]
```

### Memory Consolidation Process

Memory consolidation transfers important memories from working to long-term storage:

```
Importance(m) = α × Recency(m) + β × Frequency(m) + γ × Relevance(m)
```

Where:
- `Recency(m)` = exp(-λ × age(m))
- `Frequency(m)` = access_count(m) / total_accesses
- `Relevance(m)` = similarity to current goals

**Algorithm Implementation:**

```python
def calculate_memory_importance(memory_item: MemoryItem,
                              current_goals: List[Goal],
                              total_accesses: int,
                              recency_weight: float = 0.4,
                              frequency_weight: float = 0.3,
                              relevance_weight: float = 0.3,
                              decay_rate: float = 0.1) -> float:
    """
    Calculate importance score for memory consolidation.
    
    Args:
        memory_item: Memory item to evaluate
        current_goals: Current system goals
        total_accesses: Total memory accesses
        recency_weight: Weight for recency component
        frequency_weight: Weight for frequency component
        relevance_weight: Weight for relevance component
        decay_rate: Decay rate for recency calculation
        
    Returns:
        Importance score [0, 1]
    """
    current_time = time.time()
    
    # Recency component (exponential decay)
    age = current_time - memory_item.timestamp
    recency = math.exp(-decay_rate * age / 3600)  # Decay per hour
    
    # Frequency component
    frequency = memory_item.access_count / max(total_accesses, 1)
    
    # Relevance component (similarity to current goals)
    relevance = 0.0
    if current_goals and hasattr(memory_item, 'content_vector'):
        goal_vectors = [goal.get_vector_representation() for goal in current_goals]
        max_similarity = 0.0
        
        for goal_vector in goal_vectors:
            similarity = cosine_similarity(memory_item.content_vector, goal_vector)
            max_similarity = max(max_similarity, similarity)
        
        relevance = max_similarity
    
    # Weighted importance score
    importance = (
        recency_weight * recency +
        frequency_weight * frequency +
        relevance_weight * relevance
    )
    
    return max(0.0, min(1.0, importance))

def consolidate_memories(working_memories: List[MemoryItem],
                        long_term_storage: List[MemoryItem],
                        current_goals: List[Goal],
                        consolidation_threshold: float = 0.6,
                        max_working_size: int = 100) -> Tuple[List[MemoryItem], List[MemoryItem]]:
    """
    Consolidate memories from working to long-term storage.
    
    Args:
        working_memories: Current working memories
        long_term_storage: Long-term memory storage
        current_goals: Current system goals
        consolidation_threshold: Minimum importance for consolidation
        max_working_size: Maximum working memory size
        
    Returns:
        Tuple of (updated_working_memories, updated_long_term_storage)
    """
    if len(working_memories) <= max_working_size:
        return working_memories, long_term_storage
    
    total_accesses = sum(m.access_count for m in working_memories)
    
    # Calculate importance scores
    memory_importance = []
    for memory in working_memories:
        importance = calculate_memory_importance(
            memory, current_goals, total_accesses
        )
        memory_importance.append((importance, memory))
    
    # Sort by importance
    memory_importance.sort(key=lambda x: x[0], reverse=True)
    
    # Consolidate high-importance memories
    new_working = []
    new_long_term = list(long_term_storage)
    
    for importance, memory in memory_importance:
        if len(new_working) < max_working_size:
            new_working.append(memory)
        elif importance >= consolidation_threshold:
            # Move to long-term storage
            new_long_term.append(memory)
    
    return new_working, new_long_term
```

## Safety Validation Algorithms

### Constraint Violation Detection

Safety constraints are validated using formal verification techniques:

```
Violation = ∃c ∈ C : ¬c(S_proposed)
```

Where:
- `C` is the set of safety constraints
- `S_proposed` is the proposed system state
- `c(S)` evaluates constraint `c` on state `S`

**Algorithm Implementation:**

```python
def validate_safety_constraints(proposed_state: SystemState,
                              constraints: List[SafetyConstraint],
                              current_state: SystemState) -> ValidationResult:
    """
    Validate proposed state against safety constraints.
    
    Args:
        proposed_state: Proposed system state
        constraints: List of safety constraints
        current_state: Current system state
        
    Returns:
        Validation result with violations
    """
    violations = []
    
    for constraint in constraints:
        try:
            # Evaluate constraint
            is_satisfied = constraint.evaluate(proposed_state, current_state)
            
            if not is_satisfied:
                violation = ConstraintViolation(
                    constraint_id=constraint.constraint_id,
                    severity=constraint.severity,
                    description=constraint.description,
                    proposed_value=constraint.get_proposed_value(proposed_state),
                    allowed_range=constraint.allowed_range,
                    timestamp=time.time()
                )
                violations.append(violation)
                
        except Exception as e:
            # Constraint evaluation error
            violation = ConstraintViolation(
                constraint_id=constraint.constraint_id,
                severity="critical",
                description=f"Constraint evaluation error: {str(e)}",
                proposed_value=None,
                allowed_range=None,
                timestamp=time.time()
            )
            violations.append(violation)
    
    # Determine overall validation result
    is_valid = len(violations) == 0
    
    # Check for critical violations
    critical_violations = [v for v in violations if v.severity == "critical"]
    has_critical = len(critical_violations) > 0
    
    return ValidationResult(
        is_valid=is_valid,
        violations=violations,
        has_critical_violations=has_critical,
        validation_timestamp=time.time()
    )
```

### Risk Assessment Algorithm

Risk assessment uses probabilistic modeling:

```
Risk = P(failure) × Impact(failure)
```

Where:
- `P(failure)` is the probability of failure
- `Impact(failure)` is the expected impact of failure

**Algorithm Implementation:**

```python
def assess_adaptation_risk(proposed_adaptation: Dict,
                         current_state: SystemState,
                         historical_data: List[AdaptationResult],
                         risk_factors: Dict[str, float]) -> RiskAssessment:
    """
    Assess risk of proposed adaptation.
    
    Args:
        proposed_adaptation: Proposed adaptation changes
        current_state: Current system state
        historical_data: Historical adaptation results
        risk_factors: Risk factor weights
        
    Returns:
        Risk assessment result
    """
    # Calculate failure probability based on historical data
    failure_probability = calculate_failure_probability(
        proposed_adaptation, historical_data
    )
    
    # Calculate potential impact
    impact_score = calculate_impact_score(
        proposed_adaptation, current_state, risk_factors
    )
    
    # Overall risk score
    risk_score = failure_probability * impact_score
    
    # Risk categorization
    if risk_score < 0.2:
        risk_level = "low"
    elif risk_score < 0.5:
        risk_level = "medium"
    elif risk_score < 0.8:
        risk_level = "high"
    else:
        risk_level = "critical"
    
    return RiskAssessment(
        risk_score=risk_score,
        risk_level=risk_level,
        failure_probability=failure_probability,
        impact_score=impact_score,
        risk_factors=risk_factors,
        timestamp=time.time()
    )

def calculate_failure_probability(adaptation: Dict, 
                                historical_data: List[AdaptationResult]) -> float:
    """Calculate failure probability based on historical data."""
    if not historical_data:
        return 0.5  # Default uncertainty
    
    # Find similar adaptations
    similar_adaptations = []
    for result in historical_data:
        similarity = calculate_adaptation_similarity(adaptation, result.adaptation)
        if similarity > 0.7:  # Similarity threshold
            similar_adaptations.append(result)
    
    if not similar_adaptations:
        return 0.3  # Conservative estimate for novel adaptations
    
    # Calculate failure rate
    failures = sum(1 for result in similar_adaptations if not result.success)
    failure_rate = failures / len(similar_adaptations)
    
    # Apply temporal weighting (recent failures are more relevant)
    current_time = time.time()
    weighted_failures = 0.0
    total_weight = 0.0
    
    for result in similar_adaptations:
        age = current_time - result.timestamp
        weight = math.exp(-age / 86400)  # Decay over days
        
        if not result.success:
            weighted_failures += weight
        total_weight += weight
    
    if total_weight > 0:
        weighted_failure_rate = weighted_failures / total_weight
        return min(0.9, max(0.1, weighted_failure_rate))
    else:
        return failure_rate

def calculate_impact_score(adaptation: Dict,
                         current_state: SystemState,
                         risk_factors: Dict[str, float]) -> float:
    """Calculate potential impact of adaptation failure."""
    impact_components = {}
    
    # Performance impact
    performance_change = adaptation.get('performance_change', 0.0)
    impact_components['performance'] = abs(performance_change)
    
    # Resource impact
    resource_changes = adaptation.get('resource_changes', {})
    max_resource_change = max(abs(change) for change in resource_changes.values()) if resource_changes else 0.0
    impact_components['resource'] = max_resource_change
    
    # System stability impact
    stability_risk = adaptation.get('stability_risk', 0.0)
    impact_components['stability'] = stability_risk
    
    # Business impact
    business_criticality = current_state.metadata.get('business_criticality', 0.5)
    impact_components['business'] = business_criticality
    
    # Weighted impact score
    total_impact = 0.0
    total_weight = 0.0
    
    for component, value in impact_components.items():
        weight = risk_factors.get(component, 1.0)
        total_impact += weight * value
        total_weight += weight
    
    return total_impact / total_weight if total_weight > 0 else 0.0
```

## MCP Orchestration Algorithms

### Agent Coordination Algorithm

Agent coordination uses consensus-based decision making:

```
Decision = argmax_d Σᵢ wᵢ × Vote_i(d)
```

Where:
- `wᵢ` is the weight of agent `i`
- `Vote_i(d)` is agent `i`'s vote for decision `d`

**Algorithm Implementation:**

```python
def coordinate_agent_decision(agents: List[Agent],
                            decision_options: List[Dict],
                            context: ContextVector,
                            voting_weights: Dict[str, float]) -> Dict:
    """
    Coordinate decision making across multiple agents.
    
    Args:
        agents: List of participating agents
        decision_options: Available decision options
        context: Shared context for decision making
        voting_weights: Weights for agent votes
        
    Returns:
        Selected decision option
    """
    if not decision_options:
        return {}
    
    # Collect votes from each agent
    agent_votes = {}
    
    for agent in agents:
        if agent.status == "active":
            vote = get_agent_vote(agent, decision_options, context)
            agent_votes[agent.agent_id] = vote
    
    # Calculate weighted scores for each option
    option_scores = {}
    
    for i, option in enumerate(decision_options):
        total_score = 0.0
        total_weight = 0.0
        
        for agent_id, vote in agent_votes.items():
            agent_weight = voting_weights.get(agent_id, 1.0)
            option_score = vote.get(i, 0.0)  # Vote for option i
            
            total_score += agent_weight * option_score
            total_weight += agent_weight
        
        if total_weight > 0:
            option_scores[i] = total_score / total_weight
        else:
            option_scores[i] = 0.0
    
    # Select option with highest score
    best_option_index = max(option_scores.keys(), key=lambda k: option_scores[k])
    
    # Apply minimum consensus threshold
    min_consensus = 0.6
    if option_scores[best_option_index] >= min_consensus:
        return decision_options[best_option_index]
    else:
        # No consensus reached, return default or trigger escalation
        return {"decision": "no_consensus", "scores": option_scores}

def get_agent_vote(agent: Agent, 
                  options: List[Dict], 
                  context: ContextVector) -> Dict[int, float]:
    """Get agent's vote for decision options."""
    votes = {}
    
    for i, option in enumerate(options):
        # Calculate agent's preference for this option
        capability_match = calculate_capability_match(agent, option)
        resource_availability = calculate_resource_availability(agent, option)
        context_relevance = calculate_context_relevance(agent, option, context)
        
        # Weighted vote score
        vote_score = (
            0.4 * capability_match +
            0.3 * resource_availability +
            0.3 * context_relevance
        )
        
        votes[i] = max(0.0, min(1.0, vote_score))
    
    return votes
```

### Context Sharing Algorithm

Context sharing uses vector embeddings for efficient information transfer:

```
Relevance(c, a) = similarity(embed(c), embed(a.context))
```

Where:
- `c` is the context to share
- `a` is the receiving agent
- `embed()` creates vector embeddings

**Algorithm Implementation:**

```python
def share_context_with_agents(context: ContextVector,
                            agents: List[Agent],
                            relevance_threshold: float = 0.5) -> Dict[str, float]:
    """
    Share context with relevant agents based on similarity.
    
    Args:
        context: Context to share
        agents: List of potential recipient agents
        relevance_threshold: Minimum relevance for sharing
        
    Returns:
        Dictionary of agent_id -> relevance_score for recipients
    """
    recipients = {}
    
    for agent in agents:
        if agent.status != "active":
            continue
        
        # Calculate context relevance for agent
        relevance = calculate_agent_context_relevance(context, agent)
        
        if relevance >= relevance_threshold:
            recipients[agent.agent_id] = relevance
    
    return recipients

def calculate_agent_context_relevance(context: ContextVector, 
                                    agent: Agent) -> float:
    """Calculate relevance of context for specific agent."""
    # Capability-based relevance
    capability_relevance = 0.0
    context_capabilities = context.metadata.get('required_capabilities', [])
    
    if context_capabilities:
        matching_capabilities = set(agent.capabilities) & set(context_capabilities)
        capability_relevance = len(matching_capabilities) / len(context_capabilities)
    
    # Task-based relevance
    task_relevance = 0.0
    context_task_type = context.metadata.get('task_type', '')
    
    if context_task_type in agent.capabilities:
        task_relevance = 1.0
    elif any(cap in context_task_type for cap in agent.capabilities):
        task_relevance = 0.7
    
    # Priority-based relevance
    context_priority = context.metadata.get('priority', 1)
    priority_relevance = min(1.0, context_priority / agent.priority)
    
    # Combined relevance score
    relevance = (
        0.4 * capability_relevance +
        0.4 * task_relevance +
        0.2 * priority_relevance
    )
    
    return max(0.0, min(1.0, relevance))
```

## Performance Prediction Models

### Linear Regression Predictor

Simple linear model for short-term performance prediction:

```
P(t+1) = α + β × P(t) + γ × Δ(t) + ε
```

Where:
- `P(t)` is performance at time `t`
- `Δ(t)` is the delta at time `t`
- `α, β, γ` are learned parameters
- `ε` is noise term

**Algorithm Implementation:**

```python
def train_linear_predictor(performance_history: List[float],
                         delta_history: List[float],
                         window_size: int = 50) -> Dict[str, float]:
    """
    Train linear regression model for performance prediction.
    
    Args:
        performance_history: Historical performance values
        delta_history: Historical delta values
        window_size: Training window size
        
    Returns:
        Model parameters
    """
    if len(performance_history) < window_size or len(delta_history) < window_size:
        return {'alpha': 0.0, 'beta': 1.0, 'gamma': 0.0}
    
    # Prepare training data
    X = []
    y = []
    
    for i in range(len(performance_history) - 1):
        if i < len(delta_history):
            features = [1.0, performance_history[i], delta_history[i]]  # [bias, P(t), Δ(t)]
            target = performance_history[i + 1]
            
            X.append(features)
            y.append(target)
    
    if len(X) < 3:  # Need minimum samples
        return {'alpha': 0.0, 'beta': 1.0, 'gamma': 0.0}
    
    # Solve normal equations: θ = (X^T X)^(-1) X^T y
    X = np.array(X)
    y = np.array(y)
    
    try:
        XtX = np.dot(X.T, X)
        Xty = np.dot(X.T, y)
        
        # Add regularization to prevent overfitting
        regularization = 1e-6 * np.eye(XtX.shape[0])
        theta = np.linalg.solve(XtX + regularization, Xty)
        
        return {
            'alpha': theta[0],
            'beta': theta[1], 
            'gamma': theta[2]
        }
        
    except np.linalg.LinAlgError:
        # Fallback to simple model
        return {'alpha': 0.0, 'beta': 1.0, 'gamma': 0.0}

def predict_performance(current_performance: float,
                       current_delta: float,
                       model_params: Dict[str, float]) -> float:
    """
    Predict next performance value using linear model.
    
    Args:
        current_performance: Current performance value
        current_delta: Current delta value
        model_params: Trained model parameters
        
    Returns:
        Predicted performance value
    """
    alpha = model_params.get('alpha', 0.0)
    beta = model_params.get('beta', 1.0)
    gamma = model_params.get('gamma', 0.0)
    
    prediction = alpha + beta * current_performance + gamma * current_delta
    
    # Clamp to valid range
    return max(0.0, min(1.0, prediction))
```

### Exponential Smoothing Predictor

Exponential smoothing for trend-aware prediction:

```
S(t) = α × P(t) + (1-α) × S(t-1)
T(t) = β × (S(t) - S(t-1)) + (1-β) × T(t-1)
P(t+1) = S(t) + T(t)
```

Where:
- `S(t)` is the smoothed value
- `T(t)` is the trend component
- `α, β` are smoothing parameters

**Algorithm Implementation:**

```python
def exponential_smoothing_predict(performance_history: List[float],
                                alpha: float = 0.3,
                                beta: float = 0.1) -> float:
    """
    Predict performance using exponential smoothing with trend.
    
    Args:
        performance_history: Historical performance values
        alpha: Smoothing parameter for level
        beta: Smoothing parameter for trend
        
    Returns:
        Predicted performance value
    """
    if len(performance_history) < 2:
        return performance_history[-1] if performance_history else 0.5
    
    # Initialize
    S = [performance_history[0]]  # Smoothed values
    T = [performance_history[1] - performance_history[0]]  # Trend values
    
    # Calculate smoothed values and trends
    for i in range(1, len(performance_history)):
        s_new = alpha * performance_history[i] + (1 - alpha) * (S[i-1] + T[i-1])
        t_new = beta * (s_new - S[i-1]) + (1 - beta) * T[i-1]
        
        S.append(s_new)
        T.append(t_new)
    
    # Predict next value
    prediction = S[-1] + T[-1]
    
    # Clamp to valid range
    return max(0.0, min(1.0, prediction))
```

## Convergence Analysis

### Lyapunov Stability Analysis

SAFLA's convergence can be analyzed using Lyapunov functions:

```
V(x) = ||x - x*||²
```

Where:
- `x` is the current system state
- `x*` is the optimal state
- `V(x)` is the Lyapunov function

For stability, we need:
```
dV/dt ≤ -c × V(x)
```

Where `c > 0` is a positive constant.

**Algorithm Implementation:**

```python
def analyze_convergence(state_history: List[SystemState],
                       target_state: SystemState,
                       window_size: int = 20) -> Dict[str, float]:
    """
    Analyze system convergence using Lyapunov-like analysis.
    
    Args:
        state_history: Historical system states
        target_state: Target/optimal state
        window_size: Analysis window size
        
    Returns:
        Convergence metrics
    """
    if len(state_history) < window_size:
        return {'convergence_rate': 0.0, 'stability_measure': 0.0, 'is_converging': False}
    
    # Calculate distance to target for recent states
    recent_states = state_history[-window_size:]
    distances = []
    
    for state in recent_states:
        distance = calculate_state_distance(state, target_state)
        distances.append(distance)
    
    # Analyze convergence trend
    if len(distances) >= 2:
        # Linear regression on log distances (exponential convergence)
        log_distances = [math.log(max(d, 1e-6)) for d in distances]
        time_points = list(range(len(log_distances)))
        
        # Calculate slope (convergence rate)
        n = len(time_points)
        sum_x = sum(time_points)
        sum_y = sum(log_distances)
        sum_xy = sum(x * y for x, y in zip(time_points, log_distances))
        sum_x2 = sum(x * x for x in time_points)
        
        if n * sum_x2 - sum_x * sum_x != 0:
            slope = (n * sum_xy - sum_x * sum_y) / (n * sum_x2 - sum_x * sum_x)
            convergence_rate = -slope  # Negative slope indicates convergence
        else:
            convergence_rate = 0.0
    else:
        convergence_rate = 0.0
    
    # Stability measure (variance of recent distances)
    stability_measure = 1.0 / (1.0 + np.var(distances[-10:]) if len(distances) >= 10 else 1.0)
    
    # Convergence detection
    is_converging = (
        convergence_rate > 0.01 and  # Positive convergence rate
        distances[-1] < distances[0] and  # Overall improvement
        stability_measure > 0.7  # Stable behavior
    )
    
    return {
        'convergence_rate': convergence_rate,
        'stability_measure': stability_measure,
        'is_converging': is_converging,
        'current_distance': distances[-1],
        'distance_trend': distances
    }

def calculate_state_distance(state1: SystemState, state2: SystemState) -> float:
    """Calculate distance between two system states."""
    # Performance metrics distance
    perf1 = state1.performance_metrics
    perf2 = state2.performance_metrics
    
    perf_distance = math.sqrt(
        (perf1.accuracy - perf2.accuracy) ** 2 +
        (perf1.efficiency - perf2.efficiency) ** 2 +
        (perf1.stability - perf2.stability) ** 2 +
        (perf1.capability_score - perf2.capability_score) ** 2
    )
    
    # Resource utilization distance
    resources1 = state1.resource_utilization
    resources2 = state2.resource_utilization
    
    resource_keys = set(resources1.keys()) | set(resources2.keys())
    resource_distance = 0.0
    
    for key in resource_keys:
        val1 = resources1.get(key, 0.0)
        val2 = resources2.get(key, 0.0)
        resource_distance += (val1 - val2) ** 2
    
    resource_distance = math.sqrt(resource_distance)
    
    # Combined distance
    total_distance = math.sqrt(perf_distance ** 2 + resource_distance ** 2)
    
    return total_distance
```

## Implementation Notes

### Numerical Stability

1. **Avoid Division by Zero**: Always use small epsilon values (1e-8) in denominators
2. **Normalize Inputs**: Ensure all metrics are in [0, 1] range before calculations
3. **Regularization**: Add regularization terms to prevent overfitting in learning algorithms
4. **Gradient Clipping**: Clip gradients to prevent exploding gradients in adaptive weight learning

### Performance Optimization

1. **Vectorization**: Use NumPy operations for batch calculations
2. **Caching**: Cache frequently computed values like similarity matrices
3. **Approximation**: Use approximate algorithms for large-scale similarity search
4. **Parallel Processing**: Parallelize independent calculations across multiple cores

### Error Handling

1. **Graceful Degradation**: Provide fallback values when algorithms fail
2. **Input Validation**: Validate all inputs before processing
3. **Exception Handling**: Catch and handle numerical errors appropriately
4. **Logging**: Log algorithm performance and errors for debugging

This completes the comprehensive algorithmic documentation for SAFLA, providing the mathematical foundations and implementation details necessary for understanding and extending the system.