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Heuritmática Foundations Document

1. Introduction

1.1 Purpose

Heuritmática is a meta-decision framework designed for self-adaptive decision-making in complex and uncertain environments. It integrates Bayesian inference, reinforcement learning (RL), and quantum-inspired search (QIS) to optimize decision pathways dynamically. The framework is a key component in GAIA AIR, interfacing with Robotics Braining (RB) for adaptive robotics and IP4MLP for multi-layered predictive modeling.

Use Cases

  1. Robotic Navigation: Heuritmática optimizes pathfinding for autonomous UAVs by refining motion heuristics over time.
  2. Resource Allocation: Applied in dynamic mission planning, the framework learns optimal distributions of computational resources.
  3. Predictive Maintenance: Through heuristic-driven anomaly detection, it preemptively identifies system failures in aerospace applications.

1.2 Self-Adaptive Decision-Making

Heuritmática adapts by continuously refining heuristic parameters Θ, adjusting to changes in environmental dynamics, task objectives, and system configurations. Adaptation is achieved through:

  • Bayesian updates, which infer improved decision weights.
  • Temporal-Difference Learning (TD-Learning) for iterative improvement.
  • Quantum-inspired search, enhancing multi-path decision exploration.

2. Heuritmática as a Meta-Decision System

2.1 Decision Function Representation

The system models decision-making as an optimal function: [ f(X,T) ] where:

  • X is the current state.
  • T is the time horizon.

Heuritmática refines an approximate heuristic function: [ H(\Theta, X, T) \approx E[f(X,T) \mid \Theta] ] where H(Θ,X,T) estimates the expected optimality of an action in state X at time T.

Clarification of f(X,T): f(X,T) represents the idealized outcome metric the system aims to approximate. This could be:

  • A reward function in RL (e.g., completion time, energy efficiency).
  • A probabilistic success measure (e.g., UAV mission success probability).
  • A performance predictor (e.g., expected system uptime in predictive maintenance).

2.2 Uncertainty Mapping in Decision Selection

Heuritmática maps uncertainty into decision refinement by:

  • Using Bayesian priors to quantify initial uncertainty.
  • Adjusting weights via Bayesian posterior updates.
  • Employing RL exploration strategies to refine decision pathways.

3. Mathematical Formalization

3.1 Bayesian Learning for Heuristic Optimization

Heuritmática updates heuristic parameters Θ by computing: [ P(\Theta_t \mid T_t) = \frac{P(T_t \mid \Theta_t) P(\Theta_t)}{P(T_t)} ] where:

  • P(Θt | Tt) is the posterior belief of heuristic parameters given observations.
  • P(Tt | Θt) is the likelihood of outcomes under Θt.
  • P(Θt) is the prior belief.

Operationalization:

  • If Θ represents a linear weight set, priors can be Gaussian.
  • If Θ is a neural heuristic function, priors could be initialized as uniform distributions.
  • Non-parametric approaches (e.g., Bayesian Neural Networks) may be employed for richer adaptation.

3.2 Temporal-Difference Learning for Decision Improvement

Heuritmática refines H(Θ, X, T) using TD-learning: [ H(\Theta_{t+1}, X_t, T_t) = H(\Theta_t, X_t, T_t) + \alpha \left( R_t + \gamma H(\Theta_t, X_{t+1}, T_{t+1}) - H(\Theta_t, X_t, T_t) \right) ] where:

  • α is the learning rate.
  • γ is the discount factor.
  • R_t is the reward function (e.g., energy efficiency, safety margin).

Exploration Strategy (ε-Greedy with Exponential Decay)

The action policy follows: [ \pi(X_t) = \arg\max_a H(\Theta_t, X_t, T_t) ] with ε-greedy exploration: [ P(a = a_{random}) = \epsilon_t, \quad P(a = a_{best}) = 1 - \epsilon_t ] where ε_t decays exponentially: [ \epsilon_t = \epsilon_0 e^{-\lambda t} ]

4. Quantum-Inspired Search (QIS)

Important Note: This is a quantum-inspired classical method, leveraging probability amplitudes for enhanced search efficiency.

4.1 Heuristic Weighting via Amplitude Encoding

Each action a in state X is assigned: [ A(X, a) e^{i \Theta_a} ] where:

  • A(X, a) is a classical heuristic weight, initialized from heuristic evaluations.
  • Θ_a is an adjustment parameter, modulating search priority.

4.2 Multi-Step Heuristic Evolution

A multi-step heuristic sequence follows: [ H(\Theta, X, T) = \prod_{t=1}^{T} U(X_t, \Theta_t) H(\Theta_t, X_t, T_t) ] where U(X_t, Θ_t) is a transformation updating heuristic estimates based on prior actions.

4.3 Computational Advantage of QIS

QIS improves multi-path exploration by allowing:

  • Interference-like effects between heuristic evaluations.
  • Adaptive heuristic shifts in response to discovered optima.

5. Integration with GAIA AIR Systems

5.1 Robotics Braining (RB) Interface

  • Heuristics Θ are fed into RB's control policies, guiding robotic motion and autonomy.
  • RB feedback loops update heuristic priors, refining future predictions.

5.2 IP4MLP Meta-Learning Interface

  • IP4MLP leverages refined heuristics for adaptive learning.
  • Meta-learning adjusts Heuritmática’s priors, improving long-term learning efficiency.

Example Feedback Loops:

  • RB Example: If UAV trajectory deviates, Heuritmática updates Θ to prioritize corrective actions.
  • IP4MLP Example: If Heuritmática improves response efficiency, IP4MLP adapts its search priors accordingly.

6. Scalability and Modularity

6.1 Scalability

  • Supports distributed parallel learning for complex environments.
  • Scales across increasing state/action spaces via hierarchical heuristics.

6.2 Modularity

  • Independent optimization layers for Bayesian, RL, and QIS components.
  • Easily plugged into AI ecosystems like RB and IP4MLP.

7. Conclusion

Heuritmática provides an adaptive, multi-layered decision framework, integrating Bayesian inference, reinforcement learning, and quantum-inspired search. Through its interfaces with RB and IP4MLP, it enhances robotic intelligence and predictive modeling, supporting GAIA AIR’s vision of autonomous aerospace optimization.

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