Resonance-Guided Search: A Novel Heuristic Framework

Resonance-Guided Search (RGS) is a mathematically rigorous framework for pathfinding and optimization that incorporates adaptability considerations from conserved systems. By modulating standard distance metrics with an adaptability function, RGS guides search algorithms toward solutions that balance efficiency with robustness.

🌟 Features

• Adaptability Metric: Compute and visualize the adaptability landscape for various orbital orders and depth parameters
• Resonance-Guided Metric (RGM): Modulate standard distance metrics to guide search toward adaptable states
• Pathfinding Algorithms: Compare standard A* search with RGM-guided A* search in grid-based environments
• Visualization Tools: Generate heatmaps, profiles, and pathfinding comparisons
• Mathematical Proofs: Includes rigorous proofs for key properties of the framework
• Comprehensive Testing: Verify all theoretical claims through unit tests
• Interactive Demo: Explore the framework's behavior through an interactive Jupyter notebook

🔧 Installation

```bash

Clone the repository

git clone https://github.com/yourusername/resonance-guided-search.git cd resonance-guided-search

Create a virtual environment (optional but recommended)

python -m venv venv source venv/bin/activate # On Windows: venv\Scripts\activate

Install dependencies

pip install -r requirements.txt ```

📊 Usage

Core Functionality

```python from src.rgscore import adaptability, rgmdistance

Calculate adaptability for a specific configuration

x = 0.5 dres = 10.0 Nord = list(range(1, 13)) # Orbital orders {1, 2, ..., 12} a = adaptability(x, dres, Nord) print(f"Adaptability at x={x}: {a}")

Calculate RGM distance between two points

x1, x2 = 0.3, 0.7 dbase = abs(x1 - x2) # Base distance (e.g., Euclidean) drgm = rgmdistance(x1, x2, dres, Nord, dbase, w=2.0) print(f"Base distance: {dbase}, RGM distance: {drgm}") ```

Visualization

```python from src.rgsviz import plotadaptabilitylandscape, plotadaptability_profile

Plot adaptability landscape

fig, ax = plotadaptabilitylandscape( xrange=(0, 1), dresrange=(1, 100), Nord=list(range(1, 13)), logscaled_res=True )

Plot adaptability profiles for different d_res values

fig, ax = plotadaptabilityprofile( xrange=(0, 1), dresvalues=[1.0, 5.0, 10.0, 50.0], Nord=list(range(1, 13)) ) ```

Pathfinding

```python from src.rgspathfinding import standardastar, rgma_star

Define grid parameters

grid_size = (20, 30) start = (2, 2) goal = (17, 27)

Define mapping from grid positions to x values

def gridtox_map(row, col): return ((row * 0.1 + col * 0.05) * 2.0) % 2.0

Run standard A* search

standardpath = standardastar(gridsize, start, goal)

Run RGM-guided A* search

rgmpath = rgmastar( gridsize=gridsize, start=start, goal=goal, gridtoxmap=gridtoxmap, dres=20.0, N_ord=list(range(1, 13)), w=2.0 ) ```

Running Tests

```bash

Run all tests

python -m unittest discover tests

Run specific test file

python -m tests.testmathematicalproperties ```

Compiling the LaTeX Report

```bash

Compile the LaTeX report

bash compile_latex.sh ```

🧮 Mathematical Framework

The RGS framework is built upon two key mathematical constructs:

Adaptability Metric

The adaptability metric $A(x, d_{res})$ is defined as the average of coupling functions across a set of orbital orders:

$$A(x, d{res}) = \frac{1}{|N{ord}|} \sum{n \in N{ord}} hn(x, d{res})$$

where the coupling function $hn(x, d{res})$ for orbital order $n$ is:

$$hn(x, d{res}) = \left|\sin(n\cdot\theta(x))\right|^{d{res}/n} \cdot \left|\cos(n\cdot\phi(x, d{res}))\right|^{1/n}$$

with $\theta(x) = 2\pi(x - x0)$ and $\phi(x, d{res}) = d{res}\pi(x - x0)$.

Resonance-Guided Metric

The Resonance-Guided Metric (RGM) modifies a base distance metric by incorporating adaptability values:

$$d{rgm}(x1, x2) = d{base}(x1, x2) \cdot f{mod}\left(A(x1, d{res}), A(x2, d_{res})\right)$$

where the modulating function is:

$$f{mod}(A1, A2) = \frac{1}{1 + w \cdot \frac{A1 + A_2}{2} + \epsilon}$$

📈 Examples

Adaptability Landscapes

Pathfinding Comparison

📚 Documentation

The codebase is structured into three main modules:

• rgs_core.py: Core mathematical functions for computing adaptability and RGM
• rgs_viz.py: Visualization functions for adaptability landscapes and profiles
• rgs_pathfinding.py: Implementation of standard and RGM-guided A* search

For detailed documentation, see the full report.

📋 Project Structure

resonance-guided-search/ ├── docs/ │ ├── pdf/ │ │ └── report.pdf │ └── tex/ │ └── report.tex ├── figures/ │ ├── A_landscape_Baseline.png │ ├── A_landscape_Sparse.png │ ├── A_profiles_Baseline.png │ └── pathfinding_comparison.png ├── src/ │ ├── __init__.py │ ├── rgs_core.py │ ├── rgs_pathfinding.py │ └── rgs_viz.py ├── tests/ │ ├── __init__.py │ ├── test_adaptability_landscape.py │ ├── test_mathematical_properties.py │ └── test_rgm_pathfinding.py ├── notebooks/ │ └── rgs_interactive_demo.ipynb ├── compile_latex.sh ├── requirements.txt └── README.md

📝 Citation

If you use this framework in your research, please cite:

bibtex @article{resonance_guided_search, title={Resonance-Guided Search: A Novel Heuristic Framework Based on Adaptability in Conserved Systems}, author={Your Name}, journal={arXiv preprint}, year={2023} }

🔗 Related Work

• A* Search Algorithm
• Heuristic Search
• Fractal Geometry
• Conservation Laws in Physics

📄 License

This project is licensed under the MIT License - see the LICENSE file for details.

🤝 Contributing

Contributions are welcome! Please feel free to submit a Pull Request.