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Continuous relaxation is a bridge between discrete and continuous domains, offering a powerful way to analyze complex problems by leveraging continuous mathematics. It is widely used across fields like optimization, physics, and systems modeling.

Using Continuous Relaxation with Petri Nets

Petri nets provide a robust framework for modeling the dynamics of Tic-Tac-Toe, extending beyond traditional game-tree analysis. By incorporating continuous relaxation, this approach transforms discrete moves and outcomes into a probabilistic system, making it possible to analyze strategic nuances using differential equations.

Modeling Tic-Tac-Toe as a Petri Net

In this model:

• Places represent the grid's state, such as unclaimed cells (P00) and markers for Players X and O (e.g., X00 for Player X's mark in the top-left cell).
• Transitions represent possible moves (e.g., Player X marking X00) and win conditions (e.g., X00_X01_X02 for a row win).
• Tokens track the system's state, where a token in P00 indicates the top-left cell is unclaimed, while a token in X00 or O00 indicates it is marked by X or O.

Relaxing the System

Discrete moves are relaxed into continuous transitions, allowing the use of ODE solvers. For example:

• A transition's firing rate represents the probability of a specific move, creating a probabilistic view of gameplay.
• The win states (win_x, win_o) are modeled as accumulative probabilities derived from successful transition sequences.

Analyzing Strategies

By solving the ODEs, we can:

1. Predict Win Probabilities: The probabilities of win_x and win_o over time show the likelihood of each player achieving victory.

2. Evaluate Moves: The model ranks potential moves by their impact on the win probabilities, guiding optimal strategies.

3. Visualize Dynamics: Plots of win probabilities provide insights into how the game evolves based on current states and potential player decisions.

Conclusion

This Petri-net-based continuous relaxation approach brings a novel lens to Tic-Tac-Toe analysis. By transitioning from discrete moves to probabilistic trajectories, we uncover richer strategic insights and establish a framework applicable to more complex games and systems.

Review the code from previous post where we apply this approach using Juila and Petri.jl