Automated Approach for Solving Infinite-state Polynomial Reachability Games

Researchers have developed an automated algorithm for solving infinite-state polynomial reachability games, a class of two-player strategic games with applications in AI and reactive synthesis. The approach introduces ranking certificates as a formal proof mechanism and demonstrates the ability to solve previously intractable problems, including computing optimal strategies for the classical Cinderella-Stepmother game.

This paper addresses a fundamental problem in game theory and formal verification: determining winning strategies in infinite-state games where players have conflicting objectives. The work bridges theoretical computer science with practical automation, offering a semi-complete algorithm that runs in sub-exponential time while providing formal correctness guarantees through ranking certificates.

Reachability games have long served as models for critical AI and synthesis problems, but their infinite-state variants have resisted systematic solutions. The introduction of ranking certificates provides a sound and complete proof methodology, enabling verification that solutions are genuinely optimal. By focusing on polynomial constraints over real variables, the researchers create a tractable subset of infinite-state games amenable to automation.

The breakthrough with the Cinderella-Stepmother game—a classic game theory problem—demonstrates the method's practical power. Previous attempts lacked the ability to compute optimal strategies for arbitrary precision parameters, limiting their applicability to real-world scenarios requiring guaranteed performance bounds.

For the broader AI research community, this work strengthens the theoretical foundations of automated synthesis and verification. The methodology could enhance development of reactive systems in robotics, autonomous systems, and formal protocol verification where guaranteed winning strategies are essential. The sub-exponential complexity and demonstrated scalability on challenging benchmarks suggest the approach may handle increasingly complex real-world scenarios. Future directions likely include extending the method to stochastic variants and multi-player settings, which would further expand its practical reach in autonomous systems and game-theoretic applications.

• →Ranking certificates provide a sound and complete proof system for verifying winning strategies in infinite-state reachability games
• →The proposed algorithm is fully automated, semi-complete, and operates in sub-exponential time for polynomial game constraints
• →First successful computation of optimal strategies for the Cinderella-Stepmother game at arbitrary precision levels
• →Method has direct applications in AI reactive synthesis, formal verification, and autonomous system development
• →Algorithm demonstrates practical scalability on benchmarks previously unsolvable by existing methods