Automated Approach for Solving Infinite-state Polynomial Reachability Games

2026-05-11

AI Summary

This work addresses the decidability and automatic synthesis of winning strategies for the REACH player in infinite-state polynomial reachability games. Focusing on turn-based games defined over real-valued variable assignments, the paper proposes a fully automated algorithm based on ranking certificates—a novel application of such certificates to reachability games, serving as a sound and complete formal proof mechanism. By integrating symbolic representations of polynomial constraints with techniques from real algebraic geometry, the approach yields a semi-complete solver with sub-exponential time complexity. This method enables, for the first time, the automatic solution of Cinderella-Stepmother games at arbitrary precision and successfully handles several challenging instances beyond the capabilities of existing tools, synthesizing optimal winning strategies.

Abstract

Reachability games are two-player games played on a graph, where the objective of $\texttt{REACH}$ player is to reach the target set whereas the objective of $\texttt{SAFE}$ player is to stay away from the target set. Reachability games have important applications in artificial intelligence and reactive synthesis, and many of these applications give rise to infinite-state reachability games. In this paper, we study turn-based reachability games on infinite-state graphs defined over valuations of a finite set of real variables. We consider the problem of determining the existence of and computing a winning strategy for $\texttt{REACH}$ player. Our contributions are twofold. First, we propose ranking certificates for reachability games, a sound and complete proof rule for proving that $\texttt{REACH}$ player has a winning strategy from the specified initial state. Second, we consider polynomial reachability games, where transitions and objectives are described by polynomial constraints over real variables, and propose a fully automated algorithm for computing a winning strategy for $\texttt{REACH}$ player together with a formal correctness witness in the form of a ranking certificate. The algorithm is sound, semi-complete, and runs in sub-exponential time. Our experiments demonstrate the ability of our method to solve challenging examples from the literature that were out of the reach of existing methods. Specifically, for the classical Cinderella-Stepmother game, we are able to compute an optimal winning strategy for an arbitrary precision parameter for the first time.

Problem

• reachability games
• infinite-state systems
• polynomial constraints
• winning strategy
• real variables

Innovation

• ranking certificates
• polynomial reachability games
• infinite-state games
• automated strategy synthesis
• formal verification