Reachability in 3-VAS

Reachability analysis is central to safety-critical control, robotics, and neural network verification, but classical computational methods, such as Hamilton--Jacobi reachability and set propagation, scale poorly with state dimension. Sampling-based methods have emerged as a promising alternative, often providing finite-sample guarantees that bound the probability-mass left uncovered. However, an explicit account of how the geometry of the initial set, the dynamics, and the sampling law affect the accuracy of the estimator is not fully available in the literature. We study this by casting sampling-based reachable-set recovery as geometric support estimation over a family of problems specified by an initial set, its dynamics, and a sampling law. First, we identify two regularity properties, positive reach of the initial set's complement and Lipschitz continuity of the dynamics, that together make recovery well-posed: a probability-mass coverage guarantee can be upgraded to accuracy $r$ in Hausdorff distance. Second, we bound the resulting sample complexity: recovery is achievable with $\tilde{\mathcal{O}}\big((e^{3LT}/r)^n\big)$ samples, exponential in both the state dimension and the time horizon. Third, we show that neither can be removed: an minimax lower bound of $Ω\big((e^{LT}/r)^n\big)$ holds for every estimator, so the exponential dependence on dimension and the degradation over the horizon are both intrinsic, not artifacts of a particular method. Experiments on nonlinear systems confirm that adversarial sampling improves constants but not the scaling.

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.

The Simple Temporal Problem (STP) is a core framework for quantitative temporal constraints. As STP data can be inconsistent, we study MAXSTP: compute a maximum-cardinality consistent subset of constraints. This extension is NP-hard, and we analyze its parameterized complexity under measures that capture practically relevant instance features: the number of variables $n$ (instance scale), the maximum coefficient magnitude $k$ (numeric range), and structural parameters of the constraint graph such as treewidth $tw$ (decomposability) and vertex cover size $vc$ (density). We show that MAXSTP is W[1]-hard parameterized by $n$, implying that $n$ and parameters that depend on $n$ (including $tw$ and $vc$) are insufficient for fixed-parameter tractability. For combined parameters, we give an $O^*(k^n)$-time algorithm, yielding single-exponential solvability for fixed $k$. While $k+tw$ remains W[1]-hard, MAXSTP is in XP via an $O^*((n\cdot k)^{tw})$ algorithm. Our results suggest that MAXSTP is often computationally harder than optimizing qualitative CSPs. We verify that many such problems (including RCC-8 and Allen's algebra) are FPT when parameterized by $n$ or $tw$. However, we also demonstrate that FPT algorithms for MAXSTP are indeed possible but with other parameters such as $k + vc$.