Exact Decomposition of Value Functions for Two-Player Games in Hamilton-Jacobi Reachability

What the math gives to ML

The paper provides an exact value-function decomposition for finite-horizon, continuous-time two-player Hamilton-Jacobi reachability problems involving multiple targets and constraints. The transferable mechanism is that decomposition is valid only when the outer task-combination map satisfies a critical coordinatewise monotonicity condition; the paper also gives a counterexample showing failure without it. A neural implementation can represent primitive reachability critics separately and combine them through a certified monotone aggregator, while using Hamilton-Jacobi residuals and adversarial rollouts to test whether the decomposition remains valid. The sharp prediction is that monotone compositions preserve the composite value up to approximation error, whereas introducing a negative aggregation derivative causes a measurable residual and reachable-set mismatch.