Extending and Unifying the Fundamental Tasks of Hamilton-Jacobi Reachability Analysis

What the math gives to ML

The paper introduces generalized reach-avoid (GRA) tasks as a common primitive that strictly extends standard Hamilton-Jacobi reachability tasks and supports decomposition of composite objectives, including timed temporal-logic specifications. The transferable mechanism is a value-function algebra: solve local reach/avoid subproblems and compose their value functions rather than solving one monolithic long-horizon problem. In neural networks, this can become a bank of differentiable reachability critics for world models or policies, with temporal-logic composition implemented by explicit max/min operators and trained using Hamilton-Jacobi residuals. The strongest test is not merely return improvement, but whether composed values predict empirical success/failure boundaries and whether the residual decreases under refinement of the integration step.