What Kinds of Symbols Are Suitable for Hybrid Discrete-Continuous Planning?

The paper proposes defining the usefulness of symbols with the solving algorithm at the center, introduces the concept of "solver-sufficient symbolization," and develops a counterexample-guided symbol invention algorithm for finite-horizon piecewise-affine systems. The method can be extended to optimization objectives that use neural networks as energy functions, inducing solver-sufficient symbols from continuous optimization landscapes. Simulations and real-robot task experiments show that the resulting symbols reduce solving cost while maintaining solution quality, and improve interpretability and manipulability.