New Model Characterizes Finite-State Symbolic Controllers with Dwell

This paper delves into the theoretical underpinnings of finite-state symbolic controllers, specifically focusing on systems where visible transitions are predetermined and each visible state mandates a minimum "dwell" time. This model, termed a Destination-Labeled Self-Looping (DLSL) system with dwell, captures the visible graph and local decision logic, with dwell memory manifesting after phase expansion. A core challenge addressed is that the current visible state alone doesn't dictate whether a departure is permitted once dwell is imposed. The research identifies precisely which deterministic transducers can be realized as phase-expanded versions of DLSL systems over a fixed visible graph, characterizing them as fiber-linear graph-respecting transducers. The study also establishes that equivalent accessible realizations over the same visible graph are isomorphic, meaning the visible transduction uniquely determines the dwell vector and local decision maps. Furthermore, it proves that any graph-preserving deterministic realization enforcing specific dwell values requires a number of control states equal to the sum of those dwell values, and provides an efficient algorithm for recognizing and reconstructing such systems.