Destination-Labeled Self-Looping Systems with Dwell: Intrinsic Characterization, Realization Cost, and Recognition

We study a finite-state symbolic controller for systems in which the admissible visible transitions are fixed in advance and each visible state carries a minimum dwell requirement. The resulting model, which we call a destination-labeled self-looping system with dwell (DLSL system), records the visible graph together with local decision maps; dwell memory appears only after phase expansion. The main structural issue is that, once dwell is imposed, the current visible state no longer determines whether a departure is allowed. This leads to the converse problem: which deterministic transducers arise as phase-expanded realizations of DLSL systems over a fixed visible graph? We show that the answer is exactly the class of fiber-linear graph-respecting transducers. Under natural reachability and realizable-departure assumptions, equivalent accessible realizations over the same visible graph are isomorphic; in particular, the visible transduction determines the dwell vector and the local decision maps. We also prove that any graph-preserving deterministic realization enforcing dwell values $(d_i)$ requires exactly $\sum_i d_i$ control states. Finally, we give an $O(|Q||\Omega|)$ recognition and reconstruction procedure, and extend the analysis to an edge-entry variant in which transitions may enter interior phases of successor fibers.