Destination-Labeled Self-Looping Systems: Dwell, Cost, Recognition

Reda Belaiche submitted a paper on 29 June 2026 that formalizes destination-labeled self-looping systems with dwell, abbreviated DLSL systems, a finite-state symbolic controller model where admissible visible transitions are fixed and each visible state carries a minimum dwell requirement. The paper characterizes which deterministic transducers arise as phase-expanded realizations, proves an exact control-state cost equal to the sum of the dwell values, and supplies an O(|Q||Ω|) recognition and reconstruction procedure.

What is a DLSL system?

A destination-labeled self-looping (DLSL) system is a finite-state symbolic controller in which the visible graph of admissible transitions is fixed and each visible state has a minimum dwell requirement; dwell memory appears only after phase expansion. The model records the visible graph together with local decision maps, and the dwell constraint means that the current visible state alone no longer determines whether a departure is allowed.

How are realizations characterized?

The deterministic transducers that appear as phase-expanded realizations of DLSL systems over a fixed visible graph are exactly the class of fiber-linear graph-respecting transducers. Under natural reachability and realizable-departure assumptions, the paper proves that equivalent accessible realizations over the same visible graph are isomorphic; in particular, the visible transduction determines the dwell vector and the local decision maps.

The characterization pins down the structural relationship between visible graphs, phase expansion, and the class of transducers that encode the controller's behavior. That connection isolates which deterministic devices can be produced by enforcing local dwell constraints on a fixed visible topology.

How much does a realization cost and how is it recognized?

Any graph-preserving deterministic realization enforcing dwell values (d_i) requires exactly the sum_i d_i control states, and the paper provides an O(|Q||Ω|) recognition and reconstruction procedure. The cost theorem gives an exact count: a realization that preserves the visible graph and enforces dwell values (d_i) uses precisely ∑_i d_i control states.

The recognition routine runs in time proportional to |Q||Ω|, where Q and Ω are the objects appearing in the paper's procedure, and the author also extends the analysis to an edge-entry variant in which transitions may enter interior phases of successor fibers. That extension adapts the phase-expansion viewpoint to allow transitions that do not always land at fiber entry points.

Why it matters

The work fixes two engineering questions at once: it tells which deterministic transducers can come from dwell-constrained controllers on a fixed visible graph, and it gives an exact state-count penalty for enforcing dwell. That removes ambiguity for designers who need predictable resource budgets: the realization cost is not an upper bound but an exact sum of the dwell vector. The O(|Q||Ω|) recognition and reconstruction procedure makes the theoretical characterization algorithmically usable rather than merely existential.

What to watch

Look for follow-up work applying the O(|Q||Ω|) procedure to concrete controller synthesis or verification tasks and for further development of the edge-entry variant to cover additional practical transition patterns. The submission also signals possible comparative work that tests the fiber-linear graph-respecting characterization against other controller models.

References: Reda Belaiche, "Destination-Labeled Self-Looping Systems with Dwell: Intrinsic Characterization, Realization Cost, and Recognition," arXiv:2607.00044, submitted 29 Jun 2026.