Definition: Finite incidence controller

Fix a finite command alphabet $\mathcal{L}$, a horizon $K\in \mathbb{N}$, and a finite controller state space ${\Omega }_K$. A state records the installed packet code $p$, incidence code $j$, required permission data, a retained history of length at most $K$, and an acceptance or rejection flag.

Each command has a total deterministic update

For legal commands the following rules hold:

1. Fixed-packet flow updates only the declared flow and record data; it keeps $p$ fixed.

2. Guarded incidence changes the incidence or event code only when its exact guard holds, and keeps $p$ fixed.

3. Source replacement selects $p^{\prime }=\Lambda (t)$ from a supplied token $t$, and is admitted only in its declared regular replacement region, not at an incidence command.

Illegal commands return a declared rejection output rather than being deleted. At the history horizon, further commands use a declared overflow or stopping rule and do not create unbounded hidden memory.