The Necessity of Regulation Under Uncertainty

The Necessity of Regulation Under Uncertainty

Once a boundary exists, it must be maintained. The interior must remain distinct from the exterior despite perturbations, degradation, and environmental fluctuations. This maintenance problem has a specific structure.

Let the interior state be $\mathbf{s}^{\text{in}} \in \R^m$ and the exterior state be $\mathbf{s}^{\text{out}} \in \R^k$. The boundary mediates interactions through:

• Observations: $\mathbf{o}_t = g(\mathbf{s}^{\text{out}}_t, \mathbf{s}^{\text{in}}_t) + \bm{\epsilon}_t$
• Actions: $\mathbf{a}_t \in \mathcal{A}$ (boundary permeabilities, active transport, etc.)

The system’s persistence requires maintaining $\mathbf{s}^{\text{in}}$ within a viable region $\viable^{\text{in}}$ despite:

1. Incomplete observation of $\mathbf{s}^{\text{out}}$ (partial observability)
2. Stochastic perturbations (environmental and internal noise)
3. Degradation of the boundary itself (requiring continuous repair)
4. Finite resources (energy, raw materials)

This maintenance problem has a deep consequence: regulation requires modeling. Let $\mathcal{S}$ be a bounded system that must maintain $\mathbf{s}^{\text{in}} \in \viable^{\text{in}}$ under partial observability of $\mathbf{s}^{\text{out}}$. Any policy $\policy: \mathcal{O}^* \to \mathcal{A}$ that achieves viability with probability $p > p_{\text{random}}$ (where $p_{\text{random}}$ is the viability probability under random actions) implicitly computes a function $f: \mathcal{O}^* \to \mathcal{Z}$ where $\mathcal{Z}$ is a sufficient statistic for predicting future observations and viability-relevant outcomes.