The Algorithmic Regulator

Abstract

The regulator theorem states that, under certain conditions, any optimal controller must embody a model of the system it regulates, grounding the idea that controllers embed, explicitly or implicitly, internal models of the controlled. This principle underpins neuroscience and predictive brain theories like the Free-Energy Principle or Kolmogorov/Algorithmic Agent theory. However, the theorem is only proven in limited settings. Here, we treat the deterministic, closed, coupled world-regulator system $(W,R)$ as a single self-delimiting program $p$ via a constant-size wrapper that produces the world output string~$x$ fed to the regulator. We analyze regulation from the viewpoint of the algorithmic complexity of the output, $K(x)$. We define $R$ to be a \emph{good algorithmic regulator} if it \emph{reduces} the algorithmic complexity of the readout relative to a null (unregulated) baseline $\varnothing$, i.e., \[ \Delta = K\big(O_{W,\varnothing}\big) - K\big(O_{W,R}\big) > 0. \] We then prove that the larger $\Delta$ is, the more world-regulator pairs with high mutual algorithmic information are favored. More precisely, a complexity gap $\Delta > 0$ yields \[ \Pr\big((W,R)\mid x\big) \le C\,2^{\,M(W{:}R)}\,2^{-\Delta}, \] making low $M(W{:}R)$ exponentially unlikely as $\Delta$ grows. This is an AIT version of the idea that ``the regulator contains a model of the world.'' The framework is distribution-free, applies to individual sequences, and complements the Internal Model Principle. Beyond this necessity claim, the same coding-theorem calculus singles out a \emph{canonical scalar objective} and implicates a \emph{planner}. On the realized episode, a regulator behaves \emph{as if} it minimized the conditional description length of the readout.