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The Algorithmic Regulator

Giulio Ruffini

TL;DR

This work treats the deterministic, closed, coupled world-regulator system $(W,R) as a single self-delimiting program via a constant-size wrapper that produces the world output string $x$ fed to the regulator, and proves that the larger the complexity gap, the more world-regulator pairs with high mutual algorithmic information are favored.

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., \[ Δ= K\big(O_{W,\varnothing}\big) - K\big(O_{W,R}\big) > 0. \] We then prove that the larger $Δ$ is, the more world-regulator pairs with high mutual algorithmic information are favored. More precisely, a complexity gap $Δ> 0$ yields \[ \Pr\big((W,R)\mid x\big) \le C\,2^{\,M(W{:}R)}\,2^{-Δ}, \] making low $M(W{:}R)$ exponentially unlikely as $Δ$ 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.

The Algorithmic Regulator

TL;DR

This work treats the deterministic, closed, coupled world-regulator system x$ fed to the regulator, and proves that the larger the complexity gap, the more world-regulator pairs with high mutual algorithmic information are favored.

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 as a single self-delimiting program via a constant-size wrapper that produces the world output string~ fed to the regulator. We analyze regulation from the viewpoint of the algorithmic complexity of the output, . We define to be a \emph{good algorithmic regulator} if it \emph{reduces} the algorithmic complexity of the readout relative to a null (unregulated) baseline , i.e., We then prove that the larger is, the more world-regulator pairs with high mutual algorithmic information are favored. More precisely, a complexity gap yields making low exponentially unlikely as 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.

Paper Structure

This paper contains 42 sections, 10 theorems, 75 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Setting
  3. The Coupled World-Regulator System
  4. Probabilistic Regulator Theorems
  5. Posterior form, given the observed $x$
  6. The Good Algorithmic Regulator and Posterior with Contrast
  7. Clarifications.
  8. Inferring the Objective Function and Planner (As-If Agent)
  9. Discussion
  10. First regulator result: posterior form, given the observed $x$ (Th. \ref{['thm:first']}).
  11. Second regulator result: posterior with contrast (Th. \ref{['thm:ugar']}).
  12. Third regulator result: as-if Objective-function minimization (Th \ref{['thm:onoff-BF-uncond']}).
  13. Planner/policy representation (as‑if agent).
  14. The home thermostat.
  15. Conclusion
  16. ...and 27 more sections

Key Result

Lemma 3.1

With prefix prior $P(p)=2^{-|p|}$ and deterministic likelihood $P(x\mid p)=\mathbf{1}\{U(p)=x\}$, Consequently, by eq:coding-ugar,

Figures (2)

  • Figure 1: Regulation scenario. A) A good regulator $R$ interacts with the world $W$ so that the readout $x=O_W$ of the world’s output is clamped to a simple, highly compressible sequence (e.g., almost all zeros). B) When the regulator is turned off, the output is more complex.
  • Figure 2: To connect the IMP and the AIT formulation used here, we view the World $W$ as a box containing $E$ and $P$; the Regulator/Controller $R$ (or $C$) is a separate box. Arrows depict Forcing ($E\!\to\!P$), Ref ($E\!\to$ sum), the Error path (sum $\downarrow$ to the world boundary and $\rightarrow R$), and Control ($R\!\to\!P$).

Theorems & Definitions (20)

  • Definition 1.1: Algorithmic “internal model”
  • Lemma 3.1: Program posterior given $x$
  • proof
  • Theorem 3.1
  • Definition 3.1: Good Algorithmic Regulator, contrastive
  • Lemma 3.2: OFF run lower-bounds the world
  • proof
  • Theorem 3.2: Probabilistic regulator theorem
  • proof : Proof
  • Theorem 3.3: On/Off evidence equals unconditioned complexity gap
  • ...and 10 more