从算法复杂度角度证明调节器必须包含对世界的模型。
The Algorithmic Regulator
- 将世界与调节器视为自定界程序，用算法复杂度衡量调节效果。
- 调节器降低输出复杂度的幅度Δ越大，系统与调节器间互信息越高。
- 无需分布假设，适用于单个序列，揭示调节器的最优行为像在最小化描述长度。
调节器定理指出，在特定条件下，任何最优控制器都必须包含被调控系统的模型，这支撑了神经科学中预测性大脑理论（如自由能原理或科尔莫戈罗夫/算法代理理论）。然而该定理仅在有限场景下成立。本文将确定性、封闭、耦合的世界-调节器系统 $(W,R)$ 视为一个自定界程序 $p$，通过常数大小的包装器生成世界输出字符串 $x$ 并输入调节器。从输出的算法复杂度 $K(x)$ 视角分析调节问题。定义 $R$ 为“良好算法调节器”若其相对于无调节基线 $ ull$ 的读出复杂度降低，即 $ Δ = K(O_{W, ull}) - K(O_{W,R}) > 0 $。我们证明：Δ 越大，具有高互算法信息 $M(W{:}R)$ 的世界-调节器对越受青睐。具体地，复杂度差 $Δ>0$ 导致 $ \Pr((W,R)\mid x) \le C\,2^{M(W{:}R)}\,2^{-Δ} $，使低 $M(W{:}R)$ 的情况随 Δ 增大呈指数稀少。这是“调节器包含世界模型”的算法信息论版本。该框架无分布假设，适用于个体序列，补充了内部模型原理。此外，同一编码定理推导出一个“规范标量目标”，并暗示存在一个“规划者”。在实际实现的轨迹上，调节器行为等价于最小化读出的条件描述长度。
原文摘要 · Abstract (English)
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.
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