The Good Algorithmic Regulator Theorem: Model It, Transmit It, or Leave It in the World
Abstract
The idea that successful regulation requires an internal world model helps motivate both the generative models of active inference and the modeling engine of Kolmogorov Theory's algorithmic agent. Conant and Ashby's good regulator theorem and the internal model principle support this idea under specific assumptions. We ask when successful regulation implies that a regulator already contains information about the world. We compare finite output records of the same world with and without regulation, holding initial world conditions, disturbance inputs, and measurement rules fixed, and quantify the complexity reduction using prefix Kolmogorov complexity. Our starting point is precise information accounting, organized through an information-conserving logical construction. A computable reversible realization preserves the descriptions needed to recover the initial world and reconstruct its uncontrolled output. This realization may be auxiliary: physical determinism and reversibility are not general requirements of the resulting finite-record bounds. The Good Algorithmic Regulator Theorem (GART) bounds the reduction by the mutual algorithmic information initially shared by world and regulator plus a counterfactual residual, up to logarithmic coding overhead. The residual is the additional description needed to reconstruct the uncontrolled output given the regulated output and the regulator's initial description. A large reduction therefore requires substantial initial shared…
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