The Algorithmic Regulator

A rigorous, distribution-free proof that any regulator making a system's output compressible must share algorithmic structure with the world it controls.

The old cybernetics slogan "every good regulator must be a model of the system" has always been intuitively compelling but technically slippery. The original 1970 proof by Conant and Ashby was criticized for vague definitions, and the modern replacement — the Internal Model Principle from control theory — only works cleanly for linear, time-invariant systems with well-specified signal classes. This paper asks: can we make the core idea rigorous without those restrictions?

The answer is yes, using Algorithmic Information Theory (AIT). The key move is to measure regulation not by tracking error or stability, but by compressibility. A regulator R is "good" if the world's output string x is shorter to describe (has lower Kolmogorov complexity K(x)) when R is active than when R is switched off. The gap Δ = K(output_off) − K(output_on) is the central quantity. Intuitively: a good thermostat turns chaotic temperature fluctuations into a boring, nearly-constant signal — and boring signals compress well.

The main theorem then says: the larger Δ is, the more the universal posterior over world-regulator pairs (W, R) is concentrated on pairs that share algorithmic structure. Formally, the posterior probability of any explaining pair is bounded above by C · 2^{M(W:R)} · 2^{−Δ}, where M(W:R) is the mutual algorithmic information between world and regulator — how many bits you save describing W once you know R. This means that if Δ is large and M(W:R) is small, the pair is exponentially unlikely as an explanation of the observed data. Low shared structure becomes vanishingly improbable as regulation improves. No linearity assumptions, no probability distributions over worlds, no specific signal classes — just the observed strings.

A secondary result gives the framework a clean objective-function interpretation. By the Coding Theorem, the log-ratio of the universal probability of the on-output versus the off-output equals Δ, up to O(1). So on any realized episode, the regulator behaves as if it were minimizing K(x) — the description length of the output. Combined with the internal-model necessity result and the fact that any deterministic regulator induces a policy mapping histories to actions, the paper recovers the full "algorithmic agent" triad: a world model (M(W:R) > 0), a scalar objective (minimize K(x)), and a planner (the induced policy). The source is careful to call this a representation result, not a mechanistic claim — R need not explicitly compute K, but sustained large Δ is precisely what maximizes universal evidence for the regulated behavior.

One important caveat the paper addresses honestly: a single low-complexity output alone does not certify M(W:R) > 0. A world could be designed to produce simple output whenever the regulator outputs zeros, regardless of whether they share structure (a "chance simplification" construction is given explicitly). The contrastive design — comparing on versus off runs — is what rescues identifiability, and even then the result is probabilistic rather than deterministic. The practical recipe is concrete: pick any lossless compressor (gzip, LZ77, BDM), run the system with and without the regulator, compute two code lengths, and use their difference as evidence of model content.