The Algorithmic Regulator

Area of Science:

• Theoretical Neuroscience
• Algorithmic Information Theory
• Cybernetics

Background:

• The regulator theorem posits that optimal controllers inherently model their systems, a concept relevant to predictive brain theories.
• Existing proofs of the regulator theorem are limited in scope.
• This work extends these ideas using algorithmic information theory.

Purpose of the Study:

• To analyze regulation as data compression using algorithmic complexity.
• To formally prove that effective regulation implies an internal model of the world.
• To identify a canonical objective and planner within this framework.

Main Methods:

• Modeling the world-regulator system as a single self-delimiting program.
• Analyzing regulation via algorithmic complexity (K(x)) and mutual information (M(W:R)).
• Defining a 'good algorithmic regulator' by its ability to reduce output complexity (Δ > 0).

Main Results:

• A positive complexity gap (Δ > 0) favors world-regulator pairs with high mutual information.
• Proved that Pr((W,R)|x) ≤ C 2^M(W:R) * 2^-Δ, making low mutual information exponentially unlikely as Δ increases.
• Demonstrated that regulators act as if minimizing conditional description length.

Conclusions:

• Confirms the necessity of internal models for regulation within an algorithmic information framework.
• The approach is distribution-free and applicable to individual sequences.
• Identifies a scalar objective and planner, complementing the Internal Model Principle.