BEDS : Bayesian Emergent Dissipative Structures : A Formal Framework for Continuous Inference Under Energy Constraints

TL;DR

This paper introduces BEDS, a formal framework linking thermodynamics and continuous inference, revealing fundamental energy costs and classifying problem types beyond classical computability.

Contribution

It presents the BEDS framework that models inference under energy constraints, establishing thermodynamic costs and classifying problem types in this context.

Findings

Power required scales with dissipation rate and precision

Defines classes of problems: BEDS-attainable, maintainable, crystallizable

Proposes the G"odel-Landauer-Prigogine conjecture

Abstract

We introduce BEDS (Bayesian Emergent Dissipative Structures), a formal framework for analyzing inference systems that must maintain beliefs continuously under energy constraints. Unlike classical computational models that assume perfect memory and focus on one-shot computation, BEDS explicitly incorporates dissipation (information loss over time) as a fundamental constraint. We prove a central result linking energy, precision, and dissipation: maintaining a belief with precision $\tau$ against dissipation rate $\gamma$ requires power $P \geq \gamma k_{\rm B} T / 2$, with scaling $P \propto \gamma \cdot \tau$. This establishes a fundamental thermodynamic cost for continuous inference. We define three classes of problems -- BEDS-attainable, BEDS-maintainable, and BEDS-crystallizable -- and show these are distinct from classical decidability. We propose the G\"odel-Landauer-Prigogine conjecture, suggesting that closure pathologies across formal systems, computation, and thermodynamics share a common structure.