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A Systems-Theoretical Formalization of Closed Systems

Niloofar Shadab, Tyler Cody, Alejandro Salado, Peter Beling

TL;DR

This work addresses the lack of formal foundations for closed-system concepts in systems engineering, particularly for AI-enabled intelligent systems. It develops two formal notions—functional closure and informational closure—within a systems-theoretic and information-theoretic framework, defining context, inner/outer environments, and minimal input-output sets to relax the classical closed-system definition. The paper derives engineering constraints and design parameters (notably a minimum mutual information bound and a tunable parameter $\delta$) to realize informational closure, and clarifies how functional closure fits as a static, special case of informational closure. By linking closure concepts to multi-level abstraction in SE, the authors provide a structured approach to bound environmental interactions and guide the engineering of intelligent systems with controlled open-closed characteristics, balancing predictability, cost, and complexity.

Abstract

There is a lack of formalism for some key foundational concepts in systems engineering. One of the most recently acknowledged deficits is the inadequacy of systems engineering practices for engineering intelligent systems. In our previous works, we proposed that closed systems precepts could be used to accomplish a required paradigm shift for the systems engineering of intelligent systems. However, to enable such a shift, formal foundations for closed systems precepts that expand the theory of systems engineering are needed. The concept of closure is a critical concept in the formalism underlying closed systems precepts. In this paper, we provide formal, systems- and information-theoretic definitions of closure to identify and distinguish different types of closed systems. Then, we assert a mathematical framework to evaluate the subjective formation of the boundaries and constraints of such systems. Finally, we argue that engineering an intelligent system can benefit from appropriate closed and open systems paradigms on multiple levels of abstraction of the system. In the main, this framework will provide the necessary fundamentals to aid in systems engineering of intelligent systems.

A Systems-Theoretical Formalization of Closed Systems

TL;DR

This work addresses the lack of formal foundations for closed-system concepts in systems engineering, particularly for AI-enabled intelligent systems. It develops two formal notions—functional closure and informational closure—within a systems-theoretic and information-theoretic framework, defining context, inner/outer environments, and minimal input-output sets to relax the classical closed-system definition. The paper derives engineering constraints and design parameters (notably a minimum mutual information bound and a tunable parameter ) to realize informational closure, and clarifies how functional closure fits as a static, special case of informational closure. By linking closure concepts to multi-level abstraction in SE, the authors provide a structured approach to bound environmental interactions and guide the engineering of intelligent systems with controlled open-closed characteristics, balancing predictability, cost, and complexity.

Abstract

There is a lack of formalism for some key foundational concepts in systems engineering. One of the most recently acknowledged deficits is the inadequacy of systems engineering practices for engineering intelligent systems. In our previous works, we proposed that closed systems precepts could be used to accomplish a required paradigm shift for the systems engineering of intelligent systems. However, to enable such a shift, formal foundations for closed systems precepts that expand the theory of systems engineering are needed. The concept of closure is a critical concept in the formalism underlying closed systems precepts. In this paper, we provide formal, systems- and information-theoretic definitions of closure to identify and distinguish different types of closed systems. Then, we assert a mathematical framework to evaluate the subjective formation of the boundaries and constraints of such systems. Finally, we argue that engineering an intelligent system can benefit from appropriate closed and open systems paradigms on multiple levels of abstraction of the system. In the main, this framework will provide the necessary fundamentals to aid in systems engineering of intelligent systems.
Paper Structure (11 sections, 3 theorems, 40 equations, 3 figures)

This paper contains 11 sections, 3 theorems, 40 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background in Information Theory
  3. Mutual Information
  4. Background on Closed vs Open Systems In Systems Theory
  5. Elaboration of Terminologies:
  6. Functional Closure
  7. Functional Closure Constraint For Engineering Purposes
  8. Information Closure
  9. Functional vs Informational Closure
  10. Informational Closure constraint for Engineering Purposes
  11. Conclusion

Key Result

Proposition 1

If $S^C$ is informationally closed, joint information of $S^C_{n+1}, E^O_{n}, S^C_{n}$, equals to joint information between $S^C$ at state $n$ and state $n+1$:

Figures (3)

  • Figure 1: Interactions and relations between systems in a closed system setting are depicted. $E^O$ is the environment outside of the Context system. The context system, $S^C$, includes the system of interest, ${S^0}$, and a portion of the overall environment, the inner environment, $E^I$
  • Figure 2: Top Diagram captures all inputs and outputs between $S^0, E^I$ and $E^O$. Bottom diagram shows functional closure framing of the top diagram.
  • Figure 3: Top Diagram captures all inputs and outputs between $S^0, E^I$ and $E^O$. Bottom diagram shows informational closure framing of the top diagram.

Theorems & Definitions (9)

  • Definition 0: Systems-theoretical Closed System
  • Definition 1: Functionally Closed Context System
  • Definition 2: Interpretation of An Informationally Closed Systems Using Information
  • Proposition 1
  • proof
  • Definition 3: Interpretation of A Functionally Closed System Using Information
  • Proposition 2
  • proof
  • Theorem 1: Inequality for mutual information in closure