Toward an Algebraic Theory of Systems

TL;DR

This paper introduces a formal algebraic framework for systems with composition and connection operations, emphasizing invariance to composition order, applicable across physical, electronic, and distributed systems, including causal and timed systems.

Contribution

It formalizes the concept of system algebras with composition-order invariance and demonstrates their application to Kahn networks and causal systems, advancing the theoretical understanding of system composition.

Findings

Kahn networks form a composition-order invariant system algebra

Causal systems with time-dependent inputs are modeled within this algebraic framework

The framework captures and abstracts system behavior in various scientific contexts

Abstract

We propose the concept of a system algebra with a parallel composition operation and an interface connection operation, and formalize composition-order invariance, which postulates that the order of composing and connecting systems is irrelevant, a generalized form of associativity. Composition-order invariance explicitly captures a common property that is implicit in any context where one can draw a figure (hiding the drawing order) of several connected systems, which appears in many scientific contexts. This abstract algebra captures settings where one is interested in the behavior of a composed system in an environment and wants to abstract away anything internal not relevant for the behavior. This may include physical systems, electronic circuits, or interacting distributed systems. One specific such setting, of special interest in computer science, are functional system algebras, which capture, in the most general sense, any type of system that takes inputs and produces outputs depending on the inputs, and where the output of a system can be the input to another system. The behavior of such a system is uniquely determined by the function mapping inputs to outputs. We consider several instantiations of this very general concept. In particular, we show that Kahn networks form a functional system algebra and prove their composition-order invariance. Moreover, we define a functional system algebra of causal systems, characterized by the property that inputs can only influence future outputs, where an abstract partial order relation captures the notion of "later". This system algebra is also shown to be composition-order invariant and appropriate instantiations thereof allow to model and analyze systems that depend on time.