Once and for all: how to compose modules -- The composition calculus

TL;DR

This paper introduces a general composition calculus for modules in digital systems, emphasizing interaction and associativity, applicable across various settings, supported by theoretical results and case studies.

Contribution

It proposes a minimal set of postulates and a universal composition calculus for interacting modules, unifying diverse module composition frameworks.

Findings

The calculus exhibits key properties like associativity.

It encompasses various module composition settings.

Supported by multiple theorems and case studies.

Abstract

Computability theory is traditionally conceived as the theoretical basis of informatics. Nevertheless, numerous proposals transcend computability theory, in particular by emphasizing interaction of modules, or components, parts, constituents, as a fundamental computing feature. In a technical framework, interaction requires composition of modules. Hence, a most abstract, comprehensive theory of modules and their composition is required. To this end, we suggest a minimal set of postulates to characterize systems in the digital world that consist of interacting modules. For such systems, we suggest a calculus with a simple, yet most general composition operator which exhibits important properties, in particular associativity. We claim that this composition calculus provides not just another conceptual, formal framework, but that essentially all settings of modules and their composition can be based on this calculus. This claim is supported by a rich body of theorems, properties, special classes of modules, and case studies.