Once and for all: how to compose modules -- The composition calculus
Peter Fettke, Wolfgang Reisig
TL;DR
The paper addresses the need for a universal theory of module composition beyond traditional computability, to model the digital world of interacting components. To this end, it introduces the composition calculus based on modules with two interfaces and an associative operator $M \bullet N$ defined via merging matching labeled gates. Key contributions include formal definitions of interfaces, matches, equivalence, perfect matches, and classes of modules; proofs of associativity, cancellativity, and related properties; and connections to existing modeling frameworks. By focusing on interfaces rather than internal structure, the framework provides a flexible, scalable foundation applicable across domains and standards, enabling modular design and refinement through Levi's Lemma and related results.
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.