The Mathematical Syntax of Architectures
Christoph F. Strnadl
TL;DR
This work develops a rigorous mathematical syntax for architectures, modeling them as structures $(A, R, F)$ and formalizing views, sub-architectures, and n-tier partitions. It introduces architecture homomorphisms and the category Arch to compare architectures, and proves key results such as a precise characterization of n-tier architectures via surjective homomorphisms to elementary templates, alongside a No-Go theorem showing that modularity cannot be defined purely syntactically. The application section demonstrates compatibility with major standards (e.g., ISO 42010, ArchiMate) and situates the framework within foundational work, including Wilkinson 2018 and Dickersen 2020, while the discussion connects to model theory and outlines future refinements and extensions. The work provides a foundational, formal lens for architecture that complements semantic approaches and supports rigorous reasoning about structure, similarity, and modularity in complex systems.
Abstract
Despite several (accepted) standards, core notions typically employed in information technology or system engineering architectures lack the precise and exact foundations encountered in logic, algebra, and other branches of mathematics. In this contribution we define the syntactical aspects of the term architecture in a mathematically rigorous way. We motivate our particular choice by demonstrating (i) how commonly understood and expected properties of an architecture -- as defined by various standards -- can be suitably identified or derived within our formalization, (ii) how our concept is fully compatible with real life (business) architectures, and (iii) how our definition complements recent foundational work in this area (Wilkinson 2018, Dickersen 2020, Efatmaneshnik 2020). We furthermore develop a rigorous notion of architectural similarity based on the notion of homomorphisms allowing the class of architectures to be regarded as a category, Arch. We demonstrate the applicability of our concepts to theory by deriving theorems on the classification of certain types of architectures. Inter alia, we derive a no go theorem proving that, in contrast to n-tier architectures, one cannot sensibly define generic architectural modularity on the syntactical level alone.