Linear Software Models: Standard Modularity Highlights Residual Coupling

Abstract

Modularity — the decoupling of software units — is essential for composition of real software systems from ready-made components. But for a long time one lacked a formal theory of modularity. Recently we have been developing Linear Software Models as rigorous theoretical modularity standards based upon plain Linear Algebra. By these models, decoupling means just linear independence, within a modularity matrix. This paper applies Linear Software Models to software systems, obtaining three consequences: (1) besides decoupling, various informal notions of software engineering, such as software modules, cohesion, and single responsibility, have for the first time a well-defined formal counterpart; (2) canonical building blocks like Software Design Patterns strictly obey the Linear Software Models; (3) larger software systems obey bordered Linear Models, allowing precise location and visualization of residual coupling. The latter consequences are demonstrated by case studies of software systems from the literature. The applicability of the Linear Software Models is quantitatively shown to scale well with system size, for the given case studies.

Conceptual Integrity of Software Systems: Architecture, Abstraction and Algebra

“…[10], [11]), with respective arguments presented later on in this paper. In order to apply algebra to concepts we do not try to get an answer to Open Issue #2.…”

“…In this work we shall mainly refer to the Modularity Matrix [9], [10], [13] which is based upon linear algebra. Other matrices have been used for modular design.…”

“…Algebraic Propriety The algebraic translation of Brooks' Propriety is based upon the following ideas [10]. Instead of sets of classes (the concepts), one works with vectors of classes.…”

Conceptual Software Design: Algebraic Axioms for Conceptual Integrity

“…We aim to explain the primitive terms of these models used in the axioms of this work; for details see [8], [9], [10].…”

“…This paper claims that faithful translation of informal verbal principles to formal mathematical criteria is essential to widespread practical application of Conceptual Integrity ideas. Our proposal formulates algebraic axioms for Conceptual Integrity based upon the Modularity Matrix [10], an algebraic structure of the Linear Software Models.…”