Linear Software Models: An Algebraic Theory of Software Composition

The central goal of our software theory is to solve the composition problem of a software system of any size from sub-systems down to indivisible components either produced in house, or purchased from other manufacturers. The theory guides the gradual or agile design of the system by means of quantitative criteria for the current design quality, and by highlighting problematic coupling spots to be resolved in order to improve the software design. The theory is based upon solid results of linear algebra, and the broadly accepted wisdom that modularity is essential for understandable and maintainable software design. The main algebraic structure of the theory is the Modularity Matrix. It links structors – a generalization of classes – in the matrix columns, to their provided functionals – a generalization of methods – in the matrix rows. The theory predicts that neat modules for any given system are the blocks of a block-diagonal standard matrix. The matrix itself is the source of quantitative criteria for design quality. Module sizes are determined by the matrix eigenvectors. The theory formalizes intuitive concepts widely used in software engineering, providing precise definitions for coupling, cohesion, modules, single responsibility and canonical systems such as design patterns. This talk also touches recent research results showing that apparently different alternative theories, such as the Modularity Lattice from FCA (Formal Concept Analysis) and approaches using the Laplacian Matrix are equivalent to the Modularity Matrix. Therefore, Linear Software Models are proposed as a coherent unified algebraic theory of software composition.