Mathematical Theory of Reusability

This talk gives an introduction to my book A Generative Theory of Shape (Springer-Verlag, 550pages). The purpose of the book is to develop a generative theory that has two properties regarded as fundamental to intelligence - maximizing reusability of structure and maximizing recoverability of the generative operations. These two properties are particularly important in the representation of complex organization - which is the main concern of the book. The primary goal of the theory is the conversion of complexity into understandability. For this purpose, a mathematical theory is presented of how understandability is created in a structure. This is achieved by developing a group-theoretic approach to formalizing reusability and recoverability. To handle highly complex structure, a new class of groups is invented, called unfolding groups. These unfold structure from a maximally collapsed version of that structure. A principal aspect of the theory is that it develops a new algebraic formalization of major object-oriented concepts such as inheritance. The consequence that the book establishes a representational language for complex organizational structure, that is interoperable by virtue of the principles on which the theory is based: reusability and recoverability. The book gives extensive applications of the theory to CAD/CAM, human and machine vision, robotics, software engineering, and physics. For example, the theory is used to give new and detailed insights into the main stages of mechanical CAD/CAM: part-design, assembly and machining. And within part-design, an extensive analysis is given of sketching, alignment, dimensioning, resolution, editing, sweeping, feature-addition, and intent-management.