Algebraic simplification is examined first from the point of view of a user who needs to comprehend a large expression, and second from the point of view of a designer who wants to construct a useful and efficient system. First we describe various techniques akin to substitution. These techniques can be used to decrease the size of an expression and make it more intelligible to a user. Then we delineate the spectrum of approaches to the design of automatic simplification capabilities in an algebraic manipulation system. Systems are divided into five types. Each type provides different facilities for the manipulation and simplification of expressions. Finally we discuss some of the theoretical results related to algebraic simplification. We describe several positive results about the existence of powerful simplification algorithms and the number-theoretic conjectures on which they rely. Results about the nonexistence of algorithms for certain classes of expressions are included.

A system's architecture defines its flexibility—the ease with which changes to the system's structure may be made. Systems may possess an internal structure that is relatively complex to describe—especially after many changes. In addition, some changes may require several iterations before stakeholders converge on a final design choice. There is currently no unified mathematical theory that captures these factors. We developed metrics for flexibility, descriptive complexity, and rework potential defined on graph‐based representations of a system's structure. We applied these metrics to four idealized generic system architectures: Tree‐structured hierarchies are easy to describe and require minimal design iteration. However, they permit relatively few changes, unless the underlying architecture itself is changed. Furthermore, each such change adds significant complexity. Grid networks are more flexible than trees, but also more descriptively complex. Grids also require some degree of design iteration between stakeholders. Teams are extremely flexible but the requirement for consensus (e.g., due to cognitive limits among humans) restricts their size. Layered hierarchies possess moderate flexibility and require fewer iterations than corresponding grids and teams, although they can be difficult to describe succinctly. Our findings suggest that no architecture is ideal under all circumstances; rather, each has strengths and weaknesses that can be exploited in different environments.

Three approaches to symbolic integration in the 1960's are described.

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