One Step up the Abstraction Ladder: Combining Algebras - From Functional Pieces to a Whole

Abstract

A fundamental scientific question today is how to construct complex systems from simple parts. Science today seems mostly to analyze limited pieces of the puzzle; the combination of these pieces to form a whole is left for later or others. The lack of efficient methods to deal with the combination problem is likely the main reason. How to combine individual results is a dominant question in cognitive science or geography, where phenomena are studied from individuals and at different scales, but the results cannot be brought together. This paper proposes to use parameterized algebras much the same way that we use functional abstraction (procedures in programming languages) to create abstract building blocks which can be combined later. Algebras group operations (which are functional abstractions) and can be combined to construct more complex algebras. Algebras operate therefore at a higher level of abstraction. A table shows the parallels between procedural abstraction and the abstraction by parameterized algebras. This paper shows how algebras can be combined to form more complex pieces and compares the steps to the combination of procedures in programming. The novel contribution is to parameterize algebras and make them thus ready for reuse. The method is first explained with the familiar construction of vector space and then applied to a larger example, namely the description of geometric operations for GIS, as proposed in the current draft standard document ISO 15046 Part 7: Spatial Schema. It is shown how operations can be grouped, reused, and combined, and useful larger systems built from the pieces. The paper compares the method to combine algebras — which are independent of an implementation — with the current use of object-orientation in programming languages (and in the UML notation often used for specification). The widely used’ structural’ (or subset) polymorphism is justified by implementation considerations, but not appropriate for theory development and abstract specifications for standardization. Parametric polymorphism used for algebras avoids the contravariance of function types (which its semantically confusing consequences). Algebraic methods relate cleanly to the mathematical category theory and the method translates directly to modern functional programming or Java.