Functions & Abstractions

When defining the functionality of an algorithm for data analysis, we are generally interested in balancing its efficiency against elegance. Facilitated by abstraction, elegant solutions are considered relatively concise, and tend to avoid or hide what might appear as a clutter of low-level redundancy. An abstract function is the synthesis of other functions, be they also abstract or primitive in terms of the native tools of a computing environment. Primitives are analogous to axioms in propositional logic and abstractions are analogous to compound statements of propositions. This chapter covers some axiomatic foundations, with definitions of counting, successor, predecessor, addition, and multiplication functions. Functions are built upon the concepts of relations, with abstractions of multi-argument and curried argument functions.