Aggregation — polymorphic and polytypic

Repeating the work of Meertens [1] we show how to define in a few lines a very general “aggregation” function. It is parameterized by a binary operation ⊕ : t × t → t that performs the elementary aggregation steps on values of type t . The aggregation function, then, aggregates all t constituents of a composite ...t ... value into a single t value; its type is: ...t ... → t . The generality is three-fold. First, type t is arbitrary and may be different for different invocations; so the aggregation is polymorphic: working on many types. Second, expression ...t ... is an arbitrary regular type (built from +, ×, List and so on) and may be different for different invocations; so the aggregation is polytypic, also known as generic: working on many type structures. Third, operation ⊕ may be different for different invocations; by suitable choices and a little preprocessing and postprocessing, the function aggregates the t constituents of a given structure into their summation, or their maximum, or their count, or their average, or their first one, or their longest path, and so on. Footnote Nothing of this material is new; a far more precise and far more general elaboration with far more discussion was given by Meertens [1] in 1996. Our intention is to give a self-contained exposition that is more accessible for the uninitiated reader who wants to see the idea with a minimum of formal details. The main difference with Meerstens’ exposition is our way of dealing with functors: in our exposition the only property of functors that we use is that they map sets to sets. A minor difference is the added elaboration for lists, natural numbers, rose trees, and Maybe values. Thus, this note may be viewed as a 8-page introduction in 2011 to Meertens’ 16-page paper of 1996. Notation. In the context of a structured type, we use xs, ys, zs to vary over structured values, and x , y , z to vary over constituents of a structured value. For example, we might say “a nonempty list xs is built from a value y and a sublist zs”. Furthermore, we let f , g , h vary over functions, and t , a, b, c vary over types. (Type variable t is only more special than a, b, c in that it relates to the parameter ⊕ : t × t → t .) Finally, type varies over type expressions. Introduction. Aggregation is a frequently occurring concept in computing, and is programmed by many people over and over again for many kinds of data structures and many kinds of aggregation. Fortunately, SQL has a built-in facility for aggregation, thus freeing the user from reinventing the wheel, but unfortunately it does so for only a few kinds, namely counting, summation, maximization, minimization, and averaging, and only for sole data structure of SQL, namely tables. Meertens [1] gives one single definition of an aggregation that can be instantiated to many kinds (surpassing SQL) and that works, for arbitrary type t and operation ⊕ : t × t → t , on all of the following types and many more: