Array Theory

A pair holding a quintuple and a 2-by-3 table
of arrays R, S, …, Z. The quintuple holds the
pair (A,B), the array C,  the single holding D,
the solo holding P, and the void of array Q.

This site is in the initial stages of development. The purpose is to provide links to sources about the background, creation, implementation, and application of array theory, which is the study of the nested array as a computational model of data.

The study combines orthogonal arrangement, which is based on linear algebra, with hierarchical nesting, which is based on set theory, with principles developed in programming languages such and APL and Lisp. Array theory, abstract or conformal, is the mathematical core of two versions of the Nested Interactive Array Language, Nial, which is interpreted by Q’Nial 4 or 6.21, respectively. The focus of this site is on Version 4 and the abstract theory.

What is an array?  An array may have zero or more axes of possibly differing lengths. The valence of an array is the number of its axes. The items of an array are themselves arrays to any depth of nesting, which effectively terminates in self-containing atoms, such as numbers, truth-values, characters, phrases, and faults. A single is a nilvalent array. Since a single has no axes, it necessarily holds precisely one item, which may be any array. The single of an atom is the atom itself because atoms are self-containing.

Quartos, folios, tables, lists, and singles are arrays of valences 4, 3, 2, 1, and 0. Triples, pairs, solos, and voids are lists of lengths 3, 2, 1, and 0. The shape of an array is the list of its axis lengths. The shape of a single is therefore a void, in particular the void of 0.

An empty array, such as a void, must have at least one axis of length 0. In abstract array theory, the virtual item of an empty array may be any array, with the consequence that empty arrays of the same shape differ according to their virtual items. In conformal array theory, empty arrays of the same shape are the same.

 



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