The extension of APL to treelike data structures

The purpose of this paper is to ensure the existence of an axiom system which---contrary to [2]---leads to a uniform behaviour of all generalized arrays including basic scalars with respect to the function seal (enclose). The extension described hereafter takes up most of the ideas laid down in [1], but deviates from them in some respects while including additional tools and concepts such as---canonical tree representation of arrays---virtual indexing (indexing outside the index domain of an array)---a new concept for indexing including---virtual indexing---slice indexing, based on the outer product with respect to catenation---path indexing---an operator called composition operator (a generalization of reduction)---composition of operatorsFurthermore, the paper includes as a substantial part a simulation of a model of extended APL, as seen from an algebraic point of view. The model, intended as a working tool, was developed by the author at the Berufsförderungswerk in Heidelberg, Germany, and is contained in a workspace called HEIDELTREE.On the following pages I state the essential ideas in an informal way giving examples rather than definitions, but including the motivations for the reasoning whenever feasible in short terms.