A mathematical study of three approaches for defining nested arrays in APL is presented. Theorems exhibiting the relationships between the definitional systems are given and illustrated through graph representations. One of the approaches is used to define an APL array to be a recursive data structure equivalent to a tree structure in which all data is stored at the leaves as homogeneous arrays of numbers and characters. An extension of APL is proposed that includes new primitive functions to manipulate the nesting level of arrays and new operators to assist in the construction of data-driven algorithms.
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