Rank Mismatch in Nested Array Reduction

Question

Tacit Reduction of Mixed-Rank Arrays
When utilizing the reduction operator (e.g., +/) on nested arrays in APL, the behavior depends on whether the array is treated as a flat sequence or a nested structure of varying ranks.
A design challenge arises when composing tacit functions that must operate across a specific axis of a nested array without explicitly raveling the structure. If the reduction is applied to a nested array where elements are not of uniform rank, the resulting shape of the output may deviate from the expected axis reduction.
Given the behavior of rank-sensitive functions, how does APL resolve the axis of reduction when the target is a nested array of mixed ranks? Under what conditions will a reduction operator trigger a rank error versus implicitly flattening the nested structure?

Tasadduq Burney · Answer

Resolution of Axis Reduction in Mixed-Rank Nested Arrays
In APL, the reduction operator (+/) resolves the axis of reduction by treating a nested array as a 1D sequence of elements, regardless of the internal rank of those elements. The reduction is applied to the outermost axis. If the nested array contains elements of mixed ranks (e.g., some scalars and some vectors), APL does not implicitly flatten the internal structures; it attempts to apply the binary operator to the elements as they are encountered in the sequence.

Rank Error vs. Implicit Flattening
APL does not implicitly flatten nested structures during a reduction. The distinction between a successful operation and a rank error depends on the compatibility of the binary operator with the ranks of the elements being reduced:

Rank Error: Occurs when the binary operator (e.g., +) encounters two elements whose ranks are incompatible according to APL's rank rules. For example, attempting to add a 2D matrix to a 1D vector where the dimensions do not align for broadcasting will trigger a rank error.
Successful Reduction: Occurs if the operator can handle the mixed ranks via broadcasting or if the elements are of a type that the operator accepts (e.g., adding a scalar to a vector).

Steps to Resolve Rank Mismatches in Tacit Functions
To ensure a consistent output shape when operating across a specific axis of a nested array without manual raveling, use the following strategies:

Normalize Rank: Use the ⎕rank or ⊃ (enclose) operator to ensure all elements in the nested array are promoted to a uniform rank before the reduction.
Explicit Axis Specification: When using functions that support axis arguments, explicitly define the axis to avoid the default behavior of reducing the leftmost axis.
Conditional Flattening: If the goal is to treat the structure as a flat sequence, use the (⍴⍴X) or (⍳⍴X) logic to determine if a ⊃ (flatten/unwrap) is required before the reduction operator is applied.

Diagnostic Requirement
To provide a more precise tacit implementation, please specify: Are the nested elements of varying ranks intended to be treated as scalars (via a sum of sums) or as aligned tensors?

Rank Mismatch in Nested Array Reduction

Tacit Reduction of Mixed-Rank Arrays

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Resolution of Axis Reduction in Mixed-Rank Nested Arrays

Rank Error vs. Implicit Flattening

Steps to Resolve Rank Mismatches in Tacit Functions

Diagnostic Requirement

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