Rank polymorphism

Rank polymorphism is a concept found in array programming languages like APL, J, the Iverson-inspired language K and ShapeRank (opens in a new tab), a new statically typed, purely functional language for industrial use, that extends rank-polymorphism to streams. It refers to the ability of functions (or operators) to operate on arrays of different ranks (dimensions) in a consistent and generalized manner.

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Key Concepts

Array Rank: The rank of an array is the number of dimensions it has. For example, a scalar (single value) has rank 0, a vector (1-dimensional array) has rank 1, a matrix (2-dimensional array) has rank 2, and so on.
Function Generalization: In rank polymorphic languages, functions are designed to operate on arrays of any rank without the need for special-case handling. This means the same function can apply to scalars, vectors, matrices, and higher-dimensional arrays seamlessly.
Implicit Broadcasting: When functions operate on arrays of different ranks, the language defines rules for broadcasting or extending arrays to match ranks. This is similar to how NumPy in Python handles array operations, where lower-rank arrays are implicitly expanded to match the higher-rank arrays' dimensions.

Examples in APL/J/K

APL Example

In APL, a function can be applied to arrays of varying ranks without changing the function itself.

+ ⍳ 5  ⍝ Adding a scalar to a vector
1 2 3 4 5

Here, + is rank polymorphic and can be applied to a vector, adding the scalar 1 to each element of the vector ⍳ 5 (which generates [1, 2, 3, 4, 5]).

J Example

In J, rank polymorphism is also a core concept. Here’s an example of summing along different ranks:

+/"1 1 2 3

In this case, +/"1 means to sum along the first axis (rows) of the array, and this can apply to any higher-rank array.

K Example

K, another array language, also supports rank polymorphism:

+/ 1 2 3  ⍝ Sums the vector, result is 6
+/ 1 2 3 4 5 6  ⍝ Similarly sums the vector, result is 21

Benefits

Conciseness: Code can be much more concise because you don't need separate functions for different ranks of arrays.
Flexibility: Functions written in a rank-polymorphic style can handle a wide variety of inputs, making them more flexible and reusable.
Consistency: Operations are applied in a consistent manner across different ranks, reducing the cognitive load on the programmer.

Modern Usage

Rank polymorphism is not as prominent in mainstream languages, but the concept influences modern numerical computing libraries like NumPy, TensorFlow, and others, where operations on arrays of different ranks are handled gracefully. These libraries often use broadcasting rules to achieve similar effects.