APL Cheat Sheet

Core APL array-language notation covering arithmetic, indexing, custom dfns, and reduction/scan operators for concise array programming.

1 PageAdvancedApr 5, 2026

Basic Arithmetic & Arrays

Scalar and array arithmetic evaluated right-to-left.

apl

2 + 3          ⍝ 5⍳5             ⍝ 1 2 3 4 5  (index generator)2 × ⍳5         ⍝ 2 4 6 8 10V ← 1 2 3 4 5  ⍝ assign a vector⍴V             ⍝ 5  (shape of V)V[2]           ⍝ 2  (indexing, 1-based by default)

Core Symbols

Primitive glyphs used constantly in APL code.

←- assignment, e.g. V ← 1 2 3
⍳ (iota)- generates an index vector, e.g. ⍳5 is 1 2 3 4 5
⍴ (rho)- monadic: returns shape; dyadic: reshapes an array
⌽- reverses the elements of a vector
⍉- transposes a matrix
/ (reduce)- inserts a function between elements, e.g. +/1 2 3 sums to 6
\ (scan)- like reduce but keeps every intermediate result
⍺ / ⍵- left and right arguments inside a dfn (curly-brace function)

Custom Functions (dfns)

Defining small anonymous functions with curly braces.

apl

square ← {⍵ × ⍵}square 5        ⍝ 25add ← {⍺ + ⍵}3 add 4         ⍝ 7avg ← {(+/⍵) ÷ ≢⍵}avg 1 2 3 4     ⍝ 2.5

Reduction & Scan

Folding a function across an array's elements.

apl

+/1 2 3 4      ⍝ 10  (sum)×/1 2 3 4      ⍝ 24  (product)+\1 2 3 4      ⍝ 1 3 6 10  (running sum)⌈/3 1 4 1 5    ⍝ 5   (maximum via reduce)

Pro Tip

APL expressions evaluate strictly right-to-left with no operator precedence, so read code from right to left to understand it — `2 × 3 + 4` is `2 × (3 + 4)`, giving 14, not 10.

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