APL

1. Declaration

←: Assign
- Expressions are executed, Definitions are remembered.
String is a vector of character. It is surrounded by single quotes. e.g. 'APL'

Every data is array.
Each entry can be either number, character or array.
- integer or float are automatically dealt with.
Array can be called scalar, vector or matrix depending on their shape.
array[index[;index]*] to access its entries. index [index]* ⌷ array is also available.

It is the shape of the array. It does not consider the shape of its entries.
Scalar has the shape of empty vector ⍬.
- Scalar can contain an array which is called the enclosed vector or nested scalars.
⍴: Return the shape if monadic, Reshape if dyadic.
- It repeats the entries when the shape is larger, truncates if the shape is smaller.

It takes everything around it as its arguments, evaluated from the left.
Monadic function takes from the right, and dyadic functions take from both sides. Those are the only two options.
Scalar function pairs (or array extends if one of its argument is scalar) and then penetrate.
Mixed functions are all other functions that work on their own way.
Dfns (dynamic-function) is an inline anonymous function. It is enclosed within braces.

Tacit function (aka train) is a statement that does not have data on the right.
It is defined without the braces.

atop
- Function composition
- (f g) X = f (g X)
- X (f g) Y = f (X g Y)
  - where g is dyadic.
- ⍤: Atop. f⍤g does the same thing.

f g h fork
- (f g h) X = (f X) g (h X)
- X (f g h) Y = (X f Y) g (X h Y)
  - where f and g are dyadic.
A g h fork
- A is an array.
- (A g h) X = A g (h X)
- X (A g h) Y = A g (X h Y)
  - where h is dyadic.
⊣ and ⊢ return left argument and right argument respectively. When used in a monadic function, both return the right argument.
If the train is longer than three, then the last three become the three train recursively.

≡: Match.
≢: Tally, Not Match. The length of the leading axis.
+: Add (Complex conjugate), -: Subtract (0-), ×: Multiply (Normalize), ÷: Divide (Reciprocal), *: Exponentiate (e*), ⍟: Logarithm (e⍟), ⌈: Max (Ceiling), ⌊: Min (Floor), |: Residue (Magnitude).
∧: And, lcm. ∨: Or, gcd. ⍲, ⍱: NAND, NOR, ~: Not.
=, ≠, <, >, ≤, ≥: Dyadic. Comparison. Scalar Function. Return 0 or 1.
⍋, ⍒: Grade Up/Down. Return the indexes in sorted order. The order can be given when used dyadic.
↑: Mix. Monadic. Nest -> Rank. Filled with the type of first element of each nested array. ↓: Split. Monadic. Rank -> Nest. ⊂: Enclose. Enclose an array in a scalar. Simple scalar cannot be enclosed. ⊃: Disclose. Take the first element.
,: Catenate, Laminate.
⌽: Reverse. ⊖: Reverse, but along the first axis.
⍉: Transpose.

/: Reduce. The result is enclosed. It is evaluated from the last two elements along the last axis, since APL is right-associative.
⌿: Reduce, but along the first axis.
\: Scan. Calculate accumulations from the first element.
⍀: Scan, but along the first axis.
- L f/[N] R: L specifies the width of windows, and N specifies the index of the axis.
- A / B filters (replicates) B with A.
¨: Each.
⍨: Self, Swap. Duplicate the argument or swap the arguments.
⍣: Repeat. f⍣n repeat f n times. n can be negative for the inverse operation, or = to repeat until the fixed point is achieved.
. Dot: Inner Product. f.g apply g to each and reduce with f. +.× Vector Dot Product, Matrix Multiplication.
∘ Jot: Bind, Beside. d∘m = ⍺ d (m ⍵), D combinator. x∘d or d∘x bind x to the left or right of the dyadic function turning it into a monadic one.
∘.: Outer Product. ∘.g takes the outer product with g.
⍤: Rank. Set the rank for the next operator to work on. Atop, the two train.
⍥: Over. d⍥m Apply monadic function to both of the arguments before applying the dyadic one. S' Combinator
@: At. (m|x)@(A|m) apply m or set to x, if in A or m evaluates to 1.