Grammar

This page documents the grammar implemented by MARPLE's parser (src/marple/parser.py).

Evaluation order

APL evaluates right to left. Every function takes everything to its right as its right argument. There is no operator precedence among functions.

      2×3+4
14

This is 2×(3+4), not (2×3)+4.

Binding hierarchy (tightest first)

1. Bracket indexing -- M[i;j]
2. Operator binding -- +/, ⌽⍤1, ∘.×, f.g
3. Strand formation -- 1 2 3 (adjacent numbers form a vector)
4. Function application -- f Y (monadic) or X f Y (dyadic)
5. Assignment -- name ← value

Valence

A function is dyadic if a value appears to its left; otherwise it is monadic. The parser determines valence from context: if the token immediately left of a function glyph is a value (number, variable, or closing paren/bracket), the function is dyadic.

Tokens

Token type	Examples
Number	42, 3.14, ¯5, 1e3
String	'hello', 'it''s'
Function glyph	+ - × ÷ ⌈ ⌊ * ⍟ \| < ≤ = ≥ > ≠ ∧ ∨ ~ ⍴ ⍳ , ↑ ↓ ⌽ ⍉ ⍋ ⍒ ⊤ ⊥ ⍎ ⍕ ⌹ ○ ⌷ ≡ ≢ ∈
Operator	/ \ . ∘ ⍤ ⌶
Identifier	x, myFunc, total
Qualified name	$::io::nread, ns::func
System variable	⎕IO, ⎕CT, ⎕PP
Dfn delimiters	{ }
Dfn special names	⍵ ⍺ ∇
Guard	: (inside dfns)
Diamond	⋄ (statement separator)
Assignment	←
Brackets	[ ] ; (indexing)
Parentheses	( ) (grouping)

Statements

Multiple expressions on one line are separated by ⋄ (diamond). Each statement is parsed independently.

      a←3 ⋄ b←4 ⋄ a+b
7

Assignment

name ← expression
⎕SYSVAR ← expression

Assigns the result of expression to name or a system variable. The assigned value is the result (but the REPL suppresses display for bare assignments).

Dfn syntax

{ statement (⋄ statement)* }

Inside a dfn:

• ⍵ -- right argument
• ⍺ -- left argument
• ∇ -- self-reference (recursion)
• ⍺←expr -- default left argument
• condition : expr -- guard

Derived functions

Syntax	Meaning
f/ Y	Reduce Y with f
f\ Y	Scan Y with f
X ∘.f Y	Outer product
X f.g Y	Inner product
(f⍤k) Y	Rank operator (monadic)
X (f⍤k) Y	Rank operator (dyadic)
(⌶'path') Y	I-beam (monadic)
X (⌶'path') Y	I-beam (dyadic)

Operator-function composition

Reduce and scan can be composed with rank:

(f/⍤k) Y
(f\⍤k) Y

Bracket indexing

array[index]
array[row;col]
array[;col]
array[row;]

Bracket indexing binds tighter than function application. Empty indices select all elements along that axis.

Dyadic use of / and \

When / or \ appears between two values (not after a function glyph), they act as dyadic functions:

• X / Y -- replicate/compress
• X \ Y -- expand

Parentheses

Parentheses override the default right-to-left evaluation:

      (2+3)×4
20