Expression syntax

BQN expressions are the part of syntax that describes computations to perform. Programs are mainly made up of expressions with a little organizing material like blocks and namespaces around them. This page explains how functions, modifiers, and assignment combine with their inputs. It doesn't describe constant and array literals, which each form a single subject for grammatical purposes.

The first tutorial also covers how to build and read BQN expressions.


BQN expressions consist of subjects, functions, and modifiers arranged in sequence, with parentheses to group parts into subexpressions. Assignment arrows and can also be present and mostly act like functions. Functions can be applied to subjects or grouped into trains, while modifiers can be applied to subjects or functions. The most important kinds of application are:

left main right output name binding
w? F x Subject Function RtL, looser
F? G H Function Train
F _m Function 1-Modifier LtR, tighter
F _c_ G Function 2-Modifier

The four roles (subject, function, two kinds of modifier) describe expressions, not values. When an expression is evaluated, the value's type doesn't have to correspond to its role, and can even change from one evaluation to another. An expression's role is determined entirely by its source code, so it's fixed.

In the table, ? marks an optional left argument. If there isn't a value in that position, or it's Nothing (·), the middle function will be called with only one argument.

If you're comfortable reading BNF and want to understand things in more detail than described below, you might check the grammar specification as well.

Syntactic role

This issue is approached from a different angle in Context free grammar.

In APL, the way one part of an expression interacts with others is determined by its value. That means that to parse an expression, in general you would have to evaluate that part, get a value, check its type, and then figure out how it fits in with the rest of the expression. This is a lot of work. BQN changes things so that you can determine how to parse an expression just by looking at its source code. But because it still needs to support expressions that can evaluate to more than one possible type, BQN has to introduce a new and independent concept, called syntactic role, in order to support APL-like expressions.

Syntactic role is a property of an expression, not its value. To describe it in terms of English grammar, you might say "I like BQN", using "BQN" as an object, or "BQN scares me", using it as a subject. BQN itself isn't a subject or object, it's a programming language. Similarly you might write F g, placing f in a function role to apply it to g, or G f to use f as an argument. Maybe even in the same program, although it's unlikely.

Below, the function {𝕎𝕩} treats its left argument 𝕎 as a function and its right argument 𝕩 as a subject. With a list of functions, we can make a table of the square and square root of a few numbers:

    ט, {𝕎𝕩} 149
╵ 1 16 81  
  1  2  3  

Role spellings

The four roles are subject, function, 1-modifier, and 2-modifier, as shown in the table below. Each type has an associated role (non-operation types all correspond to subjects), and the value of an expression will often have a type that fits the role, but it doesn't have to.

BQN Names Primitives
Subject lowerCase Literals
Function UpperCase +-×÷⋆√⌊⌈|¬∧∨
1-modifier _leading ˙˜˘¨⌜⁼´˝`
2-modifier _both_ ∘○⊸⟜⌾⊘◶⎉⚇⍟⎊

Primitive tokens, since they have a fixed value, always have a role that matches their type. They're functions by default, as the modifiers have glyphs that fit specific patterns. 1-modifiers have superscript glyphs, and 2-modifiers have glyphs with an unbroken circle—that is, one without a line through it, excluding functions and .

Variable names can be written in any case and with underscores added, and these changes don't affect what identifier the name refers to. ab, aB, AB, and _a_B_ are all the same variable. However, the spelling—specifically the first and last characters—determine the variable's role. A lowercase first letter indicates a subject, and an uppercase first letter makes it a function. A leading underscore (regardless of the following character) indicates a 1-modifier, and both leading and trailing underscores makes a 2-modifier.

Besides these, character, string, and array literals always have a subject role, and the role of a block is determined by its type, which depends either on the header it has or which special variables it uses. If headerless, a block is a subject if it has no special names, but a 𝕨 or 𝕩 makes it at least a function, an 𝔽 makes it a 1- or 2-modifier, and a 𝔾 always makes it a 2-modifier.

The role of a compound expression, formed by applying an operation to some inputs, depends on the operation applied. This system is covered in the remaining sections below.


As in most programming languages, parentheses () are for grouping. The code inside a balanced set of parentheses is a single expression, which produces one value to be used by the expression that contains it—for example, in (2×3)+4, 2×3 is a subexpression evaluating to 6, so that larger expression is equivalent to 6+4. The syntactic role of a set of parentheses is also the same as that of the expression inside.


The character · is called Nothing. While it can be easier to think of it as a value, it can't be passed around in variables, and so can also be interpreted as an element of syntax. The special name 𝕨 also functions as Nothing if the block that contains it is called with one argument (the uppercase spelling 𝕎 doesn't, but instead immediately causes an error). Both · and 𝕨 have a subject role.

The following rules apply to Nothing:

For example, the expression (F 2 G ·) H I j is equivalent to H I j. But functions and arguments that will be discarded by the second rule are still evaluated, so that for example (a+1) F · increments a when run.

Nothing can only be used as an argument to a function, or the left argument in a train (it can't be the right—a train ends with a function by definition). In another position where a subject could appear, like as an operand or in a list, it causes an error: either at compile time, for ·, or when the function is called with no left argument, for 𝕨.

Kinds of application

Here is a table of the function and modifier application rules:

left main right output name
F x Subject Monadic function
w F x Subject Dyadic function
F G Function 2-train
F* G H Function 3-train
F* _m Function 1-Modifier
F* _c_ G* Function 2-Modifier

An asterisk * indicates that a subject can also be used. Since the role doesn't exist after parsing, function and subject spellings are indistinguishable in these positions. Modifier applications bind more tightly than functions, and associate left-to-right while functions associate right-to-left.


Expressions may also include assignment, which is written with or to define (similar to "declare" in many other languages) a variable and to change its definition. Assignment has a left-hand side (name below) which is usually a variable name, and a right-hand side (4) which can be any expression. The roles of the two sides have to match. It sets the value of the variable to be the result of the expression.

    name  4
⟨ 0 1 2 3 ⟩

A variable can only be defined once within a scope, and can only be changed if it has already been defined. However, it can be shadowed, meaning that an inner scope can define it even if it has a definition in an outer scope already.

    x1  {x2  x3  x}

Assignment can be used inline in an expression, and its result is always the new value of the assignment target. Function or modifier assignment must be parenthesized, while subject assignment doesn't have to be: in a subject expression, assignment arrows have the same precedence as functions.


The modification arrow can also be used to update the value of a variable by applying a function. This meaning applies only when there is no expression to the right of or that expression has a subject role, and there's a function to the left of (in terms of precedence this function is like an operand—a train must be parenthesized). The two forms of modified assignment are shown below along with equivalent expansions.

Syntax Meaning
a F a F a
a F b a a F b


The left hand side of assignment in a subject expression can be compound, so that assigned values are extracted from lists or namespaces. This is called a destructuring assignment. The most common case is list destructuring: the left hand side's written as a list, and the value on the right has to be a list of the same length. Assignments are made element-wise, and the elements can destructure things further.

    qr,s  "qr",4
⟨ "qr" ⟨ 0 1 2 3 ⟩ ⟩


⟨ 0 1 2 3 ⟩

Array destructuring using array notation [] is also possible: it's equivalent to splitting the right-hand side with <˘ and then applying list destructuring.

    [t,u]  23
╵ ⟨ 0 0 ⟩ ⟨ 0 1 ⟩ ⟨ 0 2 ⟩  
  ⟨ 1 0 ⟩ ⟨ 1 1 ⟩ ⟨ 1 2 ⟩  

⟨ ⟨ 1 0 ⟩ ⟨ 1 1 ⟩ ⟨ 1 2 ⟩ ⟩

Namespace destructuring uses an overlapping syntax, fully described in its own section. The left hand side is a list of names, or aliases tofrom.

    qr  {q2+r0.5}  q

With destructuring, you might want to discard some values from the right hand side rather than assign them any name. There's special syntax for this: use Nothing (·) for a placeholder non-name in the appropriate position, like ·y· list.

    ·  6   # Doesn't do anything


Full documentation

The double arrow exports variables from a block or program, causing the result to be a namespace. It can be used either in place of normal definition , or as a stand-alone statement with nothing to the right; in either case all the variables to its left are exported. An example with both uses, as well as namespace destructuring that uses to define variable aliases, is shown below.

aliasa, b  {