Blocks

In BQN, a block is any piece of code surrounded with curly braces {}. Blocks can be used simply to group statements, or can define functions or modifiers. They are the sole large-scale structure used to organize programs.

Blocks are most commonly used to define functions by including one of the special names for arguments, 𝕨 or 𝕩. With the operands 𝔽 or 𝔾, they can also define 1-modifiers or 2-modifiers.

↗️
    {𝕩+1} 3
4
    Γ—{𝕩𝔽𝕩} 4
16

Because they use lexical scoping, blocks can also be used to encapsulate code. If a block uses only variables that it initializes, then it has no dependence on its environment and would work the same way if defined anywhere. But it can also use external variables, defined in a containing block.

↗️
    a←b←"outer"
    { a←"inner" β‹„ aβ€Ώb }
⟨ "inner" "outer" ⟩

Headerless blocks

In the simplest case a block is just a list of statements, which are executed to evaluate the block. A block with no special names like 𝕨 or 𝕩 is called an immediate block, and is evaluated as soon as it is reached. The only think such a block does is group some statements, and create a scope for them so that definitions made there are discarded when the block finishes. Even this small amount of functionality could be useful; as an example the following program can build up an array from named components without polluting the rest of the program with those names.

↗️
    updown ← { up←↕5 β‹„ downβ†βŒ½up β‹„ up∾down }
    updown
⟨ 0 1 2 3 4 4 3 2 1 0 ⟩

An immediate block is only ever evaluated once, and can't be used for control flow in a program. Including special names in a headerless block lets us define functions and modifiers, which have a broader range of uses. All special names are listed below:

Lowercase Uppercase Meaning
𝕩 𝕏 Right argument
𝕨 π•Ž Left argument, or Nothing (Β·)
𝕀 π•Š Function self-reference
𝕗 𝔽 Left operand
π•˜ 𝔾 Right operand
𝕣 none Modifier self-reference

Of these, 𝕣 is sort of a "more special" character, as we'll discuss below. Except for 𝕣, every special name is a single character and can't have underscores added to spell it as a modifier, allowing a modifier to be applied to a special name with no spacing as in 𝕗_m, something that can't be done with ordinary names.

Arguments

The names 𝕨 and 𝕩, and their uppercase spellings, represent function arguments. As the argument to a function is typically data, it's more common to use the lowercase forms for these. Either of these names will turn an immediate block into a function (or an immediate modifier into a deferred one; see the next section). Instead of being evaluated as soon as it appears in the source, a function is evaluated when it's called, with the special names set to appropriate values. Unlike in Dyalog APL's dfns, their values can be changed like ordinary variables.

↗️
    {'c'=𝕩} "abcd"
⟨ 0 0 1 0 ⟩
    { 𝕩+↩2 β‹„ 0≍𝕩 } 3
⟨ 0 5 ⟩
    4 { βŸ¨π•©β‹„-π•¨βŸ© } 5
⟨ 5 ¯4 ⟩

A function with 𝕨 in its definition doesn't have to be called with two arguments. If it has only one, then 𝕨 is given the special value Nothing, or Β·. This is the only time a variable can ever be Nothing, as an assignment such as v←· is not allowed.

↗️
    3 { (2×𝕨)-𝕩 } 1
5
      { (2×𝕨)-𝕩 } 1
Β―1

In the second function, 𝕨 behaves just like Β·, so that the function 2×𝕨 is not evaluated and - doesn't have a left argument. It has a similar effect when used as the left argument to a function in a train.

↗️
    "abc" { (π•¨β‰βŒ½) 𝕩 } "def"
β”Œβ”€     
β•΅"abc  
  fed" 
      β”˜
          { (π•¨β‰βŒ½) 𝕩 } "def"
β”Œβ”€     
β•΅"fed" 
      β”˜

However, Β· can only be used as an argument, and not a list element or operand. Don't use 𝕨 in these ways in a function that could be called monadically. Another potential issue is that ⊸ and ⟜ don't work the way you might expect.

↗️
    { 𝕨 β‹†βŠΈ- 𝕩 } 5
143.4131591025766

Called dyadically, this function will expand to (⋆𝕨)-𝕩, so we might expect the monadic result to be -𝕩. This sort of expansion isn't right with Β· on the left. β‹†βŠΈ- taken as a whole is a function, so Β· β‹†βŠΈ- 𝕩 is just β‹†βŠΈ- 𝕩, or (⋆𝕩)-𝕩, giving the large result seen above.

Operands

The special names 𝔽 and 𝔾, and their lowercase forms, represent operands. Since operands are more often functions, they're typically shown with the uppercase spelling. If 𝔽 is present in a block then it defines a 1-modifier or 2-modifier depending on whether 𝔾 is present; if 𝔾 is there it's always a 2-modifier.

↗️
    4 {Γ—Λœπ•—}
16
    2 {𝕗+π•˜} 3
5

As shown above, modifiers without 𝕨, 𝕩, or 𝕀 behave essentially like functions with a higher precedence. These immediate modifiers take operands and return a result of any type. The result is given a function role, so it's most common to return a function, rather than a number as shown above.

↗️
    _dot_ ← {π”½Β΄βˆ˜π”Ύ}
    1β€Ώ2β€Ώ3 +_dot_Γ— 1β€Ώ0β€Ώ1
4

However, if one of these names is included, then a deferred modifier is created instead: rather than evaluate the contents when the modifier is called, the operands are bound to it to create a derived function. When this function is called, the statements in the block are evaluated.

↗️
    +{𝕩𝔽𝕩} 6
12
    2 β₯Š{βŸ¨π”½π•¨,π”Ύπ•©βŸ©}- 5
⟨ ⟨ 2 ⟩ ¯5 ⟩

The distinction between an immediate and deferred modifier only matters inside the braces. Once defined, the object is simply a modifier that can be called on operands to return a result. For a deferred modifier this result will always be a function; for an immediate modifier it could be anything.

Self-reference

If a block is assigned a name after it is created, this name can be used for recursion:

↗️
    Fact ← { 𝕩 Γ— (0⊸<)β—Ά1β€ΏFact 𝕩-1 }
    Fact 7
5040
    (Γ—Β΄1+↕) 7  # There's often a simpler solution than recursion
5040

This is somewhat unsatisfying because it is external to the function being defined, even though it doesn't depend on outside information. Instead, the special name π•Š can be used to refer to the function it appears in. This allows anonymous recursive functions to be defined.

↗️
    { 𝕩 Γ— (0⊸<)β—Ά1β€Ώπ•Š 𝕩-1 } 7
5040

For modifiers, 𝕣 refers to the containing modifier. π•Š makes the modifier a deferred modifier like 𝕨 and 𝕩 do, and refers to the derived function. For example, this tail-recursive factorial function uses the operand to accumulate a result, a task that is usually done with a second factorial_helper function in elementary Scheme.

↗️
    Fact_mod ← 1 { (0⊸<)β—ΆβŸ¨π•—, (𝕗×𝕩)_π•£βŸ© 𝕩-1 }
    Fact_mod 7
5040

Because 𝕣 only ever refers to a 1-modifier or 2-modifer, it can never make sense to refer to it as a function, and the uppercase letter ℝ is not recognized by BQN. In order to allow 𝕣 to be spelled as a 1-modifier _𝕣 or 2-modifier _𝕣_, it is treated as an ordinary identifier character, so it must be separated from letters or numbers by spaces.

Block headers

As a program becomes larger, it often becomes necessary to name inputs to blocks rather than just using special names. It can also become difficult to identify what kind of block is being defined, as it requires scanning through the block for special names. A block header, which is separated from the body of a block by a colon :, specifies the kind of block and can declare names for the block and its inputs.

Fact ← { F n:
  n Γ— (0⊸<)β—Ά1β€ΏF n-1
}

Its syntax mirrors an application of the block. As suggested by the positioning, the names given in a header apply only inside the block: for example F above is only defined inside the {} braces while Fact could be used either outside or inside. Some other possibilites are given below.

# A dyadic function that refers to itself as Func
{ l Func r:
  …

# A deferred 1-modifier with a list argument
{ Fn _apply ⟨a,b⟩:
  …

# A monadic function with no names given
{ π•Šπ•©:
  …

# An immediate or deferred 2-modifier
{ F _op_ val:
  …

In all cases special names still work just like in a headerless function. In this respect the effect of the header is the same as a series of assignments at the beginning of a function, such as the following translation of the second header above:

{ # Fn _apply ⟨a,b⟩:
  Fn ← 𝔽
  _apply ← _𝕣
  ⟨a,b⟩ ← 𝕩
  …

Unlike these assignments, the header also constrains what inputs the block can take: a monadic 1-modifier like the one above can't take a right operand or left argument, and consequently its body can't contain 𝔾 or 𝕨. Calling it with a left argument, or a right argument that isn't a two-element list, will result in an error.

Destructuring

Arguments, but not operands, allow destructuring like assignment does. While assignment only tolerates lists of variables, header destructuring also allows constants. The argument must match the given structure, including the constants where they appear, or an error results.

↗️
    Destruct ← { π•Š aβ€Ώ1β€ΏβŸ¨b,2⟩: a≍b }
    Destruct       5β€Ώ1β€ΏβŸ¨7,2⟩
⟨ 5 7 ⟩

Special names in headers

Any element of a function or modifier header can be left nameless by using the corresponding special name in that position, instead of an identifier. For example, the header 𝕨 𝔽_𝕣_𝔾 𝕩: incorporates as much vagueness as possible. It indicates a deferred 2-modifier, but provides no other information.

The name 𝕨 in this context can refer to either a left argument or no left argument, allowing a header with arguments to be used even for an ambiguous function. Recall that 𝕨 is the only token other than Β· that can have no value. If an identifier or list is given as the left argument, then the function must be called with a left argument.

Short headers

A header does not need to include all inputs, as shown by the F _op_ val: header above. The simplest case, when no inputs are given, is called a label. While it doesn't restrict the inputs, a label specifies the type of the block and gives an internal name that can be used to refer to it.

{ b:   # Block
{ π•Š:   # Function
{ _𝕣:  # 1-Modifier
{ _𝕣_: # 2-Modifier

For immediate blocks, this is the only type of header possible, and it must use an identifier as there is no applicable special name. However, the name can't be used: it doesn't make sense to refer to a value while it is still being computed!

Multiple bodies

Blocks that define functions and deferred modifiers can include more than one body, separated by semicolons ;. The body used for a particular evaluation is chosen based on the arguments the the block. One special case applies when there are exactly two bodies either without headers or with labels only: in this case, the first applies when there is one argument and the second when there are two.

↗️
    Ambiv ← { ⟨1,π•©βŸ© ; ⟨2,𝕨,π•©βŸ© }
    Ambiv 'a'
⟨ 1 'a' ⟩
    'a' Ambiv 'b'
⟨ 2 'a' 'b' ⟩

Bodies before the last two must have headers that include arguments. When a block that includes this type of header is called, its headers are checked in order for compatibility with the arguments. The first body with a compatible header is used.

↗️
    CaseAdd ← { 2π•Š3:0β€Ώ5 ; 2π•Šπ•©:⟨1,2+π•©βŸ© ; π•Šπ•©:2‿𝕩 }
    2 CaseAdd 3
⟨ 0 5 ⟩
    2 CaseAdd 4
⟨ 1 6 ⟩
      CaseAdd 4
⟨ 2 4 ⟩

If no header is compatible, the call results in an error.

↗️
    3 CaseAdd 3
ERROR

Case headers

A special rule allows for convenient case-matching syntax for one-argument functions. In any function header with one argument, the function name can be omitted as long as the argument is not a plain identifierβ€”it must be 𝕩 or a compound value like a list to distinguish it from an immediate block label.

Test ← {
  "abc": "string" ;
  ⟨2,b⟩: βŒ½π•©       ;
  5:     "number" ;
  𝕩:     "default"
}

These case-style headers function exactly the same as if they were preceded by π•Š, and can be mixed with other kinds of headers.

Predicates

Destructuring with a header is quite limited, only allowing matching structure and data with exact equality. A predicate, written with ?, allows you to test an arbitrary property before evaluating the rest of the body, and also serves as a limited kind of control flow. It can be thought of as an extension to a header, so that for example the following function requires the argument to have two elements and for the first to be less than the second before using the first body. Otherwise it moves to the next body, which is unconditional.

↗️
    CheckPair ← { π•ŠβŸ¨a,b⟩: a<b? "ok" ; "not ok" }

    CheckPair ⟨3,8⟩    # Fails destructuring
"ok"
    CheckPair ⟨1,4,5⟩  # Not a pair
"not ok"
    CheckPair ⟨3,¯1⟩   # Not ascending
"not ok"

The body where the predicate appears doesn't need to start with a header, and there can be other statements before it. In fact, ? functions just like a separator (like β‹„ or ,) with a side effect.

↗️
    { rβ†βŒ½π•© β‹„ 't'=βŠ‘r ? r ; 𝕩 }Β¨ "test"β€Ώ"this"
⟨ "tset" "this" ⟩

So r is the reversed argument, and if its first character (the last one in 𝕩) is 't' then it returns r, and otherwise we abandon that line of reasoning and return 𝕩. This sounds a lot like an if statement. And { a<b ? a ; b }, which computes a⌊b the hard way, shows how the syntax can be similar to a ternary operator. This is an immediate block with multiple bodies, something that makes sense with predicates but not headers. But ?; offers more possibilities. It can support any number of options, with multiple tests for each oneβ€”the structure below is "if _ and _ then _; else if _ then _; else _".

↗️
    Thing ← { 𝕩β‰₯3? 𝕩≀8? 2|𝕩 ; 𝕩=0? @ ; ∞ }

    (⊒ ≍ ThingΒ¨) ↕10  # Table of arguments and results
β”Œβ”€                     
β•΅ 0 1 2 3 4 5 6 7 8 9  
  @ ∞ ∞ 1 0 1 0 1 0 ∞  
                      β”˜

This structure is still constrained by the rules of block bodies: each instance of ; is a separate scope, so that variables defined before a ? don't survive past the ;.

↗️
    { 0=n←≠𝕩 ? ∞ ; n } "abc"
ERROR

This is the main drawback of predicates relative to guards in APL dfns (also written with ?), while the advantage is that it allows multiple expressions, or extra conditions, after a ?. It's not how I would have designed it if I just wanted to make a syntax for if statements, but it's a natural fit for the header system.