The BQN virtual machine and runtime

BQN's self-hosted compiler and runtime mean that only a small amount of native code is needed to run BQN on any given platform. The Javascript environment requires about 600 lines of Javascript code including system functions and performance improvements; probably around 250 would be required just to run the core language. This makes it fairly easy to port BQN to new platforms, allowing BQN to be embedded within other programming languages and interact with arrays or functions in those languages.

There's a short video introduction to the VM architecture thanks to Asher Mancinelli.

The way data is represented is part of the VM implementation: it can use native arrays or a custom data structure, depending on what the language supports. An initial implementation will be very slow, but can be improved by replacing functions from the BQN-based runtime with native code. As the VM system can be hard to work with if you're not familiar with it, I advise you to contact me to discuss implementing a VM if you are interested.


BQN source code is compiled to stack-based object code. This format is a list of numbers of unspecified precision (small precision will limit the length of list literals and number of locals per block, blocks, and constants). Previously it was encoded as bytes with the LEB128 format; while it no longer has anything to do with bytes it's called a "bytecode" because this is shorter than "object code".

Various VMs might interpret or further compile the bytecode: for example CBQN compiles to native code with function calls in x86 and interprets if this is unavailable, and the online version always compiles to JS. For reference, vm.bqn sticks to a simple design and should be easiest to read.


The complete bytecode for a program consists of the following:


Each entry in blocks is a list of the following properties:

Compilation separates blocks so that they are not nested in bytecode. A block consists of bodies, so that all compiled code is contained in some body of a block. The self-hosted compiler compiles the entire program into an immediate block, and the program is run by evaluating this block. Bodies are terminated with a RETN or RETD instruction.

When the block is evaluated depends on its type and immediateness. An immediate block (0,1) is evaluated as soon as it is pushed; a function (0,0) is evaluated when called on arguments, an immediate modifier (1 or 2, 1) is evaluated when called on operands, and a deferred modifier (1 or 2, 0) creates a derived function when called on operands and is evaluated when this derived function is called on arguments.

The last property can be a single number or a list of lists. A single number indicates the body to be executed, and is used only for blocks with exactly one body. If it's a list of lists, the length is 1 for a block without arguments and 2 or more for a block with arguments (function or deferred modifier). Each element is a list of body indices. After selecting the appropriate list, execution begins at the first body in the appropriate list, moving to the next one if a header test (SETH or PRED instruction) fails. If a test fails but there's no next body, block evaluation is an error.

The five possible cases for a function are monadic, dyadic, inverse monadic (π•ŠβΌx), inverse dyadic (wπ•ŠβΌx), and swapped-inverse dyadic (xπ•ŠΛœβΌw). The first two will always be provided, while the remaining three typically don't exist as they have to be specified with undo headers. The smallest length that covers all possible cases will be used.


Bodies in a block are separated by ;. Each entry in bodies is a list containing:

The starting index refers to the position in bytecode where execution starts in order to evaluate the block. Different bodies will always have the same set of special names, but the variables they define are unrelated, so of course they can have different counts. The given number of variables includes special names, but list of names and export mask don't.

The program's symbol list is included in the tokenization information t: it is 0βŠ‘2βŠ‘t. Since the entire program (the source code passed in one compiler call) uses this list, namespace field accesses can be performed with indices alone within a program. The symbol list is needed for cross-program access, for example if β€’BQN returns a namespace.


The following instructions are defined (those without names are tentatively reserved only). The ones emitted by the self-hosted BQN compiler are marked in the "used" column. Only those marked "X" are needed to support the compiler and self-hosted runtime. "NS" indicates instructions used only in programs with namespaces, "HE" is for headers : or predicates ?, and "HR" is for high-rank array notation [].

B Name Used Like Args Description
00 PUSH X I Push object I
01 DFND X I Localize and push block I
02 SYSV 00 I Push dynamic system value I
03 D Push return function for lexical depth D
06 POPS X Pop and discard top of stack
07 RETN X Returns top of stack
08 RETD NS 07 Return the running scope's namespace
0B LSTO X N Create length-N list
0C LSTM X 0B N Create length-N reference list
0D ARMO HR 0B N Create length-N merged array
0E ARMM HR 0D N Create length-N merged reference list
10 FN1C X Monadic function call
11 FN2C X Dyadic function call
12 FN1O X 10 Monadic call, checking for Β·
13 FN2O X 11 Dyadic call, checking for Β·
14 TR2D X Create 2-train
15 TR3D X Create 3-train
16 CHKV X Error if top of stack is Β·
17 TR3O X 15 Create 3-train, checking for Β·
1A MD1C X 1-modifier call
1B MD2C X 2-modifier call
1C MD2L Bind left operand to 2-modifier
1D MD2R Bind right operand to 2-modifier
20 VARO X D, I Push local variable I from D frames up
21 VARM X D, I Push local variable reference I from D frames up
22 VARU X 21 D, I Push and clear local variable I from D frames up
26 DYNO I Push named variable I
27 DYNM I Push named variable I reference
2A PRED HE 06 Check predicate and drop
2B VFYM HE Convert constant to matcher (for headers)
2C NOTM HE Push placeholder assignment matcher
2F SETH HE 30 Test and set header
30 SETN X Define variable
31 SETU X Change variable
32 SETM X Modify variable
33 SETC X Monadic modify variable
40 FLDO NS I Read field I from namespace
41 FLDM 40 I Push mutable field I from namespace
42 ALIM NS I Mutable target aliases field I

Stack effects for most instructions are given below. Instructions FN1O, FN2O, and TR3O are identical to FN1C, FN2C, and TR3D except that the indicated values in the higher-numbered instructions may be Β·. The non-checking instructions can be implemented using the checking ones, but avoiding the check could improve speed. VARU is identical to VARM but indicates that the local variable's value will never be used again, which may be useful for optimization.

B Name Stack effect Comments
00 PUSH β†’ (iβŠ‘consts)
01 DFND β†’ (iβŠ‘blocks) Also sets block's parent scope
06 POPS x β†’
07 RETN x β†’ x Returns from current block
08 RETD x? β†’ n Clears stack, dropping 0 or 1 value
0B LSTO x0 … xm β†’ ⟨x0 … xm⟩ N total variables (m=n-1)
0B ARMO x0 … xm β†’ [x0 … xm]
10 FN1C 𝕩 𝕀 β†’ (π•Š 𝕩) 12: 𝕩 may be Β·
11 FN2C 𝕩 𝕀 𝕨 β†’ (𝕨 π•Š 𝕩) 13: 𝕨 or 𝕩 may be Β·
14 TR2D h g β†’ (G H)
15 TR3D h g f β†’ (F G H) 17: F may be Β·
1A MD1C 𝕣 𝕗 β†’ (𝔽 _𝕣)
1B MD2C π•˜ 𝕣 𝕗 β†’ (𝔽 _𝕣_ 𝔾)
1C MD2L 𝕣 𝕗 β†’ (𝕗 _𝕣_)
1D MD2R π•˜ 𝕣 β†’ (_𝕣_ π•˜)
20 VARO β†’ x Local variable value
21 VARM β†’ r Local variable reference
2B VFYM c β†’ r Constant to match reference
2B NOTM β†’ r Constant to not-reference
30 SETN x r β†’ (r←x) r is a reference; 2F: no result
31 SETU x r β†’ (r↩x) r is a reference
32 SETM x f r β†’ (r F↩ x) r is a reference
33 SETC f r β†’ (r F↩) r is a reference
40 FLDO n β†’ n.i
42 ALIM r β†’ s Reference s gets field i and puts in r

Many instructions just call functions or modifiers or otherwise have fairly obvious implementations. Instructions to handle variables and blocks are more complicated (although very typical of bytecode representations for lexically-scoped languages) and are described in more detail below.


The bytecode representation is designed with the assumption that variables will be stored in frames, one for each time an instance of a block is run. A new frame is created when the block is evaluated (see #blocks) and in general has to be cleaned up by garbage collection, because a lexical closure might need to refer to the frame even after the corresponding block finishes. Lexical closures can form loops, so simple reference counting can leak memory, but it could be used in addition to less frequent tracing garbage collection or another strategy.

A frame is a mutable list of slots for variable values. It has slots for any special names that are available during the blocks execution followed by the local variables it defines. Special names use the ordering π•€π•©π•¨π•£π•—π•˜; the first three of these are available in non-immediate blocks while 𝕣 and 𝕗 are available in modifiers and π•˜ in 2-modifiers specifically.

When a block is pushed with DFND, an instance of the block is created, with its parent frame set to be the frame of the currently-executing block. Setting the parent frame when the block is first seen, instead of when it's evaluated, is what distinguishes lexical from dynamic scoping. If it's an immediate block, it's evaluated immediately, and otherwise it's pushed onto the stack. When the block is evaluated, its frame is initialized using any arguments passed to it, the next instruction's index is pushed onto the return stack, and execution moves to the first instruction in the block. When the RETN instruction is encountered, an index is popped from the return stack and execution returns to this location. As an alternative to maintaining an explicit return stack, a block can be implemented as a native function that creates a new execution stack and returns the value in it when the RETN instruction is reached. This approach uses the implementation language's call stack for the return stack.

Local variables are manipulated with the VARO (or VARU) and VARM instructions, which load the value of a variable and a reference to it (see the next section) respectively. These instructions reference variables by frame depth and slot index. The frame depth indicates in which frame the variable is found: the current frame has depth 0, its block's parent frame has depth 1, and so on. The slot index is an index within that frame.

Slots should be initialized with some indication they are not yet defined. The variable can be defined with SETN only if it hasn't been defined yet, and can be accessed with VARO or VARU or modified with SETU, SETM, or SETC only if it has been defined.


A variable reference indicates a particular frame slot in a way that's independent of the execution context. For example, it could be a pointer to the slot, or a reference to the frame along with the index of the slot. VARM pushes a variable reference to the stack.

A reference list is a list of variable references or reference lists. It's created with the LSTM instruction. In the Javascript VM there's no difference between a reference list and an ordinary BQN list other than the contents. The ARMM instruction makes a merged reference list, which matches an array of rank 1 or more by splitting it into cells.

The SETN, SETU, SETM, and SETC instructions set a value for a reference. If the reference is to a variable, they simply set its value. For a reference list, the value needs to be destructured. It must be a list of the same length, and each reference in the reference list is set to the corresponding element of the value list.

SETM and SETC additionally need to get the current value of a reference. For a variable reference this is its current value (with an error if it's not defined yet); for a reference list it's a list of the values of each reference in the list.

Opcode Slot must be Reads value first
SETN Unset
SETM Set Yes
SETC Set Yes


SETH is a modification of SETN for use in header destructuring. It differs in that it doesn't place its result on the stack (making it more like SETN followed by POPS), and that if the assignment fails because the reference and value don't conform then it moves on to the next eligible body in the block rather than giving an error. VFYM converts a BQN value c to a special reference: assigning a value v to it should check if v≑c but do no assignment. Only SETH needs to accept these references, and it should treat non-matching values as failing assignment. NOTM also creates a special reference, but it takes no inputs and the reference has no effectβ€”it always counts as matching but performs no assignment.

PRED drops the top value of the stack, but also checks whether it matches the number 1. If it does match, execution continues; if not, evaluation of the current body ends and evaluation moves to the next eligible body.


A namespace is the collection of variables in a particular scope, along with an association mapping some exported symbols (case- and underscore-normalized strings) to a subset of these. It can be represented using a frame for the variables, plus the variable name list and mask of exported variables from that block's properties in the bytecode. RETD finishes executing the block and returns the namespace for that execution.

The variable name list is split into a program-level list of names and a list of indices into these names: within-program field accesses can be done with the indices only while cross-program ones must use names. One way to check whether an access is cross-program is to compare the accessor's program-level name list with the one for the accessed namespace as references or pointers. Then a lookup should be performed as appropriate. A persistent lookup table is needed to make this efficient.

FLDO reads a field from a namespace. The parameter I is a program-level identifier index for this field. The VM must ensure that the field is exported, possibly by leaving unexported identifiers out of the namespace's lookup table. FLDM does the same but pushes a reference to the field, to be modified by assignment.

ALIM is used for aliased assignment such as ⟨aβ€Ώb⇐cβŸ©β†ns. It tags a reference with a namespace field, identified with a program-level index. A value assigned to the tagged reference must be a namespace. The relevant field is extracted, and then stored in the original reference.


Primitive functions and modifiers used in a program are stored in its consts array. The compiler needs to be passed a runtime with the value of every primitive so that these functions and modifiers are available.

While it's perfectly possible to implement the runtime from scratch, the pseudo-BQN files r0.bqn and r1.bqn implement the full runtime in terms of a core runtime consisting of a smaller number of much simpler functions. pr.bqn converts this file so that it can be compiled. It changes values in the core runtime to primitives and primitives to generated identifiers, so that the first 22 values in the output's consts array are exactly the core runtime, and no other primitives are required. The result is a list of two elements: first the list of all primitive values, and then a function that can be called to pass in two additional core functions used for inferred properties.

The contents of a core runtime are given below. The names given are those used in r1.bqn; the environment only provides a list of values and therefore doesn't need to use the same names. For named functions a description is given. For primitives, the implementation should match the BQN specification for that primitive on the specified domain, or the entire domain if left empty. They won't be called outside that domain except if there are bugs in the BQN sources.

Ind Name Description / restrictions
0 Type β€’Type
1 Fill Get or set the fill value for array 𝕩
2 Log ⋆⁼ (natural or base-𝕨 logarithm) for atomic arguments
3 GroupLen β‰ Β¨βŠ”π•© for a valid list 𝕩, with minimum length 𝕨
4 GroupOrd βˆΎβŠ”π•© provided 𝕨 is l GroupLen 𝕩 (any l)
5 !
6 + On one or two atoms
7 - On one or two atoms
8 Γ— On two atoms
9 Γ· On one or two atoms
10 ⋆ On one or two atoms
11 ⌊ On one atom
12 = On one value or two atoms
13 ≀ On two atoms
14 β‰’ For array 𝕩
15 β₯Š For array 𝕩 with no 𝕨 or 𝕨=β—‹(Γ—Β΄)≒𝕩
16 βŠ‘ For atom 𝕨 and list 𝕩
17 ↕ For natural number 𝕩
18 ⌜ On arrays
19 `
20 _fillBy_ 𝔽 with result fill computed using 𝔾
21 ⊘
22 ⎊
β€” Decompose β€’Decompose
β€” PrimInd Index for primitive 𝕩

To define the final two functions, call the second returned element as a function, with argument Decomposeβ€ΏPrimInd. The function PrimInd gives the index of 𝕩 in the list of all primitives (defined as glyphs in pr.bqn), or the length of the runtime if 𝕩 is not a primitive. The two functions are only needed for computing inferred properties, and are defined by default so that every value is assumed to be a primitive, and PrimInd performs a linear search over the returned runtime. If the runtime is used directly, then this means that without setting Decomposeβ€ΏPrimInd, function inferred properties will work slowly and for primitives only; if values from the runtime are wrapped then function inferred properties will not work at all. The compiler uses Under with compound right operands, so Decompose is needed to self-host.

The compiler returns a third function SetInv as well. This function is used to support inverses of non-primitives like β€’math.Sin or a block with an Undo header that the runtime has no way to identify. It's not needed for the runtime or compiler. The argument 𝕩 is a function that takes the runtime's current Inverse function (⁼ is defined as {i←Inverse𝕗⋄𝕨I𝕩}) and returns a new one. If given, 𝕨 does the same to SwapInverse, the function that inverts Fn˜. The result of SetInv is the new Inverse function.

Remember that + and - can also work on characters in some circumstances, under the rules of affine characters. Other arithmetic functions should only accept numbers. = must work on any atoms including functions and modifiers. ≀ must work on numbers and characters.

GroupLen and GroupOrd

GroupLen and GroupOrd, short for Group length and Group order, are used to implement Group (βŠ”) and also to grade small-range lists and invert permutations (the inverse of a permutation p is 1¨⊸GroupOrd p). Each of these only needs to work on lists of integers. Shown below are efficient implementations using BQN extended with the notation (iβŠ‘l) Fn↩ x meaning l ↩ Fn⟜x⌾(iβŠΈβŠ‘) l, where Fn is ⊒ if omitted. Since these special assignments only change one element of l, each can be a fast constant-time operation.

GroupLen ← {
  l ← Β―1 ⌈´ 𝕩
  r ← (l+1) β₯Š 0
  { (π•©βŠ‘r) +↩ 1 }⍟(0βŠΈβ‰€)Β¨ 𝕩
  (π•¨βŒˆβ‰ r) ↑ r

GroupOrd ← {
  # 𝕨 ≑ GroupLen 𝕩
  s ← 0 ∾ +` 𝕨
  r ← (Β―1βŠ‘s) β₯Š @   # Every element will be overwritten
  (↕≠𝕩) { ((π•©βŠ‘s)βŠ‘r)↩𝕨 β‹„ (π•©βŠ‘s)+↩1 }⍟(0βŠΈβ‰€)Β¨ 𝕩

Compiler arguments

The compiler takes the source code as 𝕩. The execution environment is passed as 𝕨, which can contain up to four values:

If 𝕨 has length greater than 4 it's assumed to be the runtime only. If the length is less than 4, empty defaults that don't define any values are used for the missing arguments.

The system-value function is passed a list of unique normalized names, meaning that each name is lowercase and contains no underscores. It should return a corresponding list of system values. Because system values are requested on each program run, a function that has access to context such as β€’path can construct appropriate system values on demand.

The variable list is used to create REPLs, but has other uses as well, such as allowing execution to take place within a surrounding scope. It consists of a list of normalized names. The corresponding depth list indicates the lexical depth of each of these, with 0 and -1 indicating that the variable should exist directly in the top-level scope. A typical interactive REPL uses only the value -1, because it allows variables to be redefinedβ€”that is, definition with ← won't fail but instead modify an existing variable. It maintains a single top-level environment to be used for all evaluations. When the programmer enters a line, it's compiled, then the environment and list of top-level names is extended according to the result.


The full BQN implementation is made up of the two components aboveβ€”virtual machine and core runtimeβ€”and the compiled runtime, compiler, and formatter. Since the compiler unlikely to work right away, I suggest initially testing the virtual machine on smaller pieces of code compiled by an existing, working, BQN implementation.

BQN sources can be compiled with cjs.bqn. It has two modes. If given a command-line argument r, c, or fmt, it compiles one of the source files. With any other command-line arguments, it will compile each one, and format it as a single line of output. The output is in a format designed for Javascript. VMs in other languages generally copy and modify cjs.bqn to work with the new language (for example cc.bqn in CBQN).


The following steps give a working BQN system, assuming a working VM and core runtime:

The compiler takes the runtime as 𝕨 and source code as 𝕩. To evaluate BQN source code, convert it into a BQN string (rank-1 array of characters), pass this string and runtime to the compiler, and evaluate the result as bytecode. Results can be formatted with the formatter for use in a REPL, or used from the implementation language.

Two formatter arguments Glyph and FmtNum are not part of the runtime. Glyph assumes 𝕩 is a primitive and returns the character (not string) that represents it, and FmtNum assumes 𝕩 is a number and returns a string representing it.


I recommend roughly the following sequence of tests to get everything working smoothly. It can be very difficult to figure out where in a VM things went wrong, so it's important to work methodically and make sure each component is all right before moving to the next. In order to run test cases before the compiler runs, I strongly recommend building an automated system to compile the test to bytecode using an existing BQN implementation, and run it with the VM being developed.

Because the compiler works almost entirely with lists of numbers, a correct fill implementation is not needed to run the compiler. Instead, you can define Fill as 0⊘⊒ and _fillBy_ as {π•˜β‹„π”½} to always use a fill element of 0.