Specification: BQN evaluation

This page describes the semantics of the code constructs whose grammar is given in grammar.md. The formation rules there are not named, and here they are identified by either the name of the term or by copying the rule entirely if there are several alternative productions.

Here we assume that the referent of each identifier, or equivalently the connections between identifiers, have been identified according to the scoping rules.

Programs and blocks

The result of parsing a valid BQN program is a PROGRAM, and the program is run by evaluating this term.

A PROGRAM or BODY is a list of STMTs, which are evaluated in program order. A result is always required for BODY nodes, and sometimes for PROGRAM nodes (for example, when loaded with •Import). If any identifiers in the node's scope are exported, or any of its statements is an EXPORT, then the result is the namespace created in order to evaluate the node. If a result is required but the namespace case doesn't apply, then the last STMT node must be an EXPR and its result is used. The statement EXPR evaluates some APL code and possibly assigns the results, while nothing evaluates any subject or Derv terms it contains but discards the results. An EXPORT statement performs no action.

A block consists of several BODY terms, some of which may have an accompanying header describing accepted inputs and how they are processed. An immediate block brImm can only have one BODY, and is evaluated by evaluating the code in it. Other types of blocks do not evaluate any BODY immediately, but instead return a function or modifier that obtains its result by evaluating a particular BODY. The BODY is identified and evaluated once the block has received enough inputs (operands or arguments), which for modifiers can take one or two calls: if two calls are required, then on the first call the operands are simply stored and no code is evaluated yet. Two calls are required if there is more than one BODY term, if the BODY contains the special names 𝕨𝕩𝕤𝕎𝕏𝕊, or if its header specifies arguments (the header-body combination is a _mCase or _cCase_). Otherwise only one is required.

To evaluate a block when enough inputs have been received, first the correct case must be identified. To do this, first each special case (FCase, _mCase, or _cCase_), excluding FCase nodes containing UndoHead, is checked in order to see if its arguments are strucurally compatible with the given arguments. That is, is headW is a subject, there must be a left argument matching that structure, and if headX is a subject, the right argument must match that structure. This means that 𝕨 not only matches any left argument but also no argument. The test for compatibility is the same as for multiple assignment described below, except that the header may contain constants, which must match the corresponding part of the given argument.If no special case matches, then an appropriate general case (FMain, _mMain, or _cMain_) is used: if there are two, the first is used with no left argument and the second with a left argument; if there are one, it is always used, and if there are none, an error results.

The only remaining step before evaluating the BODY is to bind the inputs and other names. Special names are always bound when applicable: 𝕨𝕩𝕤 if arguments are used, 𝕨 if there is a left argument, 𝕗𝕘 if operands are used, and _𝕣 and _𝕣_ for modifiers and combinators, respectively. Any names in the header are also bound, allowing multiple assignment for arguments.

If there is no left argument, but the BODY contains 𝕨 at the top level, then it is conceptually re-parsed with 𝕨 replaced by · to give a monadic version before application; this modifies the syntax tree by replacing some instances of arg with nothing. However, it also causes an error if, in a function that is called with no left argument, 𝕨 is used as an operand or list element, where nothing is not allowed by the grammar. The same effect can also be achieved dynamically by treating · as a value and checking for it during execution. If it is used as a left argument, then the function should instead be called with no left argument (and similarly in trains); it it is used as a right argument, then the function and its left argument are evaluated but rather than calling the function · is "returned" immediately; and if it is used in another context then it causes an error.

Assignment

An assignment is one of the four rules containing ASGN. It is evaluated by first evaluating the right-hand-side subExpr, FuncExpr, _m1Expr, or _m2Exp_ expression, and then storing the result in the left-hand-side identifier or identifiers. The result of the assignment expression is the result of its right-hand side. Except for subjects, only a lone identifier is allowed on the left-hand side and storage sets it equal to the result. For subjects, destructuring assignment is performed when an lhs is lhsList or lhsStr. Destructuring assignment is performed recursively by assigning right-hand-side values to the left-hand-side targets, with single-identifier assignment as the base case.

The right-hand-side value, here called v, in destructuring assignment must be a list (rank 1 array) or namespace. If it's a list, then each LHS_ENTRY node must be an LHS_ELT. The left-hand side is treated as a list of lhs targets, and matched to v element-wise, with an error if the two lists differ in length. If v is a namespace, then the left-hand side must be an lhsStr where every LHS_ATOM is an LHS_NAME, or an lhsList where every LHS_ENTRY is an LHS_NAME or lhs "⇐" LHS_NAME, so that it can be considered a list of LHS_NAME nodes some of which are also associated with lhs nodes. To perform the assignment, the value of each name is obtained from the namespace v, giving an error if v does not define that name. The value is assigned to the lhs node if present (which may be a destructuring assignment or simple subject assignment), and otherwise assigned to the same LHS_NAME node used to get it from v.

Modified assignment is the subject assignment rule lhs Derv "↩" subExpr. In this case, lhs should be evaluated as if it were a subExpr (the syntax is a subset of subExpr), and the result of the function application lhs Derv subExpr should be assigned to lhs, and is also the result of the modified assignment expression.

Expressions

We now give rules for evaluating an atom, Func, _mod1 or _mod2_ expression (the possible options for ANY). A literal or primitive sl, Fl, _ml, or _cl_ has a fixed value defined by the specification (literals and built-ins). An identifier s, F, _m, or _c_, if not preceded by atom ".", must have an associated variable due to the scoping rules, and returns this variable's value, or causes an error if it has not yet been set. If it is preceded by atom ".", then the atom node is evaluated first; its value must be a namespace, and the result is the value of the identifier's name in the namespace, or an error if the name is undefined. A parenthesized expression such as "(" _modExpr ")" simply returns the result of the interior expression. A braced construct such as BraceFunc is defined by the evaluation of the statements it contains after all parameters are accepted. Finally, a list "⟨" ⋄? ( ( EXPR ⋄ )* EXPR ⋄? )? "⟩" or ANY ( "‿" ANY )+ consists grammatically of a list of expressions. To evaluate it, each expression is evaluated in source order and their results are placed as elements of a rank-1 array. The two forms have identical semantics but different punctuation.

Rules in the table below are function and modifier evaluation.

In each case the constituent expressions are evaluated in reverse source order: Right, then Called, then Left. Then the expression's result is obtained by calling the Called value on its parameters. A left argument of nothing is not used as a parameter, leaving only a right argument in that case. The type of the Called value must be appropriate to the expression type, as indicated in the "Types" column. For function application, a data type (number, character, or array) is allowed. It is called simply by returning itself. Although the arguments are ignored in this case, they are still evaluated. A braced construct is evaluated by binding the parameter names given in columns L and R to the corresponding values. Then if all parameter levels present have been bound, its body is evaluated to give the result of application.

The following rules derive new functions or modifiers from existing ones.

As with applications, all expressions are evaluated in reverse source order before doing anything else. Then a result is formed without calling the center value. Its value in BQN is given in the rightmost column, using L, C, and R for the results of the expressions in the left, center, and right columns, respectively. For the first two rules (partial application), the given operand is bound to the 2-modifier: the result is a 1-modifier that, when called, calls the center 2-modifier with the bound operand on the same side it appeared on and the new operand on the remaining side. A train is a function that, when called, calls the right-hand function on all arguments, then the left-hand function, and calls the center function with these results as arguments. In a modifier partial application, the result will fail when applied if the center value does not have the 2-modifier type, and in a fork, it will fail if any component has a modifier type (that is, cannot be applied as a function). BQN implementations are not required to check for these types when forming the result of these expressions, but may give an error on formation even if the result will never be applied.