Direct definition (operator)

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The Del glyph ∇ is more commonly used for dfn recursion or to indicate a defined function.
∇

Direct definition (∇), or function definition, is a primitive dyadic operator that defines an anonymous function based on source strings. Its left operand is the source for the function's dyadic case and the right is the source for the monadic case. It appears in NARS and several publications by Ken Iverson: an APL81 paper with Peter Wooster,[1] Rationalized APL, and A Dictionary of APL. Definitions vary, but in all cases, the format of a source string is similar to that used in the informal direct definition notation, with ⍺ and ⍵ for the left and right arguments. Another token, _ in NARS and ∆ in Iverson's papers, is added to indicate recursion. Multiple expressions can be used by separating with ⋄. Various forms of control flow have been defined.

In NARS, operands are single strings, and one of them may be replaced with a Jot (using this operator as an operand as with Outer Product) to indicate an undefined case and create a strictly monadic or dyadic function. Conditions are indicated with colons : as in direct definition notation, but the mechanism is extended: rather than a boolean condition placed between false and true cases, it's placed first, and returns a scalar or vector of natural numbers. Each number indicates one of the following groups of statements (separated by further :s and numbered starting at 0), and these groups will then be evaluated, with the result of the last evaluated expression returned from the function as a whole.

Documentation

References

  1. ↑ Ken Iverson and Peter Wooster. A function definition operator at APL81.
APL built-ins [edit]
Primitives (Timeline) Functions
Scalar
Monadic Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
Dyadic Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
Structural Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
Selection First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
Selector Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
Computational Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators Monadic Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
Quad names Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector
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