Mix

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Mix (↑ or ⊃ in nested APLs; Disclose or Open > in flat array languages) is a primitive monadic function which reduces the depth of its argument by combining all its element arrays into a single array. The shape of its result is then the shape of the argument followed by the common shape of the argument elements. Depending on the definition of Mix, the elements may be required to share the same shape, or they may be extended in shape (with fill elements) or rank to fit a common shape. In all APLs Mix allows a simple array argument, and returns it unchanged. Mix is a left inverse to both Enclose and Split, while the function First, which is identical to Mix on scalars, is a left inverse to Enclose only.

Examples

      x ← (0 1 2) (3 4 5) (9 9)
      ⍴x ⍝ x is a nested array, i.e. array of array
3
      ↑x ⍝ note that on the third row, 0 is padded to the end so the matrix can be orthogonal (non-ragged)
0 1 2
3 4 5
9 9 0
      ⍴↑x ⍝ ↑x is a non-nested 2-d array
3 3

Example application, to find the longest common prefix: 

      lcp←{(+/∧\(∧/⊃=⊢)¨↓⍉↑⍵)↑⊃⍵}
      lcp 'flowers' 'flow' 'flip'
fl
      lcp 'choco' 'choky' 'chom'
cho

History

Early experiments in array nesting usually defined a primitive to extract the contents of a nested scalar, but did not always specify how it should behave on an array. Jim Brown's "A Generalization of APL",[1] which would become APL2, defined such a primitive, "reveal", while stating that it "may be undefined" on non-scalar arrays.

Mix (↑) was first introduced by NARS in 1981.[2] It required all elements to have matching shape, and allowed a specified axis to define where in the result array their axes would be placed. In the same year, SHARP APL introduced the concept of a boxed array, along with the function Disclose (>).[3] In SHARP, Disclose was simply defined to be the inverse of Enclose, but given function rank 0 so that it would disclose each box in an argument and combine the results. Based on SHARP's definition of Rank, the arrays to be combined were required to have the same shape, and because of its flat array model, they also had to have the same type (numeric, character, or boxed), as otherwise the result array would not be representable.

APL2, released in 1984, used the same name Disclose as SHARP APL but introduced the glyph Right Shoe (⊃). It shared NARS's function axis definition, but extended the function to allow argument elements with different shapes as long as they had the same rank. Arrays with different shapes would be padded using fill elements as with Take to bring them to a common shape large enough to contain every element.

A further extension was added to SHARP APL in 1989,[4] and featured in J as a consequence of its definition of the Rank operator. In J, result arrays in a function applied with rank can have different ranks, and results with lower rank are brought to a common rank by adding leading 1s to the shape. This change was taken up by Dyalog APL in version 14.0, which introduced Rank following J's definition, in order to make Mix and Rank consistent. However, Dyalog differs from J in that it permits mixed arrays, so while in J an array containing a mix of numbers, characters, and boxes cannot be mixed, in Dyalog any array at all can be mixed.

Language support

Mix is only available in languages which allow nesting, either by boxes or nested arrays, so it is not present in early APLs such as APL\360 and APL.SV.

Languages Name Glyph Requirements NARS-style function axis APL2-style function axis
NARS Mix ↑ Shape Yes No
A+, SHARP <19.0 Disclose > Shape and type No No
SHARP APL Disclose > Type No No
APL2, APLX Disclose ⊃ Rank No Yes
NARS2000, Kap Disclose ⊃ None No No
Dyalog APL Mix ↑ or ⊃ (ML) None Yes Yes
J Open > Type No No
ngn/apl Mix ↑ None No No
dzaima/APL Merge ↑ Rank No No
BQN Merge > Shape No No

External links

Lessons

Documentation

References

  1. ↑ Jim Brown. "A Generalization of APL" (Ph.D. thesis). 1971.
  2. ↑ Carl M. Cheney. APL*PLUS Nested Arrays System (NARS reference manual). STSC. 1981.
  3. ↑ Ken Iverson. SATN-41: Composition and Enclosure. IPSA. 1981-06-20.
  4. ↑ IPSA. "SHARP APL Release 20.0: Guide for APL Programmers".


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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