Replicate

Replicate (/, ⌿), or Copy (#) in J, is a dyadic function or monadic operator that copies each element of the right argument a given number of times, ordering the copies along a specified axis. Typically / is called Replicate while ⌿ is called "Replicate First" or an equivalent. Replicate is a widely-accepted extension of the function Compress, which requires the number of copies to be Boolean: each element is either retained (1 copy) or discarded (0 copies). Replicate with a Boolean left argument or operand may still be called "Compress".

Replicate is usually associated with Expand (\), and the two functions are related to Mask and Mesh. It is also closely related to the Indices function. It shares a glyph with Reduce even though Replicate is naturally a function and Reduce must be an operator. This incongruity is sometimes resolved by making Replicate an operator itself, and sometimes by function-operator overloading allowing both syntactic elements to coexist.

Outside of APL, filter typically provides the functionality of Compress, while Replicate has no common equivalent.

Examples

When used with a Boolean array (often called a "mask") on the left, Replicate is called Compress. It filters the right argument, returning only those elements which correspond to 1s in the provided mask.

      1 1 0 1 0 1 0 0 / 'compress'
cope

If the right argument is an array of indices generated by Iota, Replicate resembles the function Indices.

      1 1 0 0 1 / ⍳5
1 2 5

With an array of non-negative integers, Replicate copies each element of the right argument the corresponding number of times. As with Compress, these copies retain their original ordering, and the length of the result is the sum of the control array.

      0 3 0 0 2 0 1 0 2 / 'replicate'
eeeiiaee
      +/ 0 3 0 0 2 0 1 0 2
8
      ⍴ 0 3 0 0 2 0 1 0 2 / 'replicate'
8

Replicate usually allows scalar extension of the left argument, which results in every element being copied a fixed number of times.

      3 / 'replicate'
rrreeepppllliiicccaaattteee

Negative numbers

An extension introduced by NARS allows either positive or negative integers, where a negative number indicates that a fill element should be used instead of an element from the right argument. In this case the argument lengths must be equal (unless one side is a singleton). APL2 defined a different extension: negative numbers do not correspond to any element of the right argument, but still indicate that many fills should be inserted. In the APL2 extension the length of the right argument is the number of non-negative elements in the left argument. In both extensions the length of the result is the sum of the absolute value of the control array.

      0 2 ¯3 1 / ⍳4
2 2 0 0 0 4

Works in: NARS2000, Dyalog APL, APLX, ngn/apl

      0 2 ¯3 1 / ⍳3
2 2 0 0 0 3

Works in: APL2, APLX, GNU APL

The extensions are the same when the right argument is subject to singleton extension. This extension was usually supported before any extension to negative numbers, but would not typically be useful because v/s $\Leftrightarrow$ (+/v)/s where v is a non-negative integer vector and s is a singleton.

      1 ¯2 3 / 'a'
a  aaa

Works in: NARS2000, APL2, Dyalog APL, APLX, ngn/apl, GNU APL

High-rank arrays

Replicate works along a particular axis, which can be specified in languages with function axis and otherwise is the first axis for ⌿, and the last axis for / (except in A+, which uses / for the first-axis form and has no last-axis form).

      ⊢A ← 4 6⍴⎕A
ABCDEF
GHIJKL
MNOPQR
STUVWX
      1 0 0 4 0 2 / A
ADDDDFF
GJJJJLL
MPPPPRR
SVVVVXX
      0 2 1 1 ⌿ A
GHIJKL
GHIJKL
MNOPQR
STUVWX

APL2 further extends the singleton extension of the right argument, allowing it to have length 1 along the replication axis even if other axes have lengths not equal to 1.

      1 ¯2 3 / ⍪'abc'
a  aaa
b  bbb
c  ccc

Works in: APL2, Dyalog APL, APLX, ngn/apl, GNU APL

dzaima/APL expects arguments of ⌿ to have matching shape, and replicates the ravel of both.

Operator or function?

The syntax a / b is ambiguous: it may be an invocation of a dyadic function / with left argument a and right argument b, or of a monadic operator with operand a and right argument b. In early APLs there was no way to resolve this ambiguity, but with the extension of operators to allow arbitrary function operands instead of a specified set of primitive functions, the distinction becomes apparent: a function Replicate can be used as an operand while an operator Replicate cannot.

One test of Replicate's nature is to try Replicate Each with an expression such as 1 3 /¨ 'ab' 'cd'. If Replicate is implemented as an operator, it will be applied to the operand 1 3, and Each will be applied to the resulting derived function 1 3/.

      1 3 /¨ 'ab' 'cd'  
 abbb  cddd
      (1 3/)¨ 'ab' 'cd'
 abbb  cddd

Works in: SHARP APL (with ¨> in place of ¨), APL2, APLX

If Replicate is a function, then Each will apply to Replicate only, and the resulting derived function will be invoked monadically.

      1 3 /¨ 'ab' 'cd'  
 ab  cccddd
      1 3 (/¨) 'ab' 'cd'
 ab  cccddd

Works in: NARS2000, Dyalog APL, GNU APL

In early APLs such as APL\360, applying an operator to Compress will always result in a SYNTAX ERROR, because Compress is not an allowed operand of any operator. This is also the case in ngn/apl: although operators can apply to any function, Replicate cannot be used unless both arguments are immediately available. In both cases there is no way to determine whether Replicate "acts like a function" or "acts like an operator".

History

Compress was described in A Programming Language, where it was written with the symbols $/$ and $/\!\!/$. In Iverson notation compression was particularly important because Take and Drop could be performed only by compression with a prefix or suffix vector. It was included in APL\360, which changed the doubled slash to a barred slash ⌿, and allowed a specified axis and singleton extension on both sides (very briefly, singleton extension was allowed only for the right argument). The APL\360 definition continued to be included in APLs unchanged until 1980.

In 1980, Bob Bernecky introduced the extension Replicate to SHARP APL: he allowed an operand (since SHARP's Replicate is an operator) consisting of non-negative integers rather than just Booleans to indicate the number of times to copy. This extension was rapidly and widely adopted, starting with NARS in 1981, and is now a feature of the ISO/IEC 13751:2001 standard.

Two extensions to allow negative numbers in the left argument have been introduced, in each case specifying that the negative of a number indicates that many fill elements should appear in the result. In 1981 NARS specified that these fill elements replace the corresponding right argument element, so that the lengths of the left and right arguments are always equal, and extended Expand similarly. APL2, in 1984, made the opposite choice, so that the length of the right argument along the specified axis is equal to the number of non-negative elements on the left. APL2 also loosened the conformability requirements further than simply allowing singleton extension: it allowed a right argument with length 1 along the replication axis to be extended. Dyalog APL, created before APL2, adopted the NARS definition for negative elements but added APL2 conformability extension in version 13.1. Later APLX took advantage of the fact that the two negative number extensions can be distinguished by the length of the left argument, and implemented every NARS and APL2 extension.

A+ and J modified Replicate to fit leading axis theory. Rather than allow Replicate to operate on any axis they have only one Replicate function (in A+, /; in J, #) which works on the first axis—it copies major cells rather than elements. Both languages rejected the NARS extension to negative left arguments, but J introduced its own system to add fill elements by allowing complex numbers in the left argument, and removed the Expand function entirely. Arthur Whitney went on to make a more radical change in K, removing Replicate entirely in favor of Where.

Extension support

Here ">1" refers to the SHARP APL extension to non-negative integers, while "<0" refers to extension to negative integers in either NARS or APL2 style.

Language	Type	>1	<0		Conformability extension	Axis specification	Notes
			NARS	APL2
APL\360	Ambiguous	No	No		Single	Yes
SHARP APL	Operator	Yes	No		Scalar	Yes
NARS, NARS2000	Function	Yes	Yes	No	Single	Yes
Dyalog APL	Function	Yes	Yes	No	APL2 (13.1)	Yes
APL2	Operator	Yes	No	Yes	APL2	Yes
A+ (/)	Function	Yes	No		Single	No
J (#)	Function	Yes	No		Scalar	No	Complex left argument allowed
ISO/IEC 13751:2001	Function	Yes	No		Scalar	Yes
APLX	Operator	Yes	Yes	Yes	APL2	Yes
ngn/apl	Ambiguous	Yes	Yes	No	APL2	Yes	Implemented as an operator
GNU APL	Function	Yes	No	Yes	APL2	Yes
dzaima/APL (⌿)	Function	Yes	Yes	No	None	No

In each language without axis specification, there is only one form of Replicate, which always applies to the first axis or major cells—the last-axis form is discarded.

Outside of APL

While Replicate is rarely used in non-array programming languages, Compress is sometimes seen. Usually the same functionality is provided by the higher-order function filter, which an APLer might define as the monadic operator filter←{(⍺⍺¨ ⍵) / ⍵} on a vector argument.

While filter is similar to Compress, some extensions to the x86 instruction set are exactly equivalent to Compress on particular data types. In BMI2, the PEXT and PDEP instructions (parallel bit extract and deposit) are identical to Compress and Expand on the bits of a register argument. Indeed, Dyalog APL uses these instructions to implement those primitives (see Dyalog APL#Instruction set usage). The AVX-512 instructions VPCOMPRESSQ and VPEXPANDQ (and variations) are not only equivalent to Compress and Expand using a mask register for the Boolean argument and a vector register for the other argument, but are named after the APL functions. These instructions allow compression of 4-byte and 8-byte elements, and with AVX-512_VBMI2 support was added for 1-byte and 2-byte elements as well.

External Links

Lessons

• APL Cultivation

Documentation

• Dyalog
• NARS2000
• APLX
• J Dictionary, NuVoc

Other

• Marshall Lochbaum "Expanding Bits in Shrinking Time": On implementing Replicate of a Boolean array by a scalar.