Catamorphism

Catamorphism
You are encouraged to solve this task according to the task description, using any language you may know.

Reduce is a function or method that is used to take the values in an array or a list and apply a function to successive members of the list to produce (or reduce them to), a single value.

Task

Show how reduce (or foldl or foldr etc), work (or would be implemented) in your language.

See also

• Wikipedia article:   Fold
• Wikipedia article:   Catamorphism

11l

print((1..3).reduce((x, y) -> x + y))
print((1..3).reduce(3, (x, y) -> x + y))
print([1, 1, 3].reduce((x, y) -> x + y))
print([1, 1, 3].reduce(2, (x, y) -> x + y))

Output:

6
9
5
7

6502 Assembly

Works with: [Easy6502]

define catbuf $10
define catbuf_temp $12

ldx #0
ramloop:
txa
sta $00,x
inx
cpx #$10
bne ramloop	
;load zero page addresses $00-$0f with values equal
;to that address


ldx #0		;zero X
loop_cata:
lda $00,x	;load the zeroth element
clc
adc $01,x	;add the first to it.
inx		
inx		;inx twice. Otherwise the same element
		;would get added twice
sta catbuf_temp ;store in temp ram
lda catbuf	
clc
adc catbuf_temp ;add to previously stored value
sta catbuf	;store in result
cpx #$10	;is the range over?
bne loop_cata	;if not, loop again

ldx #$00
lda catbuf
sta $00,x	
;store the sum in the zeroth entry of the range

inx
lda #$00

;now clear the rest of zeropage, leaving only the sum

clear_ram:
sta $00,x
inx
cpx #$ff
bne clear_ram

ABAP

This works in ABAP version 7.40 and above.

report z_catamorphism.

data(numbers) = value int4_table( ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) ).

write: |numbers = { reduce string(
  init output = `[`
       index = 1
  for number in numbers
  next output = cond string(
         when index eq lines( numbers )
         then |{ output }, { number } ]|
         when index > 1
         then |{ output }, { number }|
         else |{ output } { number }| )
       index = index + 1 ) }|, /.

write: |sum(numbers) = { reduce int4(
  init result = 0
  for number in numbers
  next result = result + number ) }|, /.

write: |product(numbers) = { reduce int4(
  init result = 1
  for number in numbers
  next result = result * number ) }|, /.

data(strings) = value stringtab( ( `reduce` ) ( `in` ) ( `ABAP` ) ).

write: |strings = { reduce string(
  init output = `[`
       index = 1
  for string in strings
  next output = cond string(
         when index eq lines( strings )
         then |{ output }, { string } ]|
         when index > 1
         then |{ output }, { string }|
         else |{ output } { string }| )
       index = index + 1 ) }|, /.

write: |concatenation(strings) = { reduce string(
  init text = ``
  for string in strings
  next text = |{ text } { string }| ) }|, /.

Output:

numbers = [ 1, 2, 3, 4, 5 ]

sum(numbers) = 15

product(numbers) = 120

strings = [ reduce, in, ABAP ]

concatenation(strings) =  reduce in ABAP

Ada

with Ada.Text_IO;

procedure Catamorphism is
   
   type Fun is access function (Left, Right: Natural) return Natural;
   type Arr is array(Natural range <>) of Natural;
   
   function Fold_Left (F: Fun; A: Arr) return Natural is
      Result: Natural := A(A'First);
   begin
      for I in A'First+1 .. A'Last loop
	 Result := F(Result, A(I));
      end loop;
      return Result;
   end Fold_Left;
   
   function Max (L, R: Natural) return Natural is (if L > R then L else R);
   function Min (L, R: Natural) return Natural is (if L < R then L else R);     
   function Add (Left, Right: Natural) return Natural is (Left + Right);
   function Mul (Left, Right: Natural) return Natural is (Left * Right);
          
   package NIO is new Ada.Text_IO.Integer_IO(Natural);   
   
begin
   NIO.Put(Fold_Left(Min'Access, (1,2,3,4)), Width => 3);
   NIO.Put(Fold_Left(Max'Access, (1,2,3,4)), Width => 3);
   NIO.Put(Fold_Left(Add'Access, (1,2,3,4)), Width => 3);
   NIO.Put(Fold_Left(Mul'Access, (1,2,3,4)), Width => 3);
end Catamorphism;

Output:

  1  4 10 24

Aime

integer s;

s = 0;
list(1, 2, 3, 4, 5, 6, 7, 8, 9).ucall(add_i, 1, s);
o_(s, "\n");

Output:

45

ALGOL 68

# applies fn to successive elements of the array of values #
# the result is 0 if there are no values                   #
PROC reduce = ( []INT values, PROC( INT, INT )INT fn )INT:
     IF UPB values < LWB values
     THEN # no elements #
          0
     ELSE # there are some elements #
          INT result := values[ LWB values ];
          FOR pos FROM LWB values + 1 TO UPB values
          DO
              result := fn( result, values[ pos ] )
          OD;
          result
     FI; # reduce #

# test the reduce procedure #
BEGIN print( ( reduce( ( 1, 2, 3, 4, 5 ), ( INT a, b )INT: a + b ), newline ) ) # sum #
    ; print( ( reduce( ( 1, 2, 3, 4, 5 ), ( INT a, b )INT: a * b ), newline ) ) # product #
    ; print( ( reduce( ( 1, 2, 3, 4, 5 ), ( INT a, b )INT: a - b ), newline ) ) # difference #
END

Output:

        +15
       +120
        -13

APL

Reduce is a built-in APL operator, written as /.

      +/ 1 2 3 4 5 6 7
28
      ×/ 1 2 3 4 5 6 7
5040

For built-in functions, the seed value is automatically chosen to make sense.

      +/⍬
0
      ×/⍬
1
      ⌈/⍬ ⍝ this gives the minimum supported value
¯1.797693135E308

For user-supplied functions, the last element in the list is considered the seed. If F/ is called with a list of only one element, F itself is never called, and calling F/ with the empty list is an error.

      {⎕←'Input:',⍺,⍵ ⋄ ⍺+⍵}/ 1 2 3 4 5
Input: 4 5
Input: 3 9
Input: 2 12
Input: 1 14
15
      {⎕←'Input:',⍺,⍵ ⋄ ⍺+⍵}/ 1
1
      {⎕←'Input:',⍺,⍵ ⋄ ⍺+⍵}/ ⍬
DOMAIN ERROR

AppleScript

Translation of: JavaScript

Iteratively implemented foldl and foldr, using the same argument sequence as in the corresponding JavaScript array methods reduce() and reduceRight().

(Note that to obtain first-class functions from user-defined AppleScript handlers, we have to 'lift' them into script objects).

---------------------- CATAMORPHISMS ---------------------

-- the arguments available to the called function f(a, x, i, l) are
-- a: current accumulator value
-- x: current item in list
-- i: [ 1-based index in list ] optional
-- l: [ a reference to the list itself ] optional

-- foldl :: (a -> b -> a) -> a -> [b] -> a
on foldl(f, startValue, xs)
    tell mReturn(f)
        set v to startValue
        set lng to length of xs
        repeat with i from 1 to lng
            set v to |λ|(v, item i of xs, i, xs)
        end repeat
        return v
    end tell
end foldl

-- the arguments available to the called function f(a, x, i, l) are
-- a: current accumulator value
-- x: current item in list
-- i: [ 1-based index in list ] optional
-- l: [ a reference to the list itself ] optional

-- foldr :: (a -> b -> a) -> a -> [b] -> a
on foldr(f, startValue, xs)
    tell mReturn(f)
        set v to startValue
        set lng to length of xs
        repeat with i from lng to 1 by -1
            set v to |λ|(v, item i of xs, i, xs)
        end repeat
        return v
    end tell
end foldr


--- OTHER FUNCTIONS DEFINED IN TERMS OF FOLDL AND FOLDR --

-- concat :: [String] -> string
on concat(xs)
    foldl(my append, "", xs)
end concat


-- product :: Num a => [a] -> a
on product(xs)
    script
        on |λ|(a, b)
            a * b
        end |λ|
    end script
    
    foldr(result, 1, xs)
end product


-- str :: a -> String
on str(x)
    x as string
end str


-- sum :: Num a => [a] -> a
on sum(xs)
    script
        on |λ|(a, b)
            a + b
        end |λ|
    end script
    
    foldl(result, 0, xs)
end sum


--------------------------- TEST -------------------------
on run
    set xs to {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
    
    {sum(xs), product(xs), concat(map(str, xs))}
    
    --> {55, 3628800, "10987654321"}
end run


-------------------- GENERIC FUNCTIONS -------------------

-- append :: String -> String -> String
on append(a, b)
    a & b
end append


-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
    -- The list obtained by applying f
    -- to each element of xs.
    tell mReturn(f)
        set lng to length of xs
        set lst to {}
        repeat with i from 1 to lng
            set end of lst to |λ|(item i of xs, i, xs)
        end repeat
        return lst
    end tell
end map


-- Lift 2nd class handler function into 1st class script wrapper 
-- mReturn :: Handler -> Script
on mReturn(f)
    if class of f is script then
        f
    else
        script
            property |λ| : f
        end script
    end if
end mReturn

Output:

{55, 3628800, "12345678910"}

Arturo

; find the sum, with seed:0 (default)
print fold [1 2 3 4] => add

; find the product, with seed:1
print fold [1 2 3 4] .seed:1 => mul

Output:

10
24

BASIC

BASIC256

Translation of: Run BASIC

arraybase 1
global n
dim n = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

print "  +: "; " "; cat(10, "+")
print "  -: "; " "; cat(10, "-")
print "  *: "; " "; cat(10, "*")
print "  /: "; " "; cat(10, "/")
print "  ^: "; " "; cat(10, "^")
print "max: "; " "; cat(10, "max")
print "min: "; " "; cat(10, "min")
print "avg: "; " "; cat(10, "avg")
print "cat: "; " "; cat(10, "cat")
end

function min(a, b)
    if a < b then return a else return b
end function
function max(a, b)
    if a > b then return a else return b
end function

function cat(cont, op$)
    temp = n[1]
    temp$ = ""
    for i = 2 to cont
        if op$ = "+" then temp += n[i]
        if op$ = "-" then temp -= n[i]
        if op$ = "*" then temp *= n[i]
        if op$ = "/" then temp /= n[i]
        if op$ = "^" then temp = temp ^ n[i]
        if op$ = "max" then temp = max(temp, n[i])
        if op$ = "min" then temp = min(temp, n[i])
        if op$ = "avg" then temp += n[i]
        if op$ = "cat" then temp$ += string(n[i])
    next i
    if op$ = "avg" then temp /= cont
    if op$ = "cat" then temp = int(string(n[1]) + temp$)
    return temp
end function

Chipmunk Basic

Translation of: Run BASIC

Works with: Chipmunk Basic version 3.6.4

100 DIM n(10)
110 FOR i = 1 TO 10 : n(i) = i : NEXT i
120 SUB cat(cnt,op$)
130  temp = n(1)
140  FOR i = 2 TO cnt
150   IF op$ = "+" THEN temp = temp+n(i)
160   IF op$ = "-" THEN temp = temp-n(i)
170   IF op$ = "*" THEN temp = temp*n(i)
180   IF op$ = "/" THEN temp = temp/n(i)
190   IF op$ = "^" THEN temp = temp^n(i)
200   IF op$ = "max" THEN temp = FN MAX(temp,n(i))
210   IF op$ = "min" THEN temp = FN MIN(temp,n(i))
220   IF op$ = "avg" THEN temp = temp+n(i)
230   IF op$ = "cat" THEN temp$ = temp$+STR$(n(i))
240  NEXT i
250  IF op$ = "avg" THEN temp = temp/cnt
260  IF op$ = "cat" THEN temp = VAL(STR$(n(1))+temp$)
270  cat = temp
280 END SUB
290 '
300 PRINT "  +: ";cat(10,"+")
310 PRINT "  -: ";cat(10,"-")
320 PRINT "  *: ";cat(10,"*")
330 PRINT "  /: ";cat(10,"/")
340 PRINT "  ^: ";cat(10,"^")
350 PRINT "min: ";cat(10,"min")
360 PRINT "max: ";cat(10,"max")
370 PRINT "avg: ";cat(10,"avg")
380 PRINT "cat: ";cat(10,"cat")
390 END

QBasic

Works with: QBasic version 1.1

Translation of: Run BASIC

DIM SHARED n(10)
FOR i = 1 TO 10: n(i) = i: NEXT i

FUNCTION FNMIN (a, b)
IF (a < b) THEN FNMIN = a ELSE FNMIN = b
END FUNCTION
FUNCTION FNMAX (a, b)
IF (a < b) THEN FNMAX = b ELSE FNMAX = a
END FUNCTION

FUNCTION cat# (cont, op$)
temp = n(1)
FOR i = 2 TO cont
 IF op$ = "+" THEN temp = temp + n(i)
 IF op$ = "-" THEN temp = temp - n(i)
 IF op$ = "*" THEN temp = temp * n(i)
 IF op$ = "/" THEN temp = temp / n(i)
 IF op$ = "^" THEN temp = temp ^ n(i)
 IF op$ = "max" THEN temp = FNMAX(temp, n(i))
 IF op$ = "min" THEN temp = FNMIN(temp, n(i))
 IF op$ = "avg" THEN temp = temp + n(i)
NEXT i
IF op$ = "avg" THEN temp = temp / cont
cat = temp
END FUNCTION

PRINT "  +: "; " "; cat(10, "+")
PRINT "  -: "; " "; cat(10, "-")
PRINT "  *: "; " "; cat(10, "*")
PRINT "  /: "; " "; cat(10, "/")
PRINT "  ^: "; " "; cat(10, "^")
PRINT "min: "; " "; cat(10, "min")
PRINT "max: "; " "; cat(10, "max")
PRINT "avg: "; " "; cat(10, "avg")

True BASIC

SHARE n(10)
FOR i = 1 to 10
    LET n(i) = i
NEXT i

FUNCTION fnmin(a,b)
    IF (a < b) then LET fnmin = a else LET fnmin = b
END FUNCTION
FUNCTION fnmax(a,b)
    IF (a < b) then LET fnmax = b else LET fnmax = a
END FUNCTION

FUNCTION cat(cont, op$)
    LET temp = n(1)
    LET temp$ = ""
    FOR i = 2 TO cont
        IF op$ = "+" then LET temp = temp+n(i)
        IF op$ = "-" then LET temp = temp-n(i)
        IF op$ = "*" then LET temp = temp*n(i)
        IF op$ = "/" then LET temp = temp/n(i)
        IF op$ = "^" then LET temp = temp^n(i)
        IF op$ = "max" then LET temp = fnmax(temp,n(i))
        IF op$ = "min" then LET temp = fnmin(temp,n(i))
        IF op$ = "avg" then LET temp = temp+n(i)
        IF op$ = "cat" then LET temp$ = temp$ & str$(n(i))
    NEXT i
    IF op$ = "avg" then
       LET temp = temp / cont
    END IF
    IF op$ = "cat" then
       LET t$ = str$(n(1)) & temp$
       LET temp = VAL(t$)
    END IF
    LET cat = temp
END FUNCTION

PRINT "  +: "; " "; cat(10, "+")
PRINT "  -: "; " "; cat(10, "-")
PRINT "  *: "; " "; cat(10, "*")
PRINT "  /: "; " "; cat(10, "/")
PRINT "  ^: "; " "; cat(10, "^")
PRINT "min: "; " "; cat(10, "min")
PRINT "max: "; " "; cat(10, "max")
PRINT "avg: "; " "; cat(10, "avg")
PRINT "cat: "; " "; cat(10, "cat")
END

Yabasic

Translation of: Run BASIC

dim n(10)
for i = 1 to 10 : n(i) = i : next i
 
print "  +: ", " ", cat(10, "+")
print "  -: ", " ", cat(10, "-")
print "  *: ", " ", cat(10, "*")
print "  /: ", " ", cat(10, "/")
print "  ^: ", " ", cat(10, "^")
print "min: ", " ", cat(10, "min")
print "max: ", " ", cat(10, "max")
print "avg: ", " ", cat(10, "avg")
end
 
sub cat(cont,op$)
cat = n(1)
for i = 2 to cont
 if op$ = "+"  cat = cat + n(i)
 if op$ = "-"  cat = cat - n(i)
 if op$ = "*"  cat = cat * n(i) 
 if op$ = "/"  cat = cat / n(i)
 if op$ = "^"  cat = cat ^ n(i)
 if op$ = "max"  cat = max(cat,n(i))
 if op$ = "min"  cat = min(cat,n(i))
 if op$ = "avg"  cat = cat + n(i)
next i
if op$ = "avg"  cat = cat / cont
return cat
end sub

BBC BASIC

      DIM a(4)
      a() = 1, 2, 3, 4, 5
      PRINT FNreduce(a(), "+")
      PRINT FNreduce(a(), "-")
      PRINT FNreduce(a(), "*")
      END

      DEF FNreduce(arr(), op$)
      REM!Keep tmp, arr()
      LOCAL I%, tmp
      tmp = arr(0)
      FOR I% = 1 TO DIM(arr(), 1)
        tmp = EVAL("tmp " + op$ + " arr(I%)")
      NEXT
      = tmp

Output:

        15
       -13
       120

BCPL

get "libhdr"

let reduce(f, v, len, seed) = 
    len = 0 -> seed,
    reduce(f, v+1, len-1, f(!v, seed))

let start() be
$(  let add(x, y) = x+y
    let mul(x, y) = x*y
  
    let nums = table 1,2,3,4,5,6,7
    
    writef("%N*N", reduce(add, nums, 7, 0))
    writef("%N*N", reduce(mul, nums, 7, 1))
$)

Output:

28
5040

Binary Lambda Calculus

A minimal size (right) fold in lambda calculus is fold = \f\z (let go = \l.l(\h\t\z.f h (go t))z in go) which corresponds to the 69-bit BLC program

000001000110100000010110000000010111111110111001011111101111101101110

BQN

BQN has two different primitives for catamorphism:

• Fold(´): Works on lists only.
• Insert(˝): Works on arrays with higher rank.

Both of these primitives take a dyadic function, and an optional initial element.

•Show +´ 30‿1‿20‿2‿10
•Show +˝ 30‿1‿20‿2‿10
•Show tab ← (2+↕5) |⌜ 9+↕3
•Show +˝ tab

63
┌·    
· 63  
     ┘
┌─       
╵ 1 0 1  
  0 1 2  
  1 2 3  
  4 0 1  
  3 4 5  
        ┘
⟨ 9 7 12 ⟩

Bracmat

( ( fold
  =   f xs init first rest
    .   !arg:(?f.?xs.?init)
      & ( !xs:&!init
        |   !xs:%?first ?rest
          & !f$(!first.fold$(!f.!rest.!init))
        )
  )
&   out
  $ ( fold
    $ ( (=a b.!arg:(?a.?b)&!a+!b)
      . 1 2 3 4 5
      . 0
      )
    )
& (product=a b.!arg:(?a.?b)&!a*!b)
& out$(fold$(product.1 2 3 4 5.1))
);

Output:

15
120

C

#include <stdio.h>

typedef int (*intFn)(int, int);

int reduce(intFn fn, int size, int *elms)
{
    int i, val = *elms;
    for (i = 1; i < size; ++i)
        val = fn(val, elms[i]);
    return val;
}

int add(int a, int b) { return a + b; }
int sub(int a, int b) { return a - b; }
int mul(int a, int b) { return a * b; }

int main(void)
{
    int nums[] = {1, 2, 3, 4, 5};
    printf("%d\n", reduce(add, 5, nums));
    printf("%d\n", reduce(sub, 5, nums));
    printf("%d\n", reduce(mul, 5, nums));
    return 0;
}

Output:

15
-13
120

C#

var nums = Enumerable.Range(1, 10);

int summation = nums.Aggregate((a, b) => a + b);

int product = nums.Aggregate((a, b) => a * b);

string concatenation = nums.Aggregate(String.Empty, (a, b) => a.ToString() + b.ToString());

Console.WriteLine("{0} {1} {2}", summation, product, concatenation);

C++

#include <iostream>
#include <numeric>
#include <functional>
#include <vector>

int main() {
	std::vector<int> nums = { 1, 2, 3, 4, 5 };
	auto nums_added = std::accumulate(std::begin(nums), std::end(nums), 0, std::plus<int>());
	auto nums_other = std::accumulate(std::begin(nums), std::end(nums), 0, [](const int& a, const int& b) {
		return a + 2 * b;
	});
	std::cout << "nums_added: " << nums_added << std::endl;
	std::cout << "nums_other: " << nums_other << std::endl;
}

Output:

nums_added: 15
nums_other: 30

Clojure

For more detail, check Rich Hickey's blog post on Reducers.

; Basic usage
> (reduce * '(1 2 3 4 5))
120
; Using an initial value
> (reduce + 100 '(1 2 3 4 5))
115

CLU

% Reduction. 
% First type = sequence type (must support S$elements and yield R)
% Second type = right (input) datatype
% Third type = left (output) datatype 
reduce = proc [S,R,L: type] (f: proctype (L,R) returns (L),
                             id: L,
                             seq: S)
         returns (L)
         where S has elements: itertype (S) yields (R)

    for elem: R in S$elements(seq) do
        id := f(id, elem)
    end
    return(id)
end reduce

% This is necessary to get rid of the exceptions
add = proc (a,b: int) returns (int) return (a+b) end add
mul = proc (a,b: int) returns (int) return (a*b) end mul

% Usage
start_up = proc ()
    % abbreviation - reducing int->int->int function over an array[int]
    int_reduce = reduce[array[int], int, int]
    
    po: stream := stream$primary_output()
    nums: array[int] := array[int]$[1,2,3,4,5,6,7,8,9,10]
    
    % find the sum and the product using reduce
    sum: int := int_reduce(add, 0, nums)
    product: int := int_reduce(mul, 1, nums)
    
    stream$putl(po, "The sum of [1..10] is: " || int$unparse(sum))
    stream$putl(po, "The product of [1..10] is: " || int$unparse(product))
end start_up

Output:

The sum of [1..10] is: 55
The product of [1..10] is: 3628800

Common Lisp

; Basic usage
> (reduce #'* '(1 2 3 4 5))
120
; Using an initial value
> (reduce #'+ '(1 2 3 4 5) :initial-value 100)
115
; Using only a subsequence
> (reduce #'+ '(1 2 3 4 5) :start 1 :end 4)
9
; Apply a function to each element first
> (reduce #'+ '((a 1) (b 2) (c 3)) :key #'cadr)
6
; Right-associative reduction
> (reduce #'expt '(2 3 4) :from-end T)
2417851639229258349412352
; Compare with
> (reduce #'expt '(2 3 4))
4096

D

void main() {
    import std.stdio, std.algorithm, std.range, std.meta, std.numeric,
           std.conv, std.typecons;

    auto list = iota(1, 11);
    alias ops = AliasSeq!(q{a + b}, q{a * b}, min, max, gcd);

    foreach (op; ops)
        writeln(op.stringof, ": ", list.reduce!op);

    // std.algorithm.reduce supports multiple functions in parallel:
    reduce!(ops[0], ops[3], text)(tuple(0, 0.0, ""), list).writeln;
}

Output:

"a + b": 55
"a * b": 3628800
min(T1,T2,T...) if (is(typeof(a < b))): 1
max(T1,T2,T...) if (is(typeof(a < b))): 10
gcd(T): 1
Tuple!(int,double,string)(55, 10, "12345678910")

DCL

$ list = "1,2,3,4,5"
$ call reduce list "+"
$ show symbol result
$
$ numbers = "5,4,3,2,1"
$ call reduce numbers "-"
$ show symbol result
$
$ call reduce list "*"
$ show symbol result
$ exit
$
$ reduce: subroutine
$ local_list = 'p1
$ value = f$integer( f$element( 0, ",", local_list ))
$ i = 1
$ loop:
$  element = f$element( i, ",", local_list )
$  if element .eqs. "," then $ goto done
$  value = value 'p2 f$integer( element )
$  i = i + 1
$  goto loop
$ done:
$ result == value
$ exit
$ endsubroutine

Output:

$ @catamorphism
  RESULT == 15   Hex = 0000000F  Octal = 00000000017
  RESULT == -5   Hex = FFFFFFFB  Octal = 37777777773
  RESULT == 120   Hex = 00000078  Octal = 00000000170

Delphi

See Pascal.

Déjà Vu

This is a foldl:

reduce f lst init:
	if lst:
		f reduce @f lst init pop-from lst
	else:
		init

!. reduce @+ [ 1 10 200 ] 4
!. reduce @- [ 1 10 200 ] 4

Output:

215
-207

EchoLisp

;; rem : the foldX family always need an initial value
;; fold left a list 
(foldl + 0 (iota 10)) ;; 0 + 1 + .. + 9
  → 45

;; fold left a sequence 
(lib 'sequences)
(foldl * 1 [ 1 .. 10])
    → 362880 ;; 10!

;; folding left and right
(foldl / 1 ' ( 1 2 3 4))
    → 8/3
(foldr / 1 '(1 2 3 4))
    → 3/8

;;scanl gives the list (or sequence) of intermediate values :
(scanl * 1 '( 1 2 3 4 5))
   → (1 1 2 6 24 120)

Elena

ELENA 5.0 :

import system'collections;
import system'routines;
import extensions;
import extensions'text;
 
public program()
{
    var numbers := new Range(1,10).summarize(new ArrayList());
 
    var summary := numbers.accumulate(new Variable(0), (a,b => a + b));
 
    var product := numbers.accumulate(new Variable(1), (a,b => a * b));
 
    var concatenation := numbers.accumulate(new StringWriter(), (a,b => a.toPrintable() + b.toPrintable()));
 
    console.printLine(summary," ",product," ",concatenation)
}

Output:

55 362880 12345678910

Elixir

iex(1)> Enum.reduce(1..10, fn i,acc -> i+acc end)
55
iex(2)> Enum.reduce(1..10, fn i,acc -> i*acc end)
3628800
iex(3)> Enum.reduce(10..-10, "", fn i,acc -> acc <> to_string(i) end)
"109876543210-1-2-3-4-5-6-7-8-9-10"

Erlang

Translation of: Haskell

-module(catamorphism).

-export([test/0]).

test() ->
	Nums = lists:seq(1,10),
	Summation = 
		lists:foldl(fun(X, Acc) -> X + Acc end, 0, Nums),
	Product = 
		lists:foldl(fun(X, Acc) -> X * Acc end, 1, Nums),
	Concatenation = 
		lists:foldr(
			fun(X, Acc) -> integer_to_list(X) ++ Acc end, 
			"", 
			Nums),
	{Summation, Product, Concatenation}.

Output:

{55,3628800,"12345678910"}

Excel

LAMBDA

Excel provides a good number of standard catamorphisms like SUM, PRODUCT, LEN etc out of the box, but in recent builds of Excel we can write more general catamorphisms as LAMBDA expressions, and bind names to them in the WorkBook Name Manager.

Excel's compound data type is a non-empty array, for which we could write, for example, specialised column or row instances of fold, whether rightward or leftward.

Here is an example of binding the name FOLDLROW to a left fold over a row of Excel cells.

As an example of a binary operator to fold, with an accumulator, over a series of character values, we can define a custom:

updateBracketDepth(accumulator)(character) which:

1. Increments the nesting depth given a "[" character
2. reduces it given a "]" character
3. leaves the nesting depth unchanged for any other character
4. updates the accumulator no further if the nesting depth ever becomes negative.

or for a simple bracket count, we could just define a:

bracketCount(accumulator)(character) which:

1. Increments the integer accumulator value on each "[" or "]"
2. Leaves the accumulator unchanged for other characters.

(See LAMBDA: The ultimate Excel worksheet function)

Works with: Office 365 betas 2021

FOLDROW
=LAMBDA(op,
    LAMBDA(a,
        LAMBDA(xs,
            LET(
                b, op(a)(HEADROW(xs)),

                IF(1 < COLUMNS(xs),
                    FOLDROW(op)(b)(
                        TAILROW(xs)
                    ),
                    b
                )
            )
        )
    )
)


updatedBracketDepth
=LAMBDA(depth,
    LAMBDA(c,
        IF(0 <= depth,
            IF("[" = c,
                1 + depth,
                IF("]" = c,
                    depth - 1,
                    depth
                )
            ),
            depth
        )
    )
)


bracketCount
=LAMBDA(a,
    LAMBDA(c,
        IF(ISNUMBER(FIND(c, "[]", 1)),
            1 + a,
            a
        )
    )
)


HEADROW
=LAMBDA(xs,
    LET(REM, "The first item of each row in xs",

        INDEX(
            xs,
            SEQUENCE(ROWS(xs)),
            SEQUENCE(1, 1)
        )
    )
)


TAILROW
=LAMBDA(xs,
    LET(REM,"The tail of each row in the grid",
        n, COLUMNS(xs) - 1,

        IF(0 < n,
            INDEX(
                xs,
                SEQUENCE(ROWS(xs), 1, 1, 1),
                SEQUENCE(1, n, 2, 1)
            ),
            NA()
        )
    )
)


CHARSROW
=LAMBDA(s,
    MID(s,
        SEQUENCE(1, LEN(s), 1, 1),
        1
    )
)

Output:

	fx	=FOLDROW( updatedBracketDepth )( 0 )( CHARSROW(C2) )
	A	B	C
1		Final bracket nesting depth	Sample string
2		0	[simply bracketed]
3		1	[[ ]
4		-1	[ ]]
5		0	[[[ [] ]]]
6		0	[ [[[ [] ]]] [[[ ]]] [[[ [] ]]] ]
7		1	[ [[[ [ ]]] [[[ ]]] [[[ [] ]]] ]
8		-1	][ [[[ [ ]]] [[[ ]]] [[[ [] ]]] ]

Or for a simple count of bracket characters, ignoring other characters:

	fx	=FOLDROW( bracketCount )( 0 )( CHARSROW(C2) )
	A	B	C
1		Bracket character count	Sample string
2		2	[simply bracketed]
3		3	[[ ]
4		3	[ ]]
5		8	[[[ [] ]]]
6		24	[ [[[ [] ]]] [[[ ]]] [[[ [] ]]] ]
7		23	[ [[[ [ ]]] [[[ ]]] [[[ [] ]]] ]
8		24	][ [[[ [ ]]] [[[ ]]] [[[ [] ]]] ]

F#

In the REPL:

> let nums = [1 .. 10];;

val nums : int list = [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]

> let summation = List.fold (+) 0 nums;;

val summation : int = 55

> let product = List.fold (*) 1 nums;;

val product : int = 3628800

> let concatenation = List.foldBack (fun x y -> x + y) (List.map (fun i -> i.ToString()) nums) "";;

val concatenation : string = "12345678910"

Factor

{ 1 2 4 6 10 } 0 [ + ] reduce .

Output:

23

Forth

Forth has three traditions for iterating over the members of a data structure. Under the first, the data structure has words that help you navigate over it and normal Forth looping structures are used. Under the second, the data structure has dedicated looping words and you supply the code that's run for each member. Under the third, the data structure has a loop-over-members word that accepts a function to be run against each member.

There's no need to distinguish between the different kinds of looping ("this one collects function returns into a list; this one threads an accumulator between the function-calls; this one threads two accumulators through the function-calls; this one expects no return values whatsoever from the function-calls") because in Forth all that the looping words have to do is make the data stack available for the function's use. When that's the case, all of these variations, that are so important in other languages, are functionally equivalent.

Although it's possible to have a generic higher-order word that can operate under all kinds of data structures -- this just requires that one settle on an object system and then derive a collections library from it -- this is rarely done. Typically each data structure has its own looping words.

To demonstrate the above points we'll just loop over the bytes of a string.

Some helper words for these examples:

: lowercase? ( c -- f )
  [char] a [ char z 1+ ] literal within ;

: char-upcase ( c -- C )
  dup lowercase? if bl xor then ;

Using normal looping words:

: string-at ( c-addr u +n -- c )
  nip + c@ ;
: string-at! ( c-addr u +n c -- )
  rot drop  -rot  + c! ;

: type-lowercase ( c-addr u -- )
  dup 0 ?do
    2dup i string-at  dup lowercase?  if emit else drop then
  loop  2drop ;

: upcase ( 'string' -- 'STRING' )
  dup 0 ?do
    2dup 2dup  i string-at  char-upcase  i swap string-at!
  loop ;

: count-lowercase ( c-addr u -- n )
  0 -rot dup 0 ?do
    2dup i string-at  lowercase? if rot 1+ -rot then
  loop  2drop ;

Briefly, a variation:

: next-char ( a +n -- a' n' c -1 )  ( a 0 -- 0 )
  dup if 2dup  1 /string  2swap drop c@ true
  else 2drop 0 then ;

: type-lowercase ( c-addr u -- )
  begin next-char while
    dup lowercase? if emit else drop then
  repeat ;

Using dedicated looping words:

: each-char[ ( c-addr u -- )
  postpone BOUNDS postpone ?DO
  postpone I postpone C@ ;  immediate

  \ interim code: ( c -- )

: ]each-char ( -- )
  postpone LOOP ;  immediate

: type-lowercase ( c-addr u -- )
  each-char[ dup lowercase? if emit else drop then ]each-char ;

: upcase ( 'string' -- 'STRING' )
  2dup each-char[ char-upcase i c! ]each-char ;

: count-lowercase ( c-addr u -- n )
  0 -rot each-char[ lowercase? if 1+ then ]each-char ;

Using higher-order words:

: each-char ( c-addr u xt -- )
  {: xt :}  bounds ?do
    i c@ xt execute
  loop ;

: type-lowercase ( c-addr u -- )
  [: dup lowercase? if emit else drop then ;]
  each-char ;

\ producing a new string
: upcase ( 'string' -- 'STRING' )
  dup cell+ allocate throw -rot
  [: ( new-string-addr c -- new-string-addr )
    upcase over c+! ;] each-char  $@ ;

: count-lowercase ( c-addr u -- n )
  0 -rot [: lowercase? if 1+ then ;] each-char ;

In these examples COUNT-LOWERCASE updates an accumulator, UPCASE (mostly) modifies the string in-place, and TYPE-LOWERCASE performs side-effects and returns nothing to the higher-order word.

Fortran

If Fortran were to offer the ability to pass a parameter "by name", as is used in Jensen's device, then the code might be something like

      SUBROUTINE FOLD(t,F,i,ist,lst)
       INTEGER t
       BYNAME F
        DO i = ist,lst
          t = F
        END DO
      END SUBROUTINE FOLD      !Result in temp.

      temp = a(1); CALL FOLD(temp,temp*a(i),i,2,N)

Here, the function manifests as the expression that is the second parameter of subroutine FOLD, and the "by name" protocol for parameter F means that within the subroutine whenever there is a reference to F, its value is evaluated afresh in the caller's environment using the current values of temp and i as modified by the subroutine - they being passed by reference so that changes within the subroutine affect the originals. An evaluation for a different function requires merely another statement with a different expression.

Fortran however does not provide such a facility. Any parameter that is an expression is evaluated once in the caller's environment, the result placed in temporary storage, and the address of that storage location is passed to the subroutine. Repeated references to that parameter will elicit the same value. But there is special provision for passing a function to a routine, involving the special word EXTERNAL. For every different function in mind, one must diligently supply a name, and work through the overhead of declaring each such function. There is an additional word, INTRINSIC, for use when an intrinsic function (such as SIN) is to be passed as such a parameter since it will appear as its name only, and with the absence of the (...) that would be used for the function's parameters when in an arithmetic expression, it would otherwise be taken as being the name of an ordinary variable.

Here is such an arrangement, in the style of F77 though somewhat affected by F90 in that the END statement names the routine being ended. Similarly, to abate petty complaints about the types of the functions being undeclared, explicit types are specified, though unselecting the compiler diagnostic for that would match the habits of earlier compilers. Also in F90 is the MODULE protocol which involves rather more organised checking of types and additional facilities for arrays so that N need not be passed because secret additional parameters do so.

However, only programmer diligence in devising functions with the correct type of result and the correct type and number of parameters will evade mishaps. Note that the EXTERNAL statement does not specify the number or type of parameters. If the function is invoked multiple times within a subroutine, the compiler may check for consistency. This may cause trouble when some parameters are optional so that different invocations do not match.

The function's name is used as a working variable within the function (as well as it holding the function's value on exit) so that the expression F(IFOLD,A(I)) is not a recursive invocation of function IFOLD because there are no (parameters) appended to the function's name. Earlier compilers did not allow such usage so that a separate working variable would be required.

      INTEGER FUNCTION IFOLD(F,A,N)	!"Catamorphism"...
       INTEGER F	!We're working only with integers.
       EXTERNAL F	!This is a function, not an array.
       INTEGER A(*)	!An 1-D array, of unspecified size.
       INTEGER N	!The number of elements.
       INTEGER I	!A stepper.
        IFOLD = 0		!A default value.
        IF (N.LE.0) RETURN	!Dodge silly invocations.
        IFOLD = A(1)		!The function is to have two arguments.
        IF (N.EQ.1) RETURN	!So, if there is only one element, silly.
        DO I = 2,N		!Otherwise, stutter along the array.
          IFOLD = F(IFOLD,A(I))		!Applying the function.
        END DO			!On to the next element.
      END FUNCTION IFOLD!Thus, F(A(1),A(2)), or F(F(A(1),A(2)),A(3)), or F(F(F(A(1),A(2)),A(3)),A(4)), etc.

      INTEGER FUNCTION IADD(I,J)
       INTEGER I,J
        IADD = I + J
      END FUNCTION IADD

      INTEGER FUNCTION IMUL(I,J)
       INTEGER I,J
        IMUL = I*J
      END FUNCTION IMUL

      INTEGER FUNCTION IDIV(I,J)
       INTEGER I,J
        IDIV = I/J
      END FUNCTION IDIV

      INTEGER FUNCTION IVID(I,J)
       INTEGER I,J
        IVID = J/I
      END FUNCTION IVID

      PROGRAM POKE
      INTEGER ENUFF
      PARAMETER (ENUFF = 6)
      INTEGER A(ENUFF)
      PARAMETER (A = (/1,2,3,4,5,6/))
      INTEGER MSG
      EXTERNAL IADD,IMUL,IDIV,IVID	!Warn that these are the names of functions.

      MSG = 6	!Standard output.
      WRITE (MSG,1) ENUFF,A
    1 FORMAT ('To apply a function in the "catamorphic" style ',
     1 "to the ",I0," values ",/,(20I3))

      WRITE (MSG,*) "Iadd",IFOLD(IADD,A,ENUFF)
      WRITE (MSG,*) "Imul",IFOLD(IMUL,A,ENUFF)
      WRITE (MSG,*) "Idiv",IFOLD(IDIV,A,ENUFF)
      WRITE (MSG,*) "Ivid",IFOLD(IVID,A,ENUFF)
      END PROGRAM POKE

Output:

To apply a function in the "catamorphic" style to the 6 values
  1  2  3  4  5  6
 Iadd          21
 Imul         720
 Idiv           0
 Ivid           6

FreeBASIC

' FB 1.05.0 Win64

Type IntFunc As Function(As Integer, As Integer) As Integer  
 
Function reduce(a() As Integer, f As IntFunc) As Integer
   '' if array is empty or function pointer is null, return 0 say
   If UBound(a) = -1 OrElse f = 0 Then Return 0 
   Dim result As Integer = a(LBound(a))
   For i As Integer = LBound(a) + 1 To UBound(a)
     result = f(result, a(i)) 
   Next
   Return result
End Function

Function add(x As Integer, y As Integer) As Integer
  Return x + y
End Function

Function subtract(x As Integer, y As Integer) As Integer
  Return x - y
End Function

Function multiply(x As Integer, y As Integer) As Integer
  Return x * y
End Function

Function max(x As Integer, y As Integer) As Integer
  Return IIf(x > y, x, y)
End Function

Function min(x As Integer, y As Integer) As Integer
  Return IIf(x < y, x, y)
End Function

Dim a(4) As Integer = {1, 2, 3, 4, 5}
Print "Sum is        :"; reduce(a(), @add)
Print "Difference is :"; reduce(a(), @subtract)
Print "Product is    :"; reduce(a(), @multiply)
Print "Maximum is    :"; reduce(a(), @max)
Print "Minimum is    :"; reduce(a(), @min)
Print "No op is      :"; reduce(a(), 0)
Print
Print "Press any key to quit"
Sleep

Output:

Sum is        : 15
Difference is :-13
Product is    : 120
Maximum is    : 5
Minimum is    : 1
No op is      : 0

Go

package main

import (
	"fmt"
)

func main() {
	n := []int{1, 2, 3, 4, 5}

	fmt.Println(reduce(add, n))
	fmt.Println(reduce(sub, n))
	fmt.Println(reduce(mul, n))
}

func add(a int, b int) int { return a + b }
func sub(a int, b int) int { return a - b }
func mul(a int, b int) int { return a * b }

func reduce(rf func(int, int) int, m []int) int {
	r := m[0]
	for _, v := range m[1:] {
		r = rf(r, v)
	}
	return r
}

Output:

15
-13
120

Groovy

Groovy provides an "inject" method for all aggregate classes that performs a classic tail-recursive reduction, driven by a closure argument. The result of each iteration (closure invocation) is used as the accumulated valued for the next iteration. If a first argument is provided as well as a second closure argument, that first argument is used as a seed accumulator for the first iteration. Otherwise, the first element of the aggregate is used as the seed accumulator, with reduction iteration proceeding across elements 2 through n.

def vector1 = [1,2,3,4,5,6,7]
def vector2 = [7,6,5,4,3,2,1]
def map1 = [a:1, b:2, c:3, d:4]

println vector1.inject { acc, val -> acc + val }       // sum
println vector1.inject { acc, val -> acc + val*val }   // sum of squares
println vector1.inject { acc, val -> acc * val }       // product
println vector1.inject { acc, val -> acc<val?val:acc } // max
println ([vector1,vector2].transpose().inject(0) { acc, val -> acc + val[0]*val[1] }) //dot product (with seed 0)

println (map1.inject { Map.Entry accEntry, Map.Entry entry ->     // some sort of weird map-based reduction
    [(accEntry.key + entry.key):accEntry.value + entry.value ].entrySet().toList().pop()
})

Output:

28
140
5040
7
84
abcd=10

Haskell

main :: IO ()
main =
  putStrLn . unlines $
  [ show . foldr (+)    0  -- sum
  , show . foldr (*)    1  -- product
  , foldr ((++) . show) "" -- concatenation
  ] <*>
  [[1 .. 10]]

Output:

55
3628800
12345678910

and the generality of folds is such that if we replace all three of these (function, identity) combinations ((+), 0), ((*), 1) ((++), "") with the Monoid operation mappend (<>) and identity mempty, we can still obtain the same results:

import Data.Monoid

main :: IO ()
main =
  let xs = [1 .. 10]
  in (putStrLn . unlines)
       [ (show . getSum     . foldr (<>) mempty) (Sum     <$> xs)
       , (show . getProduct . foldr (<>) mempty) (Product <$> xs)
       , (show .              foldr (<>) mempty) (show    <$> xs) 
       , (show .              foldr (<>) mempty) (words
                     "Love is one damned thing after each other")
       ]

Output:

55
3628800
"12345678910"
"Loveisonedamnedthingaftereachother"

Also available are foldl1 and foldr1 which implicitly take first element as starting value. However they are not safe as they fail on empty lists.

Prelude folds work only on lists, module Data.Foldable a typeclass for more general fold - interface remains the same.

Icon and Unicon

Works in both languages:

procedure main(A)
    write(A[1],": ",curry(A[1],A[2:0]))
end

procedure curry(f,A)
    r := A[1]
    every r := f(r, !A[2:0])
    return r
end

Sample runs:

->cata + 3 1 4 1 5 9
+: 23
->cata - 3 1 4 1 5 9
-: -17
->cata \* 3 1 4 1 5 9
*: 540
->cata "||" 3 1 4 1 5 9
||: 314159

J

Solution:

    /

Example:

   +/ 1 2 3 4 5
15
   */ 1 2 3 4 5
120
   !/ 1 2 3 4 5  NB.  "n ! k" is "n choose k"
45

Insert * into 1 2 3 4 5 becomes 1 * 2 * 3 * 4 * 5

evaluated right to left

1 * 2 * 3 * 20
1 * 2 * 60
1 * 120
120

What are the implications for -/  ? For %/  ?

Java

Works with: Java version 8

import java.util.stream.Stream;

public class ReduceTask {

    public static void main(String[] args) {
        System.out.println(Stream.of(1, 2, 3, 4, 5).mapToInt(i -> i).sum());
        System.out.println(Stream.of(1, 2, 3, 4, 5).reduce(1, (a, b) -> a * b));
    }
}

Output:

15
120

JavaScript

ES5

var nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10];

function add(a, b) {
    return a + b;
}

var summation = nums.reduce(add);

function mul(a, b) {
    return a * b;
}

var product = nums.reduce(mul, 1);

var concatenation = nums.reduce(add, "");

console.log(summation, product, concatenation);

Note that the JavaScript Array methods include a right fold ( .reduceRight() ) as well as a left fold:

(function (xs) {
    'use strict';

    // foldl :: (b -> a -> b) -> b -> [a] -> b
    function foldl(f, acc, xs) {
        return xs.reduce(f, acc);
    }

    // foldr :: (b -> a -> b) -> b -> [a] -> b
    function foldr(f, acc, xs) {
        return xs.reduceRight(f, acc);
    }

    // Test folds in both directions
    return [foldl, foldr].map(function (f) {
        return f(function (acc, x) {
            return acc + (x * 2).toString() + ' ';
        }, [], xs);
    });

})([0, 1, 2, 3, 4, 5, 6, 7, 8, 9]);

Output:

["0 2 4 6 8 10 12 14 16 18 ", 
"18 16 14 12 10 8 6 4 2 0 "]

ES6

var nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10];

console.log(nums.reduce((a, b) => a + b, 0)); // sum of 1..10
console.log(nums.reduce((a, b) => a * b, 1)); // product of 1..10
console.log(nums.reduce((a, b) => a + b, '')); // concatenation of 1..10

jq

jq has an unusual and unusually powerful "reduce" control structure. A full description is beyond the scope of this short article, but an important point is that "reduce" is stream-oriented. Reduction of arrays is however trivially achieved using the ".[]" filter for converting an array to a stream of its values.

The simplest use of "reduce" can be illustrated by this definition of "factorial":

def factorial: reduce range(2;.+1) as $i (1; . * $i);

If the input is a non-negative integer, n, this will compute n!.

To understand how this works, consider "3|factorial". The computation starts by setting the implicit state variable to 1; range(2;4) will generate the sequence of values (2,3). The variable $i is set to each value in the stream in turn so that the state variable is multiplied by 2 (". * $i") and then by 3. Notice that since range/2 produces a stream, no array is ever constructed.

For a more complex illustration, see Strand sort.

The "reduce" operator is typically used within a map/reduce framework, but the implicit state variable can be any JSON entity, and so "reduce" is also a general-purpose iterative control structure, the only limitation being that it does not have the equivalent of "break". For that, the "foreach" control structure in recent versions of jq can be used.

Julia

Works with: Julia 1.2

println([reduce(op, 1:5) for op in [+, -, *]])
println([foldl(op, 1:5) for op in [+, -, *]])
println([foldr(op, 1:5) for op in [+, -, *]])

Output:

[15, -13, 120]
[15, -13, 120]
[15, 3, 120]

Kotlin

fun main(args: Array<String>) {
    val a = intArrayOf(1, 2, 3, 4, 5)
    println("Array       : ${a.joinToString(", ")}")
    println("Sum         : ${a.reduce { x, y -> x + y }}")
    println("Difference  : ${a.reduce { x, y -> x - y }}")
    println("Product     : ${a.reduce { x, y -> x * y }}")
    println("Minimum     : ${a.reduce { x, y -> if (x < y) x else y }}")
    println("Maximum     : ${a.reduce { x, y -> if (x > y) x else y }}")
}

Output:

Array       : 1, 2, 3, 4, 5
Sum         : 15
Difference  : -13
Product     : 120
Minimum     : 1
Maximum     : 5

Lambdatalk

{def nums 1 2 3 4 5}
-> nums
{S.reduce {lambda {:a :b} {+ :a :b}} {nums}}
-> 15
{S.reduce {lambda {:a :b} {- :a :b}} {nums}}
-> -13
{S.reduce {lambda {:a :b} {* :a :b}} {nums}}
-> 120
{S.reduce min {nums}}
-> 1
{S.reduce max {nums}}
-> 5

Logtalk

The Logtalk standard library provides implementations of common meta-predicates such as fold left. The example that follow uses Logtalk's native support for lambda expressions to avoid the need for auxiliary predicates.

:- object(folding_examples).

    :- public(show/0).
    show :-
        integer::sequence(1, 10, List),
        write('List: '), write(List), nl,
        meta::fold_left([Acc,N,Sum0]>>(Sum0 is Acc+N), 0, List, Sum),
        write('Sum of all elements: '), write(Sum), nl,
        meta::fold_left([Acc,N,Product0]>>(Product0 is Acc*N), 1, List, Product),
        write('Product of all elements: '), write(Product), nl,
        meta::fold_left([Acc,N,Concat0]>>(number_codes(N,NC), atom_codes(NA,NC), atom_concat(Acc,NA,Concat0)), '', List, Concat),
        write('Concatenation of all elements: '), write(Concat), nl.

:- end_object.

Output:

| ?- folding_examples::show.
List: [1,2,3,4,5,6,7,8,9,10]
Sum of all elements: 55
Product of all elements: 3628800
Concatenation of all elements: 12345678910
yes

LOLCODE

Translation of: C

HAI 1.3

HOW IZ I reducin YR array AN YR size AN YR fn
    I HAS A val ITZ array'Z SRS 0
    IM IN YR loop UPPIN YR i TIL BOTH SAEM i AN DIFF OF size AN 1
        val R I IZ fn YR val AN YR array'Z SRS SUM OF i AN 1 MKAY
    IM OUTTA YR loop
    FOUND YR val
IF U SAY SO

O HAI IM array
    I HAS A SRS 0 ITZ 1
    I HAS A SRS 1 ITZ 2
    I HAS A SRS 2 ITZ 3
    I HAS A SRS 3 ITZ 4
    I HAS A SRS 4 ITZ 5
KTHX

HOW IZ I add YR a AN YR b, FOUND YR     SUM OF a AN b, IF U SAY SO
HOW IZ I sub YR a AN YR b, FOUND YR    DIFF OF a AN b, IF U SAY SO
HOW IZ I mul YR a AN YR b, FOUND YR PRODUKT OF a AN b, IF U SAY SO

VISIBLE I IZ reducin YR array AN YR 5 AN YR add MKAY
VISIBLE I IZ reducin YR array AN YR 5 AN YR sub MKAY
VISIBLE I IZ reducin YR array AN YR 5 AN YR mul MKAY

KTHXBYE

Output:

15
-13
120

Lua

table.unpack = table.unpack or unpack -- 5.1 compatibility
local nums = {1,2,3,4,5,6,7,8,9}

function add(a,b)
   return a+b
end

function mult(a,b)
   return a*b
end

function cat(a,b)
   return tostring(a)..tostring(b)
end

local function reduce(fun,a,b,...)
   if ... then
      return reduce(fun,fun(a,b),...)
   else
      return fun(a,b)
   end
end

local arithmetic_sum = function (...) return reduce(add,...) end
local factorial5 = reduce(mult,5,4,3,2,1)

print("Σ(1..9)   : ",arithmetic_sum(table.unpack(nums)))
print("5!        : ",factorial5)
print("cat {1..9}: ",reduce(cat,table.unpack(nums)))

Output:

Σ(1..9)   : 	45
5!        : 	120
cat {1..9}: 	123456789

M2000 Interpreter

Module CheckIt {
      Function Reduce (a, f) {
            if len(a)=0 then Error "Nothing to reduce"
            if len(a)=1 then  =Array(a) : Exit
            k=each(a, 2, -1)
            m=Array(a)
            While k {
                  m=f(m, array(k))
            }
            =m
      }
      a=(1, 2, 3, 4, 5)
      Print "Array", a
      Print "Sum", Reduce(a, lambda (x,y)->x+y)
      Print "Difference", Reduce(a, lambda (x,y)->x-y)
      Print "Product", Reduce(a, lambda (x,y)->x*y)
      Print "Minimum", Reduce(a, lambda (x,y)->if(x<y->x, y))
      Print "Maximum", Reduce(a, lambda (x,y)->if(x>y->x, y))
}
CheckIt

Output:

Array               1         2         3         4          5
Sum                15
Difference        -13
Product           120
Minimum             1
Maximum             5

Maple

The left fold operator in Maple is foldl, and foldr is the right fold operator.

> nums := seq( 1 .. 10 );
                          nums := 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

> foldl( `+`, 0, nums ); # compute sum using foldl
                          55

> foldr( `*`, 1, nums ); # compute product using foldr
                          3628800

Compute the horner form of a (sorted) polynomial:

> foldl( (a,b) ->a*T+b, op(map2(op,1,[op( 72*T^5+37*T^4-23*T^3+87*T^2+44*T+29 )])));
                    ((((72 T + 37) T - 23) T + 87) T + 44) T + 29

Mathematica / Wolfram Language

Fold[f, x, {a, b, c, d}]

Output:

f[f[f[f[x, a], b], c], d]

Maxima

lreduce(f, [a, b, c, d], x0);
/* (%o1)                     f(f(f(f(x0, a), b), c), d) */

lreduce("+", [1, 2, 3, 4], 100);
/* (%o1)                                 110 */

min

Works with: min version 0.19.3

(1 2 3 4) 0 '+ reduce puts! ; sum
(1 2 3 4) 1 '* reduce puts! ; product

Output:

10
24

Modula-2

MODULE Catamorphism;
FROM InOut IMPORT WriteString, WriteCard, WriteLn;

(* Alas, there are no generic types. This function works for
   CARDINAL only - you would have to copy it and change the types
   to reduce functions of other types. *)
TYPE Reduction = PROCEDURE (CARDINAL, CARDINAL): CARDINAL;
PROCEDURE reduce(func:  Reduction; 
                 arr:   ARRAY OF CARDINAL;
                 first: CARDINAL): CARDINAL;
    VAR i: CARDINAL;
BEGIN
    FOR i := 0 TO HIGH(arr) DO
        first := func(first, arr[i]);
    END;
    RETURN first;
END reduce;

(* Demonstration *)
PROCEDURE add(a,b: CARDINAL): CARDINAL;
BEGIN RETURN a+b; END add;
PROCEDURE mul(a,b: CARDINAL): CARDINAL;
BEGIN RETURN a*b; END mul;

PROCEDURE Demonstration;
    VAR a: ARRAY [1..5] OF CARDINAL;
        i: CARDINAL;
BEGIN
    FOR i := 1 TO 5 DO a[i] := i; END;
    
    WriteString("Sum of [1..5]: ");
    WriteCard(reduce(add, a, 0), 3);
    WriteLn;
    WriteString("Product of [1..5]: ");
    WriteCard(reduce(mul, a, 1), 3);
    WriteLn;
END Demonstration;

BEGIN Demonstration;
END Catamorphism.

Output:

Sum of [1..5]:  15
Product of [1..5]: 120

Nemerle

The Nemerle.Collections namespace defines FoldLeft, FoldRight and Fold (an alias for FoldLeft) on any sequence that implements the IEnumerable[T] interface.

def seq = [1, 4, 6, 3, 7];
def sum = seq.Fold(0, _ + _); // Fold takes an initial value and a function, here the + operator

Nim

import sequtils

block:
  let
    numbers = @[5, 9, 11]
    addition = foldl(numbers, a + b)
    substraction = foldl(numbers, a - b)
    multiplication = foldl(numbers, a * b)
    words = @["nim", "is", "cool"]
    concatenation = foldl(words, a & b)

block:
  let
    numbers = @[5, 9, 11]
    addition = foldr(numbers, a + b)
    substraction = foldr(numbers, a - b)
    multiplication = foldr(numbers, a * b)
    words = @["nim", "is", "cool"]
    concatenation = foldr(words, a & b)

Oberon-2

Works with: oo2c Version 2

MODULE Catamorphism;
IMPORT
  Object,
  NPCT:Tools,
  NPCT:Args,
  IntStr,
  Out;
  
TYPE
  BinaryFunc= PROCEDURE (x,y: LONGINT): LONGINT;
  
VAR
  data: POINTER TO ARRAY OF LONGINT;
  i: LONGINT;

  PROCEDURE Sum(x,y: LONGINT): LONGINT;
  BEGIN
    RETURN x + y
  END Sum;
  
  PROCEDURE Sub(x,y: LONGINT): LONGINT;
  BEGIN
    RETURN x - y;
  END Sub;
  
  PROCEDURE Mul(x,y: LONGINT): LONGINT;
  BEGIN
    RETURN x * y;
  END Mul;
  
  PROCEDURE Reduce(x: ARRAY OF LONGINT; f: BinaryFunc): LONGINT;
  VAR
    i,res: LONGINT;
  BEGIN
    res := x[0];i := 1;
    WHILE (i < LEN(x)) DO;
      res := f(res,x[i]);
      INC(i)
    END;
    RETURN res
  END Reduce;
  
  PROCEDURE InitData(VAR x: ARRAY OF LONGINT);
  VAR
    i, j: LONGINT;
    res: IntStr.ConvResults;
    aux: Object.CharsLatin1;
  BEGIN
    i := 0;j := 1;
    WHILE (j <= LEN(x)) DO
      aux := Tools.AsString(Args.Get(j));
      IntStr.StrToInt(aux^,x[i],res);
      IF res # IntStr.strAllRight THEN
        Out.String("Incorrect format for data at index ");Out.LongInt(j,0);Out.Ln;
        HALT(1);
      END;
      INC(j);INC(i)
    END
  END InitData;
  
BEGIN
  IF Args.Number() = 1 THEN
    Out.String("Invalid number of arguments. ");Out.Ln;
    HALT(0)
  ELSE
    NEW(data,Args.Number() - 1);
    InitData(data^);
    Out.LongInt(Reduce(data^,Sum),0);Out.Ln;
    Out.LongInt(Reduce(data^,Sub),0);Out.Ln;
    Out.LongInt(Reduce(data^,Mul),0);Out.Ln
  END
END Catamorphism.

Output:

1
-11
-14400

Objeck

use Collection;

class Reducer {
  function : Main(args : String[]) ~ Nil {
    values := IntVector->New([1, 2, 3, 4, 5]);
    values->Reduce(Add(Int, Int) ~ Int)->PrintLine();
    values->Reduce(Mul(Int, Int) ~ Int)->PrintLine();
  }

  function : Add(a : Int, b : Int) ~ Int {
    return a + b;
  }
  
  function : Mul(a : Int, b : Int) ~ Int {
    return a * b;
  }
}

Output

15
120

OCaml

# let nums = [1;2;3;4;5;6;7;8;9;10];;
val nums : int list = [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]
# let sum = List.fold_left (+) 0 nums;;
val sum : int = 55
# let product = List.fold_left ( * ) 1 nums;;
val product : int = 3628800

Oforth

reduce is already defined into Collection class :

[ 1, 2, 3, 4, 5 ] reduce(#max)
[ "abc", "def", "gfi" ] reduce(#+)

PARI/GP

reduce(f, v)={
  my(t=v[1]);
  for(i=2,#v,t=f(t,v[i]));
  t
};
reduce((a,b)->a+b, [1,2,3,4,5,6,7,8,9,10])

Works with: PARI/GP version 2.8.1+

fold((a,b)->a+b, [1..10])

Pascal

Works with: Free Pascal

Should work with many pascal dialects

program reduceApp;

type
//  tmyArray = array of LongInt;
  tmyArray = array[-5..5] of LongInt;
  tmyFunc = function (a,b:LongInt):LongInt;

function add(x,y:LongInt):LongInt;
begin
  add := x+y;
end;

function sub(k,l:LongInt):LongInt;
begin
  sub := k-l;
end;

function mul(r,t:LongInt):LongInt;
begin
  mul := r*t;
end;

function reduce(myFunc:tmyFunc;a:tmyArray):LongInt;
var
  i,res : LongInt;
begin
  res := a[low(a)];
  For i := low(a)+1 to high(a) do
    res := myFunc(res,a[i]);
  reduce := res;
end;

procedure InitMyArray(var a:tmyArray);
var
  i: LongInt;
begin
  For i := low(a) to high(a) do
  begin
    //no a[i] = 0
    a[i] := i + ord(i=0);
    write(a[i],',');
  end;
  writeln(#8#32);
end;

var
  ma : tmyArray;
BEGIN
  InitMyArray(ma);
  writeln(reduce(@add,ma));
  writeln(reduce(@sub,ma));
  writeln(reduce(@mul,ma));
END.

output

-5,-4,-3,-2,-1,1,1,2,3,4,5 
1
-11
-1440

Perl

Perl's reduce function is in a standard package.

use List::Util 'reduce';

# note the use of the odd $a and $b globals
print +(reduce {$a + $b} 1 .. 10), "\n";

# first argument is really an anon function; you could also do this:
sub func { $b & 1 ? "$a $b" : "$b $a" }
print +(reduce \&func, 1 .. 10), "\n"

Phix

Translation of: C

with javascript_semantics
function add(integer a, b)  return a + b  end function
function sub(integer a, b)  return a - b  end function
function mul(integer a, b)  return a * b  end function
 
function reduce(integer rid, sequence s)
    object res = s[1]
    for i=2 to length(s) do
        res = rid(res,s[i])
    end for
    return res      
end function
 
?reduce(add,tagset(5))
?reduce(sub,tagset(5))
?reduce(mul,tagset(5))

Output:

15
-13
120

Phixmonti

include ..\Utilitys.pmt

def add + enddef
def sub - enddef 
def mul * enddef

def reduce >ps
    1 get
    swap len 2 swap 2 tolist for
        get rot swap tps exec swap
    endfor
    ps> drop
    swap
enddef


( 1 2 3 4 5 )
getid add reduce ?
getid sub reduce ?
getid mul reduce ?

PicoLisp

(de reduce ("Fun" "Lst")
   (let "A" (car "Lst")
      (for "N" (cdr "Lst")
         (setq "A" ("Fun" "A" "N")) )
      "A" ) )

(println
   (reduce + (1 2 3 4 5))
   (reduce * (1 2 3 4 5)) )
      
(bye)

PowerShell

'Filter' is a more common sequence function in PowerShell than 'reduce' or 'map', but here is one way to accomplish 'reduce':

1..5 | ForEach-Object -Begin {$result = 0} -Process {$result += $_} -End {$result}

Output:

15

Prolog

Using foldl from library(apply) and Lambda-Expressions from library(lambda)

• SWI-Prolog's library(apply) provides a `foldl/4` (the source code of which can be seen here).
• Ulrich Neumerkel wrote `library(lambda)` which can be found here. (However, SWI-Prolog's Lambda Expressions are by default based on Paulo Moura's library(yall))

:- use_module(library(lambda)).

catamorphism :-
	numlist(1,10,L),
	foldl(\XS^YS^ZS^(ZS is XS+YS), L, 0, Sum),
	format('Sum of ~w is ~w~n', [L, Sum]),
	foldl(\XP^YP^ZP^(ZP is XP*YP), L, 1, Prod),
	format('Prod of ~w is ~w~n', [L, Prod]),
	string_to_list(LV, ""),
	foldl(\XC^YC^ZC^(string_to_atom(XS, XC),string_concat(YC,XS,ZC)),
	      L, LV, Concat),
	format('Concat of ~w is ~w~n', [L, Concat]).

Output:

 ?- catamorphism.
Sum of [1,2,3,4,5,6,7,8,9,10] is 55
Prod of [1,2,3,4,5,6,7,8,9,10] is 3628800
Concat of [1,2,3,4,5,6,7,8,9,10] is 12345678910
true.

Bare Prolog

This is based on SWI Prolog 8 and has the following specificities:

• The consbox functor is [|] instead of .
• The list is terminated by the special atomic thing [] (the empty list)

% List to be folded:
%
%  +---+---+---+---[]    <-- list backbone/spine, composed of nodes, terminating in the empty list
%  |   |   |   |
%  a   b   c   d         <-- list items/entries/elements/members
%

linear foldl

% Computes "Out" as:
%
% starter value -->--f-->--f-->--f-->--f-->-- Out
%                    |     |     |     | 
%                    a     b     c     d


foldl(Foldy,[Item|Items],Acc,Result) :-    % case of nonempty list
   !,                                      % GREEN CUT for determinism
   call(Foldy,Item,Acc,AccNext),           % call Foldy(Item,Acc,AccNext)
   foldl(Foldy,Items,AccNext,Result).      % then recurse (open to tail call optimization)

foldl(_,[],Acc,Result) :-                  % case of empty list
   Acc=Result.                             % unification not in head for clarity

linear foldr

% Computes "Out" as:
%
% Out --<--f--<--f--<--f--<--f--<-- starter value
%          |     |     |     |
%          a     b     c     d

foldr(Foldy,[Item|Items],Starter,AccUp) :-    % case of nonempty list
   !,                                         % GREEN CUT for determinism
   foldr(Foldy,Items,Starter,AccUpPrev),      % recurse (NOT open to tail-call optimization)
   call(Foldy,Item,AccUpPrev,AccUp).          % call Foldy(Item,AccupPrev,AccUp) as last action

foldr(_,[],Starter,AccUp) :-                  % empty list: bounce Starter "upwards" into AccUp
   AccUp=Starter.                             % unification not in head for clarity

Unit tests

This is written using SWI-Prolog's unit testing framework.

Functions (in predicate form) of interest for our test cases:

:- use_module(library(clpfd)). % We are using #= instead of the raw "is".

foldy_len(_Item,ThreadIn,ThreadOut) :-
   succ(ThreadIn,ThreadOut).

foldy_add(Item,ThreadIn,ThreadOut) :-
   ThreadOut #= Item+ThreadIn.

foldy_mult(Item,ThreadIn,ThreadOut) :-
   ThreadOut #= Item*ThreadIn.

foldy_squadd(Item,ThreadIn,ThreadOut) :-
   ThreadOut #= Item+(ThreadIn^2).

% '[|]' is SWI-Prolog specific, replace by '.' as consbox constructor in other Prologs

foldy_build(Item,ThreadIn,ThreadOut) :-
   ThreadOut = '[|]'(Item,ThreadIn).

foldy_join(Item,ThreadIn,ThreadOut) :-
   (ThreadIn \= "")
   -> with_output_to(string(ThreadOut),format("~w,~w",[Item,ThreadIn]))
   ;  with_output_to(string(ThreadOut),format("~w",[Item])).

% '=..' ("univ") constructs a term from a list of functor and arguments

foldy_expr(Functor,Item,ThreadIn,ThreadOut) :-
   ThreadOut =.. [Functor,Item,ThreadIn].

:- begin_tests(foldr).

in([1,2,3,4,5]).

ffr(Foldy,List,Starter,AccUp) :- foldr(Foldy,List,Starter,AccUp).

test(foo_foldr_len)    :- in(L),ffr(foldy_len     , L ,  0 , R), R=5.
test(foo_foldr_add)    :- in(L),ffr(foldy_add     , L ,  0 , R), R=15.
test(foo_foldr_mult)   :- in(L),ffr(foldy_mult    , L ,  1 , R), R=120.
test(foo_foldr_build)  :- in(L),ffr(foldy_build   , L , [] , R), R=[1,2,3,4,5].
test(foo_foldr_squadd) :- in(L),ffr(foldy_squadd  , L ,  0 , R), R=507425426245.
test(foo_foldr_join)   :- in(L),ffr(foldy_join    , L , "" , R), R="1,2,3,4,5".
test(foo_foldr_expr)   :- in(L),ffr(foldy_expr(*) , L ,  1 , R), R=1*(2*(3*(4*(5*1)))).

test(foo_foldr_len_empty)    :- ffr(foldy_len     , [],  0 , R), R=0.
test(foo_foldr_add_empty)    :- ffr(foldy_add     , [],  0 , R), R=0.
test(foo_foldr_mult_empty)   :- ffr(foldy_mult    , [],  1 , R), R=1.
test(foo_foldr_build_empty)  :- ffr(foldy_build   , [], [] , R), R=[].
test(foo_foldr_squadd_empty) :- ffr(foldy_squadd  , [],  0 , R), R=0.
test(foo_foldr_join_empty)   :- ffr(foldy_join    , [], "" , R), R="".
test(foo_foldr_expr_empty)   :- ffr(foldy_expr(*) , [],  1 , R), R=1.

% library(apply) has no "foldr" so no comparison tests!

:- end_tests(foldr).


:- begin_tests(foldl).

in([1,2,3,4,5]).

ffl(Foldy,List,Starter,Result) :- foldl(Foldy,List,Starter,Result).

test(foo_foldl_len)    :- in(L),ffl(foldy_len     , L ,  0 , R), R=5.
test(foo_foldl_add)    :- in(L),ffl(foldy_add     , L,   0 , R), R=15.
test(foo_foldl_mult)   :- in(L),ffl(foldy_mult    , L,   1 , R), R=120.
test(foo_foldl_build)  :- in(L),ffl(foldy_build   , L,  [] , R), R=[5,4,3,2,1].
test(foo_foldl_squadd) :- in(L),ffl(foldy_squadd  , L,   0 , R), R=21909.
test(foo_foldl_join)   :- in(L),ffl(foldy_join    , L,  "" , R), R="5,4,3,2,1".
test(foo_foldl_expr)   :- in(L),ffl(foldy_expr(*) , L,   1 , R), R=5*(4*(3*(2*(1*1)))).

test(foo_foldl_len_empty)    :- ffl(foldy_len     , [],  0 , R), R=0.
test(foo_foldl_add_empty)    :- ffl(foldy_add     , [],  0 , R), R=0.
test(foo_foldl_mult_empty)   :- ffl(foldy_mult    , [],  1 , R), R=1.
test(foo_foldl_build_empty)  :- ffl(foldy_build   , [], [] , R), R=[].
test(foo_foldl_squadd_empty) :- ffl(foldy_squadd  , [],  0 , R), R=0.
test(foo_foldl_join_empty)   :- ffl(foldy_join    , [], "" , R), R="".
test(foo_foldl_expr_empty)   :- ffl(foldy_expr(*) , [],  1 , R), R=1.

:- end_tests(foldl).

rt :- run_tests(foldr),run_tests(foldl).

PureBasic

Procedure.i reduce(List l(),op$="+")
  If FirstElement(l())
    x=l()
    While NextElement(l())
      Select op$
        Case "+" : x+l()
        Case "-" : x-l()
        Case "*" : x*l()
      EndSelect
    Wend
  EndIf
  ProcedureReturn x
EndProcedure

NewList fold()
For i=1 To 5 : AddElement(fold()) : fold()=i : Next

Debug reduce(fold())
Debug reduce(fold(),"-")
Debug reduce(fold(),"*")

Output:

15
-13
120

Python

>>> # Python 2.X
>>> from operator import add
>>> listoflists = [['the', 'cat'], ['sat', 'on'], ['the', 'mat']]
>>> help(reduce)
Help on built-in function reduce in module __builtin__:

reduce(...)
    reduce(function, sequence[, initial]) -> value
    
    Apply a function of two arguments cumulatively to the items of a sequence,
    from left to right, so as to reduce the sequence to a single value.
    For example, reduce(lambda x, y: x+y, [1, 2, 3, 4, 5]) calculates
    ((((1+2)+3)+4)+5).  If initial is present, it is placed before the items
    of the sequence in the calculation, and serves as a default when the
    sequence is empty.

>>> reduce(add, listoflists, [])
['the', 'cat', 'sat', 'on', 'the', 'mat']
>>>

Additional example

# Python 3.X

from functools import reduce
from operator import add, mul

nums = range(1,11)

summation = reduce(add, nums)

product = reduce(mul, nums)

concatenation = reduce(lambda a, b: str(a) + str(b), nums)

print(summation, product, concatenation)

Quackery

Among its many other uses, witheach can act like reduce. In the Quackery shell (REPL):

/O> 0 ' [ 1 2 3 4 5 ] witheach +
... 1 ' [ 1 2 3 4 5 ] witheach *
...

Stack: 15 120

R

Sum the numbers in a vector:

Reduce('+', c(2,30,400,5000))
5432

Put a 0 between each pair of numbers:

Reduce(function(a,b){c(a,0,b)},  c(2,3,4,5))
2 0 3 0 4 0 5

Generate all prefixes of a string:

Reduce(paste0, unlist(strsplit("freedom", NULL)), accum=T)
"f"       "fr"      "fre"     "free"    "freed"   "freedo"  "freedom"

Filter and map:

Reduce(function(x,acc){if (0==x%%3) c(x*x,acc) else acc}, 0:22,
       init=c(), right=T)
   0   9  36  81 144 225 324 441

Racket

#lang racket
(define (fold f xs init)
  (if (empty? xs)
      init
      (f (first xs)
         (fold f (rest xs) init))))

(fold + '(1 2 3) 0)   ; the result is 6

Raku

(formerly Perl 6)

Works with: Rakudo version 2018.03

Any associative infix operator, either built-in or user-defined, may be turned into a reduce operator by putting it into square brackets (known as "the reduce metaoperator") and using it as a list operator. The operations will work left-to-right or right-to-left automatically depending on the natural associativity of the base operator.

my @list = 1..10;
say [+] @list;
say [*] @list;
say [~] @list;
say min @list;
say max @list;
say [lcm] @list;

Output:

55
3628800
12345678910
1
10
2520

In addition to the reduce metaoperator, a general higher-order function, reduce, can apply any appropriate function. Reproducing the above in this form, using the function names of those operators, we have:

my @list = 1..10;
say reduce &infix:<+>, @list;
say reduce &infix:<*>, @list;
say reduce &infix:<~>, @list;
say reduce &infix:<min>, @list;
say reduce &infix:<max>, @list;
say reduce &infix:<lcm>, @list;

Refal

$ENTRY Go {
    , 1 2 3 4 5 6 7: e.List
    = <Prout <Reduce Add e.List>>
      <Prout <Reduce Mul e.List>>;
};

Reduce {
    s.F t.I = t.I;
    s.F t.I t.J e.X = <Reduce s.F <Mu s.F t.I t.J> e.X>;
};

Output:

28
5040

REXX

This REXX example is modeled after the Raku example   (it is NOT a translation).

Also, a   list   and   show   function were added, although they aren't a catamorphism, as they don't produce or reduce the values to a   single   value, but are included here to help display the values in the list.

/*REXX program demonstrates a  method  for  catamorphism  for some simple functions.    */
@list= 1 2 3 4 5 6 7 8 9 10
                                say 'list:'     fold(@list,  "list")
                                say ' sum:'     fold(@list,  "+"   )
                                say 'prod:'     fold(@list,  "*"   )
                                say ' cat:'     fold(@list,  "||"  )
                                say ' min:'     fold(@list,  "min" )
                                say ' max:'     fold(@list,  "max" )
                                say ' avg:'     fold(@list,  "avg" )
                                say ' GCD:'     fold(@list,  "GCD" )
                                say ' LCM:'     fold(@list,  "LCM" )
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
fold: procedure;  parse arg z;  arg ,f;         z = space(z);      BIFs= 'MIN MAX LCM GCD'
      za= translate(z, f, ' ');                 zf= f"("translate(z, ',' , " ")')'
      if f== '+' | f=="*"       then interpret  "return"  za
      if f== '||'               then return  space(z, 0)
      if f== 'AVG'              then interpret  "return"  fold(z, '+')    "/"    words(z)
      if wordpos(f, BIFs)\==0   then interpret  "return"  zf
      if f=='LIST' | f=="SHOW"  then return z
      return 'illegal function:'     arg(2)
/*──────────────────────────────────────────────────────────────────────────────────────*/
GCD:  procedure;  $=;                          do j=1  for arg();    $= $ arg(j)
                                               end   /*j*/
      parse var $ x z .;    if x=0  then x= z                  /* [↑] build an arg list.*/
      x= abs(x)
                         do k=2  to words($);  y= abs( word($, k));   if y=0  then iterate
                           do until _=0;       _= x // y;      x= y;     y= _
                           end   /*until*/
                         end   /*k*/
      return x
/*──────────────────────────────────────────────────────────────────────────────────────*/
LCM:  procedure;  $=;    do j=1  for arg();     $= $ arg(j)
                         end   /*j*/
      x= abs(word($, 1))                                       /* [↑] build an arg list.*/
                         do k=2  to words($);   != abs(word($, k));  if !=0  then return 0
                         x= x*!  /  GCD(x, !)                  /*GCD does the heavy work*/
                         end   /*k*/
      return x

output:

list: 1 2 3 4 5 6 7 8 9 10
 sum: 55
prod: 3628800
 cat: 12345678910
 min: 1
 max: 10
 avg: 5.5
 GCD: 1
 LCM: 2520

Ring

n = list(10)
for i = 1 to 10
    n[i] = i
next 
 
see "  +: " + cat(10,"+") + nl+
    "  -: " + cat(10,"-") + nl +
    "  *: " + cat(10,"*") + nl +
    "  /: " + cat(10,"/") + nl+
    "  ^: " + cat(10,"^") + nl +
    "min: " + cat(10,"min") + nl+
    "max: " + cat(10,"max") + nl+
    "avg: " + cat(10,"avg") + nl +
    "cat: " + cat(10,"cat") + nl
 
func cat count,op
     cat = n[1]
     cat2 = ""
     for i = 2 to count 
         switch op 
                on "+" cat +=  n[i] 
                on "-"  cat -=  n[i]
                on "*" cat *=  n[i]
                on "/" cat /=  n[i]
                on "^" cat ^=  n[i]
                on "max" cat = max(cat,n[i])
                on "min" cat = min(cat,n[i])
                on "avg" cat +=  n[i]
                on "cat" cat2 += string(n[i])
          off
     next 
if op = "avg"  cat = cat / count ok
if op = "cat"  decimals(0) cat = string(n[1])+cat2 ok
return cat

RPL

≪ → array op 
  ≪ array 1 GET 2 
     WHILE DUP array SIZE ≤ REPEAT 
        array OVER GET ROT SWAP op EVAL 
        SWAP 1 + 
     END DROP 
≫ ≫ 'REDUCE' STO

[ 1 2 3 4 5 6 7 8 9 10 ] ≪ + ≫ REDUCE
[ 1 2 3 4 5 6 7 8 9 10 ] ≪ - ≫ REDUCE
[ 1 2 3 4 5 6 7 8 9 10 ] ≪ * ≫ REDUCE
[ 1 2 3 4 5 6 7 8 9 10 ] ≪ MAX ≫ REDUCE
[ 1 2 3 4 5 6 7 8 9 10 ] ≪ SQ + ≫ REDUCE

Output:

5: 55
4: -53
3: 3628800
2: 10
1: 385

From HP-48G models, a built-in function named STREAM performs exactly the same as the above REDUCE one, but only with lists.

Ruby

The method inject (and it's alias reduce) can be used in several ways; the simplest is to give a methodname as argument:

# sum:
p (1..10).inject(:+)
# smallest number divisible by all numbers from 1 to 20:
p (1..20).inject(:lcm) #lcm: lowest common multiple

The most versatile way uses a accumulator object (memo) and a block. In this example Pascal's triangle is generated by using an array [1,1] and inserting the sum of each consecutive pair of numbers from the previous row.

p row = [1]
10.times{p row = row.each_cons(2).inject([1,1]){|ar,(a,b)| ar.insert(-2, a+b)} }

# [1]
# [1, 1]
# [1, 2, 1]
# [1, 3, 3, 1]
# [1, 4, 6, 4, 1]
# [1, 5, 10, 10, 5, 1]
# [1, 6, 15, 20, 15, 6, 1]
# etc

Run BASIC

for i = 1 to 10 :n(i) = i:next i

print "  +: ";" ";cat(10,"+")
print "  -: ";" ";cat(10,"-")
print "  *: ";" ";cat(10,"*")
print "  /: ";" ";cat(10,"/")
print "  ^: ";" ";cat(10,"^")
print "min: ";" ";cat(10,"min")
print "max: ";" ";cat(10,"max")
print "avg: ";" ";cat(10,"avg")
print "cat: ";" ";cat(10,"cat")

function cat(count,op$)
cat = n(1)
for i = 2 to count 
 if op$ = "+" 	then cat = cat + n(i)
 if op$ = "-" 	then cat = cat - n(i)
 if op$ = "*" 	then cat = cat * n(i) 
 if op$ = "/" 	then cat = cat / n(i)
 if op$ = "^" 	then cat = cat ^ n(i)
 if op$ = "max"	then cat = max(cat,n(i))
 if op$ = "min"	then cat = min(cat,n(i))
 if op$ = "avg"	then cat = cat + n(i)
 if op$ = "cat"	then cat$ = cat$ + str$(n(i))
next i
if op$ = "avg" then cat = cat / count
if op$ = "cat" then cat = val(str$(n(1))+cat$)
end function

  +:  55
  -:  -53
  *:  3628800
  /:  2.75573205e-7
  ^:  1
min:  1
max:  10
avg:  5.5
cat:  12345678910

Rust

fn main() {
    println!("Sum: {}", (1..10).fold(0, |acc, n| acc + n));
    println!("Product: {}", (1..10).fold(1, |acc, n| acc * n));
    let chars = ['a', 'b', 'c', 'd', 'e'];
    println!("Concatenation: {}",
             chars.iter().map(|&c| (c as u8 + 1) as char).collect::<String>());
}

Output:

Sum: 45
Product: 362880
Concatenation: bcdef

Scala

object Main extends App {
  val a = Seq(1, 2, 3, 4, 5)
  println(s"Array       : ${a.mkString(", ")}")
  println(s"Sum         : ${a.sum}")
  println(s"Difference  : ${a.reduce { (x, y) => x - y }}")
  println(s"Product     : ${a.product}")
  println(s"Minimum     : ${a.min}")
  println(s"Maximum     : ${a.max}")
}

Scheme

Implementation

reduce implemented for a single list:

(define (reduce fn init lst)
  (do ((val init (fn (car rem) val)) ; accumulated value passed as second argument
       (rem lst (cdr rem)))
    ((null? rem) val)))

(display (reduce + 0 '(1 2 3 4 5))) (newline) ; => 15
(display (reduce expt 2 '(3 4))) (newline)    ; => 262144

Using SRFI 1

There is also an implementation of fold and fold-right in SRFI-1, for lists.

These take a two-argument procedure: (lambda (value acc) ...) where value is the next value in the list, and acc is the accumulated value. The initial value is used for the first value of acc.

> (import (srfi 1))
> (fold + 0 '(1 2 3 4 5))
15
> (fold expt 2 '(3 4)) ; => (expt 4 (expt 3 2))
262144
> (fold-right expt 2 '(3 4)) ; => (expt 3 (expt 4 2))
43046721

More than one list may be folded over, when the function is passed one item from each list plus the accumulated value:

> (fold + 0 '(1 2 3) '(4 5 6)) ; add up all the numbers in all the lists
21

Sidef

say (1..10 -> reduce('+'));
say (1..10 -> reduce{|a,b| a + b});

Standard ML

- val nums = [1,2,3,4,5,6,7,8,9,10];
val nums = [1,2,3,4,5,6,7,8,9,10] : int list
- val sum = foldl op+ 0 nums;
val sum = 55 : int
- val product = foldl op* 1 nums;
val product = 3628800 : int

Swift

let nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

print(nums.reduce(0, +))
print(nums.reduce(1, *))
print(nums.reduce("", { $0 + String($1) }))

Output:

55
3628800
12345678910

Tailspin

It is probably easier to just write the whole thing as an inline transform rather than create a utility.

[1..5] -> \(@: $(1); $(2..last)... -> @: $@ + $; $@!\) -> '$;
' -> !OUT::write
[1..5] -> \(@: $(1); $(2..last)... -> @: $@ - $; $@!\) -> '$;
' -> !OUT::write
[1..5] -> \(@: $(1); $(2..last)... -> @: $@ * $; $@!\) -> '$;
' -> !OUT::write

Output:

15
-13
120

If you really want to make a utility, it could look like this:

templates fold&{op:}
  @: $(1);
  $(2..last)... -> @: [$@, $] -> op;
  $@ !
end fold

templates add
  $(1) + $(2) !
end add

templates mul
  $(1) * $(2) !
end mul

[1..5] -> fold&{op:add} -> '$;
' -> !OUT::write

[1..5] -> fold&{op:mul} -> '$;
' -> !OUT::write

Output:

15
120

Tcl

Tcl does not come with a built-in fold command, but it is easy to construct:

proc fold {lambda zero list} {
    set accumulator $zero
    foreach item $list {
	set accumulator [apply $lambda $accumulator $item]
    }
    return $accumulator
}

Demonstrating:

set 1to5 {1 2 3 4 5}

puts [fold {{a b} {expr {$a+$b}}} 0 $1to5]
puts [fold {{a b} {expr {$a*$b}}} 1 $1to5]
puts [fold {{a b} {return $a,$b}} x $1to5]

Output:

15
120
x,1,2,3,4,5

Note that these particular operations would more conventionally be written as:

puts [::tcl::mathop::+ {*}$1to5]
puts [::tcl::mathop::* {*}$1to5]
puts x,[join $1to5 ,]

But those are not general catamorphisms.

uBasic/4tH

Translation of: FreeBASIC

uBasic/4tH has only got one single array so passing its address makes little sense. Instead, its bounds are passed.

For x = 1 To 5 : @(x-1) = x : Next     ' initialize array
                                       ' try different reductions
Print "Sum is        : "; FUNC(_Reduce(_add, 5))
Print "Difference is : "; FUNC(_Reduce(_subtract, 5))
Print "Product is    : "; FUNC(_Reduce(_multiply, 5))
Print "Maximum is    : "; FUNC(_Reduce(_max, 5))
Print "Minimum is    : "; FUNC(_Reduce(_min, 5))

End
                                       ' several functions
_add Param (2) : Return (a@ + b@)
_subtract Param (2) : Return (a@ - b@)
_multiply Param (2) : Return (a@ * b@)
_min Param (2) : Return (Min (a@, b@))
_max Param (2) : Return (Max (a@, b@))

_Reduce
  Param (2)                            ' function and array size
  Local (2)                            ' loop index and result
                                       ' set result and iterate array 
  d@ = @(0) : For c@ = 1 To b@-1 : d@ = FUNC(a@ (d@, @(c@))) : Next
Return (d@)

This version incorporates a "no op" as well.

Push 5, 4, 3, 2, 1: s = Used() - 1
For x = 0 To s: @(x) = Pop(): Next

Print "Sum is        : "; FUNC(_reduce(0, s, _add))
Print "Difference is : "; FUNC(_reduce(0, s, _subtract))
Print "Product is    : "; FUNC(_reduce(0, s, _multiply))
Print "Maximum is    : "; FUNC(_reduce(0, s, _max))
Print "Minimum is    : "; FUNC(_reduce(0, s, _min))
Print "No op is      : "; FUNC(_reduce(0, s, _noop))
End

_reduce
  Param (3)
  Local (2)

  If (Line(c@) = 0) + ((b@ - a@) < 1) Then Return (0)
  d@ = @(a@)
  For e@ = a@ + 1 To b@
    d@ = FUNC(c@ (d@, @(e@)))
  Next
Return (d@)
 
_add      Param (2) : Return (a@ + b@)
_subtract Param (2) : Return (a@ - b@)
_multiply Param (2) : Return (a@ * b@)
_max      Param (2) : Return (Max(a@, b@))
_min      Param (2) : Return (Min(a@, b@))

Output:

Sum is        : 15
Difference is : -13
Product is    : 120
Maximum is    : 5
Minimum is    : 1
No op is      : 0

0 OK, 0:378 

VBA

Public Sub reduce()
    s = [{1,2,3,4,5}]
    Debug.Print WorksheetFunction.Sum(s)
    Debug.Print WorksheetFunction.Product(s)
End Sub

V (Vlang)

Translation of: go

fn main() {
	n := [1, 2, 3, 4, 5]
 
	println(reduce(add, n))
	println(reduce(sub, n))
	println(reduce(mul, n))
}
 
fn add(a int, b int) int { return a + b }
fn sub(a int, b int) int { return a - b }
fn mul(a int, b int) int { return a * b }
 
fn reduce(rf fn(int, int) int, m []int) int {
	mut r := m[0]
	for  v in m[1..] {
		r = rf(r, v)
	}
	return r
}

Output:

15
-13
120

WDTE

Translated from the JavaScript ES6 example with a few modifications.

let a => import 'arrays';
let s => import 'stream';
let str => import 'strings';

# Sum of [1, 10]:
let nums => [1; 2; 3; 4; 5; 6; 7; 8; 9; 10];
a.stream nums -> s.reduce 0 + -- io.writeln io.stdout;

# As an alternative to an array, a range stream can be used. Here's the product of [1, 11):
s.range 1 11 -> s.reduce 1 * -- io.writeln io.stdout;

# And here's a concatenation:
s.range 1 11 -> s.reduce '' (str.format '{}{}') -- io.writeln io.stdout;

Wortel

You can reduce an array with the !/ operator.

!/ ^+ [1 2 3] ; returns 6

If you want to reduce with an initial value, you'll need the @fold operator.

@fold ^+ 1 [1 2 3] ; returns 7

Output:

55
3628800
12345678910

Wren

var a = [1, 2, 3, 4, 5]
var sum   = a.reduce { |acc, i| acc + i }
var prod  = a.reduce { |acc, i| acc * i }
var sumSq = a.reduce { |acc, i| acc + i*i }
System.print(a)
System.print("Sum is %(sum)")
System.print("Product is %(prod)")
System.print("Sum of squares is %(sumSq)")

Output:

[1, 2, 3, 4, 5]
Sum is 15
Product is 120
Sum of squares is 55

Zig

Works with: 0.10.x, 0.11.x, 0.12.0-dev.1591+3fc6a2f11

Reduce a slice

/// Asserts that `array`.len >= 1.
pub fn reduce(comptime T: type, comptime applyFn: fn (T, T) T, array: []const T) T {
    var val: T = array[0];
    for (array[1..]) |elem| {
        val = applyFn(val, elem);
    }
    return val;
}

Usage:

const std = @import("std");

fn add(a: i32, b: i32) i32 {
    return a + b;
}

fn mul(a: i32, b: i32) i32 {
    return a * b;
}

fn min(a: i32, b: i32) i32 {
    return @min(a, b);
}

fn max(a: i32, b: i32) i32 {
    return @max(a, b);
}

pub fn main() void {
    const arr: [5]i32 = .{ 1, 2, 3, 4, 5 };
    std.debug.print("Array: {any}\n", .{arr});
    std.debug.print(" * Reduce with add: {d}\n", .{reduce(i32, add, &arr)});
    std.debug.print(" * Reduce with mul: {d}\n", .{reduce(i32, mul, &arr)});
    std.debug.print(" * Reduce with min: {d}\n", .{reduce(i32, min, &arr)});
    std.debug.print(" * Reduce with max: {d}\n", .{reduce(i32, max, &arr)});
}

Output:

Array: { 1, 2, 3, 4, 5 }
 * Reduce with add: 15
 * Reduce with mul: 120
 * Reduce with min: 1
 * Reduce with max: 5

Reduce a vector

We use @reduce builtin function here to leverage special instructions if available, but only small set of reduce operators are available. @Vector and related builtings will use SIMD instructions if possible. If target platform does not support SIMD instructions, vectors operations will be compiled like in previous example (represented as arrays and operating with one element at a time).

const std = @import("std");

pub fn main() void {
    const vec: @Vector(5, i32) = .{ 1, 2, 3, 4, 5 };
    std.debug.print("Vec: {any}\n", .{vec});
    std.debug.print(" * Reduce with add: {d}\n", .{@reduce(.Add, vec)});
    std.debug.print(" * Reduce with mul: {d}\n", .{@reduce(.Mul, vec)});
    std.debug.print(" * Reduce with min: {d}\n", .{@reduce(.Min, vec)});
    std.debug.print(" * Reduce with max: {d}\n", .{@reduce(.Max, vec)});
}

Output:

Vec: { 1, 2, 3, 4, 5 }
 * Reduce with add: 15
 * Reduce with mul: 120
 * Reduce with min: 1
 * Reduce with max: 5

Note that std.builtin.ReduceOp.Add and std.builtin.ReduceOp.Mul operators wrap on overflow and underflow, unlike regular Zig operators, where they are considered illegal behaviour and checked in safe optimize modes. This can be demonstrated by this example (ReleaseSafe optimize mode, zig 0.11.0, Linux 6.5.11 x86_64):

const std = @import("std");

pub fn main() void {
    const vec: @Vector(2, i32) = .{ std.math.minInt(i32), std.math.minInt(i32) + 1 };
    std.debug.print("Vec: {any}\n", .{vec});
    std.debug.print(" * Reduce with .Add: {d}\n", .{@reduce(.Add, vec)});
    std.debug.print(" * Reduce with .Mul: {d}\n", .{@reduce(.Mul, vec)});

    var zero: usize = 0; // Small trick to make compiler not emit compile error for overflow below:
    std.debug.print(" * Reduce with regular add operator: {d}\n", .{vec[zero] + vec[1]});
    std.debug.print(" * Reduce with regular mul operator: {d}\n", .{vec[zero] * vec[1]});
}

Output:

Vec: { -2147483648, -2147483647 }
 * Reduce with .Add: 1
 * Reduce with .Mul: -2147483648
thread 5908 panic: integer overflow
/home/bratishkaerik/test/catamorphism.zig:10:79: 0x20c4b0 in main (catamorphism)
    std.debug.print(" * Reduce with regular add operator: {d}\n", .{vec[zero] + vec[1]});
                                                                              ^
/usr/lib64/zig/0.11.0/lib/std/start.zig:564:22: 0x20bee4 in posixCallMainAndExit (catamorphism)
            root.main();
                     ^
/usr/lib64/zig/0.11.0/lib/std/start.zig:243:5: 0x20bdc1 in _start (catamorphism)
    asm volatile (switch (native_arch) {
    ^
???:?:?: 0x0 in ??? (???)
[1]    5908 IOT instruction  ./catamorphism

For well-defined overflow/underflow behaviour you can use wrapping and saturating operators (for addition they are +% and +| respectively). With +% and *% (wrapping multiplication) operators, behaviour should be identical to .Add and .Mul reduce operators.

zkl

Most sequence objects in zkl have a reduce method.

T("foo","bar").reduce(fcn(p,n){p+n}) //--> "foobar"
"123four5".reduce(fcn(p,c){p+(c.matches("[0-9]") and c or 0)}, 0) //-->11
File("foo.zkl").reduce('+(1).fpM("0-"),0) //->5 (lines in file)

ZX Spectrum Basic

Translation of: BBC_BASIC

10 DIM a(5)
20 FOR i=1 TO 5
30 READ a(i)
40 NEXT i
50 DATA 1,2,3,4,5
60 LET o$="+": GO SUB 1000: PRINT tmp
70 LET o$="-": GO SUB 1000: PRINT tmp
80 LET o$="*": GO SUB 1000: PRINT tmp
90 STOP 
1000 REM Reduce
1010 LET tmp=a(1)
1020 FOR i=2 TO 5
1030 LET tmp=VAL ("tmp"+o$+"a(i)")
1040 NEXT i
1050 RETURN