Monads/List monad

Monads/List monad
You are encouraged to solve this task according to the task description, using any language you may know.

A Monad is a combination of a data-type with two helper functions written for that type.

The data-type can be of any kind which can contain values of some other type – common examples are lists, records, sum-types, even functions or IO streams. The two special functions, mathematically known as eta and mu, but usually given more expressive names like 'pure', 'return', or 'yield' and 'bind', abstract away some boilerplate needed for pipe-lining or enchaining sequences of computations on values held in the containing data-type.

The bind operator in the List monad enchains computations which return their values wrapped in lists. One application of this is the representation of indeterminacy, with returned lists representing a set of possible values. An empty list can be returned to express incomputability, or computational failure.

A sequence of two list monad computations (enchained with the use of bind) can be understood as the computation of a cartesian product.

The natural implementation of bind for the List monad is a composition of concat and map, which, used with a function which returns its value as a (possibly empty) list, provides for filtering in addition to transformation or mapping.

Demonstrate in your programming language the following:

1. Construct a List Monad by writing the 'bind' function and the 'pure' (sometimes known as 'return') function for that Monad (or just use what the language already has implemented)
2. Make two functions, each which take a number and return a monadic number, e.g. Int -> List Int and Int -> List String
3. Compose the two functions with bind

AppleScript

Translation of: JavaScript

We can use a list monad in AppleScript to express set comprehension for the Pythagorean triples, but the lack of nestable first class (and anonymous) functions means that the closure can only be achieved using script objects, which makes the idiom rather less direct and transparent. AppleScript is creaking at the seams here. <lang AppleScript>-- MONADIC FUNCTIONS (for list monad) ------------------------------------------

-- Monadic bind for lists is simply ConcatMap -- which applies a function f directly to each value in the list, -- and returns the set of results as a concat-flattened list

-- bind :: (a -> [b]) -> [a] -> [b] on bind(f, xs)

   -- concat :: a -> a -> [a]
   script concat
       on |λ|(a, b)
           a & b
       end |λ|
   end script
   
   foldl(concat, {}, map(f, xs))

end bind

-- Monadic return/unit/inject for lists: just wraps a value in a list -- a -> [a] on unit(a)

   [a]

end unit

-- TEST ------------------------------------------------------------------------ on run

   -- Pythagorean triples drawn from integers in the range [1..n]
   -- {(x, y, z) | x <- [1..n], y <- [x+1..n], z <- [y+1..n], (x^2 + y^2 = z^2)}
   
   pythagoreanTriples(25)
   
   --> {{3, 4, 5}, {5, 12, 13}, {6, 8, 10}, {7, 24, 25}, {8, 15, 17}, 
   --   {9, 12, 15}, {12, 16, 20}, {15, 20, 25}}
   

end run

-- pythagoreanTriples :: Int -> [(Int, Int, Int)] on pythagoreanTriples(maxInteger)

   script X
       on |λ|(X)
           script Y
               on |λ|(Y)
                   script Z
                       on |λ|(Z)
                           if X * X + Y * Y = Z * Z then
                               unit([X, Y, Z])
                           else
                               []
                           end if
                       end |λ|
                   end script
                   
                   bind(Z, enumFromTo(1 + Y, maxInteger))
               end |λ|
           end script
           
           bind(Y, enumFromTo(1 + X, maxInteger))
       end |λ|
   end script
   
   bind(X, enumFromTo(1, maxInteger))
   

end pythagoreanTriples

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

-- enumFromTo :: Int -> Int -> [Int] on enumFromTo(m, n)

   if n < m then
       set d to -1
   else
       set d to 1
   end if
   set lst to {}
   repeat with i from m to n by d
       set end of lst to i
   end repeat
   return lst

end enumFromTo

-- 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

-- map :: (a -> b) -> [a] -> [b] on map(f, 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</lang>

Output:

<lang AppleScript>{{3, 4, 5}, {5, 12, 13}, {6, 8, 10}, {7, 24, 25}, {8, 15, 17}, {9, 12, 15}, {12, 16, 20}, {15, 20, 25}}</lang>

C++

<lang cpp>#include <iostream>

1. include <vector>

using namespace std;

// std::vector can be a list monad. Use the >> operator as the bind function template <typename T> auto operator>>(const vector<T>& monad, auto f) {

   // Declare a vector of the same type that the function f returns
   vector<remove_reference_t<decltype(f(monad.front()).front())>> result;
   for(auto& item : monad)
   {
       // Apply the function f to each item in the monad. f will return a
       // new list monad containing 0 or more items. 
       const auto r = f(item);
       // Concatenate the results of f with previous results
       result.insert(result.end(), begin(r), end(r));
   }
   
   return result;

}

// The Pure function returns a vector containing one item, t auto Pure(auto t) {

   return vector{t};

}

// A function to double items in the list monad auto Double(int i) {

   return Pure(2 * i);

}

// A function to increment items auto Increment(int i) {

   return Pure(i + 1);

}

// A function to convert items to a string auto NiceNumber(int i) {

   return Pure(to_string(i) + " is a nice number\n");

}

// A function to map an item to a sequence ending at max value // for example: 497 -> {497, 498, 499, 500} auto UpperSequence = [](auto startingVal) {

   const int MaxValue = 500;
   vector<decltype(startingVal)> sequence;
   while(startingVal <= MaxValue) 
       sequence.push_back(startingVal++);
   return sequence;

};

// Print contents of a vector void PrintVector(const auto& vec) {

   cout << " ";
   for(auto value : vec)
   {
       cout << value << " ";
   }
   cout << "\n";

}

// Print the Pythagorean triples void PrintTriples(const auto& vec) {

   cout << "Pythagorean triples:\n";
   for(auto it = vec.begin(); it != vec.end();)
   {
       auto x = *it++;
       auto y = *it++;
       auto z = *it++;
       
       cout << x << ", " << y << ", " << z << "\n";
   }
   cout << "\n";

}

int main() {

   // Apply Increment, Double, and NiceNumber to {2, 3, 4} using the monadic bind 
   auto listMonad = 
       vector<int> {2, 3, 4} >> 
       Increment >> 
       Double >>
       NiceNumber;
       
   PrintVector(listMonad);
   
   // Find Pythagorean triples using the list monad.  The 'x' monad list goes
   // from 1 to the max; the 'y' goes from the current 'x' to the max; and 'z'
   // goes from the current 'y' to the max.  The last bind returns the triplet
   // if it is Pythagorean, otherwise it returns an empty list monad.
   auto pythagoreanTriples = UpperSequence(1) >> 
       [](int x){return UpperSequence(x) >>
       [x](int y){return UpperSequence(y) >>
       [x, y](int z){return (x*x + y*y == z*z) ? vector{x, y, z} : vector<int>{};};};};
   
   PrintTriples(pythagoreanTriples);

} </lang>

Output:

 6 is a nice number
 8 is a nice number
 10 is a nice number
 
Pythagorean triples:
3, 4, 5
5, 12, 13
6, 8, 10
7, 24, 25
8, 15, 17
9, 12, 15
9, 40, 41
10, 24, 26
11, 60, 61
 . . .
 . . . 
320, 336, 464
325, 360, 485
340, 357, 493

Clojure

<lang clojure> (defn bind [coll f] (apply vector (mapcat f coll))) (defn unit [val] (vector val))

(defn doubler [n] [(* 2 n)])  ; takes a number and returns a List number (def vecstr (comp vector str)) ; takes a number and returns a List string

(bind (bind (vector 3 4 5) doubler) vecstr) ; evaluates to ["6" "8" "10"] (-> [3 4 5]

 (bind doubler)
 (bind vecstr)) ; also evaluates to ["6" "8" "10"]

</lang>

Delphi

Library: System.SysUtils

Translation of: Go

<lang Delphi> program List_monad;

{$APPTYPE CONSOLE}

uses

 System.SysUtils;

type

 TmList = record
   Value: TArray<Integer>;
   function ToString: string;
   function Bind(f: TFunc<TArray<Integer>, TmList>): TmList;
 end;

function Create(aValue: TArray<Integer>): TmList; begin

 Result.Value := copy(aValue, 0, length(aValue));

end;

{ TmList }

function TmList.Bind(f: TFunc<TArray<Integer>, TmList>): TmList; begin

 Result := f(self.Value);

end;

function TmList.ToString: string; var

 i: Integer;

begin

 Result := '[ ';
 for i := 0 to length(value) - 1 do
 begin
   if i > 0 then
     Result := Result + ', ';
   Result := Result + value[i].toString;
 end;
 Result := Result + ']';

end;

function Increment(aValue: TArray<Integer>): TmList; var

 i: integer;

begin

 SetLength(Result.Value, length(aValue));
 for i := 0 to High(aValue) do
   Result.Value[i] := aValue[i] + 1;

end;

function Double(aValue: TArray<Integer>): TmList; var

 i: integer;

begin

 SetLength(Result.Value, length(aValue));
 for i := 0 to High(aValue) do
   Result.Value[i] := aValue[i] * 2;

end;

var

 ml1, ml2: TmList;

begin

 ml1 := Create([3, 4, 5]);
 ml2 := ml1.Bind(Increment).Bind(double);
 Writeln(ml1.ToString, ' -> ', ml2.ToString);
 readln;

end.</lang>

Output:

[ 3, 4, 5] -> [ 8, 10, 12]

EchoLisp

Our monadic lists will take the form (List a b c ...), ie raw lists prefixed by the List symbol. <lang scheme>

-> and ->> are the pipeline operators

(-> x f g h) = (h (g ( f x)))

(->> x f (g a) h) = (h (g a ( f x)))

(define (List.unit elem) (append '(List) elem)) (define (List.bind xs f) (List.unit (->> xs rest (map f) (map rest) (apply append)))) (define (List.lift f) (lambda(elem) (List.unit (f elem))))

(define List.square (List.lift (lambda(x) (* x x)))) (define List.cube (List.lift (lambda(x) (* x x x)))) (define List.tostr (List.lift number->string))

composition

(-> '(List 1 -2 3 -5) (List.bind List.cube) (List.bind List.tostr))

   → (List "1" "-8" "27" "-125")

or

(-> '(1 -2 3 -5) List.unit (List.bind List.cube) (List.bind List.tostr))

    → (List "1" "-8" "27" "-125")

</lang>

F#

<lang fsharp> type ListMonad() =

  member o.Bind(  (m:'a list), (f: 'a -> 'b list) ) = List.concat( List.map f m )
  member o.Return(x) = [x]
  member o.Zero()    = []

let list = ListMonad()

let pyth_triples n = list { let! x = [1..n]

                           let! y = [x..n]
                           let! z = [y..n]
                           if x*x + y*y = z*z then return (x,y,z) }

printf "%A" (pyth_triples 100) </lang>

(The original example, which follows, variously uses List.iter, a list comprehension and the Result type but doesn't define anything like a list monad.)

<lang fsharp> // Monads/List monad . Nigel Galloway: March 8th., 2021 List.iter ((+) 1>>(*) 2>>printf "%d ") [3;4;5]; printfn "";; let pT n=[for i in 1..n do for g in i+1..n do for n in g+1..n do if i*i+g*g=n*n then yield(i,g,n)] Seq.iter(printf "%A ")(pT 25) let fN g=match g<10 with false->Error "is greater than 9"|_->Ok g let fG n=match n>5 with false->Error "is less than 6" |_->Ok n let valid n=n|>Result.bind fN|>Result.bind fG let test n=match valid(Ok n) with Ok g->printfn "%d is valid" g|Error e->printfn "Error: %d %s" n e [5..10]|>List.iter test </lang>

Output:

8 10 12
(3, 4, 5) (5, 12, 13) (6, 8, 10) (7, 24, 25) (8, 15, 17) (9, 12, 15) (12, 16, 20) (15, 20, 25)
Error: 5 is less than 6
6 is valid
7 is valid
8 is valid
9 is valid
Error: 10 is greater than 9

Factor

Factor comes with an implementation of Haskell-style monads in the monads vocabulary. <lang factor>USING: kernel math monads prettyprint ; FROM: monads => do ;

{ 3 4 5 } >>= [ 1 + array-monad return ] swap call >>= [ 2 * array-monad return ] swap call .</lang> Or: <lang factor>{ 3 4 5 } [ 1 + array-monad return ] bind [ 2 * array-monad return ] bind .</lang> Or: <lang factor>{

   [ { 3 4 5 } ]
   [ 1 + array-monad return ]
   [ 2 * array-monad return ]

} do .</lang>

Output:

{ 8 10 12 }

Go

<lang go>package main

import "fmt"

type mlist struct{ value []int }

func (m mlist) bind(f func(lst []int) mlist) mlist {

   return f(m.value)

}

func unit(lst []int) mlist {

   return mlist{lst}

}

func increment(lst []int) mlist {

   lst2 := make([]int, len(lst))
   for i, v := range lst {
       lst2[i] = v + 1
   }
   return unit(lst2)

}

func double(lst []int) mlist {

   lst2 := make([]int, len(lst))
   for i, v := range lst {
       lst2[i] = 2 * v
   }
   return unit(lst2)

}

func main() {

   ml1 := unit([]int{3, 4, 5})
   ml2 := ml1.bind(increment).bind(double)
   fmt.Printf("%v -> %v\n", ml1.value, ml2.value)

}</lang>

Output:

[3 4 5] -> [8 10 12]

Haskell

Haskell has the built-in Monad type class, and the built-in list type already conforms to the Monad type class. <lang haskell>main = print $ [3,4,5] >>= (return . (+1)) >>= (return . (*2)) -- prints [8,10,12]</lang>

Or, written using do notation: <lang haskell>main = print $ do x <- [3,4,5]

                 y <- return (x+1)
                 z <- return (y*2)
                 return z</lang>

Or alternately: <lang haskell>main = print $ do x <- [3,4,5]

                 let y = x+1
                 let z = y*2
                 return z</lang>

Using the list monad to express set comprehension for Pythagorean triples: <lang haskell>pythagoreanTriples :: Integer -> [(Integer, Integer, Integer)] pythagoreanTriples n =

 [1 .. n] >>= (\x ->
 [x+1 .. n] >>= (\y ->
 [y+1 .. n] >>= (\z ->
 if x^2 + y^2 == z^2 then return (x,y,z) else [])))

main = print $ pythagoreanTriples 25</lang>

Output:

[(3,4,5),(5,12,13),(6,8,10),(7,24,25),(8,15,17),(9,12,15),(12,16,20),(15,20,25)]

Which can be written using do notation: <lang haskell>pythagoreanTriples :: Integer -> [(Integer, Integer, Integer)] pythagoreanTriples n = do x <- [1 .. n]

                         y <- [x+1 .. n]
                         z <- [y+1 .. n]
                         if x^2 + y^2 == z^2 then return (x,y,z) else []</lang>

Or directly as a list comprehension: <lang haskell>pythagoreanTriples :: Integer -> [(Integer, Integer, Integer)] pythagoreanTriples n = [(x,y,z) | x <- [1 .. n], y <- [x+1 .. n], z <- [y+1 .. n], x^2 + y^2 == z^2]</lang>

J

Note that J documentation mentions "monad" but that is an older (much older) use of the term from what is intended here. J documentation uses "box" <to describe the operation mentioned here.

That said, here is an implementation which might be adequate for the current task description:

<lang J>bind=: S:0 unit=: boxopen

m_num=: unit m_str=: unit@":</lang>

Task example:

<lang J> m_str bind m_num 5 ┌─┐ │5│ └─┘</lang>

JavaScript

<lang javascript> Array.prototype.bind = function (func) {

 return this.map(func).reduce(function (acc, a) { return acc.concat(a); });

}

Array.unit = function (elem) {

 return [elem];

}

Array.lift = function (func) {

 return function (elem) { return Array.unit(func(elem)); };

}

inc = function (n) { return n + 1; } doub = function (n) { return 2 * n; } listy_inc = Array.lift(inc); listy_doub = Array.lift(doub);

[3,4,5].bind(listy_inc).bind(listy_doub); // [8, 10, 12] </lang>

ES5 Example: Using the list monad to express set comprehension

<lang JavaScript>(function (n) {

   // ENCODING A SET COMPREHENSION IN TERMS OF A LIST MONAD

   // Pythagorean triples drawn from integers in the range [1..25]

   // Each range of integers here represents the set of possible values for the variable.
   // Where the test returns true for a particular [x, y, z] triple, we return that triple
   // to the expected data type, wrapping it using the unit or return function;

   // Where the test returns false, we return the empty list, which vanishes from the 
   // results set under concatenation, giving us a convenient encoding of filtering.

   // {(x, y, z) | x <- [1..n], y <- [x+1..n], z <- [y+1..n], (x^2 + y^2 = z^2)} 

   return bind(rng(1,     n), function (x) {
   return bind(rng(1 + x, n), function (y) {
   return bind(rng(1 + y, n), function (z) {

       return (x * x + y * y === z * z) ? unit([x, y, z]) : [];

   })})});

   // Monadic return/unit/inject for lists just wraps a value in a list
   // a -> [a]
   function unit(a) {
       return [a];
   }

   // Bind for lists is simply ConcatMap
   // which applies a function f directly to each value in the list,
   // and returns the set of results as a concat-flattened list
   // [a] -> (a -> [b]) -> [b]
   function bind(xs, f) {
       return [].concat.apply([], xs.map(f));
   }

   // we will need some ranges of integers, each expressing a range of possible values
   // [m..n]
   function rng(m, n) {
       return Array.apply(null, Array(n - m + 1))
           .map(function (x, i) {
               return m + i;
           });
   }

})(25);</lang>

Output:

[[3, 4, 5], [5, 12, 13], [6, 8, 10], [7, 24, 25], [8, 15, 17], [9, 12, 15], [12, 16, 20], [15, 20, 25]]

Julia

Julia uses the function bind for binding a channel to a task, but this can be imported and overloaded. The |> syntax in Julia can also be used to chain functions taking one argument. <lang julia>julia> unit(v) = [v...] unit (generic function with 1 method)

julia> import Base.bind

julia> bind(v, f) = f.(v) bind (generic function with 5 methods)

julia> f1(x) = x + 1 f1 (generic function with 1 method)

julia> f2(x) = 2x f2 (generic function with 1 method)

julia> bind(bind(unit([2, 3, 4]), f1), f2) 3-element Array{Int64,1}:

 6
 8
10

julia> unit([2, 3, 4]) .|> f1 .|> f2 3-element Array{Int64,1}:

 6
 8
10

</lang>

Kotlin

<lang scala>// version 1.2.10

class MList<T : Any> private constructor(val value: List<T>) {

   fun  bind(f: (List<T>) -> MList) = f(this.value)

   companion object {
       fun <T : Any> unit(lt: List<T>) = MList<T>(lt)
   }

}

fun doubler(li: List<Int>) = MList.unit(li.map { 2 * it } )

fun letters(li: List<Int>) = MList.unit(li.map { "${('@' + it)}".repeat(it) } )

fun main(args: Array<String>) {

   val iv = MList.unit(listOf(2, 3, 4))
   val fv = iv.bind(::doubler).bind(::letters)
   println(fv.value)

}</lang>

Output:

[DDDD, FFFFFF, HHHHHHHH]

Nim

a natural use of a list-wrapped return value is when there can be more than one result from a function, for example square roots have a positive and negative solution, and the inverse sine function has multiple solutions we might be interested in. <lang nim>import math,sequtils,sugar,strformat func root(x:float):seq[float] = @[sqrt(x),-sqrt(x)] func asin(x:float):seq[float] = @[arcsin(x),arcsin(x)+TAU,arcsin(x)-TAU] func format(x:float):seq[string] = @[&"{x:.2f}"]

1. 'bind' is a nim keyword, how about an infix operator instead
2. our bind is the standard map+cat

func `-->`[T,U](input: openArray[T],f: T->seq[U]):seq[U] =

 input.map(f).concat

echo [0.5] --> root --> asin --> format </lang>

Output:

@["0.79", "7.07", "-5.50", "-0.79", "5.50", "-7.07"]

Perl

With the help of the CPAN module Data::Monad, we can work with list monads. <lang perl>use strict; use feature 'say'; use Data::Monad::List;

1. Cartesian product to 'count' in binary

my @cartesian = [(

   list_flat_map_multi { scalar_list(join , @_) }
       scalar_list(0..1),
       scalar_list(0..1),
       scalar_list(0..1)

)->scalars]; say join "\n", @{shift @cartesian};

say ;

1. Pythagorean triples

my @triples = [(

   list_flat_map_multi { scalar_list(
           { $_[0] < $_[1] && $_[0]**2+$_[1]**2 == $_[2]**2 ? join(',',@_) : () }
       ) }
       scalar_list(1..10),
       scalar_list(1..10),
       scalar_list(1..10)

)->scalars];

for (@{shift @triples}) {

   say keys %$_ if keys %$_;

}</lang>

Output:

000
001
010
011
100
101
110
111

3,4,5
6,8,10

Phix

Translation of: Go

function bindf(sequence m, integer f)
    return f(m)
end function
 
function unit(sequence m)
    return m
end function
 
function increment(sequence l)
    return unit(sq_add(l,1))
end function
 
function double(sequence l)
    return unit(sq_mul(l,2))
end function
 
sequence m1 = unit({3, 4, 5}),
         m2 = bindf(bindf(m1,increment),double)
printf(1,"%v -> %v\n", {m1, m2})

Output:

{3,4,5} -> {8,10,12}

Python

<lang python>"""A List Monad. Requires Python >= 3.7 for type hints.""" from __future__ import annotations from itertools import chain

from typing import Any from typing import Callable from typing import Iterable from typing import List from typing import TypeVar

T = TypeVar("T")

class MList(List[T]):

   @classmethod
   def unit(cls, value: Iterable[T]) -> MList[T]:
       return cls(value)

   def bind(self, func: Callable[[T], MList[Any]]) -> MList[Any]:
       return MList(chain.from_iterable(map(func, self)))

   def __rshift__(self, func: Callable[[T], MList[Any]]) -> MList[Any]:
       return self.bind(func)

if __name__ == "__main__":

   # Chained int and string functions
   print(
       MList([1, 99, 4])
       .bind(lambda val: MList([val + 1]))
       .bind(lambda val: MList([f"${val}.00"]))
   )

   # Same, but using `>>` as the bind operator.
   print(
       MList([1, 99, 4])
       >> (lambda val: MList([val + 1]))
       >> (lambda val: MList([f"${val}.00"]))
   )

   # Cartesian product of [1..5] and [6..10]
   print(
       MList(range(1, 6)).bind(
           lambda x: MList(range(6, 11)).bind(lambda y: MList([(x, y)]))
       )
   )

   # Pythagorean triples with elements between 1 and 25
   print(
       MList(range(1, 26)).bind(
           lambda x: MList(range(x + 1, 26)).bind(
               lambda y: MList(range(y + 1, 26)).bind(
                   lambda z: MList([(x, y, z)])
                   if x * x + y * y == z * z
                   else MList([])
               )
           )
       )
   )

</lang>

Output:

['$2.00', '$100.00', '$5.00']
['$2.00', '$100.00', '$5.00']
[(1, 6), (1, 7), (1, 8), (1, 9), (1, 10), (2, 6), (2, 7), (2, 8), (2, 9), (2, 10), (3, 6), (3, 7), (3, 8), (3, 9), (3, 10), (4, 6), (4, 7), (4, 8), (4, 9), (4, 10), (5, 6), (5, 7), (5, 8), (5, 9), (5, 10)]
[(3, 4, 5), (5, 12, 13), (6, 8, 10), (7, 24, 25), (8, 15, 17), (9, 12, 15), (12, 16, 20), (15, 20, 25)]

Racket

Translation of: JavaScript

Vanilla Racket

Note that this also demonstrates how to use Racket's macro system to implement the do syntax.

<lang racket>#lang racket

(define (bind x f) (append-map f x)) (define return list) (define ((lift f) x) (list (f x)))

(define listy-inc (lift add1)) (define listy-double (lift (λ (x) (* 2 x))))

(bind (bind '(3 4 5) listy-inc) listy-double)

=> '(8 10 12)

(define (pythagorean-triples n)

 (bind (range 1 n)
       (λ (x)
         (bind (range (add1 x) n)
               (λ (y)
                 (bind (range (add1 y) n)
                       (λ (z)
                         (if (= (+ (* x x) (* y y)) (* z z))
                             (return (list x y z))
                             '()))))))))

(pythagorean-triples 25)

=> '((3 4 5) (5 12 13) (6 8 10) (8 15 17) (9 12 15) (12 16 20))

(require syntax/parse/define)

(define-syntax-parser do-macro

 [(_ [x {~datum <-} y] . the-rest) #'(bind y (λ (x) (do-macro . the-rest)))]
 [(_ e) #'e])

(define (pythagorean-triples* n)

 (do-macro
  [x <- (range 1 n)]
  [y <- (range (add1 x) n)]
  [z <- (range (add1 y) n)]
  (if (= (+ (* x x) (* y y)) (* z z))
      (return (list x y z))
      '())))

(pythagorean-triples* 25)

=> '((3 4 5) (5 12 13) (6 8 10) (8 15 17) (9 12 15) (12 16 20))</lang>

With functional package

The functional package has already implemented the list monad.

<lang racket>#lang racket

(require data/monad

        data/applicative)

(define (pythagorean-triples n)

 (sequence->list
  (do [x <- (range 1 n)]
      [y <- (range (add1 x) n)]
      [z <- (range (add1 y) n)]
      (if (= (+ (* x x) (* y y)) (* z z))
          (pure (list x y z))
          '()))))

(pythagorean-triples 25)

=> '((3 4 5) (5 12 13) (6 8 10) (8 15 17) (9 12 15) (12 16 20))</lang>

Raku

(formerly Perl 6) Raku does not have Monad types built in but they can be emulated/implemented without a great deal of difficulty. List Monads especially are of questionable utility in Raku. Most item types and Listy types have a Cool role in Raku. (Cool being a play on the slang term "cool" as in: "That's cool with me." (That's ok with me). So Ints are pretty much treated like one item lists for operators that work with lists. ("I work on a list." "Here's an Int." "Ok, that's cool.") Explicitly wrapping an Int into a List is worse than useless. It won't do anything Raku can't do natively, and will likely remove some functionality that it would normally have. That being said, just because it is a bad idea (in Raku) doesn't mean it can't be done.

In Raku, bind is essentially map. I'll shadow map here but again, it removes capability, not adds it. Raku also provided "hyper" operators which will descend into data structures and apply an operator / function to each member of that data structure.

Here's a simple, if contrived example. take the numbers from 0 to 9, add 3 to each, find the divisors of those sums and print the list of divisors for each sum... in base 2. Again, a bind function was implemented but it is more limited than if we just used map directly. The built in map method will work with either items or lists, here we need to implement a multi sub to handle either.

The * in the bind blocks are typically referred to as "whatever"; whatever + 3 etc. The guillemot (») is the hyper operator; descend into the data structure and apply the following operator/function to each member. <lang perl6>multi bind (@list, &code) { @list.map: &code };

multi bind ($item, &code) { $item.&code };

sub divisors (Int $int) { gather for 1 .. $int { .take if $int %% $_ } }

put (^10).&bind(* + 3).&bind(&divisors)».&bind(*.base: 2).join: "\n";</lang>

Output:

1 11
1 10 100
1 101
1 10 11 110
1 111
1 10 100 1000
1 11 1001
1 10 101 1010
1 1011
1 10 11 100 110 1100

Ring

<lang ring>

1. Project : Monads/List monad

func main()
       str = "["
       for x in [3,4,5]
            y = x+1
            z = y*2
            str = str + z + ", " 
       next
       str = left(str, len(str) -2)
       str = str + "]"
       see str + nl

</lang> Output:

[8, 10, 12]

Ruby

<lang ruby> class Array

 def bind(f)
   flat_map(&f)
 end
 def self.unit(*args)
   args
 end
 # implementing lift is optional, but is a great helper method for turning
 # ordinary funcitons into monadic versions of them.
 def self.lift(f)
   -> e { self.unit(f[e]) }
 end

end

inc = -> n { n + 1 } str = -> n { n.to_s } listy_inc = Array.lift(inc) listy_str = Array.lift(str)

Array.unit(3,4,5).bind(listy_inc).bind(listy_str) #=> ["4", "5", "6"]

1. Note that listy_inc and listy_str cannot be composed directly,
2. as they don't have compatible type signature.
3. Due to duck typing (Ruby will happily turn arrays into strings),
4. in order to show this, a new function will have to be used:

doub = -> n { 2*n } listy_doub = Array.lift(doub) [3,4,5].bind(listy_inc).bind(listy_doub) #=> [8, 10, 12]

1. Direct composition will cause a TypeError, as Ruby cannot evaluate 2*[4, 5, 6]
2. Using bind with the composition is *supposed* to fail, no matter the programming language.

comp = -> f, g {-> x {f[g[x]]}} [3,4,5].bind(comp[listy_doub, listy_inc]) #=> TypeError: Array can't be coerced into Fixnum

1. Composition needs to be defined in terms of bind

class Array

 def bind_comp(f, g)
   bind(g).bind(f)
 end

end

[3,4,5].bind_comp(listy_doub, listy_inc) #=> [8, 10, 12] </lang>

Wren

Translation of: Go

<lang ecmascript>class Mlist {

   construct new(value) { _value = value }

   value { _value }

   bind(f) { f.call(_value) }

   static unit(lst) { Mlist.new(lst) }

}

var increment = Fn.new { |lst|

   var lst2 = lst.map { |v| v + 1 }.toList
   return Mlist.unit(lst2)

}

var double = Fn.new { |lst|

   var lst2 = lst.map { |v| v * 2 }.toList
   return Mlist.unit(lst2)

}

var ml1 = Mlist.unit([3, 4, 5]) var ml2 = ml1.bind(increment).bind(double) System.print("%(ml1.value) -> %(ml2.value)")</lang>

Output:

[3, 4, 5] -> [8, 10, 12]

zkl

While I'm unsure of the utility of Monads in a dynamic type-less language, it can be done.

Translation of: Ruby

Here we create a class to do Monad like things. Unlike Ruby, we can't augment the baked in List/Array object so this more verbose. Also unlike Ruby, we can directly compose as we are applying the composition to each element (vs the list-as-object). <lang zkl>class MList{

  fcn init(xs){ var list=vm.arglist }
  fcn bind(f) { list=list.apply(f); self }
  fcn toString{ list.toString() }

}</lang> <lang zkl>inc:=Op("+",1); // '+(1) str:="toString"; MList(3,4,5).bind(inc).bind(str).println(" == (4,5,6)");

doub:=Op("*",2); MList(3,4,5).bind(inc).bind(doub).println(" == (8,10,12)");

comp:=Utils.Helpers.fcomp; // comp(f,g) == f.g == f(g(x)) MList(3,4,5).bind(comp(doub,inc)).println(" == (8,10,12)");</lang>

Output:

L("4","5","6") == (4,5,6)
L(8,10,12) == (8,10,12)
L(8,10,12) == (8,10,12)