Monads/List monad
Demonstrate in your programming language the following:
- Construct a List Monad by writing the 'bind' function and the 'unit' (sometimes known as 'return') function for that Monad (or just use what the language already has implemented)
- Make two functions, each which take a number and return a monadic number, e.g. Int -> List Int and Int -> List String
- Compose the two functions with bind
AppleScript
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>
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>
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>
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]]
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 { "${('@' + it)}".repeat(it) } )
fun main(args: Array<String>) {
val iv = MList.unit(listOf(2, 3, 4)) val fv = iv.bind(::doubler).bind(::letters) println(fv.value)
}</lang>
- Output:
[DDDD, FFFFFF, HHHHHHHH]
Perl 6
Perl 6 does not have Monad types built in but they can be emulated/implemented without a great deal of difficulty. List Monads especially are of questionable utility in Perl 6. Most item types and Listy types have a Cool role in Perl 6. (Cool being a play on the slang term "cool" as in: "That's cool with me." (That's ok with me). So Ints are pretty much treated like one item lists for operators that work with lists. ("I work on a list." "Here's an Int." "Ok, that's cool.") Explicitly wrapping an Int into a List is worse than useless. It won't do anything Perl 6 can't do natively, and will likely remove some functionality that it would normally have. That being said, just because it is a bad idea (in Perl 6) doesn't mean it can't be done.
In Perl 6, bind is essentially map. I'll shadow map here but again, it removes capability, not adds it. Perl 6 also provided "hyper" operators which will descend into data structures and apply an operator / function to each member of that data structure.
Here's a simple, if contrived example. take the numbers from 0 to 9, add 3 to each, find the divisors of those sums and print the list of divisors for each sum... in base 2. Again, a bind function was implemented but it is more limited than if we just used map directly. The built in map method will work with either items or lists, here we need to implement a multi sub to handle either.
The * in the bind blocks are typically referred to as "whatever"; whatever + 3 etc. The guillemot (») is the hyper operator; descend into the data structure and apply the following operator/function to each member. <lang perl6>multi bind (@list, &code) { @list.map: &code };
multi bind ($item, &code) { $item.&code };
sub divisors (Int $int) { gather for 1 .. $int { .take if $int %% $_ } }
put join "\n", (flat ^10).&bind(* + 3).&bind(*.&divisors)».&bind(*.base: 2);</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>
- Project : Monads/List monad
- Date : 2017/12/24
- Author : Gal Zsolt (~ CalmoSoft ~)
- Email : <calmosoft@gmail.com>
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"]
- Note that listy_inc and listy_str cannot be composed directly,
- as they don't have compatible type signature.
- Due to duck typing (Ruby will happily turn arrays into strings),
- 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]
- Direct composition will cause a TypeError, as Ruby cannot evaluate 2*[4, 5, 6]
- 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
- 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>
zkl
While I'm unsure of the utility of Monads in a dynamic type-less language, it can be done.
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)