Monads/Maybe monad

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< Monads
Revision as of 19:03, 4 February 2016 by Rdm (talk | contribs) (J draft)

Demonstrate in your programming language the following:

  1. Construct a Maybe 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)
  2. Make two functions, each which take a number and return a monadic number, e.g. Int -> Maybe Int and Int -> Maybe String
  3. Compose the two functions with bind


Clojure

<lang clojure> (defn bind [val f] (if (nil? (:value val)) val (f (:value val)))) (defn unit [val] {:value val})

(defn opt_add_3 [n] (unit (+ 3 n))) ; takes a number and returns a Maybe number (defn opt_str [n] (unit (str n)))  ; takes a number and returns a Maybe string

(bind (unit 4) opt_add_3)  ; evaluates to {:value 7} (bind (unit nil) opt_add_3)  ; evaluates to {:value nil} (bind (bind (unit 8) opt_add_3) opt_str)  ; evaluates to {:value "11"} (bind (bind (unit nil) opt_add_3) opt_str) ; evaluates to {:value nil} </lang>


EchoLisp

Our monadic Maybe elements will be pairs (boolean . value), where value is in Maybe.domain. Functions which return something not in Maybe.domain are unsafe and return (#f . input-value), If a function is given as input a (#f . value) element, it will return this element.

<lang scheme> (define (Maybe.domain? x) (or (number? x) (string? x))) (define (Maybe.unit elem (bool #t)) (cons bool elem))

f is a safe or unsafe function
(Maybe.lift f) returns a safe Maybe function which returns a Maybe element

(define (Maybe.lift f) (lambda(x) 

            (let [(u (f x))]
            (if (Maybe.domain? u) 
               (Maybe.unit u)
               (Maybe.unit x #f))))) ;; return offending x
elem = Maybe element
f is safe or unsafe (lisp) function
return Maybe element

(define (Maybe.bind f elem) (if (first elem) ((Maybe.lift f) (rest elem)) elem))

pretty-print

(define (Maybe.print elem) (if (first elem) (writeln elem ) (writeln '❌ elem)))

unsafe functions

(define (u-log x) (if (> x 0) (log x) #f)) (define (u-inv x) (if (zero? x) 'zero-div (/ x)))

(print (number->string (exp (log 3))))

(->> 3 Maybe.unit (Maybe.bind u-log) (Maybe.bind exp) (Maybe.bind number->string) Maybe.print) 

   → (#t . "3.0000000000000004")
(print (number->string (exp (log -666))))

(->> -666 Maybe.unit (Maybe.bind u-log) (Maybe.bind exp) (Maybe.bind number->string) Maybe.print) 

    → ❌     (#f . -666)
;; (print (number->string (inverse (log 1))))

(->> 1 Maybe.unit (Maybe.bind u-log) (Maybe.bind u-inv) (Maybe.bind number->string) Maybe.print) 

    →  ❌     (#f . 0)

</lang>

Hoon

<lang hoon>

- %say

|= [^ [[txt=(unit ,@tas) ~] ~]]

- %noun

|^ 

 %+  biff  txt
   ;~  biff
     m-parse
     m-double
   ==
 ++  m-parse
   |=  a=@tas
   ^-  (unit ,@ud)
   (rust (trip a) dem)
 ::
 ++  m-double
   |=  a=@ud
   ^-  (unit ,@ud)
   (some (mul a 2))
 --

</lang>

Hoon has a built-in rune, %smsg (;~) that binds gates under a monad.

++unit is Hoon's Maybe: it is either ~ (None) or [~ u=val] (Some)

++biff is the monadic bind, which %smsg uses to wire the gates together. It's defined in the standard library here. m-parse is @tas -> (unit ,@ud), so I use biff a second time in order for the program to be called with (unit ,@tas).

++rust is one of the parser combinator runners: it parses the string `a` with the rule `dem`, returning a unit with the returned value if it success or ~ if it fails. Note that Hoon's type system is complex enough to get a strongly typed result of the parsing rule, in this case an unsigned decimal (@ud) 

> +monad (some '2')
[~ 4]
> +monad (some 'a')
~
> +monad ~
~

J

It's difficult to find a useful and illustrative (but simple) example of this task in J. So we shall not try to be useful.

<lang J> NB. monad implementation: unit=: < bind=: (@>)( :: ])

NB. monad utility safeVersion=: (<@) ( ::((<_.)"_)) safeCompose=:dyad define 

 dyad def 'x`:6 bind y'/x,unit y

)

NB. unsafe functions (fail with infinite arguments) subtractFromSelf=: -~ divideBySelf=: %~

NB. wrapped functions: safeSubtractFromSelf=: subtractFromSelf safeVersion safeDivideBySelf=: divideBySelf safeVersion

NB. task example: 

    safeSubtractFromSelf bind safeDivideBySelf 1

┌─┐ │0│ └─┘ 

    safeSubtractFromSelf bind safeDivideBySelf _

┌──┐ │_.│ └──┘</lang>

JavaScript

ES5

Example: deriving and composing safe versions of reciprocal, square root and log functions.

<lang JavaScript>(function () { 

   'use strict';

   // START WITH SOME SIMPLE (UNSAFE) PARTIAL FUNCTIONS:

   // Returns Infinity if n === 0
   function reciprocal(n) {
       return 1 / n;
   }

   // Returns NaN if n < 0
   function root(n) {
       return Math.sqrt(n);
   }

   // Returns -Infinity if n === 0
   // Returns NaN if n < 0
   function log(n) {
       return Math.log(n);
   }



   // NOW DERIVE SAFE VERSIONS OF THESE SIMPLE FUNCTIONS:
   // These versions use a validity test, and return a wrapped value
   // with a boolean .isValid property as well as a .value property

   function safeVersion(f, fnSafetyCheck) {
       return function (v) {
           return maybe(fnSafetyCheck(v) ? f(v) : undefined);
       }
   }

   var safe_reciprocal = safeVersion(reciprocal, function (n) {
       return n !== 0;
   });

   var safe_root = safeVersion(root, function (n) {
       return n >= 0;
   });



   var safe_log = safeVersion(log, function (n) {
       return n > 0;
   });



   // THE DERIVATION OF THE SAFE VERSIONS USED THE 'UNIT' OR 'RETURN' 
   // FUNCTION OF THE MAYBE MONAD

   // Named maybe() here, the unit function of the Maybe monad wraps a raw value 
   // in a datatype with two elements: .isValid (Bool) and .value (Number)

   // a -> Ma
   function maybe(n) {
       return {
           isValid: (typeof n !== 'undefined'),
           value: n
       };
   }

   // THE PROBLEM FOR FUNCTION NESTING (COMPOSITION) OF THE SAFE FUNCTIONS
   // IS THAT THEIR INPUT AND OUTPUT TYPES ARE DIFFERENT

   // Our safe versions of the functions take simple numeric arguments, but return
   // wrapped results. If we feed a wrapped result as an input to another safe function,
   // it will choke on the unexpected type. The solution is to write a higher order
   // function (sometimes called 'bind' or 'chain') which handles composition, taking a 
   // a safe function and a wrapped value as arguments,

   // The 'bind' function of the Maybe monad:
   // 1. Applies a 'safe' function directly to the raw unwrapped value, and
   // 2. returns the wrapped result.

   // Ma -> (a -> Mb) -> Mb
   function bind(maybeN, mf) {
       return (maybeN.isValid ? mf(maybeN.value) : maybeN);
   }

   // Using the bind function, we can nest applications of safe_ functions,
   // without their choking on unexpectedly wrapped values returned from
   // other functions of the same kind.
   var rootOneOverFour = bind(
       bind(maybe(4), safe_reciprocal), safe_root
   ).value;

   // -> 0.5



   // We can compose a chain of safe functions (of any length) with a simple foldr/reduceRight
   // which starts by 'lifting' the numeric argument into a Maybe wrapping,
   // and then nests function applications (working from right to left)
   function safeCompose(lstFunctions, value) {
       return lstFunctions
           .reduceRight(function (a, f) {
               return bind(a, f);
           }, maybe(value));
   }

   // TEST INPUT VALUES WITH A SAFELY COMPOSED VERSION OF LOG(SQRT(1/X))

   var safe_log_root_reciprocal = function (n) {
       return safeCompose([safe_log, safe_root, safe_reciprocal], n).value;
   }

   return [-2, -1, -0.5, 0, 1 / Math.E, 1, 2, Math.E, 3, 4, 5].map(
       safe_log_root_reciprocal
   );

})();</lang>

Output:
[undefined, undefined, undefined, undefined, 0.5, 0,
-0.3465735902799726, -0.5, -0.5493061443340549,
-0.6931471805599453, -0.8047189562170503]

Ruby

OOP version using Ruby's block syntax

<lang ruby>class Maybe 

 def initialize(value)
   @value = value
 end

 def map
   if @value.nil?
     self
   else
     Maybe.new(yield @value)
   end
 end

end

Maybe.new(3).map { |n| 2*n }.map { |n| n+1 }

  1. => #<Maybe @value=7>

Maybe.new(nil).map { |n| 2*n }.map { |n| n+1 }

  1. => #<Maybe @value=nil>

Maybe.new(3).map { |n| nil }.map { |n| n+1 }

  1. => #<Maybe @value=nil>
  1. alias Maybe#new and write bind to be in line with task

class Maybe 

 class << self
   alias :unit :new
 end

 def bind
   if @value.nil?
     self
   else
     yield @value
   end
 end

end

Maybe.unit(3).bind { |n| Maybe.unit(2*n) }.bind { |n| Maybe.unit(n+1) }

  1. => #<Maybe @value=7>

Maybe.unit(nil).bind { |n| Maybe.unit(2*n) }.bind { |n| Maybe.unit(n+1) }

  1. => #<Maybe @value=nil>

</lang>

zkl

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

From the Wikipedia

Here we use the Void object as Nothing and define some functions. Sine zkl is type-less, we can consider Maybe as a native type and don't need to define it. <lang zkl>fcn bind(a,type,b){ if(type.isType(a)) b else Void } fcn just(x){ if(Deferred.isType(x)) x() else x } // force lazy evaluation fcn rtn(x) { just(x) }</lang> Since zkl is eager, add needs to gyrate a bit by creating a lazy result and evaluating that after the binds have done their bizness. <lang zkl>fcn add(mx,my){ 

  bind(mx,Int,
     bind(my,Int,
       '+.fp(mx,my))) : rtn(_)  // create a lazy mx+my to avoid eager eval

} add(1,2).println(); // two ints add(1,2.0).println(); // int and float add(self,2).println(); // class and int</lang>

Output:
3
Void
Void
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