A while back Brent Yorgey http://byorgey.wordpress.com posted a menagerie of types. A very nice survey of types in Haskell, including, among other things, the monadic types.
A reader posted a question, framed with 'you know, it'd be a lot easier for me to understand monads if you provided an implementation in Java.'[1]
Well, that reader really shouldn't have done that.
Really.
Because it got me to thinking, and dangerous things comes out of geophf's thoughts ... like monads in Java.
Let's do this.
Problem Statement
Of course, you just don't write monads for Java. You need some motivation. What's my motivation?
Well, I kept running up against the Null Object pattern, because I kept running up against null in the Java runtime, because there's a whole lot of optionality in the data sets I'm dealing with, and instead of using the Null Object, null is used to represent absence.
Good choice?
Well, it's expedient, and that's about the best I can say for that.
So I had to keep writing the Null Object pattern for each of these optional element types I encountered.
I encounter quite a few of them ... 2,100 of them and counting.
So I generalized the Null object pattern using generic types, and that worked fine ...
But I kept looking at the pattern. The generalization of the Null Object pattern is Nothing
, and a present instance is Just
that something.
I had implemented nonmonadic Maybe
.
And the problem with that is that none of the power of Maybe
, as a Monad
, is available to a generalized Null Object pattern.
Put another way: I wish to do something given that something is present.
Put anotheranother way: bind
.
So, after a long hard look ('do I really want to implement monads? Can't I just settle for 'good enough' ... that really isn't good enough?), I buckled down to do the work of implementing not just Maybe
(heh: 'Just Maybe
'), but Monad
itself.
Did it pay off? Yes, but wait and see. First the implementation. But even before that, let's review what a monad is.
What the Hell is a Monad?
Everybody seems to ask this question when first encountering monads, and everybody seems to come up with their own explanations, so we have tutorials that explain Monads as things as farranging from spacesuits to burritos.
My own explanation is to punt ... to category theory.
A monad is simply a triple (that's why it's called a monad, ... because it's a triple), defined thusly:
(T, η, μ)
Simple right?
Don't leave me!
Okay, yes. I understand you are rolling your eyes and saying 'well, that's all Greek to me.'
So allow me to explain in plain language the monad.
Monad is the triple (T, η, μ)
where
T
is the MonadFunctor
η
is the unit function that lifts an ordinary object (a unit) into the Monad functor; and,μ
is the join function that takes a composed monadM (M a)
and returns the simplified type of the composition, orM a
.
And that is (simply) what the hell a Monad is. What a monad can do ... well, there are articles galore about that, and, later, I will provide examples of what Monad gives us ... in Java, no less.
The Implementation
We need to work our way up to Monad. There's a whole framework, a way of thinking, a context, that needs to be in place for Monad to be effective or even to exist. Monad is 'happy' in the functional world, so we need to implement functions.
Or more correctly, 'functionals' ... in Haskell, they are called 'Functors.'
A functor is a container, and the function of functors is to take an ordinary function, which we write: f: a → b
(pronounced "the function f
from (type) a
to (type) b
" and means that f
takes an argument of type a
and returns a result of type b
), and give a function g: Functor a → Functor b
.
Functor basically is a container for the function fmap: Functor f ⇒ (a → b) → (f a → f b)
Well, is Functor the fundamental type? Well, yes and no.
It is, given that we have functions defined. In Java, we don't have that. We have methods, yes, but we don't have freestanding functions. Let's amend that issue:
public interface Applicable
<T1
,T2
> {public T2
apply(T1
arg)throws Failure
; };
And what's this Failure thingie? It's simply a functional exception, so:
public class Failure
extends Exception
{public Failure
(String
string) {super (string); }/// and the serialVersionUID for serialization; your IDE can define that }
I call a Function 'Applicable' because in functional languages that's what you do: apply a function on (type) a to get (type) b. And, using Java Generics terminology, type a is T1
and type b is T2
. So we're using pure Java to write pure Java code and none will be the wiser that we're actually doing Category Theory via a Haskell implementation ... in Java, right?
Right.
So, now we have functions (Applicable), we can provide the declaration of the Functor type:
public interface Functor
<F
,T
> {public <T1
>Applicable
<?extends Functor
<F
,T
>, ?extends Functor
<F
,T1
>> fmap(Applicable
<T
,T1
> f)throws Failure
;
public T
arg()throws Failure
; }
The above definition of fmap
is the same declaration (in Java) as its functional declaration: it takes a function f: a → b
and gives a function g: Functor a → Functor b
Okay, we're halfway there. A Monad
is a Functor
, and we have Functor
declared.
The unit function, η
, comes for free in Java: it's called new
.
So that leaves the join function μ
.
And defining join
for a specific monad is simple. Join for the list monad is append. Join for the Maybe
monad and the ID monad is arg
. What about join
for just Monad
?
Well, there is no definition of such, but we can declare its type:
public interface Joinable
<F
,T
>extends Functor
<F
,T
> {public Functor
<F
, ?> join()throws Failure
; }
Too easy! you complain.
Well, isn't it supposed to be? Join is simply a function that takes a functor and returns functor.
The trick is that we have to ensure that T
is of type Functor<F, ?>
, and we leave that an implementation detail for each specific monad implementing join
.
That was easy!™
Monad
So now that we have all the above, Functor, Join, and Unit, we have Monad:
... BUT WAIT!
And here's the interlude of where the power of Monad comes in: bind.
What is bind
? Bind is a function that says: do something in the monadic domain.
Really, that sounds boring, right? But it's a very powerful thing. Because once we are in the monadic domain, we can guarantee things that we cannot in Java category (if there is such).
(There is. I just declared it.)[2]
So, in the monadic domain, a function thus bound can be guaranteed to, e.g., be working with an instance and not a null.
Guaranteed.
AND! ...
Well, what is bind?
Bind is a function that takes an ordinary value and returns a result in the monadic domain:
Monad m ⇒ bind: a → m b
So the 'AND!' of this is that bind
can be chained ... meaning we can go from guarantee to guarantee, threading a safe computation from start to finish.
Monad
gives us bind
.
And the beauty of Monad
? bind
comes for free, for once join
is defined, bind
can be defined in terms of join
:
Monad m ⇒ bind (f: a → m b) (m a) = join ((fmap f) (m a))
Do you see what is happening here? We are using fmap
to lift the function f
one step higher into the monadic domain, so fmap f
gives the function Monad m ⇒ g: m a → m (m b)
so we are not actually passing in a value of type a
, we are passing in the Monad m a
, and then we get back a type of m (m b)
which then join
does it magic to simplify to the monadic type m b
.
So to the outside world, it looks like we are passing in an a
to get an m b
but this definition allows monadic chaining, for a
becomes m a
and then m (m b)
is simplified to m b
that — and here's the kicker — is passed to the next monadically bound function.
You can chain this as far as you'd like:
m a >>= f >>= g >>= ... >>= h → m z
And since the computation occurs in the monadic domain, you are guaranteed the promise of that domain.
There is the power of monadic programming.
Given that explanation, and remembering that bind
can be defined in terms of join
, let's define Monad:
public abstract class Monad
<M
,A
>implements Joinable
<M
,A
> {public <T
>Monad
<M
,T
> bind(Applicable
<A
,Monad
<M
,T
>> f)throws Failure
{Applicable
<Monad
<M
,A
>,Monad
<M
,Monad
<M
,T
>>> a = (Applicable
<Monad
<M
,A
>Monad
<M
,Monad
<M
,T
>>>) fmap(f);Monad
<M
,Monad
<M
,T
>> mmonad = a.apply(this );return (Monad
<M
,T
>) mmonad.join(); }
public Monad
<M
,A
> fail(String
ex)throws Failure
{throw new Failure
(ex); } }
The bind implementation is exactly as previously discussed: bind
is the application of the fmap
of f
with the join
ed result returned. So, we simply define join
for subclasses, usually a trivial definition, and we have the power of bind
automagically!
Now fail
is an interesting beast in subclasses, because in the monadic domain, failure is not always a bad thing, or even an exceptional one, and can be quite, and very, useful. Some examples, fail represents 'zero' in the monadic domain, so 'failure' for list is the empty list, and 'failure' for Maybe
is Nothing
. So, when failure occurs, the computation can continue gracefully and not always automatically abort from the computation, which happens with the try/catch paradigm.
There you have it: Monads in Java!
The rest of the story: ... Maybe
So, we could've ended this article right now, and you have everything you need to do monadic programming in Java. But so what? A programming paradigm isn't useful if there are no practical applications. So let's give you one: Maybe
.
Maybe
is a binary type, it is either Nothing
or Just x
where x
is whatever value that is what we're really looking for (as opposed to the usual case in Java:
So, let's define the Maybe
type in Java.
First is the protocol that extends the Monad
:
public abstract class Maybe
<A
>extends Monad
<Maybe
,A
> {
Do you see how the Maybe
type declares itself as a Monad
in its own definition? I just love that.
Anyway.
As mentioned before, fail
for Maybe
is Nothing
:
public Maybe
<A
> fail(String
ex)throws Failure
{return (Maybe
<A
>)NOTHING
; }
(NOTHING
is a constant in Maybe
to be defined later. Before we define it ('it' being NOTHING
), recall that a Monad
is a Functor
so we have to define fmap
. Let's do that now)
protected abstract <T
>Maybe
<T
> mbBind(Applicable
<A
,Monad
<Maybe
,T
>> arg)throws Failure
;
public <T
>Applicable
<Maybe
<A
>,Maybe
<T
>> fmap(final Applicable
<A
,T
> f)throws Failure
{return new Applicable
<Maybe
<A
>,Maybe
<T
>>() {public Maybe
<T
> apply(Maybe
<A
> arg)throws Failure
{Applicable
<A
,Monad
<Maybe
,T
>> liFted =new Applicable
<A
,Monad
<Maybe
,T
>>() {public Maybe
<T
> apply(A
arg)throws Failure
{return Maybe
.pure(f.apply(arg)); } };return (Maybe
<T
>)arg.mbBind(liFted); } }; }
No surprises here. fmap
lifts the function f: a → b
to Maybe m ⇒ g: m a → m b
where in this case the Functor
is a Monad
, and this case, that Monad
is Maybe
.[3]
There is the small issue(s) of what the pure
and the method mbBind
are, but these depend on the definitions of the subclasses NOTHING
and Just
, so let's define them now.
public static final Maybe
<?>NOTHING
=new Maybe
() {public String
toString() {return "Nothing"; }public Object
arg()throws Failure
{throw new Failure
("Cannot extract a value from Nothing."); }public Functor
join()throws Failure
{return this ; }protected Maybe
mbBind(Applicable
f) {return this ; } };
Trivial definition, as the Null Object pattern is trivial. But do note that the bind
operation does not perform the f
computation, but skips it, returning Nothing
, and forwarding the computation. This is expected behavior, for:
Nothing >>= f → Nothing
The definition of the internal class Just
is nearly as simple:
public final static class Just
<J
>extends Maybe
<J
> {public Just
(J
obj) { _unit = obj; }
public Maybe
<?> join()throws Failure
{try {return (Maybe
<?>)_unit; }catch (ClassCastException
ex) {throw new Failure
("Joining on a flat structure!"); } }public String
toString() {return "Just " + _unit; }public Object
arg()throws Failure
{return _unit; }protected <T
>Maybe
<T
> mbBind(Applicable
<J
,Monad
<Maybe
,T
>> f)throws Failure
{return (Maybe
<T
>)f.apply(_unit); }private final J
_unit; }
As you can see, the slight variation to NOTHING
for Just
is that join
returns the _unit
value if it's the Maybe
type (and throws a Failure
if it isn't), and mbBind
applies the monadic function f
to the Just
value.
And with the definition of Just
we get the pure
:
public static <T
>Maybe
<T
> pure(T
x) {return new Just
<T
>(x); }
And then we close out the Maybe
implementation with:
Simple.
Practical Example
Maybe
is useful for any semideterministic programming, that is: where something may be true, or it may not be. But the question I keep getting, from Java coders, is this: "I have these chains of getter methods in my web interface to my data objects to set a result, but I often have
"Fine, no problems ..." I begin, but then they interrupt me.
"No, I'm not done yet! I have like tons of these chained statements, and I don't want to abort on the first one that throws a NullPointerException
, I just want to pass through the statement with the
Weirdo stuff? Excusez moi?
I don't feel it's apropos that functionally pure languages have no (mutable) assignment, as that throws provability out the window and is therefore the root of all evil.
No matter how sorely tempted I am.
So, instead I say: "Yeah, that's a bigger problem [soto voce: so you should switch to Haskell!], but the same solution applies."[4]
So, let's go over the problem and a set of solutions.
The problem is something like:
try { foo.setD(getA().getB().getC().getD()); foo.setE(getA().getB().getC().getE());bar.setJ(getF().getG().getH().getJ()); bar.setM(getF().getG().getK().getM());
// and ... oh, more than 2000 more assignments ... 'hypothetically'
}
catch (Exception
ex) { ex.printStackTrace(); }
And the problem is this: anywhere in any chain of gets, if something goes wrong, the entire computation is stopped from that point, even if there are, oh, let's say, 1000 more valid assignments.
A big, and real, problem. Or 'challenge,' if you prefer.
Now this whole problem would go away if the Null Object pattern was used everywhere. But how to enforce that? In the Java category, you cannot. Even if you make the generic Null Object the base object, the
And if getA(), or getB(), or getC() return
Solutions
Solution 1: test until you puke
The first solution is to test for null at every turn. And you know what that looks like, but here you go. Because why? Because if you think it or code it, you've got to look at it, or I've got to look at it, so here it is:
A
a = getA();if (a !=null ) {B
b = a.getB();if (b !=null ) {C
c = b.getC();if (c !=null ) { foo.setD(c.getD()); } } }
What's the problem with this code?
Oh, no problems, just repeat that deep nesting for every single assignment? So you have [oh, let's say 'hypothetically'] 2000 of these deeply nested conditional blocks?
And, besides the fact that it entirely clutters the simple assignment:
in decision logic and algorithms that are totally unnecessary and entirely too much clutter.
All I'm doing is assigning a D
to a foo
! So why do I have to juggle all this conditional code in my head to reach that eventual assignment.
Solution 1 is bunk.
Solution 2: narrow the catch
So, solution 1 is untenable. But we need to assign where we can and skip where we can't. So how about this?
try { foo.setD(getA().getB().getC().getD()); }catch (Exception
ex) { // Intentionally empty; silently don't assign if we cannot; }
try { foo.setE(getA().getB().getC().getE()); }catch (Exception
ex) { // ditto }
try { bar.setJ(getF().getG().getH().getJ()); }catch (Exception
ex) { // ditto }
try { bar.setM(getF().getG().getK().getM()); }catch (Exception
ex) { // ditto }
And I say to this solution, meh! Granted, it's much better than solution 1, but, again, you made a computational flow described declaratively in the problem to be these set of staccato statements, having the reader of your code switch into and out of context for every statement.
Wouldn't it be nice to have all the statements together in one block, because, computationally, that's what is happening: a set of data is collected from one place and moved to another, and that is what we wish to describe by keep these assignment grouped in one block.
Solution 3: Maybe Monad
And that's what we can do with monads. It's going to look a bit different that how you're used to the usual Java fair, as we have to lift the statements in the Java category into expressions in the monadic one.
So here goes.
First of all, we need to define generic functions (not methods![5]) for getting values from objects and setting values into object in the monadic domain:
public abstract class SetterM
<Receiver
,Datum
>implements Applicable
<Datum
,Monad
<Maybe
,Receiver
>> {protected SetterM
(Receiver
r) {this .receiver = r; }// a monadic set function
public final Monad
<Maybe
,Receiver
> apply(Datum
d)throws Failure
{ set(receiver, d);return Maybe
.pure(receiver); }// subclasses implement with the particular set method being called
protected abstract void set(Receiver
r,Datum
d);
private final Receiver
receiver; }
This declares a generic setter, so to define setD
on the foo
instance, we would do the following:
SetterM
<Foo
,D
> setterDinFoo =new SetterM
<Foo
,D
>(foo) {protected void set(Foo
f,D
d) { f.setD(d); } };
The getter functional is similarly defined, with the returned value lifted into (or 'wrapped in,' if you prefer) the Maybe
type:
public abstract class GetterM
<Container
,Returned
>implements Applicable
<Container
,Monad
<Maybe
,Returned
>> {
public Monad
<Maybe
,Returned
> apply(Container
c)throws Failure {Maybe
<Returned
> ans = (Maybe
<Returned
>)Maybe
.NOTHING
;Returned
result = get(c);//
c
inside the monad is guaranteed to be an instance // butresult
has NO guarantees!
if (result !=null ) { ans =Maybe
.pure(result); // and nowans
is inside the monad it must have a value. // ... even if that value isNOTHING
}return ans; }
protected abstract Returned
get(Container
c); }
With the above declaration, a getter monadic function (again, not method) is simply defined:
GetterM
<A
,B
> getterBfromA =new GetterM
<A
,B
>() {protected B get(A a) {return a.getB(); } };
In Haskell, the bind
operator has a dataflow look to it: (>>=)
, so writing one of the 'assignments' is as simple as flowing the data to the setter:
getA >>= getB >>= getC >>= getD >>= setD foo
But we have no operator definitions, so we must soldier on using the Java syntax:
try { getterA.apply(this ).bind(getterBfromA).bind(getterCfromB).bind(getterDfromC).bind(setterDinFoo); }catch { // a superfluous catch block }
That's one of the assignment expressions.[6]
Let's walk through what happens with a couple of examples.
Let's say that A has a value of a
, but B is
getterA
forwardsJust a
togetterBfromA
getterBfromA
gets anull , converts that into aNOTHING
and forwards that togetterCfromB
getterCfromB
forwards theNOTHING
togetterDfromC
getterDfromC
forwards theNOTHING
tosetterDinFoo
setterDinFoo
simply returnsNOTHING
.
NOTHING
).
Let's counter with an example where every value has an instance:
getterA
forwardsJust a
togetterBfromA
getterBfromA
gets ab
, converts that intoJust b
and forwards that togetterCfromB
getterCfromB
gets ac
, converts that intoJust c
and forwards that togetterDfromC
getterDfromC
gets ad
, converts that intoJust d
and forwards that tosetterDinFoo
setterDinFoo
gets the wrappedd
and sets that value in theFoo
instance.
Voilà!
The beauty of this methodology is that we can put them all into one block, and every expression will be evaluated and in a typesafe manner, too, as NOTHING
will protect us from a null pointer access:
try { getterA.apply(this).bind(getterBfromA).bind(getterCfromB).bind(getterDfromC).bind(setterDinFoo); System.out.println("Hey, I've executed the first assignment"); getterA.apply(this).bind(getterBfromA).bind(getterCfromB).bind(getterEfromC).bind(setterEinFoo); System.out.println("...and the second..."); getterF.apply(this).bind(getterGfromF).bind(getterHfromG).bind(getterJfromH).bind(setterJinBar); System.out.println("...and third..."); getterF.apply(this).bind(getterGfromF).bind(getterKfromG).bind(getterMfromK).bind(setterMinBar); System.out.println("...and fourth...");// another more than 2000 of these expressions System.out.println("...and we're done with every assignment we could assign...");
}
catch { // a superfluous catch block }
And, because the monadic Maybe
works it does, every assignment that can occur will, and ones that cannot will be 'skipped' (by flowing NOTHING
through the rest of the computation, and the proof will be in the pudding ... on the standard output will be all the statements printed.
Summary
We presented an implementation of monads in Java, with a concrete example of the Monad
type: the Maybe
type. We then showed that by putting a computational set into the monadic domain, the work can be performed in a typesafe manner, even with the possible presence of
End notes
[1]  Exact verbage: Phil says: January 14, 2009 at 7:01 pm I think that this could all be easily cleared up if one of you FP guys would just show us how to write one of these monad thingies in Javaâ To which a semimocking response was: 'you know, objects would be a lot easier for me to understand if you provided an implementation in Haskell.' Exact verbage: Cory says: April 23, 2009 at 6:26 am I may be a bit late to the game here, but Phil, that can be rephrased: I think that this could all be easily cleared up if one of you OO guys would just show us how to write one of these object thingies in Haskellâ Of course you can, but it's a different type of abstraction for a different way of thinking about programmingâ I write 'semimocking,' because: 1. OOP with message passing has been implemented in a Haskelllike language: Common Lisp. The Art of the Metaobject Protocol, a wonderful book, covers the 'traditional' objectoriented programming methodology, and its implementation, quite thoroughly and simply. 
[2]  I leave it as an exercise to the reader to scope the Java category (Consult the Ωcalculus papers, Cardelli, et al;). 
[3] 
Monad <Maybe , A >Maybe <A > 
[4]  "The same solution applies!" Geddit? Haskell is an applicative language, so the same solution applies! GEDDIT? 
[5]  The distinction I raise here between methods and functions is the following: a method is enclosed in and owned by the defining class. A function, on the other hand, is a freestanding object and not necessarily contained in nor attached to an object. 
[6] 
Again, there is a distinction between an expression and a statement. An expression returns a value; a statement does not. A statement is purely imperative ... 'do something.' On the other hand, an expression can be purely functional ... 'do'ing nothing, as it were, e.g.: 3 + 4 . A pure expression has provable properties that can be attached to it, whereas it is extremely difficult to make assertions about statements in the general sense. The upshot is that having pure (nonsideeffecting) expressions in code allows us to reason about that code from a firm theoretical foundational basis. Such code is easier to test and to maintain. Sideeffecting statement code make reasoning, testing and maintaining such code rather difficult.

8 comments:
I have no words for this great post such a awesome information i got gathered. Thanks to Author.
Vee Eee Technologies
Wow. This would have probably served as a master's!
Great job!
Man I really hate how java has no annon functions or operator overloading...
Very interesting! Cool implementation
You should check out my project highJ ( http://code.google.com/p/highj ), which implements some interesting parts of Haskell's type class hierarchy including monads and arrows.
A problem, you skipped over how to write unit essentially with "you get new for free". Java does not have type constructors so you can't write unit in a generic way because you can't lift a value into the monad. This means you have to repeated write all the monad library methods (e.g. sequence and traverse) for each monad.
All 3 solutions look bad. Simply because Java is not the right language for concise and productive development any longer. Good effort explaining all this though.
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