Function Composition
In functional programming languages, nested function calls (like our `appendExclam ( toUpperCase ( trim ( sentence ) ) )`

) are called function composition .

Important
Function composition is the bread and butter of functional programming languages. In the lambda calculus, the body of a function is a single expression. Complex expressions can be created by composing functions.

Function composition

As we will see later, monads provide a solution to a problem that can arise when we compose functions. Before going on, let us therefore see variations of using function composition in different environments. Readers familiar with this important concept can jump to the next chapter.

Unix Pipes
First, it is interesting to note that the idea of function composition is similar to pipes in Unix/Linux . The output of a first command is fed as input into a second command. Then the output of the second command is fed as input into a third command, and so on. In Unix/Linux, the symbol | is used to pipe commands. Here is an example of a pipe that counts the number of files containing "page" in their name (example borrowed from How to Use Pipes on Linux ):

ls - | grep "page" | wc -l
Pipe Operator
Because pipes are useful in many contexts, some programming languages have a specific pipe operator. For example, F# uses |> to chain function calls. If Java had this operator, then function `enthuse`

could be written as:

static String enthuse ( String sentence ) {
return trim ( sentence ) |> toUpperCase |> appendExclam;
}
... which would semantically be the same, but a bit more readable than real Java that uses nested function calls:

static String enthuse ( String sentence ) {
return appendExclam ( toUpperCase ( trim ( sentence ) ) );
}
Function Composition Operator
As function composition is essential, most functional programming languages have a dedicated function composition operator that makes it trivial to compose functions.

For example, in Haskell a dot (.) is used to compose functions (derived from the ring operator symbol ∘ used in maths). The dot is itself a function whose signature is defined as follows:

(.) :: (b -> c) -> (a -> b) -> a -> c
This tells us that the function takes two functions as input (`b -> c`

and `a -> b`

), and returns another function (`a -> c`

), which is the composition of the two input functions.

Hence, to state that function `h`

is the composition of functions `f`

and `g`

, in Haskell you can simply write:

h = f . g
Note the totally different semantics of the dot operator in Haskell and object oriented languages like C#, Java, etc. In Java, `f.g`

means applying `g`

on object f (e.g. `person.name`

). In Haskell, it means composing functions `f`

and `g`

.

F# uses `>>`

to compose functions. It is defined like this:

let (>>) f g x = g(f(x))
And it is used as follows:

let h = f >> g
Note

F#'s `>>`

operator must not be confused with the Monad sequencing operator in Haskell, which also uses the symbol `>>`

.

If Java had a syntax similar to F# for function composition, then function `enthuse`

could simply be written as follows:

static String enthuse ( String sentence ) = trim >> toUpperCase >> appendExclam;
Errors, But No Exceptions
For the sake of this tutorial, suppose that our functions can fail in the following ways:

Function `trim`

fails if the input string is empty or contains only spaces (i.e., the result cannot be an empty string).

Function `toUpperCase`

fails if the input string is empty or contains characters others than letters or spaces.

Function `appendExclam`

fails if the input string is more than 20 characters long.

In idiomatic Java, an exception is thrown to indicate an error. But pure functional programming languages don't support exceptions, because a function cannot have a side-effect. A function that can fail must return error information as part of the result returned by the function. For example, the function returns a `string`

in case of success, or error data in case of an error.

So let's do that in Java.

First, we define a simple error class with an info field that describes the error:

public class SimpleError {
private final String info;
public SimpleError ( String info ) {
this .info = info;
}
public String getInfo() { return info; }
public String toString() { return info; }
}
As said already, the functions must be able to return a `string`

in case of success, or else an error object. To achieve this, we can define class `ResultOrError`

:

public class ResultOrError {
private final String result;
private final SimpleError error;
public ResultOrError ( String result ) {
this .result = result;
this .error = null;
}
public ResultOrError ( SimpleError error ) {
this .result = null;
this .error = error;
}
public String getResult() { return result; }
public SimpleError getError() { return error; }
public boolean isResult() { return error == null; }
public boolean isError() { return error != null; }
public String toString() {
if ( isResult() ) {
return " Result: " + result;
} else {
return " Error: " + error.getInfo();
}
}
}
As we can see:

The class has two immutable fields to hold either a result or an error.

There are two constructors.

The first one is used in case of success (e.g., `return new ResultOrError ( "hello");`

).

The second constructor is used in case of failure (e.g., `return new ResultOrError ( new Error ( "Something went wrong") );`

).

`isResult`

and `isError`

are utility functions.

`toString`

is overridden for debugging purposes.

Now we can rewrite the three utility functions to include error handling:

public class StringFunctions {
public static ResultOrError trim ( String string ) {
String result = string.trim();
if ( result.isEmpty() ) {
return new ResultOrError ( new SimpleError (
" String must contain non-space characters." ) );
}
return new ResultOrError ( result );
}
public static ResultOrError toUpperCase ( String string ) {
if ( ! string.matches ( " [a-zA-Z ]+" ) ) {
return new ResultOrError ( new SimpleError (
" String must contain only letters and spaces." ) );
}
return new ResultOrError ( string.toUpperCase() );
}
public static ResultOrError appendExclam ( String string ) {
if ( string.length() > 20 ) {
return new ResultOrError ( new SimpleError (
" String must not exceed 20 characters." ) );
}
return new ResultOrError ( string.concat ( " !" ) );
}
}
Note
To keep this exercise code simple, we don't check and handle `null`

values (as we would do in production code). For example, if a function is called with `null`

as input, we simply accept that a `NullPointerException`

is thrown.

What matters is that the three functions which previously returned a `string`

, now return a `ResultOrError`

object.

As a consequence, function `enthuse`

that was defined as follows:

static String enthuse ( String sentence ) {
return appendExclam ( toUpperCase ( trim ( sentence ) ) );
}
... doesn't work anymore.

Unfortunately, function composition is now invalid, because the functions now return a `ResultOrError`

object, but require a `string`

as input. The output/input types don't match anymore. The functions can't be chained anymore.

In the previous version, when the functions returned `string`

s, the output of a function could be fed as input to the next function:

Chained String Functions

But now this can't be done anymore:

However, we can still implement `enthuse`

in Java like this:

static ResultOrError enthuse ( String sentence ) {
ResultOrError trimmed = trim ( sentence );
if ( trimmed.isResult() ) {
ResultOrError upperCased = toUpperCase ( trimmed.getResult() );
if ( upperCased.isResult() ) {
return appendExclam ( upperCased.getResult() );
} else {
return upperCased;
}
} else {
return trimmed;
}
}
Not good! The initial simple one-liner has turned into an ugly monster.

We can improve a bit:

static ResultOrError enthuse_2 ( String sentence ) {
ResultOrError trimmed = trim ( sentence );
if ( trimmed.isError() ) return trimmed;
ResultOrError upperCased = toUpperCase ( trimmed.getResult() );
if ( upperCased.isError() ) return upperCased;
return appendExclam ( upperCased.getResult() );
}
This kind of code works in Java and in many other programming languages. But it's certainly not the code we want to write over and over again. Error handling code intermixed with normal flow makes code difficult to read, write, and maintain.

More importantly, we simply can't write code like this in pure functional programming languages. Remember : An expression returned by a function is made up of function composition .

It's easy to image other cases that lead to the same dilemma. And what should we do in situations where the same problem appears in many variations? Yes, we should try to find a general solution that can be used in a maximum of cases.

Monads to the rescue!

Monads provide a general solution for this kind of problem, and they have other benefits as well. As we will see later, monads enable function composition for so-called monadic functions that cannot be composed directly because their types are incompatible.

Some people say: "You could have invented monads if they didn't exist." (for example, Brian Beckman in his excellent presentation, Don't fear the Monad )

That's true!

So let's try to find a solution ourselves, temporarily ignoring the fact that a monad would solve our problem.

The 'bind' Function
In functional programming languages, everything is done with functions. So we know already that we have to create a function to solve our problem.

Let's call this function `bind`

, because its role is to bind two functions that cannot be composed directly.

Next we have to decide what should be the input of bind, and what should it return. Let's consider the case of chaining functions `trim`

and `toUppercase`

:

The logic to implement must work as follows:

If trim returns a `string`

, then `toUppercase`

can be called, because it takes a `string`

as input. So the final output will be the output of `toUppercase`

If trim returns an error, then `toUppercase`

cannot be called, and the error must simply be forwarded. So the final output will be the output of `trim`

We can deduce that bind needs two input arguments:

the result of `trim`

, which is of type `ResultOrError`

function `toUppercase`

, because if `trim`

returns a `string`

then bind must call `toUppercase`

The output type of `bind`

is easy to determine. If `trim`

returns a `string`

, then the output of `bind`

is the output of `toUppercase`

, which is of type `ResultOrError`

. If trim fails, then the output of bind is the output of `trim`

, which is also of type `ResultOrError`

. As the output type is `ResultOrError`

in both cases, the output type of bind must be `ResultOrError`

.

So now, we know the signature of bind:

In Java, this is written as:

ResultOrError bind ( ResultOrError value, Function<String, ResultOrError> function )
Implementing bind is easy, because we know exactly what needs to be done:

static ResultOrError bind ( ResultOrError value, Function<String, ResultOrError> function ) {
if ( value.isResult() ) {
return function.apply ( value.getResult() );
} else {
return value;
}
}
Function `enthuse`

can now be rewritten as follows:

static ResultOrError enthuse ( String sentence ) {
ResultOrError trimmed = trim ( sentence );
ResultOrError upperCased = bind ( trimmed, StringFunctions::toUpperCase );
ResultOrError result = bind ( upperCased, StringFunctions::appendExclam );
return result;
}
But this is still imperative code (a sequence of statements). If we did a good job, then we must be able to rewrite `enthuse`

by just using function composition. And indeed, we can do it like this:

static ResultOrError enthuse_2 ( String sentence ) {
return bind ( bind ( trim ( sentence ), StringFunctions::toUpperCase ),
StringFunctions::appendExclam );
}
At first sight, the body of `bind`

might be a bit confusing. We will change that later. The point is that we use only function composition in the body of `enthuse`

.

Notes
If you never saw this kind of code before, then please take your time to digest and fully understand what's going on here.

Understanding `bind`

is the key to understanding monads!

`bind`

is the function name used in Haskell. There are alternative names, such as: `flatMap`

, `chain`

, `andThen`

.

Does it work correctly? Let's test it. Here is a class containing `bind`

, the two variations of `enthuse`

, and some simplistic tests that cover the success and all error paths:

public class MonadTest_04 {
static ResultOrError bind
( ResultOrError value, Function<String, ResultOrError> function ) {
if ( value.isResult() ) {
return function.apply ( value.getResult() );
} else {
return value;
}
}
static ResultOrError enthuse ( String sentence ) {
ResultOrError trimmed = trim ( sentence );
ResultOrError upperCased = bind ( trimmed, StringFunctions::toUpperCase );
ResultOrError result = bind ( upperCased, StringFunctions::appendExclam );
return result;
}
static ResultOrError enthuse_2 ( String sentence ) {
return bind ( bind ( trim ( sentence ), StringFunctions::toUpperCase ),
StringFunctions::appendExclam );
}
private static void test ( String sentence ) {
System.out.println ( enthuse ( sentence ) );
System.out.println ( enthuse_2 ( sentence ) );
}
public static void tests() {
test ( " Hello bob " );
test ( " " );
test ( " hello 123" );
test ( " Krungthepmahanakhon is the capital of Thailand" );
}
}
Running function `tests`

outputs:

Result: HELLO BOB!
Result: HELLO BOB!
Error: String must contain non-space characters.
Error: String must contain non-space characters.
Error: String must contain only letters and spaces.
Error: String must contain only letters and spaces.
Error: String must not exceed 20 characters.
Error: String must not exceed 20 characters.
The `bind`

function defined above serves our specific problem. But to make it part of a monad, we will have to make it more general. We will do that soon.

Note
Using `bind`

as shown above, is the common way to solve our problem of function composition. But it's not the only way. An alternative is called Kleisli composition (out of the scope of this article).

A Monad, Finally!
Now that we have `bind`

, the remaining steps to get to the monad are easy. We just need to make some improvements in order to have a more general solution that can be applied in other cases too.

Our goal in this chapter is clear: See the pattern and improve `ResultOrError`

, so that we can finally enthuse:

"A MONAD!"

First Improvement
In the previous chapter, we defined `bind`

as an isolated function that fulfilled our specific needs. A first improvement is to move `bind`

into the `ResultOrError`

class. Function `bind`

must be part of our monad class. The reason is that the implementation of `bind`

depends on the monad that uses `bind`

. While the signature of `bind`

is always the same, different kinds of monads use different implementations .

Second Improvement
In our example code, the composed functions all take a `string`

as input, and return either a `string`

or an error. What if we have to compose functions that take an integer, and return an integer or error? Can we improve `ResultOrError`

so that it works with any type of result? Yes, we can. We just have to add a type parameter to `ResultOrError`

.

After moving `bind`

into the class and adding a type parameter, the new version now becomes:

public class ResultOrErrorMona<R> {
private final R result;
private final SimpleError error;
public ResultOrErrorMona ( R result ) {
this .result = result;
this .error = null;
}
public ResultOrErrorMona ( SimpleError error ) {
this .result = null;
this .error = error;
}
public R getResult() { return result; }
public SimpleError getError() { return error; }
public boolean isResult() { return error == null; }
public boolean isError() { return error != null; }
static <R> ResultOrErrorMona<R> bind
( ResultOrErrorMona<R> value, Function<R, ResultOrErrorMona<R>> function ) {
if ( value.isResult() ) {
return function.apply ( value.getResult() );
} else {
return value;
}
}
public String toString() {
if ( isResult() ) {
return " Result: " + result;
} else {
return " Error: " + error.getInfo();
}
}
}
Note the class name: `ResultOrErrorMona`

. This is not a typo. The class isn't a monad yet, so I call it a mona (just for fun).

Third Improvement
Suppose we have to chain the following two functions:

ResultOrError<Integer> f1 ( Integer value )
ResultOrError<String> f2 ( Integer value )
Here is a picture to illustrate this:

Our current `bind`

function is not able to handle this case, because the output types of the two functions are different (`ResultOrError<Integer>`

and `ResultOrError<String>`

). We have to make bind more general, so that functions of different value types can be chained. The signature of bind must be changed from:

static <R> Monad<R> bind ( Monad<R> monad, Function<R, Monad<R>> function )
... to

static <R1, R2> Monad<R2> bind ( Monad<R1> monad, Function<R1, Monad<R2>> function )
The implementation of `bind`

must be adapted too. Here is the new class:

public class ResultOrErrorMonad<R> {
private final R result;
private final SimpleError error;
public ResultOrErrorMonad ( R result ) {
this .result = result;
this .error = null;
}
public ResultOrErrorMonad( SimpleError error ) {
this .result = null;
this .error = error;
}
public R getResult() { return result; }
public SimpleError getError() { return error; }
public boolean isResult() { return error == null; }
public boolean isError() { return error != null; }
static <R1, R2> ResultOrErrorMonad<R2> bind
( ResultOrErrorMonad<R1> value, Function<R1, ResultOrErrorMonad<R2>> function ) {
if ( value.isResult() ) {
return function.apply ( value.result );
} else {
return new ResultOrErrorMonad<R2> ( value.error );
}
}
public String toString() {
if ( isResult() ) {
return " Result: " + result.toString();
} else {
return " Error: " + error.toString();
}
}
}
Note the class name again: `ResultOrErrorMonad`

.

Yes, now it's a monad.

Note
In the real world, we don't add a "monad" suffix for types that are monads. I called the class `ResultOrErrorMonad`

(instead of simply `ResultOrError`

) to make it clear that the class is a monad.

How can we be sure the class is indeed a monad?

While the term 'monad' has a very precise definition in mathematics (as everything in maths), the term isn't yet unequivocally defined in the world of programming languages. However, Wikipedia states a common definition . A monad consists of three parts:

A type constructor `M`

that builds up a monadic type `M T`

.

In other words, there is a type parameter for the value contained in the monad.

In our case, it's type parameter `R`

in the class declaration:

class ResultOrErrorMonad<R>
A type converter, often called `unit`

or `return`

, that embeds an object x in the monad: `unit(x) : T → M T`

In Haskell, the type converter is defined as: `return :: a -> m a`

In Java-like languages, this means there must be a constructor that takes a value of type `R`

, and returns a monad `M<R>`

that contains this value.

In our specific case, it's the constructor of class `ResultOrErrorMonad`

:

public ResultOrErrorMonad ( R result )
A combinator, typically called `bind`

(as in binding a variable) and represented with an infix operator `>>=`

, that unwraps a monadic variable, then inserts it into a monadic function/expression, resulting in a new monadic value: `(mx >>= f) : (M T, T → M U) → M U`

In Haskell, bind is defined as:` (>>=) :: m a -> (a -> m b) -> m b`

In our example, it's the `bind`

function:

<R1, R2> ResultOrErrorMonad<R2> bind
( ResultOrErrorMonad<R1> value, Function<R1, ResultOrErrorMonad<R2>> function )
Wikipedia then states: "To fully qualify as a monad though, these three parts must also respect a few laws: ..."

In Haskell , the three laws are defined as follows:

Discussing these laws is out of the scope of this article (this is an introduction to monads). The laws ensure that monads behave well in all situations. Violating them can lead to subtle and painful bugs as explained here , here , and here . As far as I know, there is currently no compiler able to enforce the monad laws. Hence, it's the developer's responsibility to verify that the monad laws are respected. Suffice to say that the above `ResultOrErrorMonad`

fulfills the monad laws.

Although we are done, there is still room for improvement.

Maximizing Reusability
Besides having a type parameter for the result value, we could also add a type parameter for the error value. This makes the monad more reusable, because users of the monad are now free to decide which type of error they want to use. For an example, you can look at F#'s Result type.

Finally, we could make the monad even more reusable by letting the user define the meaning of the two values. In our example, one value represents a result, and the other one represents an error. But we can abstract more. We can create a monad that simply holds one of two possible values - either `value_1`

or `value_2`

. And the type of each value can be freely defined by a type parameter. This is indeed a standard monad supported by some functional programming languages. In Haskell, it's called `Either`

. It's constructor is defined as follows:

data Either a b = Left a | Right b
Using our `ResultOrErrorMonad`

class as a starting point, it would be easy to create an `Either`

monad in Java.

Note
Some projects use an `Either`

monad for functions that might fail. In my opinion, using a more specific `ResultOrError`

type is a better, less error-prone option (for reasons not explained here).

An OO Bind
Now that we know how a monad works in functional programming languages, let's go back to the world of OOP (object-oriented programming). Could we create something like an OO-monad?

If we look at class `<a>ResultOrErrorMonad</a>`

, we can see that everything in this class is already standard Java, with one exception: Function `bind`

is a static member of the class. This means we can't use the dot-syntax of object methods for bind. Currently the syntax for calling bind is `bind ( v, f )`

. But if bind was a non-static member of the class, we could write `v.bind ( f )`

. This would make the syntax more readable in the case of nested function calls.

Luckily, it's easy to make `bind`

non-static.

To make the monad a bit more versatile, let's also introduce a second type parameter for error values. Then the users are not required to use `SimpleError`

- they can use their own error class.

Here is the code of a `ResultOrError`

monad, OO-style:

public class ResultOrError<R, E> {
private final R result;
private final E error;
private ResultOrError ( R result, E error ) {
this .result = result;
this .error = error;
}
public static <R, E> ResultOrError<R, E> createResult ( R result ) {
return new ResultOrError<R, E> ( result, null );
}
public static <R, E> ResultOrError<R, E> createError ( E error ) {
return new ResultOrError<R, E> ( null, error );
}
public R getResult() { return result; }
public E getError() { return error; }
public boolean isResult() { return error == null; }
public boolean isError() { return error != null; }
public <R2> ResultOrError<R2,E> bind ( Function<R, ResultOrError<R2,E>> function ) {
if ( isResult() ) {
return function.apply ( result );
} else {
return createError ( error );
}
}
public String toString() {
if ( isResult() ) {
return " Result: " + result.toString();
} else {
return " Error: " + error.toString();
}
}
}
Now the code for using bind in the body of function `enthuse`

becomes more readable. Instead of writing:

return bind ( bind ( trim ( sentence ), v -> toUpperCase(v) ), v -> appendExclam(v) );
... we can avoid the nesting and write:

return trim ( sentence ).bind ( v -> toUpperCase(v) ).bind ( v -> appendExclam(v) );
So, can monads be useful in real-world OOP environments?

Yes, they can .

But the word "can" needs to be stressed, because it depends (as so often) on what we want to achieve. Let's say, for instance, that we have some good reasons to 'not use exceptions' for error handling.

Remember the ugly error-handling code we had to write in chapter Errors, But No Exceptions :

static ResultOrError enthuse ( String sentence ) {
ResultOrError trimmed = trim ( sentence );
if ( trimmed.isResult() ) {
ResultOrError upperCased = toUpperCase ( trimmed.getResult() );
if ( upperCased.isResult() ) {
return appendExclam ( upperCased.getResult() );
} else {
return upperCased;
}
} else {
return trimmed;
}
}
Using a monad removes the boilerplate:

static ResultOrError enthuse ( String sentence ) {
return trim ( sentence ).bind ( v -> toUpperCase(v) ).bind ( v -> appendExclam(v) );
}
Nice!

Summary
The key to understanding monads is to understand `bind`

(also called `chain`

, `andThen`

, etc.). Function `bind`

is used to compose two monadic functions. A monadic function is a function that takes a value of type `T`

and returns an object that contains the value (`a -> m a`

). Monadic functions cannot be directly composed because the output type of the first function called is not compatible to the input type of the second function. `bind`

solves this problem.

Function `bind`

is useful on its own. But it's just one part of a monad.

In Java-like world, a monad is a class (type) `M`

with:

a type parameter `T`

that defines the type of the value stored in the monad (e.g. `M<T>`

)

a constructor that takes a value of type `T`

and returns a monad `M<T>`

containing the value

a `bind`

function used to compose two monadic functions

Java-like languages:
`M<T2> bind ( M<T1> monad, Function<T1, M<T2>> function )`

Haskell:
`(>>=) :: m a -> (a -> m b) -> m b`

A monad must respect three monad laws. These laws ensure that the monad behaves well in all situations.

Monads are primarily used in functional programming languages, because these languages rely on function composition. But they can also be useful in the context of other paradigms, such as object-oriented programming languages that support generic types and higher order functions.

Final Words
As hinted in its title, this article is an introduction to monads. It doesn't cover the full spectrum of monads, it doesn't show other useful examples of monads (Maybe monad, IO monad, state monad, etc.) and it completely ignores 'category theory', the mathematical background for monads. For those who want to know more, an abundance of information is already available on the net.

Hopefully, this article helps to get the gist of monads, see their beauty, and understand how they can improve code and simplify life.

"HAPPY MONADING!"

Note
This article was written in PML (Practical Markup Language). You can look at the PML code on Gitlab .

The source code examples used in this article are stored on Gitlab .

History
7^{th} January, 2021: Initial version