Simple Introduction to Monads

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

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 );
    // alternative:
    // ResultOrError upperCased = bind ( trimmed, string -> toUpperCase(string) );
    
    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 );
        // alternative:
        // ResultOrError upperCased = bind ( trimmed, string -> toUpperCase(string) );
        
        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:

  Java

  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:

  Java

  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:

  Java

  <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:

• return a >>= k = k a

• m >>= return = m

• m >>= (\x -> k x >>= h) = (m >>= k) >>= h

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

  • Java-like languages:
    M<T> create ( T value )

    

  • Haskell:
    return :: a -> m a

• 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