Harder to C++: Monads for Mortals [6], Performance of Basic Implementations

In parts 2 – 5 we have seen four different implementations of the monad. Now, it is time for a shootout. I want to see which version has best time performance and compare it with a regular task to evaluate the cost of using a monad.

The monad implementations are:

1. The first implementation that copies all its data.
2. The lazy implementation that evaluates at explicit call.
3. The references and move semantics implementation.
4. The pointer based implementation.

The Task at Hand

The challenge is to create a task that is hard to optimize by cleverly removing parts of the functionality. Imagine we have an algorithm that fills a vector with arbitrary data, which is then thrown away unused because the vector goes out of scope. We wouldn’t want the compiler to decide that it is useless to create the vector altogether and skip the code.

Further, I will make the task less predictable by extensive use of the rand() function. I know, rand() is not a great pseudo random engine. However, rand() is not used here to simulate randomness, but to get values that are available only at runtime, not at compile time – when the code optimizer runs. This way, the code cannot have optimizations based on compile time knowledge of constant values.

The algorithm that will be implemented by the functions that are ‘applied’ by the monad, and that will also be implemented by two regular functions is as follows (we will be working with the ValueObject class used in earlier episodes, but with a method added that retrieves a pointer to the payload):

1. Receive a (pointer / reference to a) ValueObject object.
2. Retrieve a char from the ValueObject object’s payload, with a random index, stored it in a global data structure.
3. Create a size for a new ValueObject object, that depends on the size of the received object, and the pseudo random generator.
4. Create a (monad with a new) ValueObject object and return (a pointer to) it.

Step 2 is to ensure that the payload is not optimized away because it will never be accessed. Step 3 serves to make the size of the payload somewhat unpredictable, and also wildly varying.

This is the version for the reference and move semantics based implementation:

	// Creates one more ValueObject object
	monad<ValueObject> onemore(const ValueObject& v)
	{
		// samples is a file scope vector<char>
		samples.push_back(*(v.getload() + rand() % (v.size()+1)));

		do
		{
			rsize = rand();
		} while (rsize == 0);

		vsize = v.size() + 1;
		rsize = rsize > vsize ? rsize / vsize : vsize / rsize;

		return monad<ValueObject>(ValueObject(rsize));
	}

Main Routine

In the main routine, we have a composition of 25 applications of the onemore function by the bind function. We execute this composition iterations times, which we call a run. We measure the time of each run. We average over the runs, and print the result tot screen.

The number of iterations and runs are given by rand(), which is seeded with a number obtained from the user.

Scoping is such that all objects are created and destroyed in measured time.

This is the code for the reference and move semantics based implementation:

{
	using namespace I3;

	srand(seed);
	unsigned int iterations = rand();
	unsigned int runs = rand() % 1000;

	cout << "Runs: " << runs << endl
		<< "Iterations per run: " << iterations << endl
		<< "Applications per iteration: 25" << endl;

	cout << "References and move semantics." << endl;
	total = 0;

	for (unsigned int i = 0; i < runs; ++i)
	{
		t.Start();

		for (unsigned int j = 0; j < iterations; ++j)
		{
			unsigned int size1 = rand();

			auto m1 = monad<ValueObject>(ValueObject(size1));

			auto m2 = m1
				| onemore | onemore | onemore | onemore | onemore
				| onemore | onemore | onemore | onemore | onemore
				| onemore | onemore | onemore | onemore | onemore
				| onemore | onemore | onemore | onemore | onemore
				| onemore | onemore | onemore | onemore | onemore;			}

		t.Stop();
		total += t.Elapsed();

		cout << ". ";
	}

	cout << endl << "Average over "
	<< runs << " differently sized runs of 25 x "
	<< iterations << " applications: " << total / runs << " ms"
	<< endl << endl;
}

As earlier, the timer code is from Simon Wybranski. The outer braces are just to assure different implementations do not interfere.

The iterations for the lazy monad differ from the above articulation in that it also makes a call: m2() to evaluate the constructed expression.

Regular Algorithms

The monad implementations are compared to regular algorithms that are functionally equivalent, but do not use monads. We will use one variant with references, and one variant with pointers and heap allocation.

The algorithm used is the same as above. The call in the main loop is terrible:

auto m2 = m(m(m(m(m(m(m(m(m(m(m(m(m(m(m(m(m(m(m(m(m(m(m(m(m(m1)
))))))))))))))))))))))));

Which also illustrates that the name onemore was abbreviated to m.

For references (&), the m function was implemented as:

	ValueObject m(const ValueObject& v)
	{
		samples.push_back(*(v.getload() + rand() % (v.size()+1)));

		do
		{
			rsize = rand();
		} while (rsize == 0);

		vsize = v.size() + 1;
		rsize = rsize > vsize ? rsize / vsize : vsize / rsize;

		// First creating a tmp, then returning it coerces move semantics
		// instead of copy semantics
		auto tmp = ValueObject(rsize);

		return tmp;
	}

Procedure

Time performance was measured using a release build, with default compiler switches for release builds, and the runs were executed outside of Visual Studio. That is, the program was started from a console window, not from Visual Studio.

Results

And now the results:

So, what do we see?

1. The monad implementation based on references and move semantics (yellow) is the fastest. But the pointer based implementation has about the same performance.
2. The regular implementations are only slightly faster (red, orange).
3. The lazy monad (blue) is very slow, as expected.

Conclusions

The choice is clear. In following episodes, we will work with the monad built on references and move semantics. If required, we might use the pointer based variant as well. I will not use the lazy monad, unless it is clear that very, very special circumstances make it a favorable choice.

I was surprised by the small difference (if at all) between the fast monad implementation and the regular algorithms; the monad covers about twice the source code as the regular function – it also includes the bind function. So (hypothesis), the cost of using the references and move semantics monad is negligible.

Next

Next time we will start looking at various types of monads and how we could build compositions of them.

Harder to C++: Monads for Mortals [5], Pointers

In part 2 we have seen a small and elegant implementation of the monad. In part 4 we have seen how references and move semantics can be used to make the monad’s performance independent of the size of the wrapped value. In this part we will see another implementation the monad based on a pointer.

Implementation of a Monad with a Smart Pointer

We have seen in part 2 that the type constructor is represented by a template in C++. In part 2 this is a simple struct, in part 3 it is a std::function holding a lambda expression – so as to implement delayed evaluation, and in this part, it is a smart pointer, the std::unique_ptr to be precise.

So, now the type constructor is:

// Monad type constructor
template<typename T>
using monad = unique_ptr<T>;

We do not use a separate unit function, the unique_ptr‘s constructor will do. The bind function is:

// Bitwise OR operator overload, Monad bind function
template<typename A, typename R>
monad<R> operator|(monad<A>& mnd, monad<R>(*func)(const A*))
{
	log("---Function bind");

	return func(mnd.get());
}

Since in the applied functions we do not manage the life cycle of the existing A object, we send in a raw pointer, not a smart pointer.

Given the simple functions we used earlier, this implementation of the monad is used as follows:

// divides arg by 2 and returns result as monadic template
monad<ValueObject> divideby2(const ValueObject* v)
{
	log("---Function divideby2");

	return monad<ValueObject>(new ValueObject(v->size() / 2));
}

// checks if arg is even and returns result as monadic template
monad<bool> even(const ValueObject* v)
{
	log("---Function even");

	return monad<bool>(new bool(v->size() % 2 == 0));
}

void valueobjectmain()
{
	{
		auto m1 = monad<ValueObject>(new ValueObject(16));

		auto m2 = m1 | divideby2 | divideby2 | divideby2 | even;

		cout << boolalpha << *m2 << endl;
	}
}

The ValueObject class is the same as used in part 4.

Running the above code results in output:

I think the most important result here is what we don’t see: calls to the move constructor and the accompanying call to the destructor of the ValueObject.

Pro’s, Con’s, and Questions

The above image of the output is nicely constrained. Functions return the unique_ptr by value, i.e. it gets copied which is ok, and the bind function takes a reference to a unique_ptr to elicit the raw pointer from. We see the calls to ValueObject destructor occur after assignment to m2, and when leaving the anonymous scope.

So, pro’s for this implementation are its simplicity, clarity, and efficiency: the overhead of a unique_ptr is comparable to the overhead of a raw pointer. Values are not copied, so performance is size independent.

You may wonder, though, whether it is a good idea to create all the monad’s values on the heap, which is not so efficient. In part 6 we take a look at what we have so far and see what works best.

Harder to C++: Monads for Mortals [3], Lazy implementation

This blog post is part 3 in a series. This part presents a simple implementation of a lazy evaluating monad. It is a minimal variation of the implementation in part 2.

Below we will see how the implementation is constructed, then an example is given that provides initial verification that the construction is indeed a monad.

Implementation of a Lazy Monad

Implementation starts again with some includes and a using statement. The first important construct is the monad type constructor, a C++ template which in this particular case takes the form of a std::function template:

// Monad type constructor: a std::function template
template<typename T>
using monad = function < T() >;

The using statement defines ‘monad’ as a type alias.

Of course, a suitably formed pointer to function object would suffice as well. The std::function template has the advantage that it looks neat, and we can instantiate it in an elegant way using a lambda expression. Thus upholding some of the functional character of the monad’s roots.

The unit function, which creates instances of the ‘monadic’ function object is as follows:

// Monad unit function, wraps a value in a lambda
template<typename V>
monad<V> unit(V val)
{
       return[=](){ return val; };
}

The unit function returns a lambda expression the returns a value, when executed (later some time, if any).

The bind function follows the description of bind operators to the letter. It evaluates the monad argument, and returns a function that applies the function argument to the result of that evaluation:

// Monad bind operator
template<typename A, typename R>
monad<R> operator|(monad<A>& mnd, monad<R>(*func)(A))
{
       // return a callable to evaluate later
       return[=]()
       {
             A unwrapped = mnd();
             monad<R> result = func(unwrapped);
             return result(); //parentheses, because we define a function that returns a value
       };
}

The returned function is not evaluated before function call is applied to it. Hence using this bind operator we can construct long and complex function compositions, which get evaluated only after we explicitly do so.

The above triple is the complete implementation of the monad. Yes I know, there should be two additional operations defined as well: the sequencing operator and the fail operator (see e.g. Mike Vanier, part 3. However, many texts discussing monads indicate that these operations are uninteresting. So, we will skip them in this series.

Let’s add the same functions as in part 2, and run a simple composition:

// divides arg by 2 and returns result as monadic template
monad<int> divideby2(int i)
{
       return unit<int>(i / 2);
}

// checks if arg is even and returns result as monadic template
monad<bool> even(int i)
{
       return unit<bool>(i % 2 == 0);
}

int _tmain(int argc, _TCHAR* argv[])
{
       SetConsoleTitle(L"Harder to C++: Monads [3]");

       // Wrap 22 in monadic template
       auto m1 = unit<int>(22);

       // Apply bind 4 times in function composition
       auto m2 = m1 | divideby2 | divideby2 | divideby2 | even;

       cout << boolalpha << m2() << endl;

       cin.get();
       return 0;
}

Note that the functions ‘divideby2’ and ‘even’ have not changed. The only change in the _tmain function is that in the output statement m2.value has been replaced by m2(). Note the parentheses, which denote lazy evaluation of the constructed composition.

Minimal Verification that Monadic Laws Hold

The minimal verification is the lazy analog to verification in the eager case. It looks like this:

int _tmain(int argc, _TCHAR* argv[])
{
       SetConsoleTitle(L"Harder to C++: Monads [3]");

       // Wrap 22 in monadic template
       auto m1 = unit<int>(22);

       // Left identity
       // First explicitly convert to function pointer (because of template)
       auto pf1 = &unit < int >;
       // then apply bind operator
       auto m2 = m1 | pf1;

       cout << "Left Identity result: " << m2() << endl;

       // Right identity
       auto m3 = m1 | divideby2;
       auto m4 = divideby2(22);

       cout << boolalpha << "Right Identity result : " << (m3() == m4()) << endl;

       // Associativity
       // Case 1: equivalent to bind(m1, divideby2);bind(m5, even)
       auto m5 = m1 | divideby2;
       auto m6 = m5 | even;

       // Case 2: equivalent to Bind(m1, divideby2;bind(..., even));
       // Create function composition
       auto f = [](int i){ return divideby2(i) | even; };

       // Explicitly convert f to function pointer
       auto pf2 = static_cast<monad<bool>(*)(int)>(f);
       auto m7 = m1 | pf2;

       cout << boolalpha << "Associativity result : " << (m6() == m7()) << endl;

       cin.get();
       return 0;
}

which generates output:

as expected (and desired).

Pros, Cons

Some texts declare that lazy implementation provides advantages over eager or strict evaluation. Advantages presented are:

1. Calculations are only made when required and in as far as required.
2. ‘Lazy’ allows he evaluation of lists that are unbounded (endless, infinite)
3. Delayed evaluation of function compositions allows for an optimization before evaluation of the composition takes place.

What to think of this?

The first advantage strikes me as odd; I usually do not write programs that make calculations which’ results are not required.

The second advantage is odd as well. My computer is as powerful as a Turing machine, hence it can process input lists of unboundedly length. Without lazy evaluation.

The third advantage could be a real advantage, but I wouldn’t know how to do this in C++; the advantage presupposes that it is possible to create code at runtime and then compile and execute it, which isn’t true for C++.

I do see a disadvantage, though, and we will get to it in detail when doing some shootouts between the various alternative monad implementations. The disadvantage is that the software does extra work. It first creates an expression, then evaluates it. So in C++, for each lazy evaluating program, there is an equivalent eager evaluating program that is faster and smaller, hence more competitive.

In the next part of this series we will work on an (eager) implementation that has the same time performance characteristics for any size value. There will be many references and move semantics. Hmm … mjummy 😉

Harder to C++: Monads for Mortals [2], First implementation

This blog post is part 2 in a series. Part 1 is a general introduction. This part discusses a first implementation of the monad in C++.

There are two fundamentally different approaches to implementing monads in C++. Let’s first discuss the approach we will not pursue.

A Way We Will Not to Implement a Monad in C++

The C++ template mechanism is a full functional language (this discovery, by Erwin Unruh, was a bit of a surprise). As a consequence C++ templates are sometimes used for template meta programming: programs can be written using (only) C++ templates and are as such evaluated / executed, at compile time. There are not many practical uses for template meta programming. One practical use was to unroll (small) loops in order to gain performance. This gain, however, can nowadays also be obtained by modern, optimizing C++ compilers.

The discovery that the C++ template mechanism constitutes a functional language, also has lead some people to implement monads in a template meta programming based functional language. Examples are the Cee Plus Plus Monad Example, and Implementing Monads for C++ Template Metaprograms. However, these implementations live in a part of C++ that is little used. Moreover, the required syntax is complex, which makes it unattractive for general use.

The above approach and the goal pursued here are, I think, trying to answer different questions. The above examples seem to try to mimic functional programming with C++, and implement monads in that context. Contrary to bringing C++ to functional programming, I want to know if bringing the functional programming construct ‘monad’ to C++ will provide me with an advantage, and to what cost.

Below we will start with a simple implementation of a monad in mainstream C++ and show that it is indeed a monad. Then we will see what we like or dislike. Note that we will not be doing any functional programming, which, in fact, is quite in the spirit of Eugenio Moggi’s intentions.

A Simple, Mainstream C++ Implementation of the Monad

Implementation starts with some includes and a using statement. The first important construct is the monad type constructor, or rather: the C++ template.

#include <iostream>;

using namespace std;

// monad type constructor
template<typename V>; struct monad
{
	// Wrapped value
	V value;
};

We saw in part 1 that the result of the unit function has the semantics of a computation. As you can see, this C++ class / struct template does not have any computations, it is a trivial computation. For one thing, we could add a function application operator to it ( operator() ), that would perform some non trivial computation, and we might indeed at some point do so.

The next item to define is the unit function, the factory:

// factory: unit function
template<typename V> monad<V> unit(V val)
{
	return monad < V > { val };
}

It is simplicity itself. An alternative could have been to implement the unit function as the monad struct constructor. That would have been more efficient, so we will do that in a number of following implementations.

I have given the bind function the form of an overload of the bitwise OR operator ” | “.

// Bitwise OR operator overload: bind function
template<typename A, typename R>
monad<R> operator| (monad<A>& mnd, monad<R>(*func)(A))
{
	return func(mnd.value);
}

I got the idea to use an operator instead of a function called ‘bind’ from the CeePlusPlus example referred to above. However, that code overloads the ” >>= ” operator, in order to make the code resemble Haskell somewhat. I prefer the ” | ” because it is the same symbol as the Unix / Linux shell pipe operator. And indeed, we will see shortly that in function composition, bind can well be considered to pipe results from function to function.

The above bind function first ‘unwraps the value’, by referencing the value of the monad object, it then applies the function ‘func’ to it, and returns the result wrapped in a new monad template object.

At this point we have a first implementation of a monad! Note how very small and simple it is. Let’s add some functions and give it a spin.

// divides arg by 2 and returns result as monadic template
monad<int> divideby2(int i)
{
	return unit<int>(i/2);
}

// checks if arg is even and returns result as monadic template
monad<bool> even(int i)
{
	return unit<bool>(i % 2 == 0);
}

int _tmain(int argc, _TCHAR* argv[])
{
	// Wrap 22 in monadic template
	auto m1 = unit<int>(22);

	// Apply bind 4 times in function composition
	auto m2 = m1 | divideby2 | divideby2 | divideby2 | even;

	cout << boolalpha << m2.value << endl;

	cin.get();
	return 0;
}

Note the elegant function composition offered by the bind operator.

Minimal Verification that Monadic Laws Hold

By means of a single example we will verify that the monadic laws (see part1) hold:

int _tmain(int argc, _TCHAR* argv[])
{
	// Wrap 22 in monadic template
	auto m1 = unit<int>(22);

	// Left identity
	// First explicitly convert to function pointer (because of template)
	auto pf1 = &unit < int >;
	// then apply bind operator
	auto m2 = m1 | pf1;

	cout << "Left Identity result: " << m2.value << endl;

	// Right identity
	auto m3 = m1 | divideby2;
	auto m4 = divideby2(22);

	cout << boolalpha << "Right Identity result : " << (m3.value == m4.value)
	     << endl;

	// Associativity
	// Case 1: equivalent to bind(m1, divideby2);bind(m5, even)
	auto m5 = m1 | divideby2;
	auto m6 = m5 | even;

	// Case 2: equivalent to Bind(m1, divideby2;bind(..., even));
	// Create function composition
	auto f = [](int i){ return divideby2(i) | even; };
	// Verbosely convert f to function pointer
	auto pf2 = static_cast<monad<bool>(*)(int)>(f);
	auto m7 = m1 | pf2;

	cout << boolalpha << "Associativity result : " << (m6.value == m7.value)
	     << endl;

	cin.get();
	return 0;
}

which generates output:

The funny thing about the associative law is that the intuitive use of the bind operator like in:

auto m2 = m1 | divideby2 | even;

is equivalent to:

auto m2 = bind(bind(m1, divideby2), even);

which is very obviously equivalent to Case 1 above. So indeed, parentheses do not matter as long as we pipe the result of applying divideby2 to m1 into event.

Pros, Cons, and Questions

So, now that we have made a start, what to think of it?

The ‘pros’ are:

• The small size
• The very elegant function composition.
• From a functional programming perspective one might add that this implementation works with immutable data.

The most visible ‘cons’ are:

• The code copies function arguments and function results, so it is inefficient, and performance for large objects will be bad.
• The const qualifier is completely absent from the code.
• It would be more efficient to make the unit function the constructor in the monadic template (as it is in the examples in Eric Lippert’s tutorial on monads in C#).

There is also this question about the bind operator:

• Why is the bind operator not a member of the monadic template like with Lippert? Note that doing so would turn the monadic template into the complete monad. Well, this is C++, and I like to follow C++ best practices as accumulated by Sutter and Alexandrescu here, see item 44 “Prefer writing nonmember nonfriend functions”. There are more reasons, having to do with composition of different types of monads. But that’s for later…

Next time: a simple implementation of the monad for lazy evaluation.

Harder to C++: Monads for Mortals [1]

The monad is a programming construct, a coding idiom, or perhaps even a design pattern. Its roots lie in Category Theory, coming to programming via programming language semantics and functional languages. I have been reading up on monads (there are multiple, closely related variants), specifically monads for C++. It turns out that the concept of a monad is quit slippery; one minute you think you have a nice understanding, the next minute you are back at zero.

It seems to me that for a thorough understanding of what monads are, you need some knowledge of Category Theory and neighboring fields in logics and algebra, and sound knowledge of the functional programming language Haskell. But alas, I’m just a simple mortal programmer. Moreover:

• I don’t want to know about Category Theory!
• I don’t care about have time for Haskell !
• I just want to write programs in C++!

An alternative route to understanding monads that suits me (simple mortal soul) better, then, is to experiment with various implementations of monads in C++, and evaluate the results, experiment with an improved implementation. And so on.

I have been busy running this experimental cycle, and will describe the condensed experiences in a number of blog posts.

Many interesting things happened, but I can’t say I found a satisfying implementation. Indeed, I have come to doubt the usefulness of monads for application in C++, despite its conceptual and syntactical beauty. Even worse, I have come to doubt the usefulness of some other additions to the C++ language, new since C++11.

But monads are hot!, They must be useful in C++! Well, let us be clear about our motive to write programs in C++. The unique selling points of C++ are (i) the fastest code, against (ii) the smallest footprint. I think people buy in on C++ for exactly these selling points.

Let’s compare with driving a car (what a boyish thing to do). Programming in C++ is then, of course, driving a formula 1 race car; fast, small, unsafe, and uncomfortable. You want comfort, drive a large American saloon, i.e. code in a language designed for developer productivity. You want to be safe, correct? Drive a Toyota Prius, i.e. program a functional language.

Of course, Formula 1 drivers want to be as safe and comfortable as possible. But not at the expense of performance!

So, the monad can be a contribution to C++, if(f) it has no time, space performance cost.

Some Background on Monads

Let’s see what monads are, where they come from, and how they got into programming.

For a C++ developer, a monad is a triple <M, u, b> where:

• M is a template (yes, m for monad 🙂 ).
• u is a factory function, in general called unitof signature a -> M<a>, that given an object or value of some type a creates a template object of type M<a>.
• b is a function application function called bind of signature (M<a>, f) -> M<b> that:
  • Retrieves the object of type a from M<a>
  • Applies a function f: a -> M<b> (from type a to a template of type M<b>)
  • Returns a template of type M<b>

This triple has to satisfy three laws in order to be a monad:

1. Left identity: bind(M<a>, u) = M<a> (so left means left argument for bind)
2. Right identity: for an f: a->M<b>, bind(M<a>, f) = f(a)
3. Associativity: bind(x, f); bind(y, g) = bind(x, f; bind(y, g))

In the description of the associative law:

• x is a variable of type M<a>
• y is the result of bind(x, f)
• the ‘;’ operator denotes left to right sequencing of operations
• f is a function of type a -> M<b> and g is a function of type b -> M<c>

Now. let’s add some intuition.

According to Eugenio Moggi, the semantics of M is that of a computation. He means to include a general class of computations, e.g. partial computations, nondeterministic computations, side-effects, exceptions, continuations, and interactive input and output, and possibly more.

Further, u is the inclusion of value a in a computation, as a parameter. So, we may think of M as a computation.

The application function b (applied to function f) is the extension of a function f from values to computations to a function from computations to computations, which first evaluates a computation and then applies f to the resulting value.

Philip Wadler took the monad to Haskell and showed it can be used to do bad things in a good way. That is, to perform imperative programming language operations in a pure functional language way. Haskell is a pure functional language. In a pure functional language all functions are pure, i.e. functions in which the input-output relation cannot be augmented by things like state, or context, or whatever; the domain – codomain relation is fixed. In a pure functional language you can thus have no side effects, or interactive input / output, etc.

The monad solves this problem by encapsulating ‘imperative operations’ in the M computation. According to Wadler a function

f: a -> M<b>

is then said to map type a onto type b, with a possible additional effect captured by M. The monad takes the ‘imperative operations’ out of the (pure) function application mechanism, and the functional composition chain, and thus saves the pure character of the pure functional language.

Does this talk about functional language has any value for C++? That is debatable. C++ is an imperative language, the idea to improve it by a construct that makes a functional language do imperative operations is preposterous. On the other hand, Bartosz Milewski is convinced that concurrency in C++ would be improved greatly if it would be structured the monadic way (think of the continuations mentioned above). Concurrency is new territory for C++, so we have to investigate this further. I intend to discuss this viewpoint somewhere in this series of articles on monads. First I want to develop a thorough understanding of monad implementations in C++.

Next in this series: a first C++ implementation of a monad.

Links to Texts on Monads

Who writes or wrote about monads, and what do they think is the merit of monads?

1. Eugenio Moggi wrote Notions of computation and monads, about. This paper introduces monads as the categorical semantics for computations. It is very hard to read, because full of funny mathematical words that have a very precise meaning which then, of course, hardly anybody knows exactly. According to Moggi, monads allow mathematicians to reason about the equivalence of a very general class of programs.
2. Philip Wadler introduced monads into Haskell in order to perform ‘imperative operations’ in a decent way. Very readable text.
3. Mike Vanier wrote an extensive introduction into monads. Hard to read because of much Haskell code. After Philip Wadler he states that monads are key in allowing programmers to write side-effecting programs in pure functional languages.
4. Bartosz Milewski is a fervent supporter of application of monads to C++, specifically in the context of concurrency. He has written much texts that might make reading Moggi’s text easier. Milewski stresses the idea that std::fucture should be monadic.
5. Eric Lippert wrote a very (very) accessible introduction to monads for C#. He saw merit of monads in the construction of LINQ-to-Objects.