Monad

Description

The Monad concept represents Applicatives with the ability to flatten nested levels of structure.

Historically, Monads are a construction coming from category theory, an abstract branch of mathematics. The functional programming community eventually discovered how Monads could be used to formalize several useful things like side effects, which led to the wide adoption of Monads in that community. However, even in a multi-paradigm language like C++, there are several constructs which turn out to be Monads, like std::optional, std::vector and others.

Everybody tries to introduce Monads with a different analogy, and most people fail. This is called the [Monad tutorial fallacy][1]. We will try to avoid this trap by not presenting a specific intuition, and we will instead present what monads are mathematically. For specific intuitions, we will let readers who are new to this concept read one of the many excellent tutorials available online. Understanding Monads might take time at first, but once you get it, a lot of patterns will become obvious Monads; this enlightening will be your reward for the hard work.

There are different ways of defining a Monad; Haskell uses a function called bind (>>=) and another one called return (it has nothing to do with C++'s return statement). They then introduce relationships that must be satisfied for a type to be a Monad with those functions. Mathematicians sometimes use a function called join and another one called unit, or they also sometimes use other category theoretic constructions like functor adjunctions and the Kleisli category.

This library uses a composite approach. First, we use the flatten function (equivalent to join) along with the lift function from Applicative (equivalent to unit) to introduce the notion of monadic function composition. We then write the properties that must be satisfied by a Monad using this monadic composition operator, because we feel it shows the link between Monads and Monoids more clearly than other approaches.

Roughly speaking, we will say that a Monad is an Applicative which also defines a way to compose functions returning a monadic result, as opposed to only being able to compose functions returning a normal result. We will then ask for this composition to be associative and to have a neutral element, just like normal function composition. For usual composition, the neutral element is the identity function id. For monadic composition, the neutral element is the lift function defined by Applicative. This construction is made clearer in the laws below.

Note

Monads are known to be a big chunk to swallow. However, it is out of the scope of this documentation to provide a full-blown explanation of the concept. The [Typeclassopedia][2] is a nice Haskell-oriented resource where more information about Monads can be found.

Minimal complete definitions

First, a Monad must be both a Functor and an Applicative. Also, an implementation of flatten or chain satisfying the laws below for monadic composition must be provided.

Note

The ap method for Applicatives may be derived from the minimal complete definition of Monad and Functor; see below for more information.

Laws

To simplify writing the laws, we use the comparison between functions. For two functions f and g, we define

f == g  if and only if  f(x) == g(x) for all x

With the usual composition of functions, we are given two functions \( f : A \to B \) and \( g : B \to C \), and we must produce a new function \( compose(g, f) : A \to C \). This composition of functions is associative, which means that

compose(h, compose(g, f)) == compose(compose(h, g), f)

Also, this composition has an identity element, which is the identity function. This simply means that

compose(f, id) == compose(id, f) == f

This is probably nothing new if you are reading the Monad laws. Now, we can observe that the above is equivalent to saying that functions with the composition operator form a Monoid, where the neutral element is the identity function.

Given an Applicative F, what if we wanted to compose two functions \( f : A \to F(B) \) and \( g : B \to F(C) \)? When the Applicative F is also a Monad, such functions taking normal values but returning monadic values are called monadic functions. To compose them, we obviously can't use normal function composition, since the domains and codomains of f and g do not match properly. Instead, we'll need a new operator – let's call it monadic_compose:

Variables
template<typename M >
constexpr auto	boost::hana::tap
	Tap inside a monadic chain. More...
constexpr auto	boost::hana::then
	Sequentially compose two monadic actions, discarding any value produced by the first but not its effects. More...

Variable Documentation

tap

template<typename M >

constexpr auto boost::hana::tap

constexpr

#include <boost/hana/fwd/tap.hpp>

Initial value:

= [](auto&& f) {
        return tag-dispatched;
    }

Tap inside a monadic chain.

Given a function f, tap<M> returns a new function which performs f on its argument and then returns the argument lifted in the M Monad. Combined with the property that chain(m, lift<M>) == m, this provides a way of executing an action inside a monadic chain without influencing its overall result. This is useful to e.g. insert debug statements or perform actions that are not tied to the chain but that need to be executed inside of it.

Note

Since C++ is not a pure language, it is possible to perform side effects inside the f function. Actually, side effects are the only reason why one might want to use tap. However, one should not rely on the side effects being done in any specific order.

Template Parameters

M	The tag (a Monad) of the monads in the tapped monadic chain.

Parameters

f	A function to be executed inside a monadic chain. It will be called as f(x), where x is a value inside the previous monad in the chain. The result of f is always discarded.

Example

// Copyright Louis Dionne 2013-2017
// Distributed under the Boost Software License, Version 1.0.
// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)
 
#include <boost/hana/assert.hpp>
#include <boost/hana/chain.hpp>
#include <boost/hana/equal.hpp>
#include <boost/hana/tap.hpp>
#include <boost/hana/tuple.hpp>
 
#include <set>
namespace hana = boost::hana;
 
 
int main() {
    // We use a sorted container because the order in which the functions
    // are called is unspecified.
    std::set<int> before, after;
 
    auto xs = hana::make_tuple(1, 2, 3)
        | hana::tap<hana::tuple_tag>([&](int x) { before.insert(x); })
        | [](auto x) { return hana::make_tuple(x, -x); }
        | hana::tap<hana::tuple_tag>([&](int x) { after.insert(x); });
 
    BOOST_HANA_RUNTIME_CHECK(before == std::set<int>{1, 2, 3});
    BOOST_HANA_RUNTIME_CHECK(after == std::set<int>{1, -1, 2, -2, 3, -3});
    BOOST_HANA_RUNTIME_CHECK(xs == hana::make_tuple(1, -1, 2, -2, 3, -3));
}

then

constexpr auto boost::hana::then

constexpr

#include <boost/hana/fwd/then.hpp>

Initial value:

= [](auto&& before, auto&& xs) -> decltype(auto) {
        return tag-dispatched;
    }

Sequentially compose two monadic actions, discarding any value produced by the first but not its effects.

Parameters

before	The first Monad in the monadic composition chain. The result of this monad is ignored, but its effects are combined with that of the second monad.
xs	The second Monad in the monadic composition chain.

Example

// Copyright Louis Dionne 2013-2017
// Distributed under the Boost Software License, Version 1.0.
// (See accompanying file LICENSE.md or copy at http://boost.org/LICENSE_1_0.txt)
 
#include <boost/hana/equal.hpp>
#include <boost/hana/then.hpp>
#include <boost/hana/tuple.hpp>
namespace hana = boost::hana;
 
 
struct undefined { };
static_assert(
    hana::then(hana::make_tuple(undefined{}, undefined{}), hana::make_tuple(1, 2, 3))
        ==
    hana::make_tuple(
        1, 2, 3,
        1, 2, 3
    )
, "");
 
int main() { }