I've seen the term Free Monad pop up every now and then for some time, but everyone just seems to use/discuss them without giving an explanation of what they are. So: what are free monads? (I'd say I'm familiar with monads and the Haskell basics, but have only a very rough knowledge of category theory.)
Haskell – What are free monads
category-theoryfree-monadhaskellmonads
Related Solutions
First: The term monad is a bit vacuous if you are not a mathematician. An alternative term is computation builder which is a bit more descriptive of what they are actually useful for.
They are a pattern for chaining operations. It looks a bit like method chaining in object-oriented languages, but the mechanism is slightly different.
The pattern is mostly used in functional languages (especially Haskell which uses monads pervasively) but can be used in any language which support higher-order functions (that is, functions which can take other functions as arguments).
Arrays in JavaScript support the pattern, so let’s use that as the first example.
The gist of the pattern is we have a type (Array
in this case) which has a method which takes a function as argument. The operation supplied must return an instance of the same type (i.e. return an Array
).
First an example of method chaining which does not use the monad pattern:
[1,2,3].map(x => x + 1)
The result is [2,3,4]
. The code does not conform to the monad pattern, since the function we are supplying as an argument returns a number, not an Array. The same logic in monad form would be:
[1,2,3].flatMap(x => [x + 1])
Here we supply an operation which returns an Array
, so now it conforms to the pattern. The flatMap
method executes the provided function for every element in the array. It expects an array as result for each invocation (rather than single values), but merges the resulting set of arrays into a single array. So the end result is the same, the array [2,3,4]
.
(The function argument provided to a method like map
or flatMap
is often called a "callback" in JavaScript. I will call it the "operation" since it is more general.)
If we chain multiple operations (in the traditional way):
[1,2,3].map(a => a + 1).filter(b => b != 3)
Results in the array [2,4]
The same chaining in monad form:
[1,2,3].flatMap(a => [a + 1]).flatMap(b => b != 3 ? [b] : [])
Yields the same result, the array [2,4]
.
You will immediately notice that the monad form is quite a bit uglier than the non-monad! This just goes to show that monads are not necessarily “good”. They are a pattern which is sometimes beneficial and sometimes not.
Do note that the monad pattern can be combined in a different way:
[1,2,3].flatMap(a => [a + 1].flatMap(b => b != 3 ? [b] : []))
Here the binding is nested rather than chained, but the result is the same. This is an important property of monads as we will see later. It means two operations combined can be treated the same as a single operation.
The operation is allowed to return an array with different element types, for example transforming an array of numbers into an array of strings or something else; as long as it still an Array.
This can be described a bit more formally using Typescript notation. An array has the type Array<T>
, where T
is the type of the elements in the array. The method flatMap()
takes a function argument of the type T => Array<U>
and returns an Array<U>
.
Generalized, a monad is any type Foo<Bar>
which has a "bind" method which takes a function argument of type Bar => Foo<Baz>
and returns a Foo<Baz>
.
This answers what monads are. The rest of this answer will try to explain through examples why monads can be a useful pattern in a language like Haskell which has good support for them.
Haskell and Do-notation
To translate the map/filter example directly to Haskell, we replace flatMap
with the >>=
operator:
[1,2,3] >>= \a -> [a+1] >>= \b -> if b == 3 then [] else [b]
The >>=
operator is the bind function in Haskell. It does the same as flatMap
in JavaScript when the operand is a list, but it is overloaded with different meaning for other types.
But Haskell also has a dedicated syntax for monad expressions, the do
-block, which hides the bind operator altogether:
do a <- [1,2,3]
b <- [a+1]
if b == 3 then [] else [b]
This hides the "plumbing" and lets you focus on the actual operations applied at each step.
In a do
-block, each line is an operation. The constraint still holds that all operations in the block must return the same type. Since the first expression is a list, the other operations must also return a list.
The back-arrow <-
looks deceptively like an assignment, but note that this is the parameter passed in the bind. So, when the expression on the right side is a List of Integers, the variable on the left side will be a single Integer – but will be executed for each integer in the list.
Example: Safe navigation (the Maybe type)
Enough about lists, lets see how the monad pattern can be useful for other types.
Some functions may not always return a valid value. In Haskell this is represented by the Maybe
-type, which is an option that is either Some value
or Nothing
.
Chaining operations which always return a valid value is of course straightforward:
streetName = getStreetName (getAddress (getUser 17))
But what if any of the functions could return Nothing
? We need to check each result individually and only pass the value to the next function if it is not Nothing
:
case getUser 17 of
Nothing -> Nothing
Just user ->
case getAddress user of
Nothing -> Nothing
Just address ->
getStreetName address
Quite a lot of repetitive checks! Imagine if the chain was longer. Haskell solves this with the monad pattern for Maybe
:
do
user <- getUser 17
addr <- getAddress user
getStreetName addr
This do
-block invokes the bind-function for the Maybe
type (since the result of the first expression is a Maybe
). The bind-function only executes the following operation if the value is Just value
, otherwise it just passes the Nothing
along.
Here the monad-pattern is used to avoid repetitive code. This is similar to how some other languages use macros to simplify syntax, although macros achieve the same goal in a very different way.
Note that it is the combination of the monad pattern and the monad-friendly syntax in Haskell which result in the cleaner code. In a language like JavaScript without any special syntax support for monads, I doubt the monad pattern would be able to simplify the code in this case.
Mutable state
Haskell does not support mutable state. All variables are constants and all values immutable. But the State
type can be used to emulate programming with mutable state:
add2 :: State Integer Integer
add2 = do
-- add 1 to state
x <- get
put (x + 1)
-- increment in another way
modify (+1)
-- return state
get
evalState add2 7
=> 9
The add2
function builds a monad chain which is then evaluated with 7 as the initial state.
Obviously this is something which only makes sense in Haskell. Other languages support mutable state out of the box. Haskell is generally "opt-in" on language features - you enable mutable state when you need it, and the type system ensures the effect is explicit. IO is another example of this.
IO
The IO
type is used for chaining and executing “impure” functions.
Like any other practical language, Haskell has a bunch of built-in functions which interface with the outside world: putStrLine
, readLine
and so on. These functions are called “impure” because they either cause side effects or have non-deterministic results. Even something simple like getting the time is considered impure because the result is non-deterministic – calling it twice with the same arguments may return different values.
A pure function is deterministic – its result depends purely on the arguments passed and it has no side effects on the environment beside returning a value.
Haskell heavily encourages the use of pure functions – this is a major selling point of the language. Unfortunately for purists, you need some impure functions to do anything useful. The Haskell compromise is to cleanly separate pure and impure, and guarantee that there is no way that pure functions can execute impure functions, directly or indirect.
This is guaranteed by giving all impure functions the IO
type. The entry point in Haskell program is the main
function which have the IO
type, so we can execute impure functions at the top level.
But how does the language prevent pure functions from executing impure functions? This is due to the lazy nature of Haskell. A function is only executed if its output is consumed by some other function. But there is no way to consume an IO
value except to assign it to main
. So if a function wants to execute an impure function, it has to be connected to main
and have the IO
type.
Using monad chaining for IO operations also ensures that they are executed in a linear and predictable order, just like statements in an imperative language.
This brings us to the first program most people will write in Haskell:
main :: IO ()
main = do
putStrLn ”Hello World”
The do
keyword is superfluous when there is only a single operation and therefore nothing to bind, but I keep it anyway for consistency.
The ()
type means “void”. This special return type is only useful for IO functions called for their side effect.
A longer example:
main = do
putStrLn "What is your name?"
name <- getLine
putStrLn "hello" ++ name
This builds a chain of IO
operations, and since they are assigned to the main
function, they get executed.
Comparing IO
with Maybe
shows the versatility of the monad pattern. For Maybe
, the pattern is used to avoid repetitive code by moving conditional logic to the binding function. For IO
, the pattern is used to ensure that all operations of the IO
type are sequenced and that IO
operations cannot "leak" to pure functions.
Summing up
In my subjective opinion, the monad pattern is only really worthwhile in a language which has some built-in support for the pattern. Otherwise it just leads to overly convoluted code. But Haskell (and some other languages) have some built-in support which hides the tedious parts, and then the pattern can be used for a variety of useful things. Like:
- Avoiding repetitive code (
Maybe
) - Adding language features like mutable state or exceptions for delimited areas of the program.
- Isolating icky stuff from nice stuff (
IO
) - Embedded domain-specific languages (
Parser
) - Adding GOTO to the language.
A bunch of them are almost completely equivalent:
mtl
uses GHC extensions, buttransformers
is Haskell 98.monads-fd
andmonads-tf
are add-ons totransformers
, using functional dependencies and type families respectively, both providing the functionality inmtl
that's missing fromtransformers
.mtl-tf
ismtl
reimplemented using type families.
So essentially, mtl
== transformers
++ monads-fd
, mtl-tf
== transformers
++ monads-tf
. The improved portability and modularity of transformers
and its associated packages is why mtl
is uncool these days, I think.
mmtl
and mtlx
both seem to be similar to and/or based on mtl
, with API differences and extra features.
MonadLib
seems to have a rather different take on matters, but I'm not familiar with it directly. Also seems to use a lot of GHC extensions, more than the others.
At a glance compose-trans
seems to be more like metaprogramming stuff for creating monad transformers. It claims to be compatible with Control.Monad.Trans
which... I guess means mtl
?
At any rate, I'd suggest the following decision algorithm:
- Do you need standard monads for a new project? Use
transformers
& co., help us laymtl
to rest. - Are you already using
mtl
in a large project?transformers
isn't completely compatible, but no one will kill you for not switching. - Does one of the other packages provide unusual functionality that you need? Might as well use it rather than rolling your own.
- Still unsatisfied? Throw them all out, download
category-extras
, and solve all the world's problems with a page and a half ofincomprehensible abstract nonsensebreathtakingly generic code.
Best Answer
Here's an even simpler answer: A Monad is something that "computes" when monadic context is collapsed by
join :: m (m a) -> m a
(recalling that>>=
can be defined asx >>= y = join (fmap y x)
). This is how Monads carry context through a sequential chain of computations: because at each point in the series, the context from the previous call is collapsed with the next.A free monad satisfies all the Monad laws, but does not do any collapsing (i.e., computation). It just builds up a nested series of contexts. The user who creates such a free monadic value is responsible for doing something with those nested contexts, so that the meaning of such a composition can be deferred until after the monadic value has been created.