You should probably mention in the introduction what the state monad is to those who don't use F#. At first blush it just seems redundant—what is a monad if not a representation of state?
- representation of "the real world" where you can send data, control things, etc;
- representation of early return error semantics, like you have with exceptions;
- encapsulation of different execution threads, so you only interface with the group;
- encapsulation of a computer architecture, so something compiled for your GPU runs on the GPU, and something compiled for your CPU run on your CPU;
- encapsulation of transactions, so you can have a multi-threaded software changing whatever shared value you want and none of that leaks to the larger software;
- data structure, like libraries that encode hardware description into them.
That list is not in any way comprehensive. It's just stuff that I can remember right now.
You can also use it for representing errors, for representing global read-only variables, for representing global write-only variables, for representing non-determinism, for representing continuations, and other things (since you can write your own).
For example, this is a really minimalistic arithmetic syntax tree interpreter in Haskell. You could write errors using the Either type:
data Term = Constant Int | Divide Term Term
eval :: Term -> Either String Int
eval (Constant n) = n
eval (Divide left right) =
case eval left of
Left errorString -> Left errorString
Right leftResult ->
case eval right of
Left errorString -> Left errorString
Right rightResult ->
if rightResult == 0
then Left "Division by zero"
else Right (div leftResult rightResult)
It's cumbersome, and there's a staircase growing. Because Either is a monad, you could rewrite the division op like this:
eval (Divide left right) = do
leftResult <- eval left
rightResult <- eval right
if rightResult == 0
then Left "Division by zero"
else Right (div leftResult rightResult)
The state monad is basically a wrapper over a pure function that takes a state and returns a tuple with the result and a new state. In an imperative language, you'd just actually change state, rather than doing that.
In an imperative language, for pseudo-random numbers, you could modify global variables every time you want a new random number. Haskell lets you do this in IO functions. But a pure way would be to return the result of the new global variable every time.
-- randomR takes a range, so randomR (1,6) produces a random number between 1 and 6
rollPair :: StdGen -> ((Int, Int), StdGen)
rollPair s0 =
let (r1, s1) = randomR (1,6) s0
(r2, s2) = randomR (1,6) s1
in ((r1, r2), s2)
It's annoying. You could use the state monad to simplify this:
rollDieS :: State StdGen Int
rollDieS = state (randomR (1,6))
rollPairS :: State StdGen (Int, Int)
rollPairS = do
r1 <- rollDieS
r2 <- rollDieS
return (r1, r2)
The state monad is basically a wrapper over a pure function that takes a state and returns a tuple with the result and a new state.
So... it's a monad? I don't understand why this is worth calling State as distinct from non-State when that doesn't seem to add any meaning. Surely there must be something implied by a State monad that is not implied by a non-State monad. That seems relevant to call out in an introduction.
The Maybe and Either monads have nothing to do with state, for example.
See this is what I'm talking about. What does `state` mean to you where monads don't have any relation? It's just an abstract structure—what makes something state or not is the context.
State can be modeled as a monad, but monads aren't a synonym for state. Just like squares are technically rectangles, but "rectangle" isn't a synonym for "square."
How do you figure? They're an abstract concept commonly used to represent values over a sequence of operations. How does that not describe state? Do you mean something very specific with the term "state" that might make it more meaningful?
If I have an Identity Int, then I really still just have an Int.
If I have a [Int] then I have a list of Ints.
If I have a Maybe Int, then I either have an Int or I have Nothing.
If I have a Reader r Int, then I have a computation taking some input of type r and producing an Int. That computation can't modify the value of r.
If I have a State s Int, I have a computation taking some initial state of type s and producing an Int. The value of s may change during the computation.
All of these are monads, but calling all of them "state" is somewhat reductive.
The Haskell wiki tutorial on monads refers to them vaguely as "strategies" - I mostly agree with that description.
I tend to think of a monad in computer science as something which is constructed from a type and a computation and wraps the return values of that computation in that type. That allows you to bake in "context" into that type which could be state in the conventional sense but it could be something else.
To make this practical, let's make a simple example of a simple somewhat useful monad with no state. Imagine you have all the regular trigonometric functions sine, cosine etc which accept radians[1] and you need versions which work in degrees. You could make a unit conversion monad which converts arguments on the way in and wraps the result in an inverse conversion monad such that if you passed it to arcsine arccosine etc on it it would know and return degrees on the way out.
Neither of those monads would have any state in the usual computer science sense, the main conversion is just multiplying all the inputs by pi/180 before calling the wrapped function and constructing a result monad type so the inverse trig functions do the opposite of that.
[1] ie the correct versions. Fight me physicists and engineers and you freaks who use gradians whoever you are[2].
Monads can be used to attach all sorts of things besides state to data. For example, you can associate read-only data, such as configuration, authenticated identies, etc. You can associate I/O methods, which can be real or mocked. You can have a place in which to scribble logs/traces (this feels like state). And yes, you can associate mutable state, of course.
The point of monads, from the point of view of programmer convenience, is that they let you have a bag of contextual stuff along with your values. Thus if you think of a server application that processes requests, you might want to carry all of:
- configuration applicable to this request
(or global configuration)
- request metadata (time received, from where,
from whom, authenticated how, etc.)
- request processing state
- logs/traces
- I/O methods, so that
- you can make all your code deterministic
and thus easier to write tests for
- so you can mock the world (see previous
item)
with each request. Thus each function that handles a request can get all of that context, and can be fully deterministic, yet it can function in a non-deterministic world.
Besides this there's everything about the Maybe ("Optional") and Error/Either monads that just makes it easy to make sure all errors and "nulls" are handled.
And what makes all of this possible (in Haskell) is the use of operators (functions) and syntax that lets one write "statements" that are actually combined into large expressions.
you might want to carry all of:
- configuration applicable to this request (or global configuration) [...]
Ok, this is super-interesting to me, as I'm currently dealing with the "configuration and interfaces as global state" pattern in a legacy application. Among other things, this makes it hell to test and refactor. Refactoring it into a functional abstraction like this seems like exactly what we need.
But ... I'm kind of struggling to figure out what "global configuration as a monad" would look like (very much not a Haskell programmer). How would something like this work in practice?
"Global configuration as a monad" is represented by the Reader monad, which is basically a wrapper over a function that takes a configuration/environment as a parameter and then returns something. It replaces a read-only global configuration.
This is Haskell, but here's a really simple example.
-- As an ordinary function:
foo :: Int -> Bool -> Int
foo n shouldAdd = if shouldAdd then n + 1 else n - 1
-- Exactly the same, but using the Reader monad
foo :: Int -> Reader Bool Int
foo n = do
shouldAdd <- ask
return (if shouldAdd then n + 1 else n - 1)
Here's the same thing, but with a record with one boolean in it:
-- Config: a record with one boolean in it,
-- and you access it with a function called `shouldAdd`
data Config = MkConfig { shouldAdd :: Bool }
foo :: Int -> Config -> Int
foo n cfg = if shouldAdd cfg then n + 1 else n - 1
foo :: Int -> Reader Config Int
foo n = do
mustAdd <- asks shouldAdd
return (if mustAdd then n + 1 else n - 1)
It's basically a way to have implicit read-only parameters. You can call a bunch of functions that take a configuration parameter without having to actually pass the configuration as a parameter explicitly. In an object oriented language, you might use classes for this (if not global variables).
In Java or C++ or other OOPish languages, say, you might make all your classes be Configurable (unless they are Configuration), which means their constructors would take a Configuration in some way, possibly with an explicit Configuration argument or with a Configurable argument whose configuration to copy. This way all your objects will know how to find configuration information.
Sure, that makes sense, and is a go-to pattern for my own projects. But to map this onto FP, making classes Configurable feels like OO version of currying. I.e., I'm taking some parameters and baking them into methods that I will call later. But does this have anything to do with monads or some monadic version of state?
It's more like `Configured<T>` -- a Functor, something not too dissimilar to `List<T>`, but a) with just one `T` in it, b) with all your configuration things in the `Configured<T>` instance.
So everywhere you deal with a value that is of some type `Configured<T>` you can then refer to all the configuration methods you'd expect of you `Configuration` types. In Java you'd say something like `this.getConfigBlah()`, and it would just work.
I'm taking some parameters and baking them into methods that I will call later.
Yes.
But does this have anything to do with monads or some monadic version of state?
In Java you'd say something like `this.getConfigBlah()`
No, the whole point is that it’s completely dissimilar to that.
But does this have anything to do with monads or some monadic version of state?
There is no monadic version of state. The State monad is a way of chaining things together (not unlike shell pipes) which is nice in Haskell because it gives access to some of the language syntactic sugar (the do notation). There is nothing particularly special there otherwise.
The key takeaway here is that you can avoid littering your global scope by explicitly passing what’s shared as functions arguments and then remove side effects if you also pass by value.
They're both, right? It's not like it specifically ignores the inputs and outputs of those operations. These are specifically entailed in the definition of the monad.
I strongly, strongly suspect there's something about how F# processes monads in the context of a do expression that explains what is the focus here.
Unfortunately I don't have the time to go through the steps now, hence why I specifically recommended modifying the introduction to give some clue about the terms involved.
An "if" statement or "map" function also has inputs and outputs and probably some internal state due to implementation details, but we don't say they represent state either. They represent operations, or a sequence of operations.
An "if" statement or "map" function also has inputs and outputs and probably some internal state due to implementation details, but we don't say they represent state either.
They absolutely are in the context of a program and not just abstract expression, which never matters. The distinction strikes me as meaningless. At the very least it's worth qualifying with "mutable state" if that's the attribute of state you care about. There's still "constant expressions" as other forms of values and "statically initialized values" as other forms of state.
It's not about monads as an abstract concept. It's specifically about the State monad.
The key idea is that you can have something ressembling states with pure functions if you pass the a representation of the initial state as an argument and have the modified state be part of what the function return. You can then use this modified state as an argument for the next function call. The State monad is just an handy way to do this chaining.
But, yes, the actual state representation is in the way the argument is encoded, that's true.
Edit: I find it hilarious that I’m the one downvoted here while all the comments saying complete non sense about monads - including the ones clearly having no clue of what a monad actually is - are not. Never change HN.
Comments
You should probably mention in the introduction what the state monad is to those who don't use F#. At first blush it just seems redundant—what is a monad if not a representation of state?
Some uses you'll find in Haskell:
- representation of "the real world" where you can send data, control things, etc;
- representation of early return error semantics, like you have with exceptions;
- encapsulation of different execution threads, so you only interface with the group;
- encapsulation of a computer architecture, so something compiled for your GPU runs on the GPU, and something compiled for your CPU run on your CPU;
- encapsulation of transactions, so you can have a multi-threaded software changing whatever shared value you want and none of that leaks to the larger software;
- data structure, like libraries that encode hardware description into them.
That list is not in any way comprehensive. It's just stuff that I can remember right now.
You can also use it for representing errors, for representing global read-only variables, for representing global write-only variables, for representing non-determinism, for representing continuations, and other things (since you can write your own).
For example, this is a really minimalistic arithmetic syntax tree interpreter in Haskell. You could write errors using the Either type:
It's cumbersome, and there's a staircase growing. Because Either is a monad, you could rewrite the division op like this: This is a good article about the either monad: https://mmhaskell.com/blog/2022/3/3/using-either-as-a-monadThe state monad is basically a wrapper over a pure function that takes a state and returns a tuple with the result and a new state. In an imperative language, you'd just actually change state, rather than doing that.
An example from this wiki page: https://en.wikibooks.org/wiki/Haskell/Understanding_monads/S...
In an imperative language, for pseudo-random numbers, you could modify global variables every time you want a new random number. Haskell lets you do this in IO functions. But a pure way would be to return the result of the new global variable every time.
It's annoying. You could use the state monad to simplify this:Just realized I should have written one of these for the Constant evaluation:
So... it's a monad? I don't understand why this is worth calling State as distinct from non-State when that doesn't seem to add any meaning. Surely there must be something implied by a State monad that is not implied by a non-State monad. That seems relevant to call out in an introduction.
I don't see why you think all monads are just like the State monad. The Maybe and Either monads have nothing to do with state, for example.
See this is what I'm talking about. What does `state` mean to you where monads don't have any relation? It's just an abstract structure—what makes something state or not is the context.
State can be modeled as a monad, but monads aren't a synonym for state. Just like squares are technically rectangles, but "rectangle" isn't a synonym for "square."
How do I think of the State monad?
- Like a Mealy machine: https://en.wikipedia.org/wiki/Mealy_machine
- Like a pure way to compose functions that use a nameless, global mutable variable.
- As a newtype wrapper over the type "s -> (a, s)" where s is the type of the state and a is the type of the computation.
monads in no way are a representation of state tho
How do you figure? They're an abstract concept commonly used to represent values over a sequence of operations. How does that not describe state? Do you mean something very specific with the term "state" that might make it more meaningful?
I think "context" is a bit more all-encompassing.
If I have an Int, then I just have an Int.
If I have an Identity Int, then I really still just have an Int.
If I have a [Int] then I have a list of Ints.
If I have a Maybe Int, then I either have an Int or I have Nothing.
If I have a Reader r Int, then I have a computation taking some input of type r and producing an Int. That computation can't modify the value of r.
If I have a State s Int, I have a computation taking some initial state of type s and producing an Int. The value of s may change during the computation.
All of these are monads, but calling all of them "state" is somewhat reductive.
The Haskell wiki tutorial on monads refers to them vaguely as "strategies" - I mostly agree with that description.
Monads can encode state but they don't have to.
I tend to think of a monad in computer science as something which is constructed from a type and a computation and wraps the return values of that computation in that type. That allows you to bake in "context" into that type which could be state in the conventional sense but it could be something else.
To make this practical, let's make a simple example of a simple somewhat useful monad with no state. Imagine you have all the regular trigonometric functions sine, cosine etc which accept radians[1] and you need versions which work in degrees. You could make a unit conversion monad which converts arguments on the way in and wraps the result in an inverse conversion monad such that if you passed it to arcsine arccosine etc on it it would know and return degrees on the way out.
Neither of those monads would have any state in the usual computer science sense, the main conversion is just multiplying all the inputs by pi/180 before calling the wrapped function and constructing a result monad type so the inverse trig functions do the opposite of that.
[1] ie the correct versions. Fight me physicists and engineers and you freaks who use gradians whoever you are[2].
[2] I think it's surveyors but I could be wrong.
Monads can be used to attach all sorts of things besides state to data. For example, you can associate read-only data, such as configuration, authenticated identies, etc. You can associate I/O methods, which can be real or mocked. You can have a place in which to scribble logs/traces (this feels like state). And yes, you can associate mutable state, of course.
The point of monads, from the point of view of programmer convenience, is that they let you have a bag of contextual stuff along with your values. Thus if you think of a server application that processes requests, you might want to carry all of:
with each request. Thus each function that handles a request can get all of that context, and can be fully deterministic, yet it can function in a non-deterministic world.Besides this there's everything about the Maybe ("Optional") and Error/Either monads that just makes it easy to make sure all errors and "nulls" are handled.
And what makes all of this possible (in Haskell) is the use of operators (functions) and syntax that lets one write "statements" that are actually combined into large expressions.
But ... I'm kind of struggling to figure out what "global configuration as a monad" would look like (very much not a Haskell programmer). How would something like this work in practice?
"Global configuration as a monad" is represented by the Reader monad, which is basically a wrapper over a function that takes a configuration/environment as a parameter and then returns something. It replaces a read-only global configuration.
This is Haskell, but here's a really simple example.
Here's the same thing, but with a record with one boolean in it: It's basically a way to have implicit read-only parameters. You can call a bunch of functions that take a configuration parameter without having to actually pass the configuration as a parameter explicitly. In an object oriented language, you might use classes for this (if not global variables).In Java or C++ or other OOPish languages, say, you might make all your classes be Configurable (unless they are Configuration), which means their constructors would take a Configuration in some way, possibly with an explicit Configuration argument or with a Configurable argument whose configuration to copy. This way all your objects will know how to find configuration information.
Sure, that makes sense, and is a go-to pattern for my own projects. But to map this onto FP, making classes Configurable feels like OO version of currying. I.e., I'm taking some parameters and baking them into methods that I will call later. But does this have anything to do with monads or some monadic version of state?
It's more like `Configured<T>` -- a Functor, something not too dissimilar to `List<T>`, but a) with just one `T` in it, b) with all your configuration things in the `Configured<T>` instance.
So everywhere you deal with a value that is of some type `Configured<T>` you can then refer to all the configuration methods you'd expect of you `Configuration` types. In Java you'd say something like `this.getConfigBlah()`, and it would just work.
Yes.
In languages that have monads: yes!
No, the whole point is that it’s completely dissimilar to that.
There is no monadic version of state. The State monad is a way of chaining things together (not unlike shell pipes) which is nice in Haskell because it gives access to some of the language syntactic sugar (the do notation). There is nothing particularly special there otherwise.
The key takeaway here is that you can avoid littering your global scope by explicitly passing what’s shared as functions arguments and then remove side effects if you also pass by value.
Yes, it is this simple.
They are an abstraction of a sequence of operations, not the state used in those operations.
They're both, right? It's not like it specifically ignores the inputs and outputs of those operations. These are specifically entailed in the definition of the monad.
I strongly, strongly suspect there's something about how F# processes monads in the context of a do expression that explains what is the focus here.
Unfortunately I don't have the time to go through the steps now, hence why I specifically recommended modifying the introduction to give some clue about the terms involved.
An "if" statement or "map" function also has inputs and outputs and probably some internal state due to implementation details, but we don't say they represent state either. They represent operations, or a sequence of operations.
They absolutely are in the context of a program and not just abstract expression, which never matters. The distinction strikes me as meaningless. At the very least it's worth qualifying with "mutable state" if that's the attribute of state you care about. There's still "constant expressions" as other forms of values and "statically initialized values" as other forms of state.
It's not about monads as an abstract concept. It's specifically about the State monad.
The key idea is that you can have something ressembling states with pure functions if you pass the a representation of the initial state as an argument and have the modified state be part of what the function return. You can then use this modified state as an argument for the next function call. The State monad is just an handy way to do this chaining.
But, yes, the actual state representation is in the way the argument is encoded, that's true.
Edit: I find it hilarious that I’m the one downvoted here while all the comments saying complete non sense about monads - including the ones clearly having no clue of what a monad actually is - are not. Never change HN.