State Monad for the Rest of Us(arialdomartini.github.io) |
State Monad for the Rest of Us(arialdomartini.github.io) |
It shows how algorithms with mutable state can be implemented with pure functions, with immutable variables only.
The series itself is an excuse to take several detours on other Functional Programming topics: currying, partial application, recursion, Functors, Applicative Functors.
The source code includes F# and C# examples. In the next weeks it will be followed by code examples showing how to apply a state monad in practice.
On first glance, your example:
(decimal | DivideByZeroException) Divide(decimal n, decimal d) => n / d;
using System.Runtime.CompilerServices;
(decimal? , DivideByZeroException?) Divide(decimal n, decimal d) {
try {
return (n / d, null);
} catch {
return (null, new DivideByZeroException());
}
}
Divide(6, 2).Switch(
(result) => Console.WriteLine($"6 /2 is {result}"),
(ex) => Console.WriteLine("This is an exception!")
);
Divide(6, 0).Switch(
(result) => Console.WriteLine($"6 /2 is {result}"),
(ex) => Console.WriteLine("This is an exception!")
);
public class DivideByZeroException : Exception;
public static class TupleExtension {
public static void Switch<T, E>(this ValueTuple<T, E> tuple, Action<T> action1, Action<E> action2) {
if (tuple.Item1 is null && tuple.Item2 is null) {
} else if (tuple.Item1 is null) {
action2(tuple.Item2);
} else {
action1(tuple.Item1);
}
}
}
(null, null)
(result, new DivideByZeroException())
(btw, you're looking through the Monads For The Rest Of Us tutorial, but the the State Monad For The Rest Of Us tutorial is the one linked to this thread)
One minor thought: where you have `A Tree is either a Leaf or a Node containing 2 branches each containing another Tree structure.`, perhaps you could have `A Tree is either a Leaf or a Node _made of_ 2 branches each containing another Tree structure.`
So that "made of" easily maps to the "of" in F#'s type definition. Just an idea.
Haskell will bring you more of the benefits from FP with expressive types; Idris will bring you even more. But you will lose the integration.
Same as F#, OCaml, etc.
I'm primarily a C# dev, but after dabbling in some F#, I concluded that most of the niceties in F# could also be done (to some extent and with some limitations) on C# as they have the same compile targets.
A week or two ago, I found myself working in the same codebase. I had to do some refactoring and, thanks to the monad I had already established, the refactoring reduced code while expanding functionality. The best refactors happen when the patterns make it easy to do.
St -> (a, St)
get :: St -> (St, St)
put :: St -> (St -> ((), St))
{-# Language DerivingVia #-}
{-# Language GADTSyntax #-}
-- get :: My St
-- put :: St -> My ()
newtype My a where
My :: (St -> (a, St)) -> My a
deriving (Functor, Applicative, Monad, MonadState St, MonadFix)
via State St St -> (a, St)
St -> (St, a)
A monad is just a type constructor and two functions respecting some rules.
It can be widely different from the State monad ranging from useful (Option) to completely weird (the naive list monad) to doing nothing (identity monad).
- 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.
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)
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.
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.
-- 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)
rollDieS :: State StdGen Int
rollDieS = state (randomR (1,6))
rollPairS :: State StdGen (Int, Int)
rollPairS = do
r1 <- rollDieS
r2 <- rollDieS
return (r1, r2) eval (Constant n) = Right n
eval (Constant n) = return nSo... 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 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.
My point is that functional programming as engineering practice and language family is broader then variants of pure lambda calculi and Haskell occupies one particular place in that broad space.
enum Tree {
Leaf,
Node(Box<Tree>, Box<Tree>),
}
fn number_of_leaves(tree: &Tree) -> u32 {
match tree {
Tree::Leaf => 1,
Tree::Node(left, right) => number_of_leaves(left) + number_of_leaves(right),
}
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn counts_the_number_of_leaves_in_a_tree() {
let tree_with_3_leaves = Tree::Node(
Box::new(Tree::Leaf),
Box::new(Tree::Node(
Box::new(Tree::Leaf),
Box::new(Tree::Leaf),
)),
);
let leaves = number_of_leaves(&tree_with_3_leaves);
assert_eq!(leaves, 3);
}
}There's a sibling comment about porting it to C#. The Rust version will be less frustrating than the C#, but it is the same kind of bad.
On Rust specifically, this happens because Rust is a low level language.
#pragma warning disable CS8509 // The switch expression does not handle all possible values.
// Justification: The switch expression is exhaustive, and Roslyn emits optimal default guard.
// In AOT, ILC should be able to statically prove by whole program view that only two subclasses of Tree exist and optimize it away.
// In practice, it seems null check and recursive nature of the call prevent it for now. It is still compact and fast codegen however.
var treeWith3Leaves = new Tree.Node(
new Tree.Leaf(),
new Tree.Node(
new Tree.Leaf(),
new Tree.Leaf()));
var leaves = NumberOfLeaves(treeWith3Leaves);
if (leaves != 3) {
throw new Exception("Huh, something went wrong.");
}
static int NumberOfLeaves(Tree tree) {
return tree switch {
Tree.Leaf => 1,
Tree.Node n => NumberOfLeaves(n.Left) + NumberOfLeaves(n.Right)
};
}
abstract record Tree {
public record Leaf: Tree;
public record Node(Tree Left, Tree Right): Tree;
}
union Tree {
Leaf;
Node(Tree Left, Tree Right);
}
record Node(Tree Left, Tree Right);
union struct Tree {
Leaf;
Node;
}
[0]: https://github.com/dotnet/csharplang/blob/18a527bcc1f0bdaf54...
The second case, if there is an exception, it would be handled as such by simply switching the order of the `if` in the extension method. None of this seems confusing nor onerous in day-to-day use.
The issue is not that they cannot be handled, it's that they can be. You included an entire extra branch and check for something that will never occur because they're legal values for your tuple type. I assume it'll be optimized out of this trivial example, but imagine writing a function to receive your tuple type in a codebase where you didn't create the tuple. ADTs prevent this entire worry because there is no way to represent illegal states.
What does it mean to receive both value and error in the context of division? What about neither?
I don't see the issue?
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.
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.
The issue is not one of usability or (direct) understandability, it’s an issue of correctness. Something like “well it works fine for me if I just remember to check if I receive both” or “everyone on my team understands that receiving both means an error occurred” just isn’t a valid response, it doesn’t address the problem.
(null, new DivideByZeroException())
(null, null)
(result, new DivideByZeroException())
This isn't a showstopper, it's just more stuff you have to keep in your head.
> if we receive both, than an exception occurred and should be handled
P(decimal ∩ DivideByZeroException) = 0 , therefore we only need 2 states, not 4.
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.
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.
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)
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) [...]
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?
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)
-- 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)
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.
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.
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 languages that have monads: yes!
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.
Yes, it is this simple.