Haskell, Monads and Purity I believe the notion that Haskell uses “monads” to enforce purity is rather misleading. It has certainly caused quite a bit of confusion! It’s very much like saying we use “rings to do arithmetic”. We don’t! We use numbers, which just happen to form a ring with arithmetic operations. The important idea is the number, not the ring. You can—and most people do—do arithmetic without understanding or even knowing about rings. Moreover, plenty of rings don’t have anything to do with arithmetic. Similarly, in Haskell, we use types to deal with effects. In particular, we use IO and State and ST , all of which happen to form monads. But it is still better to say we use “the IO type” to do IO or that we use State for state than to say we use “monads” to do IO or state. Now, how does this work? Well, I’ll give you one view on the matter, using State as a case study.

State Perhaps the most important philosophical idea is that Haskell does not “enforce” purity. Rather, Haskell as a language simply does not include any notion of effects. Base Haskell is exclusively about evaluation: going from an expression like 1 + 2 to 3 . This has no provisions for side effects; they simply don’t make any sense in this context . Side effects and mutation are not prohibited; they simply weren’t added in the first place. But let’s say that, on a flight of fancy, you decide you really want some mutable state in your otherwise-happy Haskell code. How can you do this? Here’s one option: implement the state machinery yourself. You can do this by creating a little sub-language and writing an interpreter for it. The language itself is actually just a data type; the interpreter is just a normal function which takes care of evaluating your stateful sub-program and maintaining the state. Let’s call our sub-language’s type State . A single State “action” has some value as the result, so it needs a type parameter: State a . For the sake of simplicity, let’s imagine you only ever want a single piece of state—an Int . (You could easily generalize this to any type of state, say a variable s , or, with a bit more effort, a state type that could change over time. But we’ll consider the simplest case.) So what sorts of actions do we need to support? One thing we want to do is ignore the current state: if we have an a already, we want to have a State a which just evaluate to that. It’s a way to add constants to our sub-language. We also want some way to get and set the state. Getting should just evaluate to the current value of the state; setting should take the new value to set. Since setting does not have a meaningful return value, we can just return () . Three basic actions: return some value, get the state and set the state: return ∷ a → State a get ∷ State Int set ∷ Int → State () However, these actions by themselves are rather boring. We need some way to string them together—some way to compose actions. The simplest way is to just sequence two actions: run one, get the new state and run the next one. We can call this an infix >> : (>>) ∷ State a → State b → State b When our interpreter sees a composed action like this, it will just evaluate the left one, get the resulting state and evaluate the second one using this new state. For example, set 10 >> get will result in 10 . However, this is still missing something. We can’t really use the value of the state in interesting ways. In fact, we can’t even write a simple program to increment the state value! We need some way to use arbitrary Haskell logic inside of our stateful programs. How do we do this? Well, since we want to use arbitrary logic, we’re looking to support some sort of user-supplied function embedded into our language. Since Haskell only knows how to work with a values rather than State a values, this function should take a a . Since we want to use this logic to set the state, this function should return a state action. The type we want looks something like this: a → State b This function has an interesting “shape” (for lack of a better word). By going from a normal value ( a ) to a state action ( State b ), we’re injecting normal Haskell into our state sub-language. Since we’re also transitioning from a to b , we’re enabling arbitrary logic in the function. We can now use this to write an incrementing function: addToState ∷ Int → State () addToState x = set (x + 1) Now, how do we intersperse this sort of function with our normal state actions? Well, we want to take a state action, one of these functions, combine them and get the result. The type we want, then, is: State a → (a → State b) → State b Quite similar to our existing >> function; the only difference is that the second “action” is a function now. When our interpreter sees this, it will evaluate the lefthand side, plug the result into the righthand function, get the new state action and run it with the updated state. With this, we can combine get and addToState giving us a program that increments our current state by 1 . We can use this approach to execute arbitrary logic using our state. We can call this sort of composition >>= , a fancier version of >> : (>>=) ∷ State a → (a → State b) → State b Using this >>= operator, we can write the increment program: increment = get >>= addToState and if we want to increment three times in a row, we can do this: increment3 = increment >> increment >> increment It’s important to note that the >>= doesn’t do anything; it just produces some data structure which contains both the lefthand side and the function. It’s our interpreter function that will ever do anything. increment is just a normal value of the type State () ; nothing special. This means that the order in which our expressions get evaluated does not affect how our state works—we’re just evaluating an AST; the state effects are all managed by the interpreter function. This neatly separates our stateful, observable execution from Haskell’s evaluation order which is below our level of abstraction and therefore not observable. Now whenever we want to use some mutable state, we just write our program using this State a type, combining the disparate parts using >>= . Then, to actually use this, we invoke our interpreter function. This function looks at the State a value and evaluates it, threading the changing state through each part. The exact details of how the interpreter works, or even how a State a value is represented internally, are not particularly important.

Unfortunately, using >>= and >> gets rather ugly, so we also want to throw in some syntax sugar to make using this type look like a reasonable imperative program. This will make our little sub-language bearable and the resulting code easier to read. We can o this with two core patterns: action₁ >> action₂ will get transformed into two lines of code: do action₁ action₂ and action₁ >>= \ x → action₂ will be transformed into do x ← action₁ action₂ Now our sequences of actions actually look like sequences of statements in a normal language. We’ve added state to a language where it normally simply does not exist. All using normal functions and data types, with just a bit of syntax sugar to help the medicine go down. Nifty. But not really magical. By now, you’ve probably realized that return and >>= make State a monad. The syntax sugar, of course, is do-notation. But if you’re interested in how we get mutable state, this doesn’t tell you very much. Instead, the relevant bits are what the State a type looks like (it’s basically an AST of commands) and what the interpreter does. The fact that it’s a monad is useful, but does not entirely characterize the State type.

IO Sure, we can add state by producing a little AST and explicitly evaluating it. But it’s easy to model state because it’s an entirely internal phenomenon to the program. But how do we deal with external effects—IO? This is something we fundamentally can’t express with normal Haskell types and functions. The basic idea is the same: we assemble a program and run it through an interpreter which manages the effects. The main difference is that this interpreter itself is not written in Haskell—it has to be built in. This interpreter is actually the Haskell runtime system and, by extension, the whole operating system. So, in some sense, IO is magical, but this is inevitable: to produce the actual effects, it has to talk to the operating system and ultimately the hardware directly. You simply can’t do this within normal Haskell because Haskell only knows about evaluating expressions. Talking to the OS is generally below our level of abstraction. And ultimately, this requires magic because it is magic—it’s built right into the hardware! So the runtime system is our interpreter for IO . A Haskell program has a main value which is an IO action; when you run this program, what you’re actually doing is evaluating main and then giving it to the runtime system to interpret. This system then runs the effects, intoning the appropriate incantations to make things happen in the real world. The IO type is just something the compiler and runtime system understand and run. It is a monad in basically the same way as our State type above. Conceptually, the only difference is in how it gets interpreted. In reality, of course, things aren’t implemented like this. But it’s a reasonable way of thinking about it. So the final verdict: Haskell does not “use monads to manage effects”. Rather, Haskell has types like IO to manage effects, and IO happens to form a monad. Moreover, it turns out that the monad operations are very useful to write actual programs by composing IO actions.