Multiplicities

Idris 2 is based on Quantitative Type Theory (QTT), a core language developed by Bob Atkey and Conor McBride. In practice, this means that every variable in Idris 2 has a quantity associated with it. A quantity is one of:

0, meaning that the variable is erased at run time
1, meaning that the variable is used exactly once at run time
Unrestricted, which is the same behaviour as Idris 1

We can see the multiplicities of variables by inspecting holes. For example, if we have the following skeleton definition of append on vectors…

append : Vect n a -> Vect m a -> Vect (n + m) a
append xs ys = ?append_rhs

…we can look at the hole append_rhs:

Main> :t append_rhs
 0 m : Nat
 0 a : Type
 0 n : Nat
   ys : Vect m a
   xs : Vect n a
-------------------------------------
append_rhs : Vect (plus n m) a

The 0 next to m, a and n mean that they are in scope, but there will be 0 occurrences at run-time. That is, it is guaranteed that they will be erased at run-time.

Multiplicities can be explicitly written in function types as follows:

ignoreN : (0 n : Nat) -> Vect n a -> Nat - this function cannot look at n at run time
duplicate : (1 x : a) -> (a, a) - this function must use x exactly once (so good luck implementing it, by the way. There is no implementation because it would need to use x twice!)

If there is no multiplicity annotation, the argument is unrestricted. If, on the other hand, a name is implicitly bound (like a in both examples above) the argument is erased. So, the above types could also be written as:

ignoreN : {0 a : _} -> (0 n : Nat) -> Vect n a -> Nat
duplicate : {0 a : _} -> (1 x : a) -> (a, a)

This section describes what this means for your Idris 2 programs in practice, with several examples. In particular, it describes:

Erasure - how to know what is relevant at run time and what is erased
Linearity - using the type system to encode resource usage protocols
Pattern Matching on Types - truly first class types

The most important of these for most programs, and the thing you’ll need to know about if converting Idris 1 programs to work with Idris 2, is erasure. The most interesting, however, and the thing which gives Idris 2 much more expressivity, is linearity, so we’ll start there.

Linearity

The 1 multiplicity expresses that a variable must be used exactly once. By “used” we mean either:

if the variable is a data type or primitive value, it is pattern matched against, ex. by being the subject of a case statement, or a function argument that is pattern matched against, etc.,
if the variable is a function, that function is applied (i.e. ran with an argument)

First, we’ll see how this works on some small examples of functions and data types, then see how it can be used to encode resource protocols.

Above, we saw the type of duplicate. Let’s try to write it interactively, and see what goes wrong. We’ll start by giving the type and a skeleton definition with a hole

duplicate : (1 x : a) -> (a, a)
duplicate x = ?help

Checking the type of a hole tells us the multiplicity of each variable in scope. If we check the type of ?help we’ll see that we can’t use a at run time, and we have to use x exactly once:

Main> :t help
 0 a : Type
 1 x : a
-------------------------------------
help : (a, a)

If we use x for one part of the pair…

duplicate : (1 x : a) -> (a, a)
duplicate x = (x, ?help)

…then the type of the remaining hole tells us we can’t use it for the other:

Main> :t help
 0 a : Type
 0 x : a
-------------------------------------
help : a

The same happens if we try defining duplicate x = (?help, x) (try it!).

In order to avoid parsing ambiguities, if you give an explicit multiplicity for a variable as with the argument to duplicate, you need to give it a name too. But, if the name isn’t used in the scope of the type, you can use _ instead of a name, as follows:

duplicate : (1 _ : a) -> (a, a)

The intution behind multiplicity 1 is that if we have a function with a type of the following form…

f : (1 x : a) -> b

…then the guarantee given by the type system is that if f x is used exactly once, then x is used exactly once. So, if we insist on trying to define duplicate…:

duplicate x = (x, x)

…then Idris will complain:

pmtype.idr:2:15--8:1:While processing right hand side of Main.duplicate at pmtype.idr:2:1--8:1:
There are 2 uses of linear name x

A similar intuition applies for data types. Consider the following types, Lin which wraps an argument that must be used once, and Unr which wraps an argument with unrestricted use

data Lin : Type -> Type where
     MkLin : (1 _ : a) -> Lin a

data Unr : Type -> Type where
     MkUnr : a -> Unr a

If MkLin x is used once, then x is used once. But if MkUnr x is used once, there is no guarantee on how often x is used. We can see this a bit more clearly by starting to write projection functions for Lin and Unr to extract the argument

getLin : (1 _ : Lin a) -> a
getLin (MkLin x) = ?howmanyLin

getUnr : (1 _ : Unr a) -> a
getUnr (MkUnr x) = ?howmanyUnr

Checking the types of the holes shows us that, for getLin, we must use x exactly once (Because the val argument is used once, by pattern matching on it as MkLin x, and if MkLin x is used once, x must be used once):

Main> :t howmanyLin
 0 a : Type
 1 x : a
-------------------------------------
howmanyLin : a

For getUnr, however, we still have to use val once, again by pattern matching on it, but using MkUnr x once doesn’t place any restrictions on x. So, x has unrestricted use in the body of getUnr:

Main> :t howmanyUnr
 0 a : Type
   x : a
-------------------------------------
howmanyUnr : a

If getLin has an unrestricted argument…

getLin : Lin a -> a
getLin (MkLin x) = ?howmanyLin

…then x is unrestricted in howmanyLin:

Main> :t howmanyLin
 0 a : Type
   x : a
-------------------------------------
howmanyLin : a

Remember the intuition from the type of MkLin is that if MkLin x is used exactly once, x is used exactly once. But, we didn’t say that MkLin x would be used exactly once, so there is no restriction on x.

Resource protocols

One way to take advantage of being able to express linear usage of an argument is in defining resource usage protocols, where we can use linearity to ensure that any unique external resource has only one instance, and we can use functions which are linear in their arguments to represent state transitions on that resource. A door, for example, can be in one of two states, Open or Closed

data DoorState = Open | Closed

data Door : DoorState -> Type where
     MkDoor : (doorId : Int) -> Door st

(Okay, we’re just pretending here - imagine the doorId is a reference to an external resource!)

We can define functions for opening and closing the door which explicitly describe how they change the state of a door, and that they are linear in the door

openDoor : (1 d : Door Closed) -> Door Open
closeDoor : (1 d : Door Open) -> Door Closed

Remember, the intuition is that if openDoor d is used exactly once, then d is used exactly once. So, provided that a door d has multiplicity 1 when it’s created, we know that once we call openDoor on it, we won’t be able to use d again. Given that d is an external resource, and openDoor has changed its state, this is a good thing!

We can ensure that any door we create has multiplicity 1 by creating them with a newDoor function with the following type

newDoor : (1 p : (1 d : Door Closed) -> IO ()) -> IO ()

That is, newDoor takes a function, which it runs exactly once. That function takes a door, which is used exactly once. We’ll run it in IO to suggest that there is some interaction with the outside world going on when we create the door. Since the multiplicity 1 means the door has to be used exactly once, we need to be able to delete the door when we’re finished

deleteDoor : (1 d : Door Closed) -> IO ()

So an example correct door protocol usage would be

doorProg : IO ()
doorProg
    = newDoor $ \d =>
          let d' = openDoor d
              d'' = closeDoor d' in
              deleteDoor d''

It’s instructive to build this program interactively, with holes along the way, and see how the multiplicities of d, d' etc change. For example

doorProg : IO ()
doorProg
    = newDoor $ \d =>
          let d' = openDoor d in
              ?whatnow

Checking the type of ?whatnow shows that d is now spent, but we still have to use d' exactly once:

Main> :t whatnow
 0 d : Door Closed
 1 d' : Door Open
-------------------------------------
whatnow : IO ()

Note that the 0 multiplicity for d means that we can still talk about it - in particular, we can still reason about it in types - but we can’t use it again in a relevant position in the rest of the program. It’s also fine to shadow the name d throughout

doorProg : IO ()
doorProg
    = newDoor $ \d =>
          let d = openDoor d
              d = closeDoor d in
              deleteDoor d

If we don’t follow the protocol correctly - create the door, open it, close it, then delete it - then the program won’t type check. For example, we can try not to delete the door before finishing

doorProg : IO ()
doorProg
    = newDoor $ \d =>
          let d' = openDoor d
              d'' = closeDoor d' in
              putStrLn "What could possibly go wrong?"

This gives the following error:

Door.idr:15:19--15:38:While processing right hand side of Main.doorProg at Door.idr:13:1--17:1:
There are 0 uses of linear name d''

There’s a lot more to be said about the details here! But, this shows at a high level how we can use linearity to capture resource usage protocols at the type level. If we have an external resource which is guaranteed to be used linearly, like Door, we don’t need to run operations on that resource in an IO monad, since we’re already enforcing an ordering on operations and don’t have access to any out of date resource states. This is similar to the way interactive programs work in the Clean programming language, and in fact is how IO is implemented internally in Idris 2, with a special %World type for representing the state of the outside world that is always used linearly

public export
data IORes : Type -> Type where
     MkIORes : (result : a) -> (1 x : %World) -> IORes a

export
data IO : Type -> Type where
     MkIO : (1 fn : (1 x : %World) -> IORes a) -> IO a

Having multiplicities in the type system raises several interesting questions, such as:

Can we use linearity information to inform memory management and, for example, have type level guarantees about functions which will not need to perform garbage collection?
How should multiplicities be incorporated into interfaces such as Functor, Applicative and Monad?
If we have 0, and 1 as multiplicities, why stop there? Why not have 2, 3 and more (like Granule)
What about multiplicity polymorphism, as in the Linear Haskell proposal?
Even without all of that, what can we do now?

Erasure

The 1 multiplicity give us many possibilities in the kinds of properties we can express. But, the 0 multiplicity is perhaps more important in that it allows us to be precise about which values are relevant at run time, and which are compile time only (that is, which are erased). Using the 0 multiplicity means a function’s type now tells us exactly what it needs at run time.

For example, in Idris 1 you could get the length of a vector as follows

vlen : Vect n a -> Nat
vlen {n} xs = n

This is fine, since it runs in constant time, but the trade off is that n has to be available at run time, so at run time we always need the length of the vector to be available if we ever call vlen. Idris 1 can infer whether the length is needed, but there’s no easy way for a programmer to be sure.

In Idris 2, we need to state explicitly that n is needed at run time

vlen : {n : Nat} -> Vect n a -> Nat
vlen xs = n

(Incidentally, also note that in Idris 2, names bound in types are also available in the definition without explicitly rebinding them.)

This also means that when you call vlen, you need the length available. For example, this will give an error

sumLengths : Vect m a -> Vect n a —> Nat
sumLengths xs ys = vlen xs + vlen ys

Idris 2 reports:

vlen.idr:7:20--7:28:While processing right hand side of Main.sumLengths at vlen.idr:7:1--10:1:
m is not accessible in this context

This means that it needs to use m as an argument to pass to vlen xs, where it needs to be available at run time, but m is not available in sumLengths because it has multiplicity 0.

We can see this more clearly by replacing the right hand side of sumLengths with a hole…

sumLengths : Vect m a -> Vect n a -> Nat
sumLengths xs ys = ?sumLengths_rhs

…then checking the hole’s type at the REPL:

Main> :t sumLengths_rhs
 0 n : Nat
 0 a : Type
 0 m : Nat
   ys : Vect n a
   xs : Vect m a
-------------------------------------
sumLengths_rhs : Nat

Instead, we need to give bindings for m and n with unrestricted multiplicity

sumLengths : {m, n : _} -> Vect m a -> Vect n a —> Nat
sumLengths xs ys = vlen xs + vlen xs

Remember that giving no multiplicity on a binder, as with m and n here, means that the variable has unrestricted usage.

If you’re converting Idris 1 programs to work with Idris 2, this is probably the biggest thing you need to think about. It is important to note, though, that if you have bound implicits, such as…

excitingFn : {t : _} -> Coffee t -> Moonbase t

…then it’s a good idea to make sure t really is needed, or performance might suffer due to the run time building the instance of t unnecessarily!

One final note on erasure: it is an error to try to pattern match on an argument with multiplicity 0, unless its value is inferrable from elsewhere. So, the following definition is rejected

badNot : (0 x : Bool) -> Bool
badNot False = True
badNot True = False

This is rejected with the error:

badnot.idr:2:1--3:1:Attempt to match on erased argument False in
Main.badNot

The following, however, is fine, because in sNot, even though we appear to match on the erased argument x, its value is uniquely inferrable from the type of the second argument

data SBool : Bool -> Type where
     SFalse : SBool False
     STrue  : SBool True

sNot : (0 x : Bool) -> SBool x -> Bool
sNot False SFalse = True
sNot True  STrue  = False

Experience with Idris 2 so far suggests that, most of the time, as long as you’re using unbound implicits in your Idris 1 programs, they will work without much modification in Idris 2. The Idris 2 type checker will point out where you require an unbound implicit argument at run time - sometimes this is both surprising and enlightening!

Pattern Matching on Types

One way to think about dependent types is to think of them as “first class” objects in the language, in that they can be assigned to variables, passed around and returned from functions, just like any other construct. But, if they’re truly first class, we should be able to pattern match on them too! Idris 2 allows us to do this. For example

showType : Type -> String
showType Int = "Int"
showType (List a) = "List of " ++ showType a
showType _ = "something else"

We can try this as follows:

Main> showType Int
"Int"
Main> showType (List Int)
"List of Int"
Main> showType (List Bool)
"List of something else"

Pattern matching on function types is interesting, because the return type may depend on the input value. For example, let’s add a case to showType

showType (Nat -> a) = ?help

Inspecting the type of help tells us:

Main> :t help
   a : Nat -> Type
-------------------------------------
help : String

So, the return type a depends on the input value of type Nat, and we’ll need to come up with a value to use a, for example

showType (Nat -> a) = "Function from Nat to " ++ showType (a Z)

Note that multiplicities on the binders, and the ability to pattern match on non-erased types mean that the following two types are distinct

id : a -> a
notId : {a : Type} -> a -> a

In the case of notId, we can match on a and get a function which is certainly not the identity function

notId {a = Integer} x = x + 1
notId x = x

Main> notId 93
94
Main> notId "???"
"???"

There is an important consequence of being able to distinguish between relevant and irrelevant type arguments, which is that a function is only parametric in a if a has multiplicity 0. So, in the case of notId, a is not a parameter, and so we can’t draw any conclusions about the way the function will behave because it is polymorphic, because the type tells us it might pattern match on a.

On the other hand, it is merely a coincidence that, in non-dependently typed languages, types are irrelevant and get erased, and values are relevant and remain at run time. Idris 2, being based on QTT, allows us to make the distinction between relevant and irrelevant arguments precise. Types can be relevant, values (such as the n index to vectors) can be irrelevant.

For more details on multiplicities, see Idris 2: Quantitative Type Theory in Action.