Posted on 2017-01-17
by Oleg Grenrus

I have been wondering about linear logic for a while, and after reading Edsko de Vries' blog post: Linearity, Uniqueness, and Haskell, I finally got to dump my thoughts on the topic.

Edsko starts with two example functions, where the second one is following:

```
frugal :: a -> (a, a)
frugal x = (x, x)
```

In Haskell functions can reuse their arguments, as `frugal`

does. Later he shows that linear setting we can forbid such functions.

A taste of linear logic by Philip Wadler is a very nice introduction to the intuitionistic linear logic. One of the take aways, is that in linear logic, there are two kind of pairs, *"ands"*. And there's still single "or".

So to recap, Take `A`

to be the proposition "I have ten zlotys", `B`

to be the proposition "I have a pizza", and `C`

to be the proposition "I have a cake". We can say

*A ⊢ B*: for ten zlotys I may buy a pizza*A ⊢ C*: for ten zlotys I may buy a cake

And there are three ways to "pair" the terms:

*A, A ⊢ B ⊗ C*: for twenty zlotys I can buy both a pizza*and*a cake,*A ⊢ B & C*: for ten zlotys I can buy whichever I choose from a cake*and*a pizza,*A ⊢ B ⊕ C*: for ten zlotys I can buy either a pizza*or*a cake, but I don't have a choice.

In natural language we'd might say: whichver I choose from a cake *or* a pizza. That's confusing.

In the Edsko's post, the pairs are assumed to be of the first kind: ⊗, tensors. We cannot write `fst`

or `snd`

variants for the tensor, but

```
uncurry :: (a -o b -o c) -o a ⊗ b -o c
uncurry f p = case p of (x, y) -> f x y
```

is ok.

However, if we work with the second kind of the pair, "with", we cannot write `uncurry`

, but can have `fst`

and `snd`

:

```
fst :: a & b -o a
fst (x, _) = a
snd :: a & b -o b
snd (_, y) = y
```

And with this pair we can write something like:

```
haveAndEat :: Vector Double -o Vector Double & Vector Double
heveAndEat arr = (write 0 2.3 arr, arr)
-- one should really have different constructor for tensor and with
```

There is no problem, as a callee cannot use both sides of the with, they have to choose only (and exactly) one half.

I.e. we can write `frugal`

as

```
frugal :: a -o a & a
frugal x = (x, x)
```

IMHO `with`

is a very interesting construction. It seems it cannot exist in a strict language: only when we apply `fst`

or `snd`

will force it. In addition, when we force one half, the thunk for the other can be released immediately.

I guess this can be useful in non-backtracking infinite solution-space traversals, by using *with* we could be sure we don't introduce space leaks by accidentally retaining other paths.

Recall a state monad

`newtype State a = State { runState :: s -> (a, s) }`

We can have linear version of it, and the `Monad`

instance would type-check:

```
newtype LinearState a = LinearState { runLinearState :: s -o a ⊗ s }
instance Monad LinearState where
return x = \s -> (x, s)
m >>= k = LinearState $ \s -> case runLinearState m s of
(a, s') -> runLinearState (k a) s'
```

The interesting part, is that this type isn't an instance of `MonadState`

. We cannot write `get`

or `put`

. And we cannot write `MonadReader`

with `ask = get`

either. But we can have a different class:

```
class Monad m => MonadLinearState s m | m -> s where
linearModify :: (s -o s) -o m ()
instance MonadLinearState s (LinearState s) where
linearModify f = LinearState $ \s -> ((), f s)
```

Also `LinearWriter`

, i.e. the one without `listen`

or `pass`

.

AFAICS, we could have `MonadLinearState RealWorld# IO`

, without problems!

It seems, that if we had linearity, then we can use linearity to handle resources. We can write `bracket`

like functions, where we have to return the token at the end:

```
-- Not sure about the arrows, should they be lollipops?
withFile :: FilePath -> (Handle -o IO (Handle ⊗ a)) -> IO a
getFileContents :: Handle -o IO (Handle & String)
lineCount :: FilePath -> IO Int
lineCount fp = withFile fp $ \h -> do
(h, contents) <- getFileContents h
return (h, length $ lines contents)
```

That's something where `LinearStateT`

will make code nicer. Or we can change `getFileContents :: Handle OPENED -o IO (Handle CLOSED ⊗ String)`

, or actually we'd need to have

```
withFile
:: FilePath
-> (forall s. Handle s OPENED -o IO (Handle s CLOSED ⊗ a))
-> IO a
```

The type variable `s`

will index our `Handle`

, so don't pass a wrong one when we have nested `withFile`

s.

But then we couldn't use `do`

notation, as state variable will change. We'd need indexed monads, oh dear.

At this point, I have to mention, that linear types would give us great power, but explaining this kind of IO (or any "magic" involcing them) to a beginner won't be easy.

FWIW, `withMutableArray`

from Edsko's post is simpler example of this idea of handling resources.

```
phadej @pl \xs -> map f (filter g xs)
lambdabot map f . filter g -- looks linear
phadej @pl \x -> x * x
lambdabot join (*) -- but join isn't (Reader)
phadej @pl \p -> (fst p, snd p)
lambdabot id -- sometimes pl is smart
phadej @pl \p -> (fst p, snd p + 2)
lambdabot liftM2 (,) fst ((2 +) . snd) -- sometimes it isn't
phadej @pl \p -> uncurry (\x y -> (x, y + 2)) p
lambdabot second (2 +) -- so you have to help it
```

TL;DR concatative languages are linear by default, as there are special operators to contract and weaken!

It's probablty not obvious, but the type

`traverse :: (Applicative f, Traversable t) => (a -> f b) -> t a -> f (t b)`

is ok to linearise into:

```
linearTraverse
:: (LinearApplicative f, LinearTraversable t)
=> (a -o f b) -> t a -o f (t b)
```

Or can we reuse `Applicative f`

? Hard to say at this point. What arrows should be there? Polymorphic in arrows, with some relations?

In this post I used different syntax than in used by Edsko, the one I learned from watching lectures and reading papers on linear logic. And I have to agree, it's hard to evaluate how well type system will work in practice. Will we need *with*, how higher order functions or polymorphism will fit etc. I'm also looking forward how Haskell could support linearity, and what cool stuff we can do with it!

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