LGtk/Semantics

The semantics of LGtk is given by a reference implementation. The reference implementation is given in three stages: lenses, references and effects.

Lenses

LGtk use simple lenses defined in the data-lens package:

newtype Lens a b = Lens { runLens :: a -> Store b a }

This data type is isomorphic to (a -> b, b -> a -> a), the well-know lens data type. The isomorphism is established by the following operations:

getL :: Lens a b -> a -> b

setL :: Lens a b -> b -> a -> a

lens :: (a -> b) -> (b -> a -> a) -> Lens a b

Lens laws

The three well-known laws for lenses:

• get-set: set k (get k a) a === a
• set-get: get k (set k b a) === b
• set-set: set k b2 (set k b1 a) === set k b2 a

Impure lenses, i.e. lenses which brake lens laws are allowed in certain places. Those places are explicitly marked and explained in this overview.

References

Motivation

Let s :: * be a type.

Consider the three types Lens s :: * -> *, State s :: * -> *, Reader s :: * -> * with their type class instances and operations.

This structure is useful in practice (see this use case). We have the following goals:

• Define a structure similar to (Lens s, State s, Reader s) in which s is not accessable.
• Extend the defined structure with operations which help modularity.

The first goal is justified by our solution for the second goal. The second goal is justified by the fact that a global state is not easy to maintain explicitly.

Types

We keep the types (Lens s, State s, Reader s) but give them a more restricted API. There are several ways to do this in Haskell. LGtk defines type classes with associated types, but that is just a technical detail.

Instead of giving a concrete implementation in Haskell, suppose that

• s is a fixed arbitrary type,
• Ref :: * -> * ~ Lens s; references are lenses from s to the type of the referred value,
• R :: * -> * ~ Reader s; the reference reading monad is the reader monad over s,
• S :: * -> * ~ State s; the reference modifying monad is the state monad over s.

The three equality constraints are not exposed in the API, of course.

Operations

Exposed operations of Ref, S and R are the following:

• The Monad instance of R and S

• The monad morphism between R and S

liftReadPart :: R a -> S a
liftReadPart = gets . runReader

• Reference read

readRef :: Ref a -> R a
readRef = reader . getL

• Reference write

writeRef :: Ref a -> a -> S ()
writeRef = modify . setL r

• Lens application on a reference

lensMap :: Lens a b -> Ref a -> Ref b
lensMap = (.)

• Reference join

joinRef :: R (Ref a) -> Ref a
joinRef = Lens . join . (runLens .) . runReader

• The unit reference

unitRef :: Ref ()
unitRef = lens (const ()) (const id)

Note that the identity lens is not a reference because that would leak the program state s.

Reference creation

New reference creation is our first operation wich helps modularity.

New reference creation with a given initial value extends the state. For example, if the state is (1, 'c') :: (Int, Char), extending the state with True :: Bool would yield the state ((1, 'c'), True) :: ((Int, Char), Bool).

We could model the type change of the state with an indexed monad, but that would complicate both the API and the implementation too.

Instead of changing the type of the state, we use an extensible state, an abstract data type S with the operations

empty :: S

extend :: a -> State S (Lens S a)

such that the following laws hold:

• extend v >> return () === return ()

• extend v >>= liftReadPart . readRef === return v

The first law sais that extend has no side-effect, i.e. extending the state does not change the values of existing references. The second law sais that extending the state with value v creates a reference with inital value v.

Question: Is there a data type with such operations?

The answer is yes, but we should guarantee linear usage of the state. The (constructive) existence proof is given in the next section.

The linear usage of state is guaranteed with refereces API (check the definitions), which means that we have a solution.

Can this extensible state be implemented efficiently? However this question is not relevant for the semantics, we will see that there is an efficient implementation with MVars.

Reference creation API

Let S be an extensible state as specified before. Let s = S in the definition of references.

• C :: (* -> *) -> * -> * ~ StateT S, the reference creating monad is the state monad transformer over S.

The equality constraint is not exposed in the API. The following functions are exposed:

• New reference creation

newRef :: Monad m => a -> C m (Ref a)
newRef v = mapStateT (return . runIdentity) $ extend v

• Lift reference read and write operations

liftWrite :: Monad m => S a -> C m a
liftWrite = mapStateT (return . runIdentity)

• Derived MonadTrans instance of C to be able to lift operations in the m monad.

Note that C is parameterized by a monad because in this way it is easier to add other effects later.

Running

The API is completed with the following function:

• Run the reference creation monad

runC :: Monad m => C m a -> m a
runC x = runStateT x empty

Lens-stacks

TODO

Effects

TODO