# LGtk/Semantics

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The semantics of LGtk is given by a reference implementation. | The semantics of LGtk is given by a reference implementation. | ||

− | The reference implementation is given in three stages: lenses, references and effects. | + | The reference implementation is given in three stages: lenses, references and effects. [http://www.reddit.com/r/haskell/comments/1fjerf/lens_chains_lgtk_semantics_first_part/ Comments on Reddit] |

== Lenses == | == Lenses == | ||

Line 28: | Line 28: | ||

The three well-known laws for lenses: | The three well-known laws for lenses: | ||

− | * get-set: <hask> | + | * get-set: <hask>setL k (getL k a) a</hask> === <hask>a</hask> |

− | * set-get: <hask> | + | * set-get: <hask>getL k (setL k b a)</hask> === <hask>b</hask> |

− | * set-set: <hask> | + | * set-set: <hask>setL k b2 (setL k b1 a)</hask> === <hask>setL k b2 a</hask> |

Impure lenses, i.e. lenses which | Impure lenses, i.e. lenses which | ||

− | + | break lens laws are allowed in certain places. Those places are explicitly marked and explained in this overview. (TODO) | |

== References == | == References == | ||

Line 41: | Line 41: | ||

Let <hask>s :: *</hask> be a type. | Let <hask>s :: *</hask> be a type. | ||

− | Consider the three types <hask>Lens s :: * -> *</hask>, <hask>State s :: * -> *</hask>, <hask>Reader s :: * -> *</hask> with their type class instances and operations. This structure is useful in practice ([http://www.haskellforall.com/2013/05/program-imperatively-using-haskell.html see this use case]). | + | Consider the three types <hask>Lens s :: * -> *</hask>, <hask>State s :: * -> *</hask>, <hask>Reader s :: * -> *</hask> with their type class instances and operations. This rich structure is useful in practice ([http://www.haskellforall.com/2013/05/program-imperatively-using-haskell.html see this use case]). |

We have the following goals: | We have the following goals: | ||

− | * Define a structure similar to <hask>(Lens s, State s, Reader s)</hask> in which <hask>s</hask> is not | + | * Define a structure similar to <hask>(Lens s, State s, Reader s)</hask> in which <hask>s</hask> is not accessible. |

* Extend the defined structure with operations which help modularity. | * Extend the defined structure with operations which help modularity. | ||

Line 112: | Line 112: | ||

Note that the identity lens is '''not''' a reference because that would leak the program state <hask>s</hask>. | Note that the identity lens is '''not''' a reference because that would leak the program state <hask>s</hask>. | ||

+ | |||

+ | ==== Reference laws ==== | ||

+ | |||

+ | Laws for <hask>liftReadPart</hask> (derived from the properties of the <hask>Reader</hask> monad): | ||

+ | |||

+ | * <hask>liftReadPart m >> return ()</hask> === <hask>return ()</hask> | ||

+ | |||

+ | * <hask>liftM2 (,) (liftReadPart m) (liftReadPart m)</hask> === <hask>liftM (\a -> (a, a)) (liftReadPart m)</hask> | ||

+ | |||

+ | Laws for <hask>readRef</hask> and <hask>writeRef</hask> (derived from lens laws): | ||

+ | |||

+ | * <hask>liftReadPart (readRef r) >>= writeRef r</hask> === <hask>return ()</hask> | ||

+ | |||

+ | * <hask>writeRef r a >> liftReadPart (readRef r)</hask> === <hask>return a</hask> | ||

+ | |||

+ | * <hask>writeRef r a >> writeRef r a'</hask> === <hask>writeRef r a'</hask> | ||

+ | |||

+ | Laws for lens application: | ||

+ | |||

+ | * <hask>lensMap id r</hask> === <hask>r</hask> | ||

+ | |||

+ | * <hask>lensMap k (lensMap l r)</hask> === <hask>lensMap (k . l) r</hask> | ||

+ | |||

+ | * <hask>readRef (lensMap k r)</hask> === <hask>liftM (getL k) (readRef r)</hask> | ||

+ | |||

+ | * <hask>writeRef (lensMap k r) a</hask> === <hask>liftReadPart (readRef r) >>= writeRef r . setL k a</hask> | ||

+ | |||

+ | Laws for reference join: | ||

+ | |||

+ | * <hask>joinRef (return r)</hask> === <hask>r</hask> | ||

+ | |||

+ | * <hask>readRef (joinRef m)</hask> === <hask>m >>= readRef</hask> | ||

+ | |||

+ | * <hask>writeRef (joinRef m) a</hask> === <hask>liftReadPart m >>= flip writeRef a</hask> | ||

+ | |||

+ | Laws for the unit refernce: | ||

+ | |||

+ | * <hask>readRef unitRef</hask> === <hask>return ()</hask> | ||

=== Reference creation === | === Reference creation === | ||

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Let <hask>s</hask> = <hask>S</hask> in the definition of references. | Let <hask>s</hask> = <hask>S</hask> in the definition of references. | ||

− | * <hask>C :: (* -> *) -> * -> *</hask> ~ <hask>StateT S</hask>, the '''reference creating monad''' is the state monad transformer over <hask>S</hask>. | + | * <hask>C :: (* -> *) -> * -> *</hask> ~ <hask>StateT S</hask>, the '''reference creating monad''', is the state monad transformer over <hask>S</hask>. |

The equality constraint is not exposed in the API. | The equality constraint is not exposed in the API. | ||

Line 161: | Line 199: | ||

<haskell> | <haskell> | ||

newRef :: Monad m => a -> C m (Ref a) | newRef :: Monad m => a -> C m (Ref a) | ||

− | newRef | + | newRef = mapStateT (return . runIdentity) . extend |

</haskell> | </haskell> | ||

Line 167: | Line 205: | ||

<haskell> | <haskell> | ||

− | + | liftMod :: Monad m => M a -> C m a | |

− | + | liftMod = mapStateT (return . runIdentity) | |

</haskell> | </haskell> | ||

Line 186: | Line 224: | ||

</haskell> | </haskell> | ||

− | === | + | === Dependent reference creation === |

==== Motivation ==== | ==== Motivation ==== | ||

− | + | With <hask>newRef</hask> the program state can be extended in an independent way (the new state piece is independent from the others). This is not always enough for modular design. | |

− | Suppose that we have a reference <hask>r :: Ref (Maybe Int)</hask> and we would like to expose an editor of it to the user. The exposed editor would | + | Suppose that we have a reference <hask>r :: Ref (Maybe Int)</hask> and we would like to expose an editor of it to the user. The exposed editor would contain a checkbox and an integer entry field. If the checkbox is unchecked, the value of <hask>r</hask> should be <hask>Nothing</hask>, otherwise it should be <hask>Just i</hask> where <hask>i</hask> is the value of the entry field. |

− | The problem is that we cannot remember the value of the entry field if the checkbox is unchecked! | + | The problem is that we cannot remember the value of the entry field if the checkbox is unchecked! Dependent reference creation give a nice solution to this problem. |

− | ==== | + | ==== Specification ==== |

− | + | Suppose that we have an abstract data type <hask>S</hask> with the operations | |

<haskell> | <haskell> | ||

Line 210: | Line 248: | ||

such that the following laws hold: | such that the following laws hold: | ||

− | * Law 1: <hask>extend'</hask> | + | * Law 1: <hask>extend' r k a0 >> return ()</hask> === <hask>return ()</hask> |

− | <hask>extend' r k a0 | + | * Law 2: <hask>liftM (k .) $ extend' r k a0</hask> === <hask>return r</hask> |

− | * Law | + | * Law 3: <hask>extend' r k a0 >>= readRef</hask> === <hask>readRef r >>= setL k a0</hask> |

− | <hask> | + | Law 1 sais that <hask>extend'</hask> has no side-effect. Law 2 sais that <hask>k</hask> applied on the result of <hask>extend' r k a0</hask> behaves as <hask>r</hask>. Law 3 defines the initial value of the newly defined reference. |

− | + | ||

− | + | ||

− | + | ||

− | <hask>extend' r k a0 | + | |

==== Usage ==== | ==== Usage ==== | ||

Line 239: | Line 273: | ||

<hask>extend' r maybeLens (False, 0)</hask> | <hask>extend' r maybeLens (False, 0)</hask> | ||

− | This operations returns <hask>q :: Lens S (Bool, Int)</hask>. If we connect <hask>fstLens q</hask> to a checkbox and <hask>sndLens q</hask> to an integer entry field, we get the expected behaviour as described in the motivation. | + | This operations returns <hask>q :: Lens S (Bool, Int)</hask>. If we connect <hask>fstLens . q</hask> to a checkbox and <hask>sndLens . q</hask> to an integer entry field, we get the expected behaviour as described in the motivation. |

<hask>extend'</hask> introduces a dependent local state because <hask>q</hask> is automatically updated if <hask>r</hask> changes (and vice-versa). | <hask>extend'</hask> introduces a dependent local state because <hask>q</hask> is automatically updated if <hask>r</hask> changes (and vice-versa). | ||

− | Backward lens application can solve | + | Backward lens application can solve long-standing problems with application of lenses, see [[LGtk/ADT_lenses]]. |

− | ==== | + | ==== Reference creation API (revised) ==== |

Let <hask>S</hask> be an extensible state as specified above. | Let <hask>S</hask> be an extensible state as specified above. | ||

Line 259: | Line 293: | ||

<haskell> | <haskell> | ||

extRef :: Monad m => Ref b -> Lens a b -> a -> C m (Ref a) | extRef :: Monad m => Ref b -> Lens a b -> a -> C m (Ref a) | ||

− | extRef r k | + | extRef r k = mapStateT (return . runIdentity) . extend' r k |

</haskell> | </haskell> | ||

Line 271: | Line 305: | ||

* Derived <hask>MonadTrans</hask> instance of <hask>C</hask> to be able to lift operations in the <hask>m</hask> monad. | * Derived <hask>MonadTrans</hask> instance of <hask>C</hask> to be able to lift operations in the <hask>m</hask> monad. | ||

− | Note that <hask>newRef</hask> can be defined in terms of <hask>extRef</hask> so <hask>extRef</hask> is more | + | Note that <hask>newRef</hask> can be defined in terms of <hask>extRef</hask> so <hask>extRef</hask> is more powerful than <hask>newRef</hask>: |

<haskell> | <haskell> | ||

Line 278: | Line 312: | ||

</haskell> | </haskell> | ||

− | + | ==== Existence proof of <hask>S</hask> ==== | |

− | ==== | + | |

We prove constructively, by giving a reference implementation, that <hask>S</hask> exists. With this we also prove that <hask>S</hask> defined in the previous section exists because <hask>extend</hask> can be defined in terms of <hask>extend'</hask> (easy exercise: give the definition). | We prove constructively, by giving a reference implementation, that <hask>S</hask> exists. With this we also prove that <hask>S</hask> defined in the previous section exists because <hask>extend</hask> can be defined in terms of <hask>extend'</hask> (easy exercise: give the definition). | ||

− | + | Overview: | |

− | + | The idea behind the implementation is that <hask>S</hask> not only stores a gradually extending state with type <hask>(a'', (a', (a, ...))</hask> but also a chain of lenses with type <hask>Lens (a'', (a', (a, ...))) (a', (a, ...))</hask>, <hask>Lens (a', (a, ...)) (a, ...)</hask>, <hask>Lens (a, ...) ...</hask>, <hask>...</hask>. | |

− | + | When a new reference is created, both the state and the lens-chain are extended. The dependency between the newly created state part and the old state parts can be encoded into the new lens in the lens-chain. | |

+ | |||

+ | When a previously created reference (i.e. a lens) is accessed with a state after several extensions, a proper prefix of the lens-chain makes possible to convert the program state such that it fits the previously created reference. | ||

+ | |||

+ | |||

+ | Reference implementation: | ||

+ | |||

+ | Definition of <hask>S</hask>: | ||

<haskell> | <haskell> | ||

− | + | type S = [Part] | |

− | + | ||

</haskell> | </haskell> | ||

− | |||

− | |||

<haskell> | <haskell> | ||

− | + | data Part | |

− | + | = forall a | |

− | + | . Part | |

− | + | { selfAdjustment :: S -> a -> a -- does not change (static) | |

+ | , statePart :: a -- variable | ||

+ | } | ||

</haskell> | </haskell> | ||

+ | |||

+ | Note that instead of lenses, self-adjusting functions are stored in state parts, which is a simplification in the implementation. | ||

Definition of <hask>empty</hask>: | Definition of <hask>empty</hask>: | ||

Line 307: | Line 348: | ||

<haskell> | <haskell> | ||

empty :: S | empty :: S | ||

− | empty = S () | + | empty = [] |

+ | </haskell> | ||

+ | |||

+ | Auxiliary defintion: Add a state part and adjust its local state. | ||

+ | |||

+ | <haskell> | ||

+ | snoc :: S -> Part -> S | ||

+ | s `snoc` Part f a | ||

+ | = s ++ [Part f (f s a)] | ||

</haskell> | </haskell> | ||

Line 314: | Line 363: | ||

<haskell> | <haskell> | ||

extend' :: Lens S b -> Lens a b -> a -> State S (Lens S a) | extend' :: Lens S b -> Lens a b -> a -> State S (Lens S a) | ||

− | extend' r1 r2 | + | extend' r1 r2 a = do |

+ | -- get number of state parts | ||

+ | n <- gets length | ||

+ | -- add a properly initialized new state part | ||

+ | modify (`snoc` Part (setL r2 . getL r1) a) | ||

+ | -- return a lens which accesses n+1+k state parts (k depends on future extensions) | ||

+ | return $ Lens $ mkStore (setL r1 . getL r2) . splitAt n | ||

+ | </haskell> | ||

− | + | <haskell> | |

− | + | mkStore :: (a -> S -> S) -> (S, S) -> Store a S | |

− | + | mkStore g (s, Part f a: ps) -- (previous state parts, self state part: next state parts) | |

− | + | = store | |

− | + | -- set self state part, | |

− | + | -- adjust previously added state parts (by g), | |

− | + | -- adjust recently added state parts (by re-adding them with snoc) | |

− | + | (\a -> foldl snoc (g a s ++ [Part f (unsafeCoerce a)]) ps) | |

− | + | -- get self state part | |

− | + | (unsafeCoerce a) | |

</haskell> | </haskell> | ||

− | + | If the values of <hask>S</hask> are used linearly, types will match at runtime so the use of <hask>unsafeCoerce</hask> is safe. | |

+ | |||

+ | === Summary === | ||

+ | |||

+ | '''References''' is a structure with the abstract data types | ||

+ | |||

+ | * <hask>Ref :: * -> *</hask> -- reference | ||

+ | * <hask>R :: * -> *</hask> -- reference reading monad | ||

+ | * <hask>M :: * -> *</hask> -- reference modifying monad | ||

+ | * <hask>C :: (* -> *) -> * -> *</hask> -- reference creating monad transformer | ||

+ | |||

+ | where | ||

+ | |||

+ | * <hask>R</hask> and <hask>M</hask> are monads, | ||

+ | * <hask>C</hask> is a monad transformer, | ||

+ | * <hask>liftReadPart :: R a -> M a</hask> is a monad morphism between <hask>R</hask> and <hask>M</hask>, | ||

+ | * <hask>liftMod :: Monad m => M a -> C m a</hask> is a monad morphism between <hask>M</hask> and <hask>C m</hask>, | ||

+ | |||

+ | with the operations | ||

+ | |||

+ | * <hask>unitRef :: Ref ()</hask> -- unit reference | ||

+ | * <hask>lensMap :: Lens a b -> Ref a -> Ref b</hask> -- lens application on a reference | ||

+ | * <hask>joinRef :: R (Ref a) -> Ref a</hask> -- reference join | ||

+ | * <hask>readRef :: Ref a -> R a</hask> -- reference read | ||

+ | * <hask>writeRef :: Ref a -> a -> M ()</hask> -- reference write | ||

+ | * <hask>extRef :: Monad m => Ref b -> Lens a b -> a -> C m (Ref a)</hask> -- reference creation | ||

+ | * <hask>runC :: Monad m => C m a -> m a</hask> -- running | ||

+ | |||

+ | such that the following laws hold | ||

+ | |||

+ | * <hask>m >> return ()</hask> === <hask>return ()</hask>, for all <hask>m :: R a</hask> | ||

+ | * <hask>liftM2 (,) m m</hask> === <hask>liftM (\a -> (a, a)) m</hask>, for all <hask>m :: R a</hask> | ||

+ | |||

+ | * <hask>readRef unitRef</hask> === <hask>return ()</hask> | ||

+ | |||

+ | * <hask>lensMap id r</hask> === <hask>r</hask> | ||

+ | * <hask>lensMap k (lensMap l r)</hask> === <hask>lensMap (k . l) r</hask> | ||

+ | * <hask>readRef (lensMap k r)</hask> === <hask>liftM (getL k) (readRef r)</hask> | ||

+ | * <hask>writeRef (lensMap k r) a</hask> === <hask>liftReadPart (readRef r) >>= writeRef r . setL k a</hask> | ||

+ | |||

+ | * <hask>joinRef (return r)</hask> === <hask>r</hask> | ||

+ | * <hask>readRef (joinRef m)</hask> === <hask>m >>= readRef</hask> | ||

+ | * <hask>writeRef (joinRef m) a</hask> === <hask>liftReadPart m >>= flip writeRef a</hask> | ||

+ | |||

+ | * <hask>liftReadPart (readRef r) >>= writeRef r</hask> === <hask>return ()</hask> | ||

+ | * <hask>writeRef r a >> liftReadPart (readRef r)</hask> === <hask>return a</hask> | ||

+ | * <hask>writeRef r a >> writeRef r a'</hask> === <hask>writeRef r a'</hask> | ||

− | + | * <hask>extRef r k a >> return ()</hask> === <hask>return ()</hask> | |

+ | * <hask>liftM (k .) (extRef r k a)</hask> === <hask>return r</hask> | ||

+ | * <hask>extRef r k a >>= liftMod . liftReadPart . readRef</hask> === <hask>liftMod (liftReadPart (readRef r >>= setL k a))</hask> | ||

== Effects == | == Effects == | ||

TODO | TODO |

## Latest revision as of 23:00, 7 June 2013

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

## Contents |

## [edit] 1 Lenses

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

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

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

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

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

### [edit] 1.1 Lens laws

The three well-known laws for lenses:

- get-set: ===setL k (getL k a) aa
- set-get: ===getL k (setL k b a)b
- set-set: ===setL k b2 (setL k b1 a)setL k b2 a

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

## [edit] 2 References

### [edit] 2.1 Motivation

LetWe have the following goals:

- Define a structure similar to in which(Lens s, State s, Reader s)is not accessible.s
- 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 convenient to maintain explicitly.

### [edit] 2.2 Types

We keep the typesInstead of giving a concrete implementation in Haskell, suppose that

- is a fixed arbitrary type,s
- ~Ref :: * -> *;Lens s
**references**are lenses fromto the type of the referred value,s - ~R :: * -> *; theReader s
**reference reading monad**is the reader monad over,s - ~M :: * -> *; theState s
**reference modifying monad**is the state monad over.s

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

### [edit] 2.3 Operations

Exposed operations of- The instance ofMonadandRM

- The monad morphism between andRM

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

- Reference read

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

- Reference write

writeRef :: Ref a -> a -> M () 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)

**not**a reference because that would leak the program state

#### [edit] 2.3.1 Reference laws

Laws for- ===liftReadPart m >> return ()return ()

- ===liftM2 (,) (liftReadPart m) (liftReadPart m)liftM (\a -> (a, a)) (liftReadPart m)

- ===liftReadPart (readRef r) >>= writeRef rreturn ()

- ===writeRef r a >> liftReadPart (readRef r)return a

- ===writeRef r a >> writeRef r a'writeRef r a'

Laws for lens application:

- ===lensMap id rr

- ===lensMap k (lensMap l r)lensMap (k . l) r

- ===readRef (lensMap k r)liftM (getL k) (readRef r)

- ===writeRef (lensMap k r) aliftReadPart (readRef r) >>= writeRef r . setL k a

Laws for reference join:

- ===joinRef (return r)r

- ===readRef (joinRef m)m >>= readRef

- ===writeRef (joinRef m) aliftReadPart m >>= flip writeRef a

Laws for the unit refernce:

- ===readRef unitRefreturn ()

### [edit] 2.4 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 isWe 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

`empty :: S`

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

such that the following laws hold:

- ===extend v >> return ()return ()

- ===extend v >>= liftReadPart . readRefreturn 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.

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

Can this extensible state be implemented efficiently? Although this question is not relevant for the semantics, we will see that there is an efficient implementation with#### [edit] 2.4.1 Reference creation API

Let- ~C :: (* -> *) -> * -> *, theStateT S
**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 = mapStateT (return . runIdentity) . extend

- Lift reference modifying operations

liftMod :: Monad m => M a -> C m a liftMod = mapStateT (return . runIdentity)

- Derived instance ofMonadTransto be able to lift operations in theCmonad.m

### [edit] 2.5 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

### [edit] 2.6 Dependent reference creation

#### [edit] 2.6.1 Motivation

WithThe problem is that we cannot remember the value of the entry field if the checkbox is unchecked! Dependent reference creation give a nice solution to this problem.

#### [edit] 2.6.2 Specification

Suppose that we have an abstract data type`empty :: S`

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

such that the following laws hold:

- Law 1: ===extend' r k a0 >> return ()return ()

- Law 2: ===liftM (k .) $ extend' r k a0return r

- Law 3: ===extend' r k a0 >>= readRefreadRef r >>= setL k a0

#### [edit] 2.6.3 Usage

Law 2 is the most interesting, it sais that we can**apply a lens backwards**with

**dependent local state**.

Consider the following pure lens:

maybeLens :: Lens (Bool, a) (Maybe a) maybeLens = lens (\(b, a) -> if b then Just a else Nothing) (\x (_, a) -> maybe (False, a) (\a' -> (True, a')) x)

Backward lens application can solve long-standing problems with application of lenses, see LGtk/ADT_lenses.

#### [edit] 2.6.4 Reference creation API (revised)

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

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

- Dependent new reference creation

extRef :: Monad m => Ref b -> Lens a b -> a -> C m (Ref a) extRef r k = mapStateT (return . runIdentity) . extend' r k

- Lift reference modifying operations

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

- Derived instance ofMonadTransto be able to lift operations in theCmonad.m

newRef :: Monad m => a -> C m (Ref a) newRef = extRef unitRef (lens (const ()) (const id))

#### [edit] 2.6.5 Existence proof of S

We prove constructively, by giving a reference implementation, that Overview:

The idea behind the implementation is thatWhen a new reference is created, both the state and the lens-chain are extended. The dependency between the newly created state part and the old state parts can be encoded into the new lens in the lens-chain.

When a previously created reference (i.e. a lens) is accessed with a state after several extensions, a proper prefix of the lens-chain makes possible to convert the program state such that it fits the previously created reference.

Reference implementation:

type S = [Part]

data Part = forall a . Part { selfAdjustment :: S -> a -> a -- does not change (static) , statePart :: a -- variable }

Note that instead of lenses, self-adjusting functions are stored in state parts, which is a simplification in the implementation.

Definition ofempty :: S empty = []

Auxiliary defintion: Add a state part and adjust its local state.

snoc :: S -> Part -> S s `snoc` Part f a = s ++ [Part f (f s a)]

extend' :: Lens S b -> Lens a b -> a -> State S (Lens S a) extend' r1 r2 a = do -- get number of state parts n <- gets length -- add a properly initialized new state part modify (`snoc` Part (setL r2 . getL r1) a) -- return a lens which accesses n+1+k state parts (k depends on future extensions) return $ Lens $ mkStore (setL r1 . getL r2) . splitAt n

mkStore :: (a -> S -> S) -> (S, S) -> Store a S mkStore g (s, Part f a: ps) -- (previous state parts, self state part: next state parts) = store -- set self state part, -- adjust previously added state parts (by g), -- adjust recently added state parts (by re-adding them with snoc) (\a -> foldl snoc (g a s ++ [Part f (unsafeCoerce a)]) ps) -- get self state part (unsafeCoerce a)

### [edit] 2.7 Summary

**References** is a structure with the abstract data types

- -- referenceRef :: * -> *
- -- reference reading monadR :: * -> *
- -- reference modifying monadM :: * -> *
- -- reference creating monad transformerC :: (* -> *) -> * -> *

where

- andRare monads,M
- is a monad transformer,C
- is a monad morphism betweenliftReadPart :: R a -> M aandR,M
- is a monad morphism betweenliftMod :: Monad m => M a -> C m aandM,C m

with the operations

- -- unit referenceunitRef :: Ref ()
- -- lens application on a referencelensMap :: Lens a b -> Ref a -> Ref b
- -- reference joinjoinRef :: R (Ref a) -> Ref a
- -- reference readreadRef :: Ref a -> R a
- -- reference writewriteRef :: Ref a -> a -> M ()
- -- reference creationextRef :: Monad m => Ref b -> Lens a b -> a -> C m (Ref a)
- -- runningrunC :: Monad m => C m a -> m a

such that the following laws hold

- ===m >> return (), for allreturn ()m :: R a
- ===liftM2 (,) m m, for allliftM (\a -> (a, a)) mm :: R a

- ===readRef unitRefreturn ()

- ===lensMap id rr
- ===lensMap k (lensMap l r)lensMap (k . l) r
- ===readRef (lensMap k r)liftM (getL k) (readRef r)
- ===writeRef (lensMap k r) aliftReadPart (readRef r) >>= writeRef r . setL k a

- ===joinRef (return r)r
- ===readRef (joinRef m)m >>= readRef
- ===writeRef (joinRef m) aliftReadPart m >>= flip writeRef a

- ===liftReadPart (readRef r) >>= writeRef rreturn ()
- ===writeRef r a >> liftReadPart (readRef r)return a
- ===writeRef r a >> writeRef r a'writeRef r a'

- ===extRef r k a >> return ()return ()
- ===liftM (k .) (extRef r k a)return r
- ===extRef r k a >>= liftMod . liftReadPart . readRefliftMod (liftReadPart (readRef r >>= setL k a))

## [edit] 3 Effects

TODO