> {-# LANGUAGE > GeneralizedNewtypeDeriving, > FunctionalDependencies, > MultiParamTypeClasses, > OverlappingInstances, > IncoherentInstances, > FlexibleInstances, > FlexibleContexts, > TypeOperators, > TypeFamilies, > RankNTypes > #-} > > module CFLP.Control.Strategy ( > > Enumerable(..), Ctx, Res, Strategy(..), Transformer(..), StrategyT(..), > > Monadic(..), inside > > ) where > > import Control.Monad.State > > import CFLP.Control.Monad.UpdateResults of a non-deterministic computation can be enumerated if the strategy is an instance of the `Enumerable` class.

> class Enumerable s > where > enumeration :: s a -> [a]A strategy provides an operation choose that is used for constructing non-deterministic choices. We provide the following type families used when defining strategies: * `Ctx s` specifies the evaluation context of a strategy. Strategy transformer can extend the evaluation context later. * `Res s` specifies the monad in which results are returned by normal-form computations.

> type family Ctx (s :: * -> *) > type family Res (s :: * -> *) :: * -> *If we use the monad transformer `UpdateT` to add updateable state to a monad, then the base monad is used to return results when computing normal forms.

> type instance Res (UpdateT c m) = mA strategy is parameterised by an evlauation context of type `c`. The type of this context may be different from `Ctx s` because a strategy can be transformed. The strategy operations need to be able to access the final evaluation context after applying all transformations that may extend the context. A value of type `s a` represents a CFLP computation that yields a value of type `a` where the strategy `s` is used for evaluation.

> class Strategy c s whereStrategies provide the following operations:

> emptyContext :: s c -> Ctx syields an empty context of the associated type `Ctx s`. The argument of type `s c` is sometimes necessary to satisfy the type checker. It must always be safe to pass undefined, i.e., the argument must be ignored.

> choose :: c -> Int -> [s a] -> s ais used to cconstruct non-deterministic choices of computations. We pass an evaluation context of type `c` and a unique label of type `Int` that identifies the constructed choice.

> isNarrowed :: c -> Int -> s Boolspecifies whether a choice identified by the given label of type `Int` is already narrowed w.r.t. the given evaluation context of type `c`. The result is returned as computation of type `s Bool` rather than only `Bool`, again, to support the type checker occasionally. A Default Instance ------------------ Usually, strategies are instances of `MonadPlus`. In fact we can wrap every instance of `MonadPlus` to make it an instance of `Strategy` for arbitrary evaluation contexts `c` (that are ignored).

> newtype Monadic m a = Monadic { fromMonadic :: m a } > deriving (Monad, MonadPlus, Enumerable, MonadUpdate c)We use generalized newtype deriving to lift some type-class instances over the newtype constructor. The type instances for `Ctx` and `Res` are given as follows:

> type instance Ctx (Monadic m) = () > type instance Res (Monadic s) = Monadic (Res s)Implementing the strategy operations with `MonadPlus` instances is straigh-forward.

> instance MonadPlus m => Strategy c (Monadic m) whereThe operation

> emptyContext _ = ()yields `()` because stores are ignored by (untransformed) monadic strategies.

> choose _ _ [x] = x > choose _ _ xs = foldr mplus mzero xsuses `mplus` and `mzero` to construct the choice. Note that these are the automatically derived operations.

> isNarrowed _ _ = return Falsealways returns `False` because we ignore evaluation contexts here. As we cannot derive an instance for the class `Update` automatically (the newtype occurs twice in the instance head), we need to define the lifting manually.

> instance Update s m m' => Update s (Monadic m) (Monadic m') > where > updateState x = StateT (Monadic . runStateT (updateState (fromMonadic x)))Transforming Evaluation Contexts and Strategies =============================================== Evaluation contexts can be extended by strategy transformers using context transformers that are an instance of the class `Transformer`.

> class Transformer t whereThis class provides operations

> project :: t a -> ato get the transformed value, and

> replace :: t a -> a -> t ato replace the wrapped value with a different one. The operation

> inside :: (Monad m, Transformer t) => (a -> m a) -> t a -> m (t a) > inside f t = liftM (replace t) (f (project t))transforms the base value of a transformed value with a monadic update operation. Strategy Transformers --------------------- A strategy transformer is used to add functionality to strategies. Transformers are parameterised by an evaluation context just like strategies. If `t` is the type of a strategy transformer and `s` the type of a strategy, then a value of type `t s a` represents a computation that yields a value of type `a` using the strategy `s` transformed by `t`.

> class StrategyT c t whereStrategy transformers provide the following operations:

> liftStrategy :: c -> s a -> t s alifts a computation using the base strategy to one that uses the transformed strategy. The parameter of type `c` is only used to help the type checker. It must be safe to pass `undefined` here.

> baseStrategy :: c -> t s a -> s atakes a computation that uses the transformed strategy and yields one that uses the base strategy. The first parameter is used to support type checking and must be ignored.

> extendContext :: t s c -> Ctx s -> Ctx (t s)may extend the evaluation context of the base strategy. Again, the first argument must be ignored by instantiations of `StrategyT`.

> extendChoices :: (Monad s, MonadUpdate c s) > => c -> Int -> [t s a] -> [t s a] > extendChoices _ _ = idcan be used to update the evaluation context of the alternatives of non-deterministic choices. We provide a default implementation for instances of `StrategyT` that do not want to alter choices.

> alterNarrowed :: Monad s => c -> Int -> t s Bool -> t s Bool > alterNarrowed _ _ = idcan be used to alter the predicate that determines whether the choice identified by the given label is narrowed w.r.t. the given context. The result of the original strategy is passed as an additional argument. We provide a default implementation for instances that do not want to alter this predicate. If we have a strategy transformer and a strategy that is an updateable monad, then the transformed strategy is again a strategy.

> instance (Monad s, MonadUpdate c s, Strategy c s, StrategyT c t) > => Strategy c (t s) > whereThe operation `emptyContext` calls the corresponding operation of the base strategy and extends the result accoring to the transformer.

> emptyContext c = extendContext c (emptyContext (undefined `forBaseOf` c))Here, we give a hint to the type checker using the type-constrained `const` function `forBaseOf` (defined below). The operation `choose` extends the choices according to the transformer and calls the `choose` operation of the base strategy with the resulting choices.

> choose c n = liftStrategy c > . choose c n > . map (baseStrategy c) > . extendChoices c nFinally, the predicate `isNarrowed` is altered according the strategy transformer.

> isNarrowed c n = alterNarrowed c n (liftStrategy c (isNarrowed c n))The function `forBaseOf` is a `const` function with a specialised type that supports the type checker in finding the corresponding operation `emptyContext` for the base strategy.

> forBaseOf :: s a -> t s a -> s a > forBaseOf = constUniform Liftings ---------------- We can define instances for the type classes `MonadUpdate` and `Update` for strategies that are transformed with arbitrary strategy transformers. The instances simply use the lifting functions where appropriate and use type-constrained `const` functions where necessary.

> instance (MonadUpdate c s, StrategyT c t) => MonadUpdate c (t s) > where > update upd = liftStrategy (undefined `forContextOf` upd) (update upd) > > forContextOf :: c -> (forall m . MonadPlus m => c -> m c) -> c > forContextOf = const > > instance (Update c s s', StrategyT c t) => Update c (t s) (t s') > where > updateState x > = let y = StateT (liftStrategy (undefined `forStateOf` y) . > runStateT (updateState > (baseStrategy (undefined `forStateOf` y) x))) > in y > > forStateOf :: c -> StateT c m a -> c > forStateOf = const