generic-random 0.1.1.0 → 0.2.0.0

raw patch · 18 files changed

+1864/−1258 lines, 18 filesdep ~basePVP ok

version bump matches the API change (PVP)

Dependency ranges changed: base

API changes (from Hackage documentation)

Files

• README.md +63/−3
• bench/binaryTree.hs +3/−3
• generic-random.cabal +12/−8
• src/Data/Random/Generics.hs +0/−302
• src/Data/Random/Generics/Internal.hs +0/−146
• src/Data/Random/Generics/Internal/Oracle.hs +0/−539
• src/Data/Random/Generics/Internal/Solver.hs +0/−65
• src/Data/Random/Generics/Internal/Types.hs +0/−191
• src/Generic/Random/Boltzmann.hs +215/−0
• src/Generic/Random/Data.hs +313/−0
• src/Generic/Random/Generic.hs +30/−0
• src/Generic/Random/Internal/Common.hs +39/−0
• src/Generic/Random/Internal/Data.hs +146/−0
• src/Generic/Random/Internal/Generic.hs +286/−0
• src/Generic/Random/Internal/Oracle.hs +499/−0
• src/Generic/Random/Internal/Solver.hs +66/−0
• src/Generic/Random/Internal/Types.hs +191/−0
• test/tree.hs +1/−1

@@ -1,13 +1,17 @@ Generic random generators [![Hackage](https://img.shields.io/hackage/v/generic-random.svg)](https://hackage.haskell.org/package/generic-random) [![Build Status](https://travis-ci.org/Lysxia/generic-random.svg)](https://travis-ci.org/Lysxia/generic-random) ========================= +`Generic.Random.Data`+---------------------+ Define sized random generators for almost any type.  ```haskell     {-# LANGUAGE DeriveDataTypeable #-}+     import Data.Data     import Test.QuickCheck-    import Data.Random.Generics+    import Generic.Random.Data      data Term = Lambda Int Term | App Term Term | Var Int       deriving (Show, Data)@@ -25,11 +29,67 @@ - Objects of the same size (number of constructors) occur with the same   probability (see Duchon et al., references below). - Implements rejection sampling and pointing.-- Works with QuickCheck and MonadRandom.-- Can be extended or modified with user defined generators.+- Uses `Data.Data` generics.+- Works with QuickCheck and MonadRandom, but also similar user-defined monads+  for randomness (just implement `MonadRandomLike`).+- Can be tweaked somewhat with user defined generators. +`Generic.Random.Generic`+------------------------++Say goodbye to `Constructor <$> arbitrary <*> arbitrary <*> arbitrary`-boilerplate.++```haskell+    {-# LANGUAGE DataKinds #-}+    {-# LANGUAGE DeriveGeneric #-}+    {-# LANGUAGE TypeApplications #-}++    import GHC.Generics ( Generic )+    import Test.QuickCheck+    import Generic.Random.Generic++    data Tree a = Leaf | Node (Tree a) a (Tree a)+      deriving (Show, Generic)++    instance Arbitrary a => Arbitrary (Tree a) where+      arbitrary = genericArbitrary' @'Z++    -- Equivalent to+    -- > arbitrary =+    -- >   sized $ \n ->+    -- >     if n == 0 then+    -- >       return Leaf+    -- >     else+    -- >       oneof+    -- >         [ return Leaf+    -- >         , Node <$> arbitrary <*> arbitrary <*> arbitrary+    -- >         ]++    main = sample (arbitrary :: Gen (Tree ()))+```++- User-specified distribution of constructors.+- A simple (optional) strategy to ensure termination: `Test.QuickCheck.Gen`'s+  size parameter decreases at every recursive `genericArbitrary'` call; when it+  reaches zero, sample directly from a finite set of finite values.+- Uses `GHC.Generics` generics.+- Just for QuickCheck's `arbitrary`.+- More flexible than `Generic.Random.Data`'s Boltzmann samplers, which compute+  fixed weights for a given target size and concrete type, but with a less+  regular distribution.++`Generic.Random.Boltzmann`+--------------------------++An experimental interface to obtain Boltzmann samplers from an applicative+specification of a combinatorial system.++No documentation (yet).+ References ----------++Papers about Boltzmann samplers, used in `Generic.Random.Data`:  - The core theory of Boltzmann samplers is described in   [Boltzmann Samplers for the Random Generation of Combinatorial Structures](http://algo.inria.fr/flajolet/Publications/DuFlLoSc04.pdf),

@@ -14,9 +14,9 @@ import Test.QuickCheck.Gen import Test.QuickCheck.Random import Control.Exception ( evaluate )-import Data.Random.Generics-import Data.Random.Generics.Internal-import Data.Random.Generics.Internal.Types+import Generic.Random.Data+import Generic.Random.Internal.Data+import Generic.Random.Internal.Types  data T = N T T | L   deriving (Eq, Ord, Show, Data, Generic)

@@ -1,7 +1,7 @@ name:                generic-random-version:             0.1.1.0+version:             0.2.0.0 synopsis:            Generic random generators-description:         Please see the README below.+description:         Please see the README. homepage:            http://github.com/lysxia/generic-random license:             MIT license-file:        LICENSE@@ -12,16 +12,20 @@ build-type:          Simple extra-source-files:  README.md cabal-version:       >=1.10-tested-with:         GHC == 7.10.3+tested-with:         GHC == 8.0.1  library   hs-source-dirs:      src   exposed-modules:-    Data.Random.Generics-    Data.Random.Generics.Internal-    Data.Random.Generics.Internal.Oracle-    Data.Random.Generics.Internal.Solver-    Data.Random.Generics.Internal.Types+    Generic.Random.Boltzmann+    Generic.Random.Data+    Generic.Random.Generic+    Generic.Random.Internal.Common+    Generic.Random.Internal.Data+    Generic.Random.Internal.Generic+    Generic.Random.Internal.Oracle+    Generic.Random.Internal.Solver+    Generic.Random.Internal.Types   build-depends:     base >= 4.8 && < 5,     containers,

@@ -1,302 +0,0 @@--- | Generic Boltzmann samplers.------ Here, the words "/sampler/" and "/generator/" are used interchangeably.------ Given an algebraic datatype:------ > data A = A1 B C | A2 D------ a Boltzmann sampler is recursively defined by choosing a constructor with--- some fixed distribution, and /independently/ generating values for the--- corresponding fields with the same method.------ A key component is the aforementioned distribution, defined for every type--- such that the resulting generator produces a finite value in the end. These--- distributions are obtained from a precomputed object called /oracle/, which--- we will not describe further here.------ Oracles depend on the target size of the generated data (except for singular--- samplers), and can be fairly expensive to compute repeatedly, hence some of--- the functions below attempt to avoid (re)computing too many of them even--- when the required size changes.------ When these functions are specialized, oracles are memoized and will be--- reused for different sizes.--module Data.Random.Generics (-  Size',-  -- * Main functions-  -- $sized-  generatorSR,-  generatorP,-  generatorPR,-  generatorR,-  -- ** Fixed size-  -- $fixed-  generatorP',-  generatorPR',-  generatorR',-  generator',-  -- * Generators with aliases-  -- $aliases-  generatorSRWith,-  generatorPWith,-  generatorPRWith,-  generatorRWith,-  -- ** Fixed size-  generatorPWith',-  generatorPRWith',-  generatorRWith',-  generatorWith',-  -- * Other generators-  -- $other-  Points,-  generatorM,-  generatorMR,-  generator_,-  generatorR_,-  -- * Auxiliary definitions-  -- ** Type classes-  MonadRandomLike (..),-  AMonadRandom (..),-  -- ** Alias-  alias,-  aliasR,-  coerceAlias,-  coerceAliases,-  Alias (..),-  AliasR,-  ) where--import Data.Data-import Data.Random.Generics.Internal-import Data.Random.Generics.Internal.Types---- * Main functions---- $sized------ === Suffixes------ [@S@] Singular sampler.------     This works with recursive tree-like structures, as opposed to (lists of)---     structures with bounded size. More precisely, the generating function of---     the given type should have a finite radius of convergence, with a---     singularity of a certain kind (see Duchon et al., reference in the---     README), so that the oracle can be evaluated at that point.------     This has the advantage of using the same oracle for all size parameters,---     which simply specify a target size interval.------ [@P@] Generator of pointed values.------     It usually has a flatter distribution of sizes than a simple Boltzmann---     sampler, making it an efficient alternative to rejection sampling.------     It also works on more types, particularly lists and finite types,---     but relies on multiple oracles.------ [@R@] Rejection sampling.------     These generators filter out values whose sizes are not within some---     interval. In the first two sections, that interval is implicit:---     @[(1-'epsilon')*size', (1+'epsilon')*size']@, for @'epsilon' = 0.1@.------     The generator restarts as soon as it has produced more constructors than---     the upper bound, this strategy is called /ceiled rejection sampling/.------ = Pointing------ The /pointing/ of a type @t@ is a derived type whose values are essentially--- values of type @t@, with one of their constructors being "pointed".--- Alternatively, we may turn every constructor into variants that indicate--- the position of points.------ @---   -- Original type---   data Tree = Node Tree Tree | Leaf---   -- Pointing of Tree---   data Tree'---     = Tree' Tree -- Point at the root---     | Node'0 Tree' Tree -- Point to the left---     | Node'1 Tree Tree' -- Point to the right--- @------ Pointed values are easily mapped back to the original type by erasing the--- point. Pointing makes larger values occur much more frequently, while--- preserving the uniformness of the distribution conditionally to a fixed--- size.------- | @---   'generatorSR' :: Int -> 'Gen' a---   'asMonadRandom' . 'generatorSR' :: 'MonadRandom' m => Int -> m a--- @------ Singular ceiled rejection sampler.-generatorSR :: (Data a, MonadRandomLike m) => Size' -> m a-generatorSR = generatorSRWith []---- | @---   'generatorP' :: Int -> 'Gen' a---   'asMonadRandom' . 'generatorP' :: 'MonadRandom' m => Int -> m a--- @------ Generator of pointed values.--generatorP :: (Data a, MonadRandomLike m) => Size' -> m a-generatorP = generatorPWith []---- | Pointed generator with rejection.-generatorPR :: (Data a, MonadRandomLike m) => Size' -> m a-generatorPR = generatorPRWith []---- | Generator with rejection and dynamic average size.-generatorR :: (Data a, MonadRandomLike m) => Size' -> m a-generatorR = generatorRWith []---- ** Fixed size---- $fixed--- The @'@ suffix indicates functions which do not do any--- precomputation before passing the size parameter.------ This means that oracles are computed from scratch for every size value,--- which may incur a significant overhead.---- | Pointed generator.-generatorP' :: (Data a, MonadRandomLike m) => Size' -> m a-generatorP' = generatorPWith' []---- | Pointed generator with rejection.-generatorPR' :: (Data a, MonadRandomLike m) => Size' -> m a-generatorPR' = generatorPRWith' []---- | Ceiled rejection sampler with given average size.-generatorR' :: (Data a, MonadRandomLike m) => Size' -> m a-generatorR' = generatorRWith' []---- | Basic boltzmann sampler with no optimization.-generator' :: (Data a, MonadRandomLike m) => Size' -> m a-generator' = generatorWith' []---- * Generators with aliases---- $aliases--- Boltzmann samplers can normally be defined only for types @a@ such that:------ - they are instances of 'Data';--- - the set of types of subterms of values of type @a@ is finite;--- - and all of these types have at least one finite value (i.e., values with---   finitely many constructors).------ Examples of misbehaving types are:------ - @a -> b -- Not Data@--- - @data E a = L a | R (E [a]) -- Contains a, [a], [[a]], [[[a]]], etc.@--- - @data I = C I -- No finite value@------ = Alias------ The 'Alias' type works around these limitations ('AliasR' for rejection--- samplers).--- This existential wrapper around a user-defined function @f :: a -> m b@--- makes @generic-random@ view occurences of the type @b@ as @a@ when--- processing a recursive system of types, possibly stopping some infinite--- unrolling of type definitions. When a value of type @b@ needs to be--- generated, it generates an @a@ which is passed to @f@.------ @---   let---     as = ['aliasR' $ \\() -> return (L []) :: 'Gen' (E [[Int]])]---   in---     'generatorSRWith' as 'asGen' :: 'Size' -> 'Gen' (E Int)--- @------ Another use case is to plug in user-defined generators where the default is--- not satisfactory, for example, to get positive @Int@s:------ @---   let---     as = ['alias' $ \\() -> 'choose' (0, 100) :: 'Gen' Int)]---   in---     'generatorPWith' as 'asGen' :: 'Size' -> 'Gen' [Int]--- @--generatorSRWith-  :: (Data a, MonadRandomLike m) => [AliasR m] -> Size' -> m a-generatorSRWith aliases =-  generatorR_ aliases 0 Nothing . tolerance epsilon--generatorPRWith-  :: (Data a, MonadRandomLike m) => [AliasR m] -> Size' -> m a-generatorPRWith aliases size' =-  generatorMR aliases 1 size' (tolerance epsilon size')--generatorPWith-  :: (Data a, MonadRandomLike m) => [Alias m] -> Size' -> m a-generatorPWith aliases = generatorM aliases 1--generatorRWith-  :: (Data a, MonadRandomLike m) => [AliasR m] -> Size' -> m a-generatorRWith aliases size' =-  generatorMR aliases 0 size' (tolerance epsilon size')---- ** Fixed size--generatorPWith'-  :: (Data a, MonadRandomLike m) => [Alias m] -> Size' -> m a-generatorPWith' aliases = generator_ aliases 1 . Just--generatorPRWith'-  :: (Data a, MonadRandomLike m) => [AliasR m] -> Size' -> m a-generatorPRWith' aliases size' =-  generatorR_ aliases 1 (Just size') (tolerance epsilon size')--generatorRWith'-  :: (Data a, MonadRandomLike m) => [AliasR m] -> Size' -> m a-generatorRWith' aliases size' =-  generatorR_ aliases 0 (Just size') (tolerance epsilon size')--generatorWith'-  :: (Data a, MonadRandomLike m) => [Alias m] -> Size' -> m a-generatorWith' aliases = generator_ aliases 0 . Just---- * Other generators---- $other Used in the implementation of the generators above.--- These also allow to apply pointing more than once.------ === Suffixes------ [@M@] Sized generators are memoized for some sparsely chosen values of--- sizes. Subsequently supplied sizes are approximated by the closest larger--- value. This strategy avoids recomputing too many oracles. Aside from--- singular samplers, all other generators above not marked by @'@ use this.------ [@_@] If the size parameter is @Nothing@, produces the singular generator--- (associated with the suffix @S@); otherwise the generator produces values--- with average size equal to the given value.--generatorM-  :: (Data a, MonadRandomLike m)-  => [Alias m] -> Points -> Size' -> m a-generatorM = memo make apply--generatorMR-  :: (Data a, MonadRandomLike m)-  => [AliasR m] -> Points -> Size' -> (Size', Size') -> m a-generatorMR = memo makeR applyR---- | Boltzmann sampler without rejection.-generator_-  :: (Data a, MonadRandomLike m)-  => [Alias m] -> Points -> Maybe Size' -> m a-generator_ aliases = apply (make aliases [])---- | Boltzmann sampler with rejection.-generatorR_-  :: (Data a, MonadRandomLike m)-  => [AliasR m] -> Points -> Maybe Size' -> (Size', Size') -> m a-generatorR_ aliases = applyR (makeR aliases [])

@@ -1,146 +0,0 @@-{-# LANGUAGE RecordWildCards, DeriveFunctor #-}-module Data.Random.Generics.Internal where--import Control.Arrow ( (&&&) )-import Control.Applicative-import Data.Data-import Data.Foldable-import Data.Maybe-import qualified Data.HashMap.Lazy as HashMap-import Data.Random.Generics.Internal.Oracle-import Data.Random.Generics.Internal.Types---- | Sized generator.-data SG r = SG-  { minSize :: Size-  , maxSizeM :: Maybe Size-  , runSG :: Points -> Maybe Double -> r-  , runSmallG :: Points -> r-  } deriving Functor---- | Number of pointing iterations.-type Points = Int--rangeSG :: SG r -> (Size, Maybe Size)-rangeSG = minSize &&& maxSizeM---- | For documentation.-applySG :: SG r -> Points -> Maybe Double -> r-applySG SG{..} k sizeM-  | Just minSize == maxSizeM = runSG k (fmap fromIntegral maxSizeM)-  | Just size <- sizeM, size <= fromIntegral minSize =-      error "Target size too small."-  | Just True <- liftA2 ((<=) . fromIntegral) maxSizeM sizeM =-      error "Target size too large."-  | Nothing <- sizeM, Just _ <- maxSizeM =-      error "Cannot make singular sampler for finite type."-  | otherwise = runSG k sizeM---- * Helper functions--make :: (Data a, MonadRandomLike m)-  => [Alias m] -> proxy a -> SG (m a)-make aliases a =-  SG minSize maxSizeM make' makeSmall-  where-    dd = collectTypes aliases a-    t = typeRep a-    i = case index dd #! t of-      Left j -> fst (xedni' dd #! j)-      Right i -> i-    minSize = natToInt $ fst (lTerm dd #! i)-    maxSizeM = HashMap.lookup i (degree dd)-    make' k sizeM = getGenerator dd' generators a k-      where-        dd' = dds !! k-        oracle = makeOracle dd' t sizeM-        generators = makeGenerators dd' oracle-    makeSmall k = getSmallGenerator dd' (smallGenerators dd') a-      where-        dd' = dds !! k-    dds = iterate point dd--makeR :: (Data a, MonadRandomLike m)-  => [AliasR m] -> proxy a-  -> SG ((Size, Size) -> m a)-makeR aliases a = fmap (flip runRejectT) (make aliases a)---- | The size of a value is its number of constructors.------ Here, however, the 'Size'' type is interpreted differently to make better--- use of QuickCheck's size parameter provided by the 'Test.QuickCheck.sized'--- combinator, so that we generate non-trivial data even at very small size--- values.------ For infinite types, with objects of unbounded sizes @> minSize@, given a--- parameter @delta :: 'Size''@, the produced values have an average size close--- to @minSize + delta@.------ For example, values of type @Either () [Bool]@ have at least two constructors,--- so------ @---   'generator' delta :: 'Gen' (Either () [Bool])--- @------ will target sizes close to @2 + delta@;--- the offset becomes less noticeable as @delta@ grows to infinity.------ For finite types with sizes in @[minSize, maxSize]@, the target expected--- size is obtained by clamping a 'Size'' to @[0, 99]@ and applying an affine--- mapping.-type Size' = Int--rescale :: SG r -> Size' -> Double-rescale (SG minSize (Just maxSize) _ _) size' =-  fromIntegral minSize + fromIntegral (min 99 size' * (maxSize - minSize)) / 100-rescale (SG minSize Nothing _ _) size' = fromIntegral (minSize + size')--apply :: SG r -> Points -> Maybe Size' -> r-apply sg k (Just 0) = runSmallG sg k-apply sg k size' = runSG sg k (fmap (rescale sg) size')--applyR :: SG ((Size, Size) -> r) -> Points -> Maybe Size' -> (Size', Size') -> r-applyR sg k size' = apply sg k size' . rescaleInterval sg--rescaleInterval :: SG r -> (Size', Size') -> (Size, Size)-rescaleInterval sg (a', b') = (a, b)-  where-    a = (clamp . floor .rescale sg) a'-    b = (clamp . ceiling . rescale sg) b'-    clamp x-      | Just maxSize <- maxSizeM sg, x >= 100 = maxSize-      | otherwise = x---- | > 'epsilon' = 0.1------ Default approximation ratio.-epsilon :: Double-epsilon = 0.1---- | > (size * (1 - epsilon), size * (1 + epsilon))-tolerance :: Double -> Int -> (Int, Int)-tolerance epsilon size = (size - delta, size + delta)-  where-    delta = ceiling (fromIntegral size * epsilon)---- * Auxiliary definitions--memo-  :: (t -> [t2] -> SG r)-  -> (SG r -> t1 -> Maybe Int -> a)-  -> t -> t1 -> Int -> a-memo make apply aliases k = generators-  where-    sg = make aliases []-    generators = sparseSized (apply sg k . Just) (99 <$ maxSizeM sg)---- Oracles are computed only for sizes that are a power of two away from--- the minimum size of the datatype @minSize + 2 ^ e@.-sparseSized :: (Int -> a) -> Maybe Int -> Int -> a-sparseSized f maxSizeM =-  maybe a0 snd . \size' -> find ((>= size') . fst) as-  where-    as = [ (s, f s) | s <- ss ]-    ss = 0 : maybe id (takeWhile . (>)) maxSizeM [ 2 ^ e | e <- [ 0 :: Int ..] ]-    a0 = f (fromJust maxSizeM)

@@ -1,539 +0,0 @@-{-# LANGUAGE FlexibleContexts, GADTs, RankNTypes, ScopedTypeVariables #-}-{-# LANGUAGE DeriveGeneric, ImplicitParams #-}-{-# LANGUAGE RecordWildCards, DeriveDataTypeable #-}-module Data.Random.Generics.Internal.Oracle where--import Control.Applicative-import Control.Monad-import Control.Monad.Fix-import Control.Monad.Reader-import Control.Monad.State-import Data.Bifunctor-import Data.Data-import Data.Hashable ( Hashable )-import Data.HashMap.Lazy ( HashMap )-import qualified Data.HashMap.Lazy as HashMap-import Data.Maybe ( fromJust, isJust )-import Data.Monoid-import qualified Data.Vector as V-import qualified Data.Vector.Storable as S-import GHC.Generics ( Generic )-import Numeric.AD-import Data.Random.Generics.Internal.Types-import Data.Random.Generics.Internal.Solver---- | We build a dictionary which reifies type information in order to--- create a Boltzmann generator.------ We denote by @n@ (or 'count') the number of types in the dictionary.------ Every type has an index @0 <= i < n@; the variable @X i@ represents its--- generating function @C_i(x)@, and @X (i + k*n)@ the GF of its @k@-th--- "pointing" @C_i[k](x)@; we have------ @---   C_i[0](x) = C_i(x)---   C_i[k+1](x) = x * C_i[k]'(x)--- @------ where @C_i[k]'@ is the derivative of @C_i[k]@. See also 'point'.------ The /order/ (or /valuation/) of a power series is the index of the first--- non-zero coefficient, called the /leading coefficient/.--data DataDef m = DataDef-  { count :: Int -- ^ Number of registered types-  , points :: Int -- ^ Number of iterations of the pointing operator-  , index :: HashMap TypeRep (Either Aliased Ix) -- ^ Map from types to indices-  , xedni :: HashMap Ix SomeData' -- ^ Inverse map from indices to types-  , xedni' :: HashMap Aliased (Ix, Alias m) -- ^ Inverse map to aliases-  , types :: HashMap C [(Integer, Constr, [C'])]-  -- ^ Structure of types and their pointings (up to 'points', initially 0)-  ---  -- Primitive types and empty types are mapped to an empty constructor list, and-  -- can be distinguished using 'Data.Data.dataTypeRep' on the 'SomeData'-  -- associated to it by 'xedni'.-  ---  -- The integer is a multiplicity which can be > 1 for pointings.-  , lTerm :: HashMap Ix (Nat, Integer)-  -- ^ Leading term @a * x ^ u@ of the generating functions @C_i[k](x)@ in the-  -- form (u, a).-  ---  -- [Order @u@] Smallest size of objects of a given type.-  -- [Leading coefficient @a@] number of objects of smallest size.-  , degree :: HashMap Ix Int-  -- ^ Degrees of the generating functions, when applicable: greatest size of-  -- objects of a given type.-  } deriving Show---- | A pair @C i k@ represents the @k@-th "pointing" of the type at index @i@,--- with generating function @C_i[k](x)@.-data C = C Ix Int-  deriving (Eq, Ord, Show, Generic)--instance Hashable C--data AC = AC Aliased Int-  deriving (Eq, Ord, Show, Generic)--instance Hashable AC--type C' = (Maybe Aliased, C)--newtype Aliased = Aliased Int-  deriving (Eq, Ord, Show, Generic)--instance Hashable Aliased--type Ix = Int--data Nat = Zero | Succ Nat-  deriving (Eq, Ord, Show)--instance Monoid Nat where-  mempty = Zero-  mappend (Succ n) = Succ . mappend n-  mappend Zero = id--natToInt :: Nat -> Int-natToInt Zero = 0-natToInt (Succ n) = 1 + natToInt n--infinity :: Nat-infinity = Succ infinity--dataDef :: [Alias m] -> DataDef m-dataDef as = DataDef-  { count = 0-  , points = 0-  , index = index-  , xedni = HashMap.empty-  , xedni' = xedni'-  , types = HashMap.empty-  , lTerm = HashMap.empty-  , degree = HashMap.empty-  } where-    xedni' = HashMap.fromList (fmap (\(i, a) -> (i, (-1, a))) as')-    index = HashMap.fromList (fmap (\(i, a) -> (ofType a, Left i)) as')-    as' = zip (fmap Aliased [0 ..]) as-    ofType (Alias f) = typeRep (f undefined)---- | Find all types that may be types of subterms of a value of type @a@.------ This will loop if there are infinitely many such types.-collectTypes :: Data a => [Alias m] -> proxy a -> DataDef m-collectTypes as a = collectTypesM a `execState` dataDef as---- | Primitive datatypes have @C(x) = x@: they are considered as--- having a single object (@lCoef@) of size 1 (@order@)).-primOrder :: Int-primOrder = 1--primOrder' :: Nat-primOrder' = Succ Zero--primlCoef :: Integer-primlCoef = 1---- | The type of the first argument of 'Data.Data.gunfold'.-type GUnfold m = forall b r. Data b => m (b -> r) -> m r---- | Type of 'xedni''.-type AMap m = HashMap Aliased (Ix, Alias m)--collectTypesM :: Data a => proxy a-  -> State (DataDef m) (Either Aliased Ix, ((Nat, Integer), Maybe Int))-collectTypesM a = chaseType a (const id)--chaseType :: Data a => proxy a-  -> ((Maybe (Alias m), Ix) -> AMap m -> AMap m)-  -> State (DataDef m) (Either Aliased Ix, ((Nat, Integer), Maybe Int))-chaseType a k = do-  let t = typeRep a-  dd@DataDef{..} <- get-  let-    lookup i r =-      let-        lTerm_i = lTerm #! i-        degree_i = HashMap.lookup i degree-      in return (r, (lTerm_i, degree_i))-  case HashMap.lookup t index of-    Nothing -> do-      let i = count-      put dd-        { count = i + 1-        , index = HashMap.insert t (Right i) index-        , xedni = HashMap.insert i (someData' a) xedni-        , xedni' = k (Nothing, i) xedni'-        }-      traverseType a i -- Updates lTerm and degree-    Just (Right i) -> do-      put dd { xedni' = k (Nothing, i) xedni' }-      lookup i (Right i)-    Just (Left j) ->-      case xedni' #! j of-        (-1, Alias f) -> do-          (_, ld) <- chaseType (ofType f) $ \(alias, i) ->-            let-              alias' = case alias of-                Nothing -> Alias f-                Just (Alias g) -> Alias (composeCastM f g)-            in-            k (Just alias', i) . HashMap.insert j (i, alias')-          return (Left j, ld)-        (i, _) -> lookup i (Left j)-  where-    ofType :: (m a -> m b) -> m a-    ofType _ = undefined---- | Traversal of the definition of a datatype.-traverseType-  :: Data a => proxy a -> Ix-  -> State (DataDef m) (Either Aliased Ix, ((Nat, Integer), Maybe Int))-traverseType a i = do-  let d = withProxy dataTypeOf a-  mfix $ \ ~(_, (lTerm_i0, _)) -> do-    modify $ \dd@DataDef{..} -> dd-      { lTerm = HashMap.insert i lTerm_i0 lTerm-      }-    (types_i, ld@(_, degree_i)) <- traverseType' a d-    modify $ \dd@DataDef{..} -> dd-      { types = HashMap.insert (C i 0) types_i types-      , degree = maybe id (HashMap.insert i) degree_i degree-      }-    return (Right i, ld)--traverseType'-  :: Data a => proxy a -> DataType-  -> State (DataDef m)-      ([(Integer, Constr, [(Maybe Aliased, C)])], ((Nat, Integer), Maybe Int))-traverseType' a d | isAlgType d = do-  let-    constrs = dataTypeConstrs d-    collect-      :: GUnfold (StateT-        ([Either Aliased Ix], (Nat, Integer), Maybe Int)-        (State (DataDef m)))-    collect mkCon = do-      f <- mkCon-      let ofType :: (b -> a) -> Proxy b-          ofType _ = Proxy-          b = ofType f-      (j, (lTerm_, degree_)) <- lift (collectTypesM b)-      modify $ \(js, lTerm', degree') ->-        (j : js, lMul lTerm_ lTerm', liftA2 (+) degree_ degree')-      return (withProxy f b)-  tlds <- forM constrs $ \constr -> do-    (js, lTerm', degree') <--      gunfold collect return constr `proxyType` a-        `execStateT` ([], (Zero, 1), Just 1)-    dd <- get-    let-      c (Left j) = (Just j, C (fst (xedni' dd #! j)) 0)-      c (Right i) = (Nothing, C i 0)-    return ((1, constr, [ c j | j <- js]), lTerm', degree')-  let-    (types_i, ls, ds) = unzip3 tlds-    lTerm_i = first Succ (lSum ls)-    degree_i = maxDegree ds-  return (types_i, (lTerm_i, degree_i))-traverseType' _ _ =-  return ([], ((primOrder', primlCoef), Just primOrder))---- | If @(u, a)@ represents a power series of leading term @a * x ^ u@, and--- similarly for @(u', a')@, this finds the leading term of their sum.------ The comparison of 'Nat' is unrolled here for maximum laziness.-lPlus :: (Nat, Integer) -> (Nat, Integer) -> (Nat, Integer)-lPlus (Zero, lCoef) (Zero, lCoef') = (Zero, lCoef + lCoef')-lPlus (Zero, lCoef) _ = (Zero, lCoef)-lPlus _ (Zero, lCoef') = (Zero, lCoef')-lPlus (Succ order, lCoef) (Succ order', lCoef') =-  first Succ $ lPlus (order, lCoef) (order', lCoef')---- | Sum of a list of series.-lSum :: [(Nat, Integer)] -> (Nat, Integer)-lSum [] = (infinity, 0)-lSum ls = foldl1 lPlus ls---- | Leading term of a product of series.-lMul :: (Nat, Integer) -> (Nat, Integer) -> (Nat, Integer)-lMul (order, lCoef) (order', lCoef') = (order <> order', lCoef * lCoef')--lProd :: [(Nat, Integer)] -> (Nat, Integer)-lProd = foldl lMul (Zero, 1)--maxDegree :: [Maybe Int] -> Maybe Int-maxDegree = foldl (liftA2 max) (Just minBound)---- | Pointing operator.------ Populates a 'DataDef' with one more level of pointings.--- ('collectTypes' produces a dictionary at level 0.)------ The "pointing" of a type @t@ is a derived type whose values are essentially--- values of type @t@, with one of their constructors being "pointed".--- Alternatively, we may turn every constructor into variants that indicate--- the position of points.------ @---   -- Original type---   data Tree = Node Tree Tree | Leaf---   -- Pointing of Tree---   data Tree'---     = Tree' Tree -- Point at the root---     | Node'0 Tree' Tree -- Point to the left---     | Node'1 Tree Tree' -- Point to the right---   -- Pointing of the pointing---   -- Notice that the "points" introduced by both applications of pointing---   -- are considered different: exchanging their positions (when different)---   -- produces a different tree.---   data Tree''---     = Tree'' Tree' -- Point 2 at the root, the inner Tree' places point 1---     | Node'0' Tree' Tree -- Point 1 at the root, point 2 to the left---     | Node'1' Tree Tree' -- Point 1 at the root, point 2 to the right---     | Node'0'0 Tree'' Tree -- Points 1 and 2 to the left---     | Node'0'1 Tree' Tree' -- Point 1 to the left, point 2 to the right---     | Node'1'0 Tree' Tree' -- Point 1 to the right, point 2 to the left---     | Node'0'1 Tree Tree'' -- Points 1 and 2 to the right--- @------ If we ignore points, some constructors are equivalent. Thus we may simply--- calculate their multiplicity instead of duplicating them.------ Given a constructor with @c@ arguments @C x_1 ... x_c@, and a sequence--- @p_0 + p_1 + ... + p_c = k@ corresponding to a distribution of @k@ points--- (@p_0@ are assigned to the constructor @C@ itself, and for @i > 0@, @p_i@--- points are assigned within the @i@-th subterm), the multiplicity of the--- constructor paired with that distribution is the multinomial coefficient--- @multinomial k [p_1, ..., p_c]@.--point :: DataDef m -> DataDef m-point dd@DataDef{..} = dd-  { points = points'-  , types = foldl g types [0 .. count-1]-  } where-    points' = points + 1-    g types i = HashMap.insert (C i points') (types' i) types-    types' i = types #! C i 0 >>= h-    h (_, constr, js) = do-      ps <- partitions points' (length js)-      let-        mult = multinomial points' ps-        js' = zipWith (\(j', C i _) p -> (j', C i p)) js ps-      return (mult, constr, js')---- | An oracle gives the values of the generating functions at some @x@.-type Oracle = HashMap C Double---- | Find the value of @x@ such that the average size of the generator--- for the @k-1@-th pointing is equal to @size@, and produce the associated--- oracle. If the size is @Nothing@, find the radius of convergence.------ The search evaluates the generating functions for some values of @x@ in--- order to run a binary search. The evaluator is implemented using Newton's--- method, the convergence of which has been shown for relevant systems in--- /Boltzmann Oracle for Combinatorial Systems/,--- C. Pivoteau, B. Salvy, M. Soria.-makeOracle :: DataDef m -> TypeRep -> Maybe Double -> Oracle-makeOracle dd0 t size' =-  seq v-  HashMap.fromList (zip cs (S.toList v))-  where-    -- We need the next pointing to capture the average size in an equation.-    dd@DataDef{..} = if isJust size' then point dd0 else dd0-    cs = flip C <$> [0 .. points] <*> [0 .. count - 1]-    m = count * (points + 1)-    k = points - 1-    i = case index #! t of-      Left j -> fst (xedni' #! j)-      Right i -> i-    checkSize _ (Just ys) | S.any (< 0) ys = False-    -- There may be solutions outside of the radius-    -- of convergence, but with negative components.-    checkSize (Just size) (Just ys) =-      size >= size_-      where-        size_ = ys S.! j' / ys S.! j-        j = dd ? C i k-        j' = dd ? C i (k + 1)-    checkSize Nothing (Just _) = True-    checkSize _ Nothing = False-    -- Equations defining C_i(x) for all types with indices i-    phis :: Num a => V.Vector (a -> V.Vector a -> a)-    phis = V.fromList [ phi dd c (types #! c) | c <- listCs dd ]-    eval' :: Double -> Maybe (S.Vector Double)-    eval' x = fixedPoint defSolveArgs phi' (S.replicate m 0)-      where-        phi' :: (Mode a, Scalar a ~ Double) => V.Vector a -> V.Vector a-        phi' y = fmap (\f -> f (auto x) y) phis-    v = fromJust (search eval' (checkSize size'))---- | Generating function definition. This defines a @Phi_i[k]@ function--- associated with the @k@-th pointing of the type at index @i@, such that:------ > C_i[k](x)--- >   = Phi_i[k](x, C_0[0](x), ..., C_(n-1)[0](x),--- >              ..., C_0[k](x), ..., C_(n-1)[k](x))------ Primitive datatypes have @C(x) = x@: they are considered as--- having a single object ('lCoef') of size 1 ('order')).-phi :: Num a => DataDef m -> C -> [(Integer, constr, [C'])]-  -> a -> V.Vector a -> a-phi DataDef{..} (C i _) [] =-  case xedni #! i of-    SomeData a ->-      case (dataTypeRep . withProxy dataTypeOf) a of-        AlgRep _ -> \_ _ -> 0-        _ -> \x _ -> fromInteger primlCoef * x ^ primOrder-phi dd@DataDef{..} _ tyInfo = f-  where-    f x y = x * (sum . fmap (toProd y)) tyInfo-    toProd y (w, _, js) =-      fromInteger w * product [ y V.! (dd ? j) | (_, j) <- js ]---- | Maps a key representing a type @a@ (or one of its pointings) to a--- generator @m a@.-type Generators m = (HashMap AC (SomeData m), HashMap C (SomeData m))---- | Build all involved generators at once.-makeGenerators-  :: forall m. MonadRandomLike m-  => DataDef m -> Oracle -> Generators m-makeGenerators DataDef{..} oracle =-  seq oracle-  (generatorsL, generatorsR)-  where-    f (C i _) tyInfo = case xedni #! i of-      SomeData a -> SomeData $ incr >>-        case tyInfo of-          [] -> defGen-          _ -> frequencyWith doubleR (fmap g tyInfo) `proxyType` a-    g :: Data a => (Integer, Constr, [C']) -> (Double, m a)-    g (v, constr, js) =-      ( fromInteger v * w-      , gunfold generate return constr `runReaderT` gs)-      where-        gs = fmap (\(j', i) -> m j' i) js-        m = maybe (generatorsR #!) m'-        m' j (C _ k) = (generatorsL #! AC j k)-        w = product $ fmap ((oracle #!) . snd) js-    h (j, (i, Alias f)) k =-      (AC j k, applyCast f (generatorsR #! C i k))-    generatorsL = HashMap.fromList (liftA2 h (HashMap.toList xedni') [0 .. points])-    generatorsR = HashMap.mapWithKey f types--type SmallGenerators m =-  (HashMap Aliased (SomeData m), HashMap Ix (SomeData m))---- | Generators of values of minimal sizes.-smallGenerators-  :: forall m. MonadRandomLike m => DataDef m -> SmallGenerators m-smallGenerators DataDef{..} = (generatorsL, generatorsR)-  where-    f i (SomeData a) = SomeData $ incr >>-      case types #! C i 0 of-        [] -> defGen-        tyInfo ->-          let gs = (tyInfo >>= g (fst (lTerm #! i))) in-          frequencyWith integerR gs `proxyType` a-    g :: Data a => Nat -> (Integer, Constr, [C']) -> [(Integer, m a)]-    g minSize (_, constr, js) =-      guard (minSize == Succ size) *>-      [(weight, gunfold generate return constr `runReaderT` gs)]-      where-        (size, weight) = lProd [ lTerm #! i | (_, C i _) <- js ]-        gs = fmap lookup js-        lookup (j', C i _) = maybe (generatorsR #! i) (generatorsL #!) j'-    h (j, (i, Alias f)) = (j, applyCast f (generatorsR #! i))-    generatorsL = (HashMap.fromList . fmap h . HashMap.toList) xedni'-    generatorsR = HashMap.mapWithKey f xedni--generate :: Applicative m => GUnfold (ReaderT [SomeData m] m)-generate rest = ReaderT $ \(g : gs) ->-  rest `runReaderT` gs <*> unSomeData g--defGen :: (Data a, MonadRandomLike m) => m a-defGen = gen-  where-    gen =-      let dt = withProxy dataTypeOf gen in-      case dataTypeRep dt of-        IntRep -> fromConstr . mkIntegralConstr dt <$> int-        FloatRep -> fromConstr . mkRealConstr dt <$> double-        CharRep -> fromConstr . mkCharConstr dt <$> char-        AlgRep _ -> error "Cannot generate for empty type."-        NoRep -> error "No representation."---- * Short operators--(?) :: DataDef m -> C -> Int-dd ? C i k = i + k * count dd---- | > dd ? (listCs dd !! i) = i-listCs :: DataDef m -> [C]-listCs dd = liftA2 (flip C) [0 .. points dd] [0 .. count dd - 1]--ix :: C -> Int-ix (C i _) = i---- | > dd ? (dd ?! i) = i-(?!) :: DataDef m -> Int -> C-dd ?! j = C i k-  where (k, i) = j `divMod` count dd--getGenerator :: (Functor m, Data a)-  => DataDef m -> Generators m -> proxy a -> Int -> m a-getGenerator dd (l, r) a k = unSomeData $-  case index dd #! typeRep a of-    Right i -> (r #! C i k)-    Left j -> (l #! AC j k)--getSmallGenerator :: (Functor m, Data a)-  => DataDef m -> SmallGenerators m -> proxy a -> m a-getSmallGenerator dd (l, r) a = unSomeData $-  case index dd #! typeRep a of-    Right i -> (r #! i)-    Left j -> (l #! j)---- * General helper functions--frequencyWith-  :: (Show r, Ord r, Num r, Monad m) => (r -> m r) -> [(r, m a)] -> m a-frequencyWith _ [(_, a)] = a-frequencyWith randomR as = randomR total >>= select as-  where-    total = (sum . fmap fst) as-    select ((w, a) : as) x-      | x < w = a-      | otherwise = select as (x - w)-    select _ _ = (snd . head) as-    -- That should not happen in theory, but floating point might be funny.--(#!) :: (Eq k, Hashable k)-  => HashMap k v -> k -> v-(#!) = (HashMap.!)---- | @partitions k n@: lists of non-negative integers of length @n@ with sum--- less than or equal to @k@.-partitions :: Int -> Int -> [[Int]]-partitions _ 0 = [[]]-partitions k n = do-  p <- [0 .. k]-  (p :) <$> partitions (k - p) (n - 1)---- | Multinomial coefficient.------ > multinomial n ps == factorial n `div` product [factorial p | p <- ps]-multinomial :: Int -> [Int] -> Integer-multinomial _ [] = 1-multinomial n (p : ps) = binomial n p * multinomial (n - p) ps---- | Binomial coefficient.------ > binomial n k == factorial n `div` (factorial k * factorial (n-k))-binomial :: Int -> Int -> Integer-binomial = \n k -> pascal !! n !! k-  where-    pascal = [1] : fmap nextRow pascal-    nextRow r = zipWith (+) (0 : r) (r ++ [0])

@@ -1,65 +0,0 @@--- | Solve systems of equations--{-# LANGUAGE RecordWildCards #-}-{-# LANGUAGE RankNTypes, FlexibleContexts, TypeFamilies #-}-module Data.Random.Generics.Internal.Solver where--import Control.Applicative-import Data.AEq ( (~==) )-import Numeric.AD.Mode-import Numeric.AD.Mode.Forward-import Numeric.LinearAlgebra-import qualified Data.Vector as V-import qualified Data.Vector.Storable as S--data SolveArgs = SolveArgs-  { accuracy :: Double-  , numIterations :: Int-  } deriving (Eq, Ord, Show)--defSolveArgs :: SolveArgs-defSolveArgs = SolveArgs 1e-8 20--findZero-  :: SolveArgs-  -> (forall s. V.Vector (AD s (Forward R)) -> V.Vector (AD s (Forward R)))-  -> Vector R-  -> Maybe (Vector R)-findZero SolveArgs{..} f = newton numIterations-  where-    newton 0 _ = Nothing-    newton n x-      | norm_y == 1/0 = Nothing-      | norm_y > accuracy = newton (n - 1) (x - jacobian <\> y)-      | otherwise = Just x-      where-        norm_y = norm_Inf y-        jacobian = (fromRows . V.toList . fmap (V.convert . snd)) yj-        y = (V.convert . fmap fst) yj-        yj = jacobian' f (S.convert x)--fixedPoint-  :: SolveArgs-  -> (forall a. (Mode a, Scalar a ~ R) => V.Vector a -> V.Vector a)-  -> Vector R-  -> Maybe (Vector R)-fixedPoint args f = findZero args (liftA2 (V.zipWith (-)) f id)---- | Assuming @p . f@ is satisfied only for positive values in some interval--- @(0, r]@, find @f r@.-search :: (Double -> a) -> (a -> Bool) -> a-search f p = search' e0 (0 : [2 ^ n | n <- [0 .. 100 :: Int]])-  where-    search' y (x : xs@(x' : _))-      | p y' = search' y' xs-      | otherwise = search'' y x x'-      where y' = f x'-    search' _ _ = error "Solution not found. Uncontradictable predicate?"-    search'' y x x'-      | x ~== x' = y-      | p y_ = search'' y_ x_ x'-      | otherwise = search'' y x x_-      where-        x_ = (x + x') / 2-        y_ = f x_-    e0 = error "Solution not found. Unsatisfiable predicate?"

@@ -1,191 +0,0 @@-{-# LANGUAGE RankNTypes, GADTs, ScopedTypeVariables, ImplicitParams #-}-{-# LANGUAGE TypeOperators, GeneralizedNewtypeDeriving #-}-module Data.Random.Generics.Internal.Types where--import Control.Monad.Random-import Control.Monad.Trans-import Data.Coerce-import Data.Data-import Data.Function-import Test.QuickCheck--data SomeData m where-  SomeData :: Data a => m a -> SomeData m--type SomeData' = SomeData Proxy---- | Dummy instance for debugging.-instance Show (SomeData m) where-  show _ = "SomeData"--data Alias m where-  Alias :: (Data a, Data b) => !(m a -> m b) -> Alias m--type AliasR m = Alias (RejectT m)---- | Dummy instance for debugging.-instance Show (Alias m) where-  show _ = "Alias"---- | Main constructor for 'Alias'.-alias :: (Monad m, Data a, Data b) => (a -> m b) -> Alias m-alias = Alias . (=<<)---- | Main constructor for 'AliasR'.-aliasR :: (Monad m, Data a, Data b) => (a -> m b) -> AliasR m-aliasR = Alias . (=<<) . fmap lift---- | > coerceAlias :: Alias m -> Alias (AMonadRandom m)-coerceAlias :: Coercible m n => Alias m -> Alias n-coerceAlias = coerce---- | > coerceAliases :: [Alias m] -> [Alias (AMonadRandom m)]-coerceAliases :: Coercible m n => [Alias m] -> [Alias n]-coerceAliases = coerce---- | > composeCast f g = f . g-composeCastM :: forall a b c d m-  . (Typeable b, Typeable c)-  => (m c -> d) -> (a -> m b) -> (a -> d)-composeCastM f g | Just Refl <- eqT :: Maybe (b :~: c) = f . g-composeCastM _ _ = castError ([] :: [b]) ([] :: [c])--castM :: forall a b m-  . (Typeable a, Typeable b)-  => m a -> m b-castM a | Just Refl <- eqT :: Maybe (a :~: b) = a-castM a = let x = castError a x in x--unSomeData :: Typeable a => SomeData m -> m a-unSomeData (SomeData a) = castM a--applyCast :: (Typeable a, Data b) => (m a -> m b) -> SomeData m -> SomeData m-applyCast f = SomeData . f . unSomeData--castError :: (Typeable a, Typeable b)-  => proxy a -> proxy' b -> c-castError a b = error $ unlines-  [ "Error trying to cast"-  , "  " ++ show (typeRep a)-  , "to"-  , "  " ++ show (typeRep b)-  ]--withProxy :: (a -> b) -> proxy a -> b-withProxy f _ =-  f (error "This should not be evaluated\n")--reproxy :: proxy a -> Proxy a-reproxy _ = Proxy--proxyType :: m a -> proxy a -> m a-proxyType = const--someData' :: Data a => proxy a -> SomeData'-someData' = SomeData . reproxy---- | Size as the number of constructors.-type Size = Int---- | Internal transformer for rejection sampling.------ > ReaderT Size (StateT Size (MaybeT m)) a-newtype RejectT m a = RejectT-  { unRejectT :: forall r. Size -> Size -> m r -> (Size -> a -> m r) -> m r-  }--instance Functor (RejectT m) where-  fmap f (RejectT go) = RejectT $ \maxSize size retry cont ->-    go maxSize size retry $ \size a -> cont size (f a)--instance Applicative (RejectT m) where-  pure a = RejectT $ \_maxSize size _retry cont ->-    cont size a-  RejectT f <*> RejectT x = RejectT $ \maxSize size retry cont ->-    f maxSize size retry $ \size f_ ->-      x maxSize size retry $ \size x_ ->-        cont size (f_ x_)--instance Monad (RejectT m) where-  RejectT x >>= f = RejectT $ \maxSize size retry cont ->-    x maxSize size retry $ \size x_ ->-      unRejectT (f x_) maxSize size retry cont--instance MonadTrans RejectT where-  lift m = RejectT $ \_maxSize size _retry cont ->-    m >>= cont size---- | Set lower bound-runRejectT :: Monad m => (Size, Size) -> RejectT m a -> m a-runRejectT (minSize, maxSize) (RejectT m) = fix $ \go ->-  m maxSize 0 go $ \size a ->-    if size < minSize then-      go-    else-      return a---runRejectT (minSize, maxSize) (RejectT m) = fix $ \go -> do---  x' <- runMaybeT (m `runReaderT` maxSize `runStateT` 0)---  case x' of---    Just (x, size) | size >= minSize -> return x---    _ -> go--newtype AMonadRandom m a = AMonadRandom-  { asMonadRandom :: m a-  } deriving (Functor, Applicative, Monad)--instance MonadTrans AMonadRandom where-  lift = AMonadRandom---- ** Dictionaries---- | @'MonadRandomLike' m@ defines basic components to build generators,--- allowing the implementation to remain abstract over both the--- 'Test.QuickCheck.Gen' type and 'MonadRandom' instances.------ For the latter, the wrapper 'AMonadRandom' is provided to avoid--- overlapping instances.-class Monad m => MonadRandomLike m where-  -- | Called for every constructor. Counter for ceiled rejection sampling.-  incr :: m ()-  incr = return ()--  -- | @doubleR upperBound@: generates values in @[0, upperBound]@.-  doubleR :: Double -> m Double--  -- | @integerR upperBound@: generates values in @[0, upperBound-1]@.-  integerR :: Integer -> m Integer--  -- | Default @Int@ generator.-  int :: m Int--  -- | Default @Double@ generator.-  double :: m Double--  -- | Default @Char@ generator.-  char :: m Char--instance MonadRandomLike Gen where-  doubleR x = choose (0, x)-  integerR x = choose (0, x-1)-  int = arbitrary-  double = arbitrary-  char = arbitrary--instance MonadRandomLike m => MonadRandomLike (RejectT m) where-  incr = RejectT $ \maxSize size retry cont ->-    if size >= maxSize then-      retry-    else-      cont (size + 1) ()-  doubleR = lift . doubleR-  integerR = lift . integerR-  int = lift int-  double = lift double-  char = lift char--instance MonadRandom m => MonadRandomLike (AMonadRandom m) where-  doubleR x = lift $ getRandomR (0, x)-  integerR x = lift $ getRandomR (0, x-1)-  int = lift getRandom-  double = lift getRandom-  char = lift getRandom

@@ -0,0 +1,215 @@+-- | Applicative interface to define recursive structures and derive Boltzmann+-- samplers.+--+-- Given the recursive structure of the types, and how to combine generators,+-- the library takes care of computing the oracles and setting the right+-- distributions.++{-# LANGUAGE FlexibleContexts, FlexibleInstances, GADTs, RankNTypes, ScopedTypeVariables #-}+{-# LANGUAGE DeriveFunctor, DeriveGeneric, ImplicitParams #-}+{-# LANGUAGE RecordWildCards, DeriveDataTypeable #-}+{-# LANGUAGE TypeFamilies, MultiParamTypeClasses #-}+{-# LANGUAGE TypeApplications #-}+module Generic.Random.Boltzmann where++import Control.Applicative+import Control.Monad+import Data.Bifunctor+import Data.Coerce+import Data.Function+import Data.Foldable+import Data.List+import Data.Maybe+import Data.Vector ( Vector )+import qualified Data.Vector as V+import qualified Numeric.AD as AD+import Generic.Random.Internal.Common+import Generic.Random.Internal.Solver+import Generic.Random.Internal.Types++class Embed f m where+  emap :: (m a -> m b) -> f a -> f b+  -- | A natural transformation between @f@ and @m@?+  embed :: m a -> f a++-- | 'Applicative' defines a product, 'Alternative' defines an addition,+-- with scalar multiplication we get a module.+--+-- This typeclass allows to directly tweak weights in the oracle by+-- chosen factors.+class (Alternative f, Num (Scalar f)) => Module f where+  type Scalar f :: *++  -- | Scalar embedding.+  scalar :: Scalar f -> f ()+  scalar x = x <.> pure ()++  -- | Scalar multiplication.+  (<.>) :: Scalar f -> f a -> f a+  x <.> f = scalar x *> f++infixr 3 <.>++type Endo a = a -> a++data System f a c = System+  { dim :: Int+  , sys' :: f () -> Vector (f a) -> (Vector (f a), c)+  } deriving (Functor)++sys :: System f a c -> f () -> Vector (f a) -> Vector (f a)+sys = (fmap . fmap . fmap) fst sys'++newtype ConstModule r a = ConstModule { unConstModule :: r }++instance Functor (ConstModule r) where+  fmap _ (ConstModule r) = ConstModule r++instance Num r => Embed (ConstModule r) m where+  emap _ (ConstModule r) = ConstModule r+  embed _ = ConstModule 1++instance Num r => Applicative (ConstModule r) where+  pure _ = ConstModule 1+  ConstModule x <*> ConstModule y = ConstModule (x * y)++instance Num r => Alternative (ConstModule r) where+  empty = ConstModule 0+  ConstModule x <|> ConstModule y = ConstModule (x + y)++instance Num r => Module (ConstModule r) where+  type Scalar (ConstModule r) = r+  scalar = ConstModule+  x <.> ConstModule r = ConstModule (x * r)++solve+  :: forall b c+  . (forall a. Num a => System (ConstModule a) b c)+  -> Double -> Maybe (Vector Double)+solve s x = fixedPoint defSolveArgs phi' (V.replicate (dim (s @Int)) 0)+  where+    phi' :: forall a. (AD.Mode a, AD.Scalar a ~ Double) => Endo (Vector a)+    phi' = coerce (sys s (scalar (AD.auto x)) :: Endo (Vector (ConstModule a b)))++sizedGenerator+  :: forall b c m+  . MonadRandomLike m+  => (forall f. (Module f, Embed f m) => System (Pointiful f) b c)+  -> Int  -- ^ Index of type+  -> Int  -- ^ Points+  -> Maybe Double  -- ^ Expected size (or singular sampler)+  -> m b+sizedGenerator s i k size' = fst (sfix s' x oracle) V.! j+  where+    (x, oracle) = solveSized s i k size'+    s' = point (k + 1) s+    j = i * (k + 2) + k++solveSized+  :: forall b c+  . (forall a. Num a => System (Pointiful (ConstModule a)) b c)+  -> Int  -- ^ Index of type+  -> Int  -- ^ Points+  -> Maybe Double  -- ^ Expected size (or singular sampler)+  -> (Double, Vector Double)+solveSized s i k size' =+  fmap fromJust (search (solve s') (checkSize size'))+  where+    s' :: forall a. Num a => System (ConstModule a) b c+    s' = point (k + 1) s+    j = i * (k + 2) + k+    j' = i * (k + 2) + k + 1+    checkSize _ (Just ys) | V.any (< 0) ys = False+    checkSize (Just size) (Just ys) = size >= ys V.! j' / ys V.! j+    checkSize Nothing (Just _) = True+    checkSize _ Nothing = False++newtype Weighted m a = Weighted [(Double, m a)]++weighted :: Double -> m a -> Weighted m a+weighted x a = Weighted [(x, a)]++runWeighted :: MonadRandomLike m => Weighted m a -> (Double, m a)+runWeighted (Weighted [a]) = a+runWeighted (Weighted as) = (sum (fmap fst as), frequencyWith doubleR as)++instance Functor m => Functor (Weighted m) where+  fmap f (Weighted as) = Weighted ((fmap . fmap . fmap) f as)++instance MonadRandomLike m => Embed (Weighted m) m where+  emap f = Weighted . (: []) . fmap f . runWeighted+  embed m = Weighted [(1, m)]++instance MonadRandomLike m => Applicative (Weighted m) where+  pure a = Weighted [(1, pure a)]+  f' <*> a' = Weighted [(u * v, f <*> a)]+    where+      (u, f) = runWeighted f'+      (v, a) = runWeighted a'++instance MonadRandomLike m => Alternative (Weighted m) where+  empty = Weighted []+  Weighted as <|> Weighted bs = Weighted (as ++ bs)++instance MonadRandomLike m => Module (Weighted m) where+  type Scalar (Weighted m) = Double+  scalar x = Weighted [(x, pure ())]+  x <.> Weighted as = Weighted (fmap (first (x *)) as)++sfix+  :: MonadRandomLike m+  => System (Weighted m) b c -> Double -> Vector Double -> (Vector (m b), c)+sfix s x oracle =+  fix $+    (first . fmap) (snd . runWeighted) .+    sys' s (scalar x) .+    V.zipWith weighted oracle .+    fst++data Pointiful f a = Pointiful [f a] | Zero (f a)++instance Functor f => Functor (Pointiful f) where+  fmap f (Pointiful v) = Pointiful ((fmap . fmap) f v)+  fmap f (Zero x) = Zero (fmap f x)++instance Embed f m => Embed (Pointiful f) m where+  emap f (Pointiful v) = Pointiful ((fmap . emap) f v)+  emap f (Zero x) = Zero (emap f x)+  embed = Zero . embed++instance Module f => Applicative (Pointiful f) where+  pure a = Zero (pure a)+  Zero f <*> Zero x = Zero (f <*> x)+  Zero f <*> Pointiful xs = Pointiful (fmap (f <*>) xs)+  Pointiful fs <*> Zero x = Pointiful (fmap (<*> x) fs)+  Pointiful fs <*> Pointiful xs = Pointiful (convolute fs xs)+    where+      convolute fs xs = zipWith3 sumOfProducts [0 ..] (inits' fs) (inits' xs)+      inits' = tail . inits+      sumOfProducts k f x = asum (zipWith3 (times k) [0 ..] f (reverse x))+      times k k1 f x = fromInteger (binomial k k1) <.> f <*> x++instance Module f => Alternative (Pointiful f) where+  empty = Zero empty+  Pointiful xs <|> Pointiful ys = Pointiful (zipWith (<|>) xs ys)+  Pointiful (x : xs) <|> Zero y = Pointiful ((x <|> y) : xs)+  Zero x <|> Pointiful (y : ys) = Pointiful ((x <|> y) : ys)+  Zero x <|> Zero y = Zero (x <|> y)+  Pointiful [] <|> m = m+  m <|> Pointiful [] = m++instance Module f => Module (Pointiful f) where+  type Scalar (Pointiful f) = Scalar f+  scalar = Zero . scalar++unPointiful :: Alternative f => Pointiful f a -> [f a]+unPointiful (Pointiful as) = as+unPointiful (Zero a) = a : repeat empty++point :: Module f => Int -> System (Pointiful f) b c -> System f b c+point k s = System ((k + 1) * dim s) $ \x ->+  first flatten . sys' s (Pointiful (repeat x)) . resize+  where+    flatten = join . fmap (V.fromList . take (k + 1) . unPointiful)+    resize v = V.generate (dim s) $ \i ->+      Pointiful [v V.! j | j <- [i * (k + 1) .. i * (k + 1) + k]]

@@ -0,0 +1,313 @@+-- | Generic Boltzmann samplers.+--+-- Here, the words "/sampler/" and "/generator/" are used interchangeably.+--+-- Given an algebraic datatype:+--+-- > data A = A1 B C | A2 D+--+-- a Boltzmann sampler is recursively defined by choosing a constructor with+-- some fixed distribution, and /independently/ generating values for the+-- corresponding fields with the same method.+--+-- A key component is the aforementioned distribution, defined for every type+-- such that the resulting generator produces a finite value in the end. These+-- distributions are obtained from a precomputed object called /oracle/, which+-- we will not describe further here.+--+-- Oracles depend on the target size of the generated data (except for singular+-- samplers), and can be fairly expensive to compute repeatedly, hence some of+-- the functions below attempt to avoid (re)computing too many of them even+-- when the required size changes.+--+-- When these functions are specialized, oracles are memoized and will be+-- reused for different sizes.++module Generic.Random.Data (+  Size',+  -- * Main functions+  -- $sized+  generatorSR,+  generatorP,+  generatorPR,+  generatorR,+  -- ** Fixed size+  -- $fixed+  generatorP',+  generatorPR',+  generatorR',+  generator',+  -- * Generators with aliases+  -- $aliases+  generatorSRWith,+  generatorPWith,+  generatorPRWith,+  generatorRWith,+  -- ** Fixed size+  generatorPWith',+  generatorPRWith',+  generatorRWith',+  generatorWith',+  -- * Other generators+  -- $other+  Points,+  generatorM,+  generatorMR,+  generator_,+  generatorR_,+  -- * Auxiliary definitions+  -- ** Type classes+  MonadRandomLike (..),+  AMonadRandom (..),+  -- ** Alias+  alias,+  aliasR,+  coerceAlias,+  coerceAliases,+  Alias (..),+  AliasR,+  ) where++import Data.Data+import Generic.Random.Internal.Data+import Generic.Random.Internal.Types++-- * Main functions++-- $sized+--+-- === Suffixes+--+-- [@S@] Singular sampler.+--+--     This works with recursive tree-like structures, as opposed to (lists of)+--     structures with bounded size. More precisely, the generating function of+--     the given type should have a finite radius of convergence, with a+--     singularity of a certain kind (see Duchon et al., reference in the+--     README), so that the oracle can be evaluated at that point.+--+--     This has the advantage of using the same oracle for all size parameters,+--     which simply specify a target size interval.+--+-- [@P@] Generator of pointed values.+--+--     It usually has a flatter distribution of sizes than a simple Boltzmann+--     sampler, making it an efficient alternative to rejection sampling.+--+--     It also works on more types, particularly lists and finite types,+--     but relies on multiple oracles.+--+-- [@R@] Rejection sampling.+--+--     These generators filter out values whose sizes are not within some+--     interval. In the first two sections, that interval is implicit:+--     @[(1-'epsilon')*size', (1+'epsilon')*size']@, for @'epsilon' = 0.1@.+--+--     The generator restarts as soon as it has produced more constructors than+--     the upper bound, this strategy is called /ceiled rejection sampling/.+--+-- = Pointing+--+-- The /pointing/ of a type @t@ is a derived type whose values are essentially+-- values of type @t@, with one of their constructors being "pointed".+-- Alternatively, we may turn every constructor into variants that indicate+-- the position of points.+--+-- @+--   -- Original type+--   data Tree = Node Tree Tree | Leaf+--   -- Pointing of Tree+--   data Tree'+--     = Tree' Tree -- Point at the root+--     | Node'0 Tree' Tree -- Point to the left+--     | Node'1 Tree Tree' -- Point to the right+-- @+--+-- Pointed values are easily mapped back to the original type by erasing the+-- point. Pointing makes larger values occur much more frequently, while+-- preserving the uniformness of the distribution conditionally to a fixed+-- size.+--++-- | @+--   'generatorSR' :: Int -> 'Gen' a+--   'asMonadRandom' . 'generatorSR' :: 'MonadRandom' m => Int -> m a+-- @+--+-- Singular ceiled rejection sampler.+generatorSR :: (Data a, MonadRandomLike m) => Size' -> m a+generatorSR = generatorSRWith []++-- | @+--   'generatorP' :: Int -> 'Gen' a+--   'asMonadRandom' . 'generatorP' :: 'MonadRandom' m => Int -> m a+-- @+--+-- Generator of pointed values.++generatorP :: (Data a, MonadRandomLike m) => Size' -> m a+generatorP = generatorPWith []++-- | Pointed generator with rejection.+generatorPR :: (Data a, MonadRandomLike m) => Size' -> m a+generatorPR = generatorPRWith []++-- | Generator with rejection and dynamic average size.+generatorR :: (Data a, MonadRandomLike m) => Size' -> m a+generatorR = generatorRWith []++-- ** Fixed size++-- $fixed+-- The @'@ suffix indicates functions which do not do any+-- precomputation before passing the size parameter.+--+-- This means that oracles are computed from scratch for every size value,+-- which may incur a significant overhead.++-- | Pointed generator.+generatorP' :: (Data a, MonadRandomLike m) => Size' -> m a+generatorP' = generatorPWith' []++-- | Pointed generator with rejection.+generatorPR' :: (Data a, MonadRandomLike m) => Size' -> m a+generatorPR' = generatorPRWith' []++-- | Ceiled rejection sampler with given average size.+generatorR' :: (Data a, MonadRandomLike m) => Size' -> m a+generatorR' = generatorRWith' []++-- | Basic boltzmann sampler with no optimization.+generator' :: (Data a, MonadRandomLike m) => Size' -> m a+generator' = generatorWith' []++-- * Generators with aliases++-- $aliases+-- Boltzmann samplers can normally be defined only for types @a@ such that:+--+-- - they are instances of 'Data';+-- - the set of types of subterms of values of type @a@ is finite;+-- - and all of these types have at least one finite value (i.e., values with+--   finitely many constructors).+--+-- Examples of misbehaving types are:+--+-- - @a -> b -- Not Data@+-- - @data E a = L a | R (E [a]) -- Contains a, [a], [[a]], [[[a]]], etc.@+-- - @data I = C I -- No finite value@+--+-- = Alias+--+-- The 'Alias' type works around these limitations ('AliasR' for rejection+-- samplers).+-- This existential wrapper around a user-defined function @f :: a -> m b@+-- makes @generic-random@ view occurences of the type @b@ as @a@ when+-- processing a recursive system of types, possibly stopping some infinite+-- unrolling of type definitions. When a value of type @b@ needs to be+-- generated, it generates an @a@ which is passed to @f@.+--+-- @+--   let+--     as = ['aliasR' $ \\() -> return (L []) :: 'Gen' (E [[Int]])]+--   in+--     'generatorSRWith' as 'asGen' :: 'Size' -> 'Gen' (E Int)+-- @+--+-- Another use case is to plug in user-defined generators where the default is+-- not satisfactory, for example, to generate positive @Int@s:+--+-- @+--   let+--     as = ['alias' $ \\() -> 'choose' (0, 100) :: 'Gen' Int)]+--   in+--     'generatorPWith' as 'asGen' :: 'Size' -> 'Gen' [Int]+-- @+--+-- or to modify the weights assigned to some types. In particular, in some+-- cases it seems preferable to make @String@ (and @Text@) have the same weight+-- as @Int@ and @()@.+--+-- @+--   let+--     as = ['alias' $ \\() -> arbitrary :: 'Gen' String]+--   in+--     'generatorPWith' as 'asGen' :: 'Size' -> 'Gen' (Either Int String)+-- @++generatorSRWith+  :: (Data a, MonadRandomLike m) => [AliasR m] -> Size' -> m a+generatorSRWith aliases =+  generatorR_ aliases 0 Nothing . tolerance epsilon++generatorPRWith+  :: (Data a, MonadRandomLike m) => [AliasR m] -> Size' -> m a+generatorPRWith aliases size' =+  generatorMR aliases 1 size' (tolerance epsilon size')++generatorPWith+  :: (Data a, MonadRandomLike m) => [Alias m] -> Size' -> m a+generatorPWith aliases = generatorM aliases 1++generatorRWith+  :: (Data a, MonadRandomLike m) => [AliasR m] -> Size' -> m a+generatorRWith aliases size' =+  generatorMR aliases 0 size' (tolerance epsilon size')++-- ** Fixed size++generatorPWith'+  :: (Data a, MonadRandomLike m) => [Alias m] -> Size' -> m a+generatorPWith' aliases = generator_ aliases 1 . Just++generatorPRWith'+  :: (Data a, MonadRandomLike m) => [AliasR m] -> Size' -> m a+generatorPRWith' aliases size' =+  generatorR_ aliases 1 (Just size') (tolerance epsilon size')++generatorRWith'+  :: (Data a, MonadRandomLike m) => [AliasR m] -> Size' -> m a+generatorRWith' aliases size' =+  generatorR_ aliases 0 (Just size') (tolerance epsilon size')++generatorWith'+  :: (Data a, MonadRandomLike m) => [Alias m] -> Size' -> m a+generatorWith' aliases = generator_ aliases 0 . Just++-- * Other generators++-- $other Used in the implementation of the generators above.+-- These also allow to apply pointing more than once.+--+-- === Suffixes+--+-- [@M@] Sized generators are memoized for some sparsely chosen values of+-- sizes. Subsequently supplied sizes are approximated by the closest larger+-- value. This strategy avoids recomputing too many oracles. Aside from+-- singular samplers, all other generators above not marked by @'@ use this.+--+-- [@_@] If the size parameter is @Nothing@, produces the singular generator+-- (associated with the suffix @S@); otherwise the generator produces values+-- with average size equal to the given value.++generatorM+  :: (Data a, MonadRandomLike m)+  => [Alias m] -> Points -> Size' -> m a+generatorM = memo make apply++generatorMR+  :: (Data a, MonadRandomLike m)+  => [AliasR m] -> Points -> Size' -> (Size', Size') -> m a+generatorMR = memo makeR applyR++-- | Boltzmann sampler without rejection.+generator_+  :: (Data a, MonadRandomLike m)+  => [Alias m] -> Points -> Maybe Size' -> m a+generator_ aliases = apply (make aliases [])++-- | Boltzmann sampler with rejection.+generatorR_+  :: (Data a, MonadRandomLike m)+  => [AliasR m] -> Points -> Maybe Size' -> (Size', Size') -> m a+generatorR_ aliases = applyR (makeR aliases [])

@@ -0,0 +1,30 @@+-- | Simple 'GHC.Generics'-based 'arbitrary' generators.+--+-- Here is an example. Define your type.+--+-- > data Tree a = Leaf a | Node (Tree a) (Tree a)+--+-- Derive 'GHC.Generics.Generic'.+--+-- >   deriving 'Generic'  -- Turn on the DeriveGeneric extension+--+-- Pick an arbitrary implementation.+--+-- > instance Arbitrary a => Arbitrary (Tree a) where+-- >   arbitrary = genericArbitraryFrequency [9, 8]+--+-- @arbitrary :: 'Gen' (Tree a)@ picks a @Leaf@ with probability 9\/17, or a+-- @Node@ with probability 8\/17, and recursively fills their fields with+-- @arbitrary@.++module Generic.Random.Generic+  ( genericArbitrary+  , genericArbitraryFrequency+  , genericArbitraryFrequency'+  , genericArbitrary'+  , Nat (..)+  , BaseCases'+  , BaseCases+  ) where++import Generic.Random.Internal.Generic

@@ -0,0 +1,39 @@+-- | General helper functions++module Generic.Random.Internal.Common where++frequencyWith+  :: (Ord r, Num r, Monad m) => (r -> m r) -> [(r, m a)] -> m a+frequencyWith _ [(_, a)] = a+frequencyWith randomR as = randomR total >>= select as+  where+    total = (sum . fmap fst) as+    select ((w, a) : as) x+      | x < w = a+      | otherwise = select as (x - w)+    select _ _ = (snd . head) as+    -- That should not happen in theory, but floating point might be funny.++-- | @partitions k n@: lists of non-negative integers of length @n@ with sum+-- less than or equal to @k@.+partitions :: Int -> Int -> [[Int]]+partitions _ 0 = [[]]+partitions k n = do+  p <- [0 .. k]+  (p :) <$> partitions (k - p) (n - 1)++-- | Binomial coefficient.+--+-- > binomial n k == factorial n `div` (factorial k * factorial (n-k))+binomial :: Int -> Int -> Integer+binomial = \n k -> pascal !! n !! k+  where+    pascal = [1] : fmap nextRow pascal+    nextRow r = zipWith (+) (0 : r) (r ++ [0])++-- | Multinomial coefficient.+--+-- > multinomial n ps == factorial n `div` product [factorial p | p <- ps]+multinomial :: Int -> [Int] -> Integer+multinomial _ [] = 1+multinomial n (p : ps) = binomial n p * multinomial (n - p) ps

@@ -0,0 +1,146 @@+{-# LANGUAGE RecordWildCards, DeriveFunctor #-}+module Generic.Random.Internal.Data where++import Control.Arrow ( (&&&) )+import Control.Applicative+import Data.Data+import Data.Foldable+import Data.Maybe+import qualified Data.HashMap.Lazy as HashMap+import Generic.Random.Internal.Oracle+import Generic.Random.Internal.Types++-- | Sized generator.+data SG r = SG+  { minSize :: Size+  , maxSizeM :: Maybe Size+  , runSG :: Points -> Maybe Double -> r+  , runSmallG :: Points -> r+  } deriving Functor++-- | Number of pointing iterations.+type Points = Int++rangeSG :: SG r -> (Size, Maybe Size)+rangeSG = minSize &&& maxSizeM++-- | For documentation.+applySG :: SG r -> Points -> Maybe Double -> r+applySG SG{..} k sizeM+  | Just minSize == maxSizeM = runSG k (fmap fromIntegral maxSizeM)+  | Just size <- sizeM, size <= fromIntegral minSize =+      error "Target size too small."+  | Just True <- liftA2 ((<=) . fromIntegral) maxSizeM sizeM =+      error "Target size too large."+  | Nothing <- sizeM, Just _ <- maxSizeM =+      error "Cannot make singular sampler for finite type."+  | otherwise = runSG k sizeM++-- * Helper functions++make :: (Data a, MonadRandomLike m)+  => [Alias m] -> proxy a -> SG (m a)+make aliases a =+  SG minSize maxSizeM make' makeSmall+  where+    dd = collectTypes aliases a+    t = typeRep a+    i = case index dd #! t of+      Left j -> fst (xedni' dd #! j)+      Right i -> i+    minSize = natToInt $ fst (lTerm dd #! i)+    maxSizeM = HashMap.lookup i (degree dd)+    make' k sizeM = getGenerator dd' generators a k+      where+        dd' = dds !! k+        oracle = makeOracle dd' t sizeM+        generators = makeGenerators dd' oracle+    makeSmall k = getSmallGenerator dd' (smallGenerators dd') a+      where+        dd' = dds !! k+    dds = iterate point dd++makeR :: (Data a, MonadRandomLike m)+  => [AliasR m] -> proxy a+  -> SG ((Size, Size) -> m a)+makeR aliases a = fmap (flip runRejectT) (make aliases a)++-- | The size of a value is its number of constructors.+--+-- Here, however, the 'Size'' type is interpreted differently to make better+-- use of QuickCheck's size parameter provided by the 'Test.QuickCheck.sized'+-- combinator, so that we generate non-trivial data even at very small size+-- values.+--+-- For infinite types, with objects of unbounded sizes @> minSize@, given a+-- parameter @delta :: 'Size''@, the produced values have an average size close+-- to @minSize + delta@.+--+-- For example, values of type @Either () [Bool]@ have at least two constructors,+-- so+--+-- @+--   'generator' delta :: 'Gen' (Either () [Bool])+-- @+--+-- will target sizes close to @2 + delta@;+-- the offset becomes less noticeable as @delta@ grows to infinity.+--+-- For finite types with sizes in @[minSize, maxSize]@, the target expected+-- size is obtained by clamping a 'Size'' to @[0, 99]@ and applying an affine+-- mapping.+type Size' = Int++rescale :: SG r -> Size' -> Double+rescale (SG minSize (Just maxSize) _ _) size' =+  fromIntegral minSize + fromIntegral (min 99 size' * (maxSize - minSize)) / 100+rescale (SG minSize Nothing _ _) size' = fromIntegral (minSize + size')++apply :: SG r -> Points -> Maybe Size' -> r+apply sg k (Just 0) = runSmallG sg k+apply sg k size' = runSG sg k (fmap (rescale sg) size')++applyR :: SG ((Size, Size) -> r) -> Points -> Maybe Size' -> (Size', Size') -> r+applyR sg k size' = apply sg k size' . rescaleInterval sg++rescaleInterval :: SG r -> (Size', Size') -> (Size, Size)+rescaleInterval sg (a', b') = (a, b)+  where+    a = (clamp . floor .rescale sg) a'+    b = (clamp . ceiling . rescale sg) b'+    clamp x+      | Just maxSize <- maxSizeM sg, x >= 100 = maxSize+      | otherwise = x++-- | > 'epsilon' = 0.1+--+-- Default approximation ratio.+epsilon :: Double+epsilon = 0.1++-- | > (size * (1 - epsilon), size * (1 + epsilon))+tolerance :: Double -> Int -> (Int, Int)+tolerance epsilon size = (size - delta, size + delta)+  where+    delta = ceiling (fromIntegral size * epsilon)++-- * Auxiliary definitions++memo+  :: (t -> [t2] -> SG r)+  -> (SG r -> t1 -> Maybe Int -> a)+  -> t -> t1 -> Int -> a+memo make apply aliases k = generators+  where+    sg = make aliases []+    generators = sparseSized (apply sg k . Just) (99 <$ maxSizeM sg)++-- Oracles are computed only for sizes that are a power of two away from+-- the minimum size of the datatype @minSize + 2 ^ e@.+sparseSized :: (Int -> a) -> Maybe Int -> Int -> a+sparseSized f maxSizeM =+  maybe a0 snd . \size' -> find ((>= size') . fst) as+  where+    as = [ (s, f s) | s <- ss ]+    ss = 0 : maybe id (takeWhile . (>)) maxSizeM [ 2 ^ e | e <- [ 0 :: Int ..] ]+    a0 = f (fromJust maxSizeM)

@@ -0,0 +1,286 @@+{-# LANGUAGE FlexibleContexts, FlexibleInstances, MultiParamTypeClasses #-}+{-# LANGUAGE TypeApplications, TypeOperators #-}+{-# LANGUAGE DeriveFunctor, GeneralizedNewtypeDeriving #-}+{-# LANGUAGE AllowAmbiguousTypes, ScopedTypeVariables #-}+{-# LANGUAGE DataKinds, KindSignatures #-}+{-# LANGUAGE ConstraintKinds #-}+module Generic.Random.Internal.Generic where++import Control.Applicative+import Data.Coerce+import GHC.Generics hiding ( S )+import Test.QuickCheck++-- * Random generators++-- | Pick a constructor with uniform probability, and fill its fields+-- recursively.+--+-- An equivalent definition for @Tree@ is:+--+-- > genericArbitrary :: Arbitrary a => Gen (Tree a)+-- > genericArbitrary =+-- >   oneof+-- >     [ Leaf <$> arbitrary                -- Uses Arbitrary a+-- >     , Node <$> arbitrary <*> arbitrary  -- Uses Arbitrary (Tree a)+-- >     ]+--+-- Note that for many types, 'genericArbitrary' tends to produce big values.+-- For instance for @Tree a@ values are finite but the average number of+-- @Leaf@ and @Node@ constructors is infinite.++genericArbitrary :: (Generic a, GA Unsized (Rep a)) => Gen a+genericArbitrary = ($ repeat 1) . unFreq . fmap to $ ga @Unsized+++-- | This allows to specify the probability distribution of constructors+-- as a list of weights, in the same order as the data type definition.+--+-- An equivalent definition for @Tree@ is:+--+-- > genericArbitraryFrequency :: Arbitrary a => [Int] -> Gen (Tree a)+-- > genericArbitraryFrequency [x, y] =+-- >   frequency+-- >     [ (x, Leaf <$> arbitrary)+-- >     , (y, Node <$> arbitrary <*> arbitrary)+-- >     ]++genericArbitraryFrequency+  :: (Generic a, GA Unsized (Rep a))+  => [Int]  -- ^ List of weights for every constructor+  -> Gen a+genericArbitraryFrequency = unFreq . fmap to $ ga @Unsized+++-- | The size parameter of 'Gen' is divided among the fields of the chosen+-- constructor.  When it reaches zero, the generator selects a finite term+-- whenever it can find any of the given type.+--+-- The type of 'genericArbitraryFrequency'' has an ambiguous @n@ parameter; it+-- is a type-level natural number of type 'Nat'. That number determines the+-- maximum /depth/ of terms that can be used to end recursion.+--+-- You'll need the @TypeApplications@ and @DataKinds@ extensions.+--+-- > genericArbitraryFrequency' @n weights+--+-- With @n ~ ''Z'@, the generator looks for a simple nullary constructor.  If none+-- exist at the current type, as is the case for our @Tree@ type, it carries on+-- as in 'genericArbitraryFrequency'.+--+-- > genericArbitraryFrequency' @'Z :: Arbitrary a => [Int] -> Gen (Tree a)+-- > genericArbitraryFrequency' @'Z [x, y] =+-- >   frequency+-- >     [ (x, Leaf <$> arbitrary)+-- >     , (y, scale (`div` 2) $ Node <$> arbitrary <*> arbitrary)+-- >     ]+-- >     -- 2 because Node is 2-ary.+--+-- Here is another example:+--+-- > data Tree' = Leaf1 | Leaf2 | Node3 Tree' Tree' Tree'+-- >   deriving Generic+-- >+-- > instance Arbitrary Tree' where+-- >   arbitrary = genericArbitraryFrequency' @'Z [1, 2, 3]+--+-- 'genericArbitraryFrequency'' is equivalent to:+--+-- > genericArbitraryFrequency' @'Z :: [Int] -> Gen Tree'+-- > genericArbitraryFrequency' @'Z [x, y, z] =+-- >   sized $ \n ->+-- >     if n == 0 then+-- >       -- If the size parameter is zero, the non-nullary alternative is discarded.+-- >       frequency $+-- >         [ (x, return Leaf1)+-- >         , (y, return Leaf2)+-- >         ]+-- >     else+-- >       frequency $+-- >         [ (x, return Leaf1)+-- >         , (y, return Leaf2)+-- >         , (z, resize (n `div` 3) node)+-- >         ]+-- >         -- 3 because Node3 is 3-ary+-- >   where+-- >     node = Node3 <$> arbitrary <*> arbitrary <*> arbitrary+--+-- To increase the chances of termination when no nullary constructor is directly+-- available, such as in @Tree@, we can pass a larger depth @n@. The effectiveness+-- of this parameter depends on the concrete type the generator is used for.+--+-- For instance, if we want to generate a value of type @Tree ()@, there is a+-- value of depth 1 (represented by @''S' ''Z'@) that we can use to end+-- recursion: @Leaf ()@.+--+-- > genericArbitraryFrequency' @('S 'Z) :: [Int] -> Gen (Tree ())+-- > genericArbitraryFrequency' @('S 'Z) [x, y] =+-- >   sized $ \n ->+-- >     if n == 0 then+-- >       return (Leaf ())+-- >     else+-- >       frequency+-- >         [ (x, Leaf <$> arbitrary)+-- >         , (y, scale (`div` 2) $ Node <$> arbitrary <*> arbitrary)+-- >         ]+--+-- Because the argument of @Tree@ must be inspected in order to discover+-- values of type @Tree ()@, we incur some extra constraints if we want+-- polymorphism.+--+-- @FlexibleContexts@ and @UndecidableInstances@ are also required.+--+-- > instance (Arbitrary a, Generic a, BaseCases 'Z (Rep a))+-- >   => Arbitrary (Tree a) where+-- >   arbitrary = genericArbitraryFrequency' @('S 'Z) [1, 2]+--+-- A synonym is provided for brevity.+--+-- > instance (Arbitrary a, BaseCases' 'Z a) => Arbitrary (Tree a) where+-- >   arbitrary = genericArbitraryFrequency' @('S 'Z) [1, 2]++genericArbitraryFrequency'+  :: forall (n :: Nat) a+  . (Generic a, GA (Sized n) (Rep a))+  => [Int]  -- ^ List of weights for every constructor+  -> Gen a+genericArbitraryFrequency' = unFreq . fmap to $ ga @(Sized n)+++-- | Like 'genericArbitraryFrequency'', but with uniformly distributed+-- constructors.++genericArbitrary'+  :: forall (n :: Nat) a. (Generic a, GA (Sized n) (Rep a)) => Gen a+genericArbitrary' = ($ repeat 1) . unFreq . fmap to $ ga @(Sized n)+++-- * Internal++newtype Freq sized a = Freq { unFreq :: [Int] -> Gen a }+  deriving Functor++instance Applicative (Freq sized) where+  pure = Freq . pure . pure+  Freq f <*> Freq x = Freq (liftA2 (<*>) f x)++newtype Gen' sized a = Gen' { unGen' :: Gen a }+  deriving (Functor, Applicative)++data Sized :: Nat -> *+data Unsized++liftGen :: Gen a -> Freq sized a+liftGen = Freq . const++-- | Generic Arbitrary+class GA sized f where+  ga :: Freq sized (f p)++instance GA sized U1 where+  ga = pure U1++instance Arbitrary c => GA sized (K1 i c) where+  ga = liftGen . fmap K1 $ arbitrary++instance GA sized f => GA sized (M1 i c f) where+  ga = fmap M1 ga++instance (GASum (Sized n) f, GASum (Sized n) g, BaseCases n f, BaseCases n g)+  => GA (Sized n) (f :+: g) where+  ga = frequency' gaSum baseCases+    where+      frequency' :: [Gen' sized a] -> Tagged n [[a]] -> Freq sized a+      frequency' as (Tagged a0s) = Freq $ \ws ->+        let+          units = [(w, elements a0) | (w, a0@(_ : _)) <- zip ws a0s]+        in+          sized $ \sz -> frequency $+            if sz == 0 && not (null units) then+              units+            else+              [(w, a) | (w, Gen' a) <- zip ws as]++instance (GASum Unsized f, GASum Unsized g) => GA Unsized (f :+: g) where+  ga = frequency' gaSum+    where+      frequency' :: [Gen' sized a] -> Freq sized a+      frequency' as = Freq $ \ws -> frequency+        [(w, a) | (w, Gen' a) <- zip ws as]++instance (GA Unsized f, GA Unsized g) => GA Unsized (f :*: g) where+  ga = liftA2 (:*:) ga ga++instance (GAProduct f, GAProduct g) => GA (Sized n) (f :*: g) where+  ga = constScale' a+    where+      constScale' :: Gen' Unsized a -> Freq (Sized n) a+      constScale' = Freq . const . scale (`div` arity) . unGen'+      (arity, a) = gaProduct+++gArbitrarySingle :: forall sized f p . GA sized f => Gen' sized (f p)+gArbitrarySingle = Gen' (unFreq (ga :: Freq sized (f p)) [1])+++class GASum sized f where+  gaSum :: [Gen' sized (f p)]++instance (GASum sized f, GASum sized g) => GASum sized (f :+: g) where+  gaSum = (fmap . fmap) L1 gaSum ++ (fmap . fmap) R1 gaSum++instance GA sized f => GASum sized (M1 i c f) where+  gaSum = [gArbitrarySingle]+++class GAProduct f where+  gaProduct :: (Int, Gen' Unsized (f p))++instance GA Unsized f => GAProduct (M1 i c f) where+  gaProduct = (1, gArbitrarySingle)++instance (GAProduct f, GAProduct g) => GAProduct (f :*: g) where+  gaProduct = (m + n, liftA2 (:*:) a b)+    where+      (m, a) = gaProduct+      (n, b) = gaProduct+++newtype Tagged (a :: Nat) b = Tagged { unTagged :: b }++-- | Peano-encoded natural numbers.+data Nat = Z | S Nat++-- | A @BaseCases n ('Rep' a)@ constraint basically provides the list of values+-- of type @a@ with depth at most @n@.+class BaseCases (n :: Nat) f where+  baseCases :: Tagged n [[f p]]++-- | For convenience.+type BaseCases' n a = (Generic a, BaseCases n (Rep a))++baseCases' :: forall n f p. BaseCases n f => Tagged n [f p]+baseCases' = (Tagged . concat . unTagged) (baseCases @n)++instance BaseCases n U1 where+  baseCases = Tagged [[U1]]++instance BaseCases n f => BaseCases n (M1 i c f) where+  baseCases = (coerce :: Tagged n [[f p]] -> Tagged n [[M1 i c f p]]) baseCases++instance BaseCases 'Z (K1 i c) where+  baseCases = Tagged [[]]++instance (Generic c, BaseCases n (Rep c)) => BaseCases ('S n) (K1 i c) where+  baseCases = (Tagged . (fmap . fmap) (K1 . to) . unTagged) (baseCases @n)++instance (BaseCases n f, BaseCases n g) => BaseCases n (f :+: g) where+  baseCases = Tagged $+    (fmap . fmap) L1 (unTagged (baseCases @n)) +++    (fmap . fmap) R1 (unTagged (baseCases @n))++instance (BaseCases n f, BaseCases n g) => BaseCases n (f :*: g) where+  baseCases = Tagged+    [ liftA2 (:*:)+        (unTagged (baseCases' @n))+        (unTagged (baseCases' @n)) ]

@@ -0,0 +1,499 @@+{-# LANGUAGE FlexibleContexts, GADTs, RankNTypes, ScopedTypeVariables #-}+{-# LANGUAGE DeriveGeneric, ImplicitParams #-}+{-# LANGUAGE RecordWildCards, DeriveDataTypeable #-}+module Generic.Random.Internal.Oracle where++import Control.Applicative+import Control.Monad+import Control.Monad.Fix+import Control.Monad.Reader+import Control.Monad.State+import Data.Bifunctor+import Data.Data+import Data.Hashable ( Hashable )+import Data.HashMap.Lazy ( HashMap )+import qualified Data.HashMap.Lazy as HashMap+import Data.Maybe ( fromJust, isJust )+import Data.Monoid+import qualified Data.Vector as V+import GHC.Generics ( Generic )+import Numeric.AD+import Generic.Random.Internal.Common+import Generic.Random.Internal.Solver+import Generic.Random.Internal.Types++-- | We build a dictionary which reifies type information in order to+-- create a Boltzmann generator.+--+-- We denote by @n@ (or 'count') the number of types in the dictionary.+--+-- Every type has an index @0 <= i < n@; the variable @X i@ represents its+-- generating function @C_i(x)@, and @X (i + k*n)@ the GF of its @k@-th+-- "pointing" @C_i[k](x)@; we have+--+-- @+--   C_i[0](x) = C_i(x)+--   C_i[k+1](x) = x * C_i[k]'(x)+-- @+--+-- where @C_i[k]'@ is the derivative of @C_i[k]@. See also 'point'.+--+-- The /order/ (or /valuation/) of a power series is the index of the first+-- non-zero coefficient, called the /leading coefficient/.++data DataDef m = DataDef+  { count :: Int -- ^ Number of registered types+  , points :: Int -- ^ Number of iterations of the pointing operator+  , index :: HashMap TypeRep (Either Aliased Ix) -- ^ Map from types to indices+  , xedni :: HashMap Ix SomeData' -- ^ Inverse map from indices to types+  , xedni' :: HashMap Aliased (Ix, Alias m) -- ^ Inverse map to aliases+  , types :: HashMap C [(Integer, Constr, [C'])]+  -- ^ Structure of types and their pointings (up to 'points', initially 0)+  --+  -- Primitive types and empty types are mapped to an empty constructor list, and+  -- can be distinguished using 'Data.Data.dataTypeRep' on the 'SomeData'+  -- associated to it by 'xedni'.+  --+  -- The integer is a multiplicity which can be > 1 for pointings.+  , lTerm :: HashMap Ix (Nat, Integer)+  -- ^ Leading term @a * x ^ u@ of the generating functions @C_i[k](x)@ in the+  -- form (u, a).+  --+  -- [Order @u@] Smallest size of objects of a given type.+  -- [Leading coefficient @a@] number of objects of smallest size.+  , degree :: HashMap Ix Int+  -- ^ Degrees of the generating functions, when applicable: greatest size of+  -- objects of a given type.+  } deriving Show++-- | A pair @C i k@ represents the @k@-th "pointing" of the type at index @i@,+-- with generating function @C_i[k](x)@.+data C = C Ix Int+  deriving (Eq, Ord, Show, Generic)++instance Hashable C++data AC = AC Aliased Int+  deriving (Eq, Ord, Show, Generic)++instance Hashable AC++type C' = (Maybe Aliased, C)++newtype Aliased = Aliased Int+  deriving (Eq, Ord, Show, Generic)++instance Hashable Aliased++type Ix = Int++data Nat = Zero | Succ Nat+  deriving (Eq, Ord, Show)++instance Monoid Nat where+  mempty = Zero+  mappend (Succ n) = Succ . mappend n+  mappend Zero = id++natToInt :: Nat -> Int+natToInt Zero = 0+natToInt (Succ n) = 1 + natToInt n++infinity :: Nat+infinity = Succ infinity++dataDef :: [Alias m] -> DataDef m+dataDef as = DataDef+  { count = 0+  , points = 0+  , index = index+  , xedni = HashMap.empty+  , xedni' = xedni'+  , types = HashMap.empty+  , lTerm = HashMap.empty+  , degree = HashMap.empty+  } where+    xedni' = HashMap.fromList (fmap (\(i, a) -> (i, (-1, a))) as')+    index = HashMap.fromList (fmap (\(i, a) -> (ofType a, Left i)) as')+    as' = zip (fmap Aliased [0 ..]) as+    ofType (Alias f) = typeRep (f undefined)++-- | Find all types that may be types of subterms of a value of type @a@.+--+-- This will loop if there are infinitely many such types.+collectTypes :: Data a => [Alias m] -> proxy a -> DataDef m+collectTypes as a = collectTypesM a `execState` dataDef as++-- | Primitive datatypes have @C(x) = x@: they are considered as+-- having a single object (@lCoef@) of size 1 (@order@)).+primOrder :: Int+primOrder = 1++primOrder' :: Nat+primOrder' = Succ Zero++primlCoef :: Integer+primlCoef = 1++-- | The type of the first argument of 'Data.Data.gunfold'.+type GUnfold m = forall b r. Data b => m (b -> r) -> m r++-- | Type of 'xedni''.+type AMap m = HashMap Aliased (Ix, Alias m)++collectTypesM :: Data a => proxy a+  -> State (DataDef m) (Either Aliased Ix, ((Nat, Integer), Maybe Int))+collectTypesM a = chaseType a (const id)++chaseType :: Data a => proxy a+  -> ((Maybe (Alias m), Ix) -> AMap m -> AMap m)+  -> State (DataDef m) (Either Aliased Ix, ((Nat, Integer), Maybe Int))+chaseType a k = do+  let t = typeRep a+  dd@DataDef{..} <- get+  let+    lookup i r =+      let+        lTerm_i = lTerm #! i+        degree_i = HashMap.lookup i degree+      in return (r, (lTerm_i, degree_i))+  case HashMap.lookup t index of+    Nothing -> do+      let i = count+      put dd+        { count = i + 1+        , index = HashMap.insert t (Right i) index+        , xedni = HashMap.insert i (someData' a) xedni+        , xedni' = k (Nothing, i) xedni'+        }+      traverseType a i -- Updates lTerm and degree+    Just (Right i) -> do+      put dd { xedni' = k (Nothing, i) xedni' }+      lookup i (Right i)+    Just (Left j) ->+      case xedni' #! j of+        (-1, Alias f) -> do+          (_, ld) <- chaseType (ofType f) $ \(alias, i) ->+            let+              alias' = case alias of+                Nothing -> Alias f+                Just (Alias g) -> Alias (composeCastM f g)+            in+            k (Just alias', i) . HashMap.insert j (i, alias')+          return (Left j, ld)+        (i, _) -> lookup i (Left j)+  where+    ofType :: (m a -> m b) -> m a+    ofType _ = undefined++-- | Traversal of the definition of a datatype.+traverseType+  :: Data a => proxy a -> Ix+  -> State (DataDef m) (Either Aliased Ix, ((Nat, Integer), Maybe Int))+traverseType a i = do+  let d = withProxy dataTypeOf a+  mfix $ \ ~(_, (lTerm_i0, _)) -> do+    modify $ \dd@DataDef{..} -> dd+      { lTerm = HashMap.insert i lTerm_i0 lTerm+      }+    (types_i, ld@(_, degree_i)) <- traverseType' a d+    modify $ \dd@DataDef{..} -> dd+      { types = HashMap.insert (C i 0) types_i types+      , degree = maybe id (HashMap.insert i) degree_i degree+      }+    return (Right i, ld)++traverseType'+  :: Data a => proxy a -> DataType+  -> State (DataDef m)+      ([(Integer, Constr, [(Maybe Aliased, C)])], ((Nat, Integer), Maybe Int))+traverseType' a d | isAlgType d = do+  let+    constrs = dataTypeConstrs d+    collect+      :: GUnfold (StateT+        ([Either Aliased Ix], (Nat, Integer), Maybe Int)+        (State (DataDef m)))+    collect mkCon = do+      f <- mkCon+      let ofType :: (b -> a) -> Proxy b+          ofType _ = Proxy+          b = ofType f+      (j, (lTerm_, degree_)) <- lift (collectTypesM b)+      modify $ \(js, lTerm', degree') ->+        (j : js, lMul lTerm_ lTerm', liftA2 (+) degree_ degree')+      return (withProxy f b)+  tlds <- forM constrs $ \constr -> do+    (js, lTerm', degree') <-+      gunfold collect return constr `proxyType` a+        `execStateT` ([], (Zero, 1), Just 1)+    dd <- get+    let+      c (Left j) = (Just j, C (fst (xedni' dd #! j)) 0)+      c (Right i) = (Nothing, C i 0)+    return ((1, constr, [ c j | j <- js]), lTerm', degree')+  let+    (types_i, ls, ds) = unzip3 tlds+    lTerm_i = first Succ (lSum ls)+    degree_i = maxDegree ds+  return (types_i, (lTerm_i, degree_i))+traverseType' _ _ =+  return ([], ((primOrder', primlCoef), Just primOrder))++-- | If @(u, a)@ represents a power series of leading term @a * x ^ u@, and+-- similarly for @(u', a')@, this finds the leading term of their sum.+--+-- The comparison of 'Nat' is unrolled here for maximum laziness.+lPlus :: (Nat, Integer) -> (Nat, Integer) -> (Nat, Integer)+lPlus (Zero, lCoef) (Zero, lCoef') = (Zero, lCoef + lCoef')+lPlus (Zero, lCoef) _ = (Zero, lCoef)+lPlus _ (Zero, lCoef') = (Zero, lCoef')+lPlus (Succ order, lCoef) (Succ order', lCoef') =+  first Succ $ lPlus (order, lCoef) (order', lCoef')++-- | Sum of a list of series.+lSum :: [(Nat, Integer)] -> (Nat, Integer)+lSum [] = (infinity, 0)+lSum ls = foldl1 lPlus ls++-- | Leading term of a product of series.+lMul :: (Nat, Integer) -> (Nat, Integer) -> (Nat, Integer)+lMul (order, lCoef) (order', lCoef') = (order <> order', lCoef * lCoef')++lProd :: [(Nat, Integer)] -> (Nat, Integer)+lProd = foldl lMul (Zero, 1)++maxDegree :: [Maybe Int] -> Maybe Int+maxDegree = foldl (liftA2 max) (Just minBound)++-- | Pointing operator.+--+-- Populates a 'DataDef' with one more level of pointings.+-- ('collectTypes' produces a dictionary at level 0.)+--+-- The "pointing" of a type @t@ is a derived type whose values are essentially+-- values of type @t@, with one of their constructors being "pointed".+-- Alternatively, we may turn every constructor into variants that indicate+-- the position of points.+--+-- @+--   -- Original type+--   data Tree = Node Tree Tree | Leaf+--   -- Pointing of Tree+--   data Tree'+--     = Tree' Tree -- Point at the root+--     | Node'0 Tree' Tree -- Point to the left+--     | Node'1 Tree Tree' -- Point to the right+--   -- Pointing of the pointing+--   -- Notice that the "points" introduced by both applications of pointing+--   -- are considered different: exchanging their positions (when different)+--   -- produces a different tree.+--   data Tree''+--     = Tree'' Tree' -- Point 2 at the root, the inner Tree' places point 1+--     | Node'0' Tree' Tree -- Point 1 at the root, point 2 to the left+--     | Node'1' Tree Tree' -- Point 1 at the root, point 2 to the right+--     | Node'0'0 Tree'' Tree -- Points 1 and 2 to the left+--     | Node'0'1 Tree' Tree' -- Point 1 to the left, point 2 to the right+--     | Node'1'0 Tree' Tree' -- Point 1 to the right, point 2 to the left+--     | Node'0'1 Tree Tree'' -- Points 1 and 2 to the right+-- @+--+-- If we ignore points, some constructors are equivalent. Thus we may simply+-- calculate their multiplicity instead of duplicating them.+--+-- Given a constructor with @c@ arguments @C x_1 ... x_c@, and a sequence+-- @p_0 + p_1 + ... + p_c = k@ corresponding to a distribution of @k@ points+-- (@p_0@ are assigned to the constructor @C@ itself, and for @i > 0@, @p_i@+-- points are assigned within the @i@-th subterm), the multiplicity of the+-- constructor paired with that distribution is the multinomial coefficient+-- @multinomial k [p_1, ..., p_c]@.++point :: DataDef m -> DataDef m+point dd@DataDef{..} = dd+  { points = points'+  , types = foldl g types [0 .. count-1]+  } where+    points' = points + 1+    g types i = HashMap.insert (C i points') (types' i) types+    types' i = types #! C i 0 >>= h+    h (_, constr, js) = do+      ps <- partitions points' (length js)+      let+        mult = multinomial points' ps+        js' = zipWith (\(j', C i _) p -> (j', C i p)) js ps+      return (mult, constr, js')++-- | An oracle gives the values of the generating functions at some @x@.+type Oracle = HashMap C Double++-- | Find the value of @x@ such that the average size of the generator+-- for the @k-1@-th pointing is equal to @size@, and produce the associated+-- oracle. If the size is @Nothing@, find the radius of convergence.+--+-- The search evaluates the generating functions for some values of @x@ in+-- order to run a binary search. The evaluator is implemented using Newton's+-- method, the convergence of which has been shown for relevant systems in+-- /Boltzmann Oracle for Combinatorial Systems/,+-- C. Pivoteau, B. Salvy, M. Soria.+makeOracle :: DataDef m -> TypeRep -> Maybe Double -> Oracle+makeOracle dd0 t size' =+  seq v+  HashMap.fromList (zip cs (V.toList v))+  where+    -- We need the next pointing to capture the average size in an equation.+    dd@DataDef{..} = if isJust size' then point dd0 else dd0+    cs = flip C <$> [0 .. points] <*> [0 .. count - 1]+    m = count * (points + 1)+    k = points - 1+    i = case index #! t of+      Left j -> fst (xedni' #! j)+      Right i -> i+    checkSize _ (Just ys) | V.any (< 0) ys = False+    -- There may be solutions outside of the radius+    -- of convergence, but with negative components.+    checkSize (Just size) (Just ys) =+      size >= size_+      where+        size_ = ys V.! j' / ys V.! j+        j = dd ? C i k+        j' = dd ? C i (k + 1)+    checkSize Nothing (Just _) = True+    checkSize _ Nothing = False+    -- Equations defining C_i(x) for all types with indices i+    phis :: Num a => V.Vector (a -> V.Vector a -> a)+    phis = V.fromList [ phi dd c (types #! c) | c <- listCs dd ]+    eval' :: Double -> Maybe (V.Vector Double)+    eval' x = fixedPoint defSolveArgs phi' (V.replicate m 0)+      where+        phi' :: (Mode a, Scalar a ~ Double) => V.Vector a -> V.Vector a+        phi' y = fmap (\f -> f (auto x) y) phis+    v = (fromJust . snd) (search eval' (checkSize size'))++-- | Generating function definition. This defines a @Phi_i[k]@ function+-- associated with the @k@-th pointing of the type at index @i@, such that:+--+-- > C_i[k](x)+-- >   = Phi_i[k](x, C_0[0](x), ..., C_(n-1)[0](x),+-- >              ..., C_0[k](x), ..., C_(n-1)[k](x))+--+-- Primitive datatypes have @C(x) = x@: they are considered as+-- having a single object ('lCoef') of size 1 ('order')).+phi :: Num a => DataDef m -> C -> [(Integer, constr, [C'])]+  -> a -> V.Vector a -> a+phi DataDef{..} (C i _) [] =+  case xedni #! i of+    SomeData a ->+      case (dataTypeRep . withProxy dataTypeOf) a of+        AlgRep _ -> \_ _ -> 0+        _ -> \x _ -> fromInteger primlCoef * x ^ primOrder+phi dd@DataDef{..} _ tyInfo = f+  where+    f x y = x * (sum . fmap (toProd y)) tyInfo+    toProd y (w, _, js) =+      fromInteger w * product [ y V.! (dd ? j) | (_, j) <- js ]++-- | Maps a key representing a type @a@ (or one of its pointings) to a+-- generator @m a@.+type Generators m = (HashMap AC (SomeData m), HashMap C (SomeData m))++-- | Build all involved generators at once.+makeGenerators+  :: forall m. MonadRandomLike m+  => DataDef m -> Oracle -> Generators m+makeGenerators DataDef{..} oracle =+  seq oracle+  (generatorsL, generatorsR)+  where+    f (C i _) tyInfo = case xedni #! i of+      SomeData a -> SomeData $ incr >>+        case tyInfo of+          [] -> defGen+          _ -> frequencyWith doubleR (fmap g tyInfo) `proxyType` a+    g :: Data a => (Integer, Constr, [C']) -> (Double, m a)+    g (v, constr, js) =+      ( fromInteger v * w+      , gunfold generate return constr `runReaderT` gs)+      where+        gs = fmap (\(j', i) -> m j' i) js+        m = maybe (generatorsR #!) m'+        m' j (C _ k) = (generatorsL #! AC j k)+        w = product $ fmap ((oracle #!) . snd) js+    h (j, (i, Alias f)) k =+      (AC j k, applyCast f (generatorsR #! C i k))+    generatorsL = HashMap.fromList (liftA2 h (HashMap.toList xedni') [0 .. points])+    generatorsR = HashMap.mapWithKey f types++type SmallGenerators m =+  (HashMap Aliased (SomeData m), HashMap Ix (SomeData m))++-- | Generators of values of minimal sizes.+smallGenerators+  :: forall m. MonadRandomLike m => DataDef m -> SmallGenerators m+smallGenerators DataDef{..} = (generatorsL, generatorsR)+  where+    f i (SomeData a) = SomeData $ incr >>+      case types #! C i 0 of+        [] -> defGen+        tyInfo ->+          let gs = (tyInfo >>= g (fst (lTerm #! i))) in+          frequencyWith integerR gs `proxyType` a+    g :: Data a => Nat -> (Integer, Constr, [C']) -> [(Integer, m a)]+    g minSize (_, constr, js) =+      guard (minSize == Succ size) *>+      [(weight, gunfold generate return constr `runReaderT` gs)]+      where+        (size, weight) = lProd [ lTerm #! i | (_, C i _) <- js ]+        gs = fmap lookup js+        lookup (j', C i _) = maybe (generatorsR #! i) (generatorsL #!) j'+    h (j, (i, Alias f)) = (j, applyCast f (generatorsR #! i))+    generatorsL = (HashMap.fromList . fmap h . HashMap.toList) xedni'+    generatorsR = HashMap.mapWithKey f xedni++generate :: Applicative m => GUnfold (ReaderT [SomeData m] m)+generate rest = ReaderT $ \(g : gs) ->+  rest `runReaderT` gs <*> unSomeData g++defGen :: (Data a, MonadRandomLike m) => m a+defGen = gen+  where+    gen =+      let dt = withProxy dataTypeOf gen in+      case dataTypeRep dt of+        IntRep -> fromConstr . mkIntegralConstr dt <$> int+        FloatRep -> fromConstr . mkRealConstr dt <$> double+        CharRep -> fromConstr . mkCharConstr dt <$> char+        AlgRep _ -> error "Cannot generate for empty type."+        NoRep -> error "No representation."++-- * Short operators++(?) :: DataDef m -> C -> Int+dd ? C i k = i + k * count dd++-- | > dd ? (listCs dd !! i) = i+listCs :: DataDef m -> [C]+listCs dd = liftA2 (flip C) [0 .. points dd] [0 .. count dd - 1]++ix :: C -> Int+ix (C i _) = i++-- | > dd ? (dd ?! i) = i+(?!) :: DataDef m -> Int -> C+dd ?! j = C i k+  where (k, i) = j `divMod` count dd++getGenerator :: Data a => DataDef m -> Generators m -> proxy a -> Int -> m a+getGenerator dd (l, r) a k = unSomeData $+  case index dd #! typeRep a of+    Right i -> (r #! C i k)+    Left j -> (l #! AC j k)++getSmallGenerator :: Data a => DataDef m -> SmallGenerators m -> proxy a -> m a+getSmallGenerator dd (l, r) a = unSomeData $+  case index dd #! typeRep a of+    Right i -> (r #! i)+    Left j -> (l #! j)++(#!) :: (Eq k, Hashable k)+  => HashMap k v -> k -> v+(#!) = (HashMap.!)

@@ -0,0 +1,66 @@+-- | Solve systems of equations++{-# LANGUAGE RecordWildCards #-}+{-# LANGUAGE RankNTypes, FlexibleContexts, TypeFamilies #-}+module Generic.Random.Internal.Solver where++import Control.Applicative+import Data.AEq ( (~==) )+import Numeric.AD.Mode+import Numeric.AD.Mode.Forward+import Numeric.LinearAlgebra+import qualified Data.Vector as V+import qualified Data.Vector.Storable as S++data SolveArgs = SolveArgs+  { accuracy :: Double+  , numIterations :: Int+  } deriving (Eq, Ord, Show)++defSolveArgs :: SolveArgs+defSolveArgs = SolveArgs 1e-8 20++findZero+  :: SolveArgs+  -> (forall s. V.Vector (AD s (Forward R)) -> V.Vector (AD s (Forward R)))+  -> Vector R+  -> Maybe (Vector R)+findZero SolveArgs{..} f = newton numIterations+  where+    newton 0 _ = Nothing+    newton n x+      | norm_y == 1/0 = Nothing+      | norm_y > accuracy = newton (n - 1) (x - jacobian <\> y)+      | otherwise = Just x+      where+        norm_y = norm_Inf y+        jacobian = (fromRows . V.toList . fmap (V.convert . snd)) yj+        y = (V.convert . fmap fst) yj+        yj = jacobian' f (S.convert x)++fixedPoint+  :: SolveArgs+  -> (forall a. (Mode a, Scalar a ~ R) => V.Vector a -> V.Vector a)+  -> V.Vector R+  -> Maybe (V.Vector R)+fixedPoint args f =+  fmap S.convert . findZero args (liftA2 (V.zipWith (-)) f id) . S.convert++-- | Assuming @p . f@ is satisfied only for positive values in some interval+-- @(0, r]@, find @f r@.+search :: (Double -> a) -> (a -> Bool) -> (Double, a)+search f p = search' e0 (0 : [2 ^ n | n <- [0 .. 100 :: Int]])+  where+    search' y (x : xs@(x' : _))+      | p y' = search' y' xs+      | otherwise = search'' y x x'+      where y' = f x'+    search' _ _ = error "Solution not found. Uncontradictable predicate?"+    search'' y x x'+      | x ~== x' = (x, y)+      | p y_ = search'' y_ x_ x'+      | otherwise = search'' y x x_+      where+        x_ = (x + x') / 2+        y_ = f x_+    e0 = error "Solution not found. Unsatisfiable predicate?"

@@ -0,0 +1,191 @@+{-# LANGUAGE RankNTypes, GADTs, ScopedTypeVariables, ImplicitParams #-}+{-# LANGUAGE TypeOperators, GeneralizedNewtypeDeriving #-}+module Generic.Random.Internal.Types where++import Control.Monad.Random+import Control.Monad.Trans+import Data.Coerce+import Data.Data+import Data.Function+import Test.QuickCheck++data SomeData m where+  SomeData :: Data a => m a -> SomeData m++type SomeData' = SomeData Proxy++-- | Dummy instance for debugging.+instance Show (SomeData m) where+  show _ = "SomeData"++data Alias m where+  Alias :: (Data a, Data b) => !(m a -> m b) -> Alias m++type AliasR m = Alias (RejectT m)++-- | Dummy instance for debugging.+instance Show (Alias m) where+  show _ = "Alias"++-- | Main constructor for 'Alias'.+alias :: (Monad m, Data a, Data b) => (a -> m b) -> Alias m+alias = Alias . (=<<)++-- | Main constructor for 'AliasR'.+aliasR :: (Monad m, Data a, Data b) => (a -> m b) -> AliasR m+aliasR = Alias . (=<<) . fmap lift++-- | > coerceAlias :: Alias m -> Alias (AMonadRandom m)+coerceAlias :: Coercible m n => Alias m -> Alias n+coerceAlias = coerce++-- | > coerceAliases :: [Alias m] -> [Alias (AMonadRandom m)]+coerceAliases :: Coercible m n => [Alias m] -> [Alias n]+coerceAliases = coerce++-- | > composeCast f g = f . g+composeCastM :: forall a b c d m+  . (Typeable b, Typeable c)+  => (m c -> d) -> (a -> m b) -> (a -> d)+composeCastM f g | Just Refl <- eqT :: Maybe (b :~: c) = f . g+composeCastM _ _ = castError ([] :: [b]) ([] :: [c])++castM :: forall a b m+  . (Typeable a, Typeable b)+  => m a -> m b+castM a | Just Refl <- eqT :: Maybe (a :~: b) = a+castM a = let x = castError a x in x++unSomeData :: Typeable a => SomeData m -> m a+unSomeData (SomeData a) = castM a++applyCast :: (Typeable a, Data b) => (m a -> m b) -> SomeData m -> SomeData m+applyCast f = SomeData . f . unSomeData++castError :: (Typeable a, Typeable b)+  => proxy a -> proxy' b -> c+castError a b = error $ unlines+  [ "Error trying to cast"+  , "  " ++ show (typeRep a)+  , "to"+  , "  " ++ show (typeRep b)+  ]++withProxy :: (a -> b) -> proxy a -> b+withProxy f _ =+  f (error "This should not be evaluated\n")++reproxy :: proxy a -> Proxy a+reproxy _ = Proxy++proxyType :: m a -> proxy a -> m a+proxyType = const++someData' :: Data a => proxy a -> SomeData'+someData' = SomeData . reproxy++-- | Size as the number of constructors.+type Size = Int++-- | Internal transformer for rejection sampling.+--+-- > ReaderT Size (StateT Size (MaybeT m)) a+newtype RejectT m a = RejectT+  { unRejectT :: forall r. Size -> Size -> m r -> (Size -> a -> m r) -> m r+  }++instance Functor (RejectT m) where+  fmap f (RejectT go) = RejectT $ \maxSize size retry cont ->+    go maxSize size retry $ \size a -> cont size (f a)++instance Applicative (RejectT m) where+  pure a = RejectT $ \_maxSize size _retry cont ->+    cont size a+  RejectT f <*> RejectT x = RejectT $ \maxSize size retry cont ->+    f maxSize size retry $ \size f_ ->+      x maxSize size retry $ \size x_ ->+        cont size (f_ x_)++instance Monad (RejectT m) where+  RejectT x >>= f = RejectT $ \maxSize size retry cont ->+    x maxSize size retry $ \size x_ ->+      unRejectT (f x_) maxSize size retry cont++instance MonadTrans RejectT where+  lift m = RejectT $ \_maxSize size _retry cont ->+    m >>= cont size++-- | Set lower bound+runRejectT :: Monad m => (Size, Size) -> RejectT m a -> m a+runRejectT (minSize, maxSize) (RejectT m) = fix $ \go ->+  m maxSize 0 go $ \size a ->+    if size < minSize then+      go+    else+      return a+--runRejectT (minSize, maxSize) (RejectT m) = fix $ \go -> do+--  x' <- runMaybeT (m `runReaderT` maxSize `runStateT` 0)+--  case x' of+--    Just (x, size) | size >= minSize -> return x+--    _ -> go++newtype AMonadRandom m a = AMonadRandom+  { asMonadRandom :: m a+  } deriving (Functor, Applicative, Monad)++instance MonadTrans AMonadRandom where+  lift = AMonadRandom++-- ** Dictionaries++-- | @'MonadRandomLike' m@ defines basic components to build generators,+-- allowing the implementation to remain abstract over both the+-- 'Test.QuickCheck.Gen' type and 'MonadRandom' instances.+--+-- For the latter, the wrapper 'AMonadRandom' is provided to avoid+-- overlapping instances.+class Monad m => MonadRandomLike m where+  -- | Called for every constructor. Counter for ceiled rejection sampling.+  incr :: m ()+  incr = return ()++  -- | @doubleR upperBound@: generates values in @[0, upperBound]@.+  doubleR :: Double -> m Double++  -- | @integerR upperBound@: generates values in @[0, upperBound-1]@.+  integerR :: Integer -> m Integer++  -- | Default @Int@ generator.+  int :: m Int++  -- | Default @Double@ generator.+  double :: m Double++  -- | Default @Char@ generator.+  char :: m Char++instance MonadRandomLike Gen where+  doubleR x = choose (0, x)+  integerR x = choose (0, x-1)+  int = arbitrary+  double = arbitrary+  char = arbitrary++instance MonadRandomLike m => MonadRandomLike (RejectT m) where+  incr = RejectT $ \maxSize size retry cont ->+    if size >= maxSize then+      retry+    else+      cont (size + 1) ()+  doubleR = lift . doubleR+  integerR = lift . integerR+  int = lift int+  double = lift double+  char = lift char++instance MonadRandom m => MonadRandomLike (AMonadRandom m) where+  doubleR x = lift $ getRandomR (0, x)+  integerR x = lift $ getRandomR (0, x-1)+  int = lift getRandom+  double = lift getRandom+  char = lift getRandom

@@ -4,7 +4,7 @@ import Data.Foldable import Data.List import Test.QuickCheck-import Data.Random.Generics+import Generic.Random.Data  data T = N T T | L   deriving (Eq, Ord, Show, Data)