content: ---
  layout: post
  title: "Binary Search Trees"
  date: 2013-02-15 16:12
  comments: true
  external-url:
  categories: abstract-refinements
  author: Niki Vazou
  published: false
  ---
  
  In this example, we show how we can use multi-parameter abstract refinements 
  to encode ordering invarants on tree structures.
  
  \begin{code}
  module Map where
  \end{code}
  
  Take for example the following refined type used to encode functional maps (from Data.Map):
  
  \begin{code}
  {-@ 
    data Map k a <l :: root:k -> k -> Prop, r :: root:k -> k -> Prop>
        = Tip 
        | Bin (sz    :: Size) 
              (key   :: k) 
              (value :: a) 
              (left  :: Map <l, r> (k <l key>) a) 
              (right :: Map <l, r> (k <r key>) a) 
    @-}
  
  data Map k a = Tip
               | Bin Size k a (Map k a) (Map k a)
  
  type Size    = Int
  \end{code}
  
  The abstract refinements `l` and `r` relate each `key` of the tree with `all` the keys in the `left` and `right` subtrees of `key`, as those keys are respectively of type `k <l key>` and `k <r key>`.
  
  Thus, if we instantiate the refinements with the following predicates
  
  \begin{code}
  {-@ type BST k a     = Map <\r -> {v:k | v < r }, \r -> {v:k | v > r }> k a @-}
  {-@ type MinHeap k a = Map <\r -> {v:k | r <= v}, \r -> {v:k | r <= v}> k a @-}
  {-@ type MaxHeap k a = Map <\r -> {v:k | r >= v}, \r -> {v:k | r >= v}> k a @-}
  \end{code}
  
  then `BST k v`, `MinHeap k v` and `MaxHeap k v` denote exactly binary-search-ordered, min-heap-ordered, and max-heap-ordered trees (with keys and values of types `k` and `v`).  
  
  We can use the above types to automatically verify ordering properties of complex libraries.
  
  For example, we cab use `BST` to prove that `Data.Map`'s `insert` and `delete` functions return a binary search tree:
  
  \begin{code}
  {-@ insert :: Ord k => k:k -> a:a -> t:BST k a -> BST k a @-}
  insert :: Ord k => k -> a -> Map k a -> Map k a
  insert kx x t
    = case t of 
       Tip -> singleton kx x
       Bin sz ky y l r
           -> case compare kx ky of
                LT -> balance ky y (insert kx x l) r
                GT -> balance ky y l (insert kx x r)
                EQ -> Bin sz kx x l r
  
  {-@ delete :: (Ord k) => k:k -> t:BST k a -> BST k a @-}
  delete :: Ord k => k -> Map k a -> Map k a
  delete k t 
    = case t of 
        Tip -> Tip
        Bin _ kx x l r
            -> case compare k kx of 
                 LT -> balance kx x (delete k l) r
                 GT -> balance kx x l (delete k r)
                 EQ -> glue kx l r
  \end{code}
  
  
  Below are the functions used by `insert` and `delete`:
  
  \begin{code}
  singleton :: k -> a -> Map k a
  singleton k x
    = Bin 1 k x Tip Tip
  
  
  glue :: k -> Map k a -> Map k a -> Map k a
  glue k Tip r = r
  glue k l Tip = l
  glue k l r
    | size l > size r = let (km1, vm, lm) = deleteFindMax l in balance km1 vm lm r
    | otherwise       = let (km2, vm, rm) = deleteFindMin r in balance km2 vm l rm
  
  deleteFindMax t 
    = case t of
        Bin _ k x l Tip -> (k, x, l)
        Bin _ k x l r -> let (km3, vm, rm) = deleteFindMax r in 
                         (km3, vm, (balance k x l rm))
        Tip           -> (error ms, error ms, Tip)
    where ms = "Map.deleteFindMax : can not return the maximal element of an empty Map"   
  
  
  deleteFindMin t 
    = case t of
        Bin _ k x Tip r -> (k, x, r)
        Bin _ k x l r -> let (km4, vm, lm) = deleteFindMin l in 
                         (km4, vm, (balance k x lm r))
        Tip             -> (error ms, error ms, Tip)
    where ms = "Map.deleteFindMin : can not return the maximal element of an empty Map"   
  
  
  -------------------------------------------------------------------------------
  --------------------------------- BALANCE -------------------------------------
  -------------------------------------------------------------------------------
  
  delta, ratio :: Int
  delta = 5
  ratio = 2
  
  balance :: k -> a -> Map k a -> Map k a -> Map k a 
  balance k x l r 
    | sizeL + sizeR <= 1   = Bin sizeX k x l r
    | sizeR >= delta*sizeL = rotateL k x l r
    | sizeL >= delta*sizeR = rotateR k x l r
    | otherwise            = Bin sizeX k x l r
    where sizeL = size l
          sizeR = size r
          sizeX = sizeL + sizeR + 1
  
  -- rotate
  rotateL :: a -> b -> Map a b -> Map a b -> Map a b
  rotateL k x l r@(Bin _ _ _ ly ry) 
    | size ly < ratio*size ry  = singleL k x l r
    | otherwise                = doubleL k x l r
  rotateL _ _ _ Tip = error "rotateL Tip"
  
  rotateR :: a -> b -> Map a b -> Map a b -> Map a b
  rotateR k x l@(Bin _ _ _ ly ry) r
    | size ry < ratio*size ly  = singleR k x l r
    | otherwise                = doubleR k x l r
  rotateR _ _ _ Tip = error "rotateR Tip"
  
  -- basic rotations
  singleL, singleR :: a -> b -> Map a b -> Map a b -> Map a b
  singleL k1 x1 t1 (Bin _ k2 x2 t2 t3) = bin k2 x2 (bin k1 x1 t1 t2) t3
  singleL _  _  _ Tip = error "sinlgeL Tip"
  singleR k1 x1 (Bin _ k2 x2 t1 t2) t3 = Bin 0 k2 x2 t1 (Bin 0 k1 x1 t2 t3)
  singleR _  _  _ Tip = error "sinlgeR Tip"
  
  doubleL, doubleR :: a -> b -> Map a b -> Map a b -> Map a b
  doubleL k1 x1 t1 (Bin _ k2 x2 (Bin _ k3 x3 t2 t3) t4)
   =bin k3 x3 (bin k1 x1 t1 t2) (bin k2 x2 t3 t4)
  doubleL _ _ _ _ = error "doubleL" 
  doubleR k1 x1 (Bin _ k2 x2 t1 (Bin _ k3 x3 t2 t3)) t4 
    = bin k3 x3 (bin k2 x2 t1 t2) (bin k1 x1 t3 t4)
  doubleR _ _ _ _ = error "doubleR" 
  
  bin :: k -> a -> Map k a -> Map k a -> Map k a
  bin k x l r 
    = Bin (size l + size r + 1) k x l r
  
  size :: Map k a -> Int
  size t 
    = case t of 
        Tip            -> 0
        Bin sz _ _ _ _ -> sz
  \end{code}
  