Tutorial_05_Datatypes.lhs

Refined Datatypes {#refineddatatypes}
=================


\begin{comment}
\begin{code}
{-@ LIQUID "--short-names"    @-}
{-@ LIQUID "--no-termination" @-}

module Tutorial_05_Datatypes
       (
         -- * Sparse: Data
         Sparse (..)

         -- * Sparse: Functions
       , dotProd, dotProd', plus, fromList

         -- * Sparse: Examples
       , okSP, badSP, test1, test2

          -- * OrdList: Data
       , IncList  (..)

          -- * OrdList: Examples
       , okList, badList

          -- * OrdList: Functions
       ,  insertSort, insertSort', mergeSort, quickSort

          -- * BST: Data
       , BST (..)

          -- * BST: Functions
       , mem, add, delMin, del, bstSort, toBST, toIncList

          -- * BST: Examples
       , okBST, badBST

       )
      where

import Prelude      hiding (abs, length, min)
import Data.List    (foldl')
import Data.Vector  hiding (singleton, foldl', foldr, fromList, (++))
import Data.Maybe   (fromJust)

dotProd, dotProd' :: Vector Int -> Sparse Int -> Int
test1 :: Sparse String
test2 :: Sparse Int

{-@ die :: {v:_ | false} -> a @-}
die msg = error msg

-- {-@ fail badSP @-}
-- {-@ fail test1 @-}
-- {-@ fail test2 @-}
-- {-@ fail badList @-}
-- {-@ ignore append @-}
-- {-@ fail badBST @-}
-- {-@ ignore delMin @-}


\end{code}
\end{comment}


So far, we have seen how to refine the types of *functions*, to
specify, for example, pre-conditions on the inputs, or post-conditions
on the outputs. Very often, we wish to define *datatypes* that satisfy
certain invariants. In these cases, it is handy to be able to directly
refine the `data` definition, making it impossible to create
illegal inhabitants.

Sparse Vectors Revisited {#autosmart}
-------------------------------------

As our first example of a refined datatype, let's revisit the
sparse vector representation that we [saw earlier](#sparsetype).
The `SparseN` type alias we used got the job done, but is not
pleasant to work with because we have no way of determining
the *dimension* of the sparse vector. Instead, let's create a new
datatype to represent such vectors:

\begin{code}
data Sparse a = SP { spDim   :: Int
                   , spElems :: [(Int, a)] }
\end{code}

\noindent
Thus, a sparse vector is a pair of a dimension and a list of
index-value tuples. Implicitly, all indices *other* than those
in the list have the value `0` or the equivalent value type `a`.

\newthought{Legal}
`Sparse` vectors satisfy two crucial properties.
First, the dimension stored in `spDim` is non-negative.
Second, every index in `spElems` must be valid, i.e.
between `0` and the dimension. Unfortunately, Haskell's
type system does not make it easy to ensure that
*illegal vectors are not representable*.^[The standard
approach is to use abstract types and
[smart constructors][smart-ctr-wiki] but even
then there is only the informal guarantee that the
smart constructor establishes the right invariants.]

\newthought{Data Invariants} LiquidHaskell lets us enforce
these invariants with a refined data definition:

\begin{code}
{-@ data Sparse a = SP { spDim   :: Nat
                       , spElems :: [(Btwn 0 spDim, a)]} @-}
\end{code}

\noindent Where, as before, we use the aliases:

\begin{code}
{-@ type Nat        = {v:Int | 0 <= v}            @-}
{-@ type Btwn Lo Hi = {v:Int | Lo <= v && v < Hi} @-}
\end{code}

\newthought{Refined Data Constructors} The refined data
definition is internally converted into refined types
for the data constructor `SP`:

~~~~~{.spec}
-- Generated Internal representation
data Sparse a where
  SP :: spDim:Nat
     -> spElems:[(Btwn 0 spDim, a)]
     -> Sparse a
~~~~~

\noindent In other words, by using refined input types for `SP`
we have automatically converted it into a *smart* constructor that
ensures that *every* instance of a `Sparse` is legal.
Consequently, LiquidHaskell verifies:

\begin{code}
okSP :: Sparse String
okSP = SP 5 [ (0, "cat")
            , (3, "dog") ]
\end{code}

\noindent but rejects, due to the invalid index:

\begin{code}
badSP :: Sparse String
badSP = SP 5 [ (0, "cat")
             , (6, "dog") ]
\end{code}

\newthought{Field Measures} It is convenient to write an alias
for sparse vectors of a given size `N`. We can use the field name
`spDim` as a *measure*, like `vlen`. That is, we can use `spDim`
inside refinements^[Note that *inside* a refined `data` definition,
a field name like `spDim` refers to the value of the field, but *outside*
it refers to the field selector measure or function.]

\begin{code}
{-@ type SparseN a N = {v:Sparse a | spDim v == N} @-}
\end{code}

\newthought{Sparse Products}
Let's write a function to compute a sparse product

\begin{code}
{-@ dotProd :: x:Vector Int -> SparseN Int (vlen x) -> Int @-}
dotProd x (SP _ y) = go 0 y
  where
    go sum ((i, v) : y') = go (sum + (x ! i) * v) y'
    go sum []            = sum
\end{code}

\noindent
LiquidHaskell verifies the above by using the specification
to conclude that for each tuple `(i, v)` in the list `y`, the
value of `i` is within the bounds of the vector `x`, thereby
proving `x ! i` safe.

\newthought{Folded Product} We can port the `fold`-based product
to our new representation:

\begin{code}
{-@ dotProd' :: x:Vector Int -> SparseN Int (vlen x) -> Int @-}
dotProd' x (SP _ y) = foldl' body 0 y
  where
    body sum (i, v) = sum + (x ! i)  * v
\end{code}

\noindent As before, LiquidHaskell checks the above by
[automatically instantiating refinements](#sparsetype)
for the type parameters of `foldl'`, saving us a fair
bit of typing and enabling the use of the elegant
polymorphic, higher-order combinators we know and love.

<div class="hwex" id="Sanitization"> \singlestar
Invariants are all well and good for data computed
*inside* our programs. The only way to ensure the
legality of data coming from *outside*, i.e. from
the "real world", is to write a sanitizer that will
check the appropriate invariants before constructing
a `Sparse` vector. Write the specification and
implementation of a sanitizer `fromList`, so that
the following typechecks:
</div>

\hint You need to check that *all* the indices in
`elts` are less than `dim`; the easiest way is to
compute a new `Maybe [(Int, a)]` which is `Just`
the original pairs if they are valid, and `Nothing`
otherwise.

\begin{code}
fromList          :: Int   -> [(Int, a)] -> Maybe (Sparse a)
fromList dim elts = undefined

{-@ test1 :: SparseN String 3 @-}
test1     = fromJust $ fromList 3 [(0, "cat"), (2, "mouse")]
\end{code}

<div class="hwex" id="Addition">
Write the specification and implementation
of a function `plus` that performs the addition of two `Sparse`
vectors of the *same* dimension, yielding an output of that dimension.
When you are done, the following code should typecheck:
</div>

\begin{code}
plus     :: (Num a) => Sparse a -> Sparse a -> Sparse a
plus x y = undefined

{-@ test2 :: SparseN Int 3 @-}
test2    = plus vec1 vec2
  where
    vec1 = SP 3 [(0, 12), (2, 9)]
    vec2 = SP 3 [(0, 8),  (1, 100)]
\end{code}

Ordered Lists {#orderedlists}
--------------

As a second example of refined data types, let's consider a
different problem: representing *ordered* sequences. Here's
a type for sequences that mimics the classical list:




\begin{code}
data IncList a =
    Emp
  | (:<) { hd :: a, tl :: IncList a }

infixr 9 :<
\end{code}

\noindent
The Haskell type above does not state that the elements
are in order of course, but we can specify that requirement
by refining *every* element in `tl` to be *greater than* `hd`:

\begin{code}
{-@ data IncList a =
        Emp
      | (:<) { hd :: a, tl :: IncList {v:a | hd <= v}}  @-}
\end{code}

\newthought{Refined Data Constructors} Once again,
the refined data definition is internally converted
into a "smart" refined data constructor

~~~~~{.spec}
-- Generated Internal representation
data IncList a where
  Emp  :: IncList a
  (:<) :: hd:a -> tl:IncList {v:a | hd <= v} -> IncList a
~~~~~


\noindent which ensures that we can only create legal ordered lists.

\begin{code}
okList  = 1 :< 2 :< 3 :< Emp      -- accepted by LH

badList = 2 :< 1 :< 3 :< Emp      -- rejected by LH
\end{code}

\noindent
It's all very well to *specify* ordered lists.
Next, let's see how it's equally easy to *establish*
these invariants by implementing several textbook
sorting routines.

\newthought{Insertion Sort}
First, let's implement insertion sort, which converts an ordinary
list `[a]` into an ordered list `IncList a`.

\begin{code}
insertSort        :: (Ord a) => [a] -> IncList a
insertSort []     = Emp
insertSort (x:xs) = insert x (insertSort xs)
\end{code}

The hard work is done by `insert` which places an element into
the correct position of a sorted list. LiquidHaskell infers that
if you give `insert` an element and a sorted list, it returns a
sorted list.

\begin{code}
insert             :: (Ord a) => a -> IncList a -> IncList a
insert y Emp       = y :< Emp
insert y (x :< xs)
  | y <= x         = y :< x :< xs
  | otherwise      = x :< insert y xs
\end{code}

<div class="hwex" id="Insertion Sort">
Complete the implementation of the function below to
use `foldr` to eliminate the explicit recursion in `insertSort`.
</div>

\begin{code}
insertSort'     :: (Ord a) => [a] -> IncList a
insertSort' xs  = foldr f b xs
  where
     f          = undefined    -- Fill this in
     b          = undefined    -- Fill this in
\end{code}

\newthought{Merge Sort} Similarly, it is easy to write merge sort,
by implementing the three steps. First, we write a function that
*splits* the input into two equal sized halves:

\begin{code}
split          :: [a] -> ([a], [a])
split (x:y:zs) = (x:xs, y:ys)
  where
    (xs, ys)   = split zs
split xs       = (xs, [])
\end{code}

\noindent
Second, we need a function that *combines* two ordered lists

\begin{code}
merge         :: (Ord a) => IncList a -> IncList a -> IncList a
merge xs  Emp = xs
merge Emp ys  = ys
merge (x :< xs) (y :< ys)
  | x <= y    = x :< merge xs (y :< ys)
  | otherwise = y :< merge (x :< xs) ys
merge _ _     = Emp
\end{code}

\noindent
Finally, we compose the above steps to divide (i.e. `split`)
and conquer (`sort` and `merge`) the input list:

\begin{code}
mergeSort :: (Ord a) => [a] -> IncList a
mergeSort []  = Emp
mergeSort [x] = x :< Emp
mergeSort xs  = merge (mergeSort ys) (mergeSort zs)
  where
    (ys, zs)  = split xs
\end{code}

<div class="hwex" id="QuickSort"> \doublestar
Why is the following implementation of `quickSort`
rejected by LiquidHaskell? Modify it so it is accepted.
</div>

\hint Think about how `append` should behave so that the
`quickSort` has the desired property. That is, suppose
that `ys` and `zs` are already in *increasing order*.
Does that mean that `append x ys zs` are *also* in
increasing order? No! What other requirement do you need?
bottle that intuition into a suitable *specification* for
`append` and then ensure that the code satisfies that
specification.

\begin{code}
quickSort           :: (Ord a) => [a] -> IncList a
quickSort []        = Emp
quickSort (x:xs)    = append x lessers greaters
  where
    lessers         = quickSort [y | y <- xs, y < x ]
    greaters        = quickSort [z | z <- xs, z >= x]

{-@ append :: x:a -> IncList a
                  -> IncList a
                  -> IncList a
  @-}
append z Emp       ys = z :< ys
append z (x :< xs) ys = x :< append z xs ys
\end{code}

Ordered Trees {#binarysearchtree}
---------------------------------

As a last example of refined data types, let us consider binary search ordered
trees, defined thus:

\begin{code}
data BST a = Leaf
           | Node { root  :: a
                  , left  :: BST a
                  , right :: BST a }
\end{code}

\newthought{Binary Search Trees}
enjoy the [property][bst-wiki]
that each `root` lies (strictly) between the elements belonging in the
`left` and `right` subtrees hanging off the root. The ordering
invariant makes it easy to check whether a certain value occurs in the
tree.  If the tree is empty i.e. a `Leaf`, then the value does not occur
in the tree.  If the given value is at the root then the value does
occur in the tree.  If it is less than (respectively greater than) the
root, we recursively check whether the value occurs in the left
(respectively right) subtree.

<div class="marginfigure"
  id="fig:bst"
  caption="A Binary Search Tree with values between 1 and 9.
           Each root's value lies between the values appearing
           in its left and right subtrees."
  height="200px"
  file="img/bst.png">
</div>

Figure [auto](#fig:bst) shows a binary search tree whose nodes
are labeled with a subset of values from `1` to `9`.
We might represent such a tree with the Haskell value:

\begin{code}
okBST :: BST Int
okBST =  Node 6
             (Node 2
                 (Node 1 Leaf Leaf)
                 (Node 4 Leaf Leaf))
             (Node 9
                 (Node 7 Leaf Leaf)
                 Leaf)
\end{code}

\newthought{Refined Data Type} The Haskell type says nothing about the
ordering invariant, and hence, cannot prevent us from creating illegal
`BST` values that violate the invariant. We can remedy this with a
refined data definition that captures the invariant. The aliases `BSTL`
and `BSTR` denote `BST`s with values less than and greater than some `X`,
respectively.^[We could also just *inline* the definitions
of `BSTL` and `BSTR` into that of `BST` but they will be
handy later.]

\begin{code}
{-@ data BST a    = Leaf
                  | Node { root  :: a
                         , left  :: BSTL a root
                         , right :: BSTR a root } @-}

{-@ type BSTL a X = BST {v:a | v < X}             @-}
{-@ type BSTR a X = BST {v:a | X < v}             @-}
\end{code}


\newthought{Refined Data Constructors} As before, the above data definition
creates a refined smart constructor for `BST`

~~~~~{.spec}
data BST a where
  Leaf :: BST a
  Node :: r:a -> BST {v:a| v < r}
       -> BST {v:a | r < v}
       -> BST a
~~~~~

\noindent which *prevents* us from creating illegal trees

\begin{code}
badBST =  Node 66
             (Node 4
                 (Node 1 Leaf Leaf)
                 (Node 69 Leaf Leaf))  -- Out of order, rejected
             (Node 99
                 (Node 77 Leaf Leaf)
                 Leaf)
\end{code}

<div class="hwex" id="Duplicates">
Can a `BST Int` contain duplicates?
</div>

\newthought{Membership}
Lets write some functions to create and manipulate
these trees. First, a function to check whether a value
is in a `BST`:

\begin{code}
mem                 :: (Ord a) => a -> BST a -> Bool
mem _ Leaf          = False
mem k (Node k' l r)
  | k == k'         = True
  | k <  k'         = mem k l
  | otherwise       = mem k r
\end{code}

\newthought{Singleton} Next, another easy warm-up: a function to create
a `BST` with a single given element:

\begin{code}
one   :: a -> BST a
one x = Node x Leaf Leaf
\end{code}

\newthought{Insertion} Lets write a function that adds an
element to a `BST`.^[While writing this exercise
I inadvertently swapped the `k` and `k'` which caused
LiquidHaskell to protest.]

\begin{code}
add                  :: (Ord a) => a -> BST a -> BST a
add k' Leaf          = one k'
add k' t@(Node k l r)
  | k' < k           = Node k (add k' l) r
  | k  < k'          = Node k l (add k' r)
  | otherwise        = t
\end{code}

\newthought{Minimum} For our next trick, let's write a function to delete the
*minimum* element from a `BST`. This function will return a *pair* of outputs
-- the smallest element and the remainder of the tree. We can say that the
output element is indeed the smallest, by saying that the remainder's
elements exceed the element. To this end, let's define a helper type:
^[This helper type approach is rather verbose.
We should be able to just use plain old pairs
and specify the above requirement as a *dependency*
between the pairs' elements. Later, we will see how
to do so using [abstract refinements][vazou13].]


\begin{code}
data MinPair a = MP { mElt :: a, rest :: BST a }
\end{code}

\noindent We can specify that `mElt` is indeed smaller than all
the elements in `rest` via the data type refinement:

\begin{code}
{-@ data MinPair a = MP { mElt :: a, rest :: BSTR a mElt} @-}
\end{code}

\noindent Finally, we can write the code to compute `MinPair`

\begin{code}
delMin                 :: (Ord a) => BST a -> MinPair a
delMin (Node k Leaf r) = MP k r
delMin (Node k l r)    = MP k' (Node k l' r)
  where
    MP k' l'           = delMin l
delMin Leaf            = die "Don't say I didn't warn ya!"
\end{code}



<div class="hwex" id="Delete"> Use `delMin` to complete the
implementation of `del` which *deletes* a given element from
a `BST`, if it is present.
</div>

\begin{code}
del                   :: (Ord a) => a -> BST a -> BST a
del k' t@(Node k l r) = undefined
del _  Leaf           = Leaf
\end{code}

\begin{comment}
**FIXME** See issue about using RAW FIELDS vs PATTERN MATCHING.
\end{comment}

<div class="hwex" id="Safely Deleting Minimum"> \singlestar
The function `delMin` is only sensible for non-empty trees.
[Read ahead](#usingmeasures) to learn how to specify and verify that
it is only called with such trees, and then apply that technique here
to verify the call to `die` in `delMin`.
</div>

\begin{comment}
**FIXME**  I shouldn't have to read a future chapter to complete an exercise in this chapter.
\end{comment}

<div class="hwex" id="BST Sort">
Complete the implementation of `toIncList` to obtain a `BST`
based sorting routine `bstSort`.
</div>

\begin{code}
bstSort   :: (Ord a) => [a] -> IncList a
bstSort   = toIncList . toBST

toBST     :: (Ord a) => [a] -> BST a
toBST     = foldr add Leaf

toIncList :: BST a -> IncList a
toIncList (Node x l r) = undefined
toIncList Leaf         = undefined
\end{code}

\hint This exercise will be a lot easier *after* you finish the
`quickSort` exercise. Note that the signature for `toIncList`
does not use `Ord` and so you *cannot* (and *need not*) use
a sorting procedure to implement it.


Recap
-----

In this chapter we saw how LiquidHaskell lets you refine data
type definitions to capture sophisticated invariants. These
definitions are internally represented by refining the types
of the data constructors, automatically making them "smart"  in
that they preclude the creation of illegal values that violate
the invariants. We will see much more of this handy technique
in future chapters.

One recurring theme in this chapter was that we had to create new
versions of standard datatypes, just in order to specify certain
invariants.  For example, we had to write a special list type, with
its own *copies* of nil and cons. Similarly, to implement `delMin` we
had to create our own pair type.

\newthought{This duplication} of types is quite tedious.
There should be a way to just slap the desired invariants
on to *existing* types, thereby facilitating their reuse.
In a few chapters, we will see how to achieve this reuse
by [abstracting refinements][vazou13] from the definitions of
datatypes or functions in the same way we abstract
the element type `a` from containers like `[a]` or `BST a`.