Perfect depth binary tree.

• Only has elements at the leaf of the tree
• A tree of depth d has 2^d elements.

"treplicate d a" returns a tree of depth d, and has 2^d copies of a.

>>> treplicate (SNat :: SNat 3) 6
<<<6,6>,<6,6>>,<<6,6>,<6,6>>>
>>> treplicate d3 6
<<<6,6>,<6,6>>,<<6,6>,<6,6>>>

"trepeat a" creates a tree with as many copies of a as demanded by the context.

>>> trepeat 6 :: RTree 2 Int
<<6,6>,<6,6>>

Extract the first element of a tree

The first element is defined to be the bottom-left leaf.

>>> thead $ BR (BR (LR 1) (LR 2)) (BR (LR 3) (LR 4))
1

Extract the last element of a tree

The last element is defined to be the bottom-right leaf.

>>> tlast $ BR (BR (LR 1) (LR 2)) (BR (LR 3) (LR 4))
4

"indexTree t n" returns the n'th element of t.

The bottom-left leaf had index 0, and the bottom-right leaf has index 2^d-1, where d is the depth of the tree

>>> indexTree (BR (BR (LR 1) (LR 2)) (BR (LR 3) (LR 4))) 0
1
>>> indexTree (BR (BR (LR 1) (LR 2)) (BR (LR 3) (LR 4))) 2
3
>>> indexTree (BR (BR (LR 1) (LR 2)) (BR (LR 3) (LR 4))) 14
*** Exception: Clash.Sized.Vector.(!!): index 14 is larger than maximum index 3
...

Generate a tree of indices, where the depth of the tree is determined by the context.

>>> tindices :: RTree 3 (Index 8)
<<<0,1>,<2,3>>,<<4,5>,<6,7>>>

"replaceTree n a t" returns the tree t where the n'th element is replaced by a.

The bottom-left leaf had index 0, and the bottom-right leaf has index 2^d-1, where d is the depth of the tree

>>> replaceTree 0 5 (BR (BR (LR 1) (LR 2)) (BR (LR 3) (LR 4)))
<<5,2>,<3,4>>
>>> replaceTree 2 7 (BR (BR (LR 1) (LR 2)) (BR (LR 3) (LR 4)))
<<1,2>,<7,4>>
>>> replaceTree 9 6 (BR (BR (LR 1) (LR 2)) (BR (LR 3) (LR 4)))
<<1,2>,<3,*** Exception: Clash.Sized.Vector.replace: index 9 is larger than maximum index 3
...

"tmap f t" is the tree obtained by apply f to each element of t, i.e.,

tmap f (BR (LR a) (LR b)) == BR (LR (f a)) (LR (f b))

tzipWith generalizes tzip by zipping with the function given as the first argument, instead of a tupling function. For example, "tzipWith (+)" applied to two trees produces the tree of corresponding sums.

tzipWith f (BR (LR a1) (LR b1)) (BR (LR a2) (LR b2)) == BR (LR (f a1 a2)) (LR (f b1 b2))

tzip takes two trees and returns a tree of corresponding pairs.

tunzip transforms a tree of pairs into a tree of first components and a tree of second components.

Reduce a tree to a single element

A dependently typed fold over trees.

As an example of when you might want to use dtfold we will build a population counter: a circuit that counts the number of bits set to '1' in a BitVector. Given a vector of n bits, we only need we need a data type that can represent the number n: Index (n+1). Index k has a range of [0 .. k-1] (using ceil(log2(k)) bits), hence we need Index n+1. As an initial attempt we will use tfold, because it gives a nice (log2(n)) tree-structure of adders:

populationCount :: (KnownNat (2^d), KnownNat d, KnownNat (2^d+1))
                => BitVector (2^d) -> Index (2^d+1)
populationCount = tfold (resize . bv2i . pack) (+) . v2t . bv2v

The "problem" with this description is that all adders have the same bit-width, i.e. all adders are of the type:

(+) :: Index (2^d+1) -> Index (2^d+1) -> Index (2^d+1).

This is a "problem" because we could have a more efficient structure: one where each layer of adders is precisely wide enough to count the number of bits at that layer. That is, at height d we want the adder to be of type:

Index ((2^d)+1) -> Index ((2^d)+1) -> Index ((2^(d+1))+1)

We have such an adder in the form of the add function, as defined in the instance ExtendingNum instance of Index. However, we cannot simply use fold to create a tree-structure of adds:

>>> :{
let populationCount' :: (KnownNat (2^d), KnownNat d, KnownNat (2^d+1))
                     => BitVector (2^d) -> Index (2^d+1)
    populationCount' = tfold (resize . bv2i . pack) add . v2t . bv2v
:}

<interactive>:...
    • Couldn't match type ‘(((2 ^ d) + 1) + ((2 ^ d) + 1)) - 1’
                     with ‘(2 ^ d) + 1’
      Expected type: Index ((2 ^ d) + 1)
                     -> Index ((2 ^ d) + 1) -> Index ((2 ^ d) + 1)
        Actual type: Index ((2 ^ d) + 1)
                     -> Index ((2 ^ d) + 1)
                     -> AResult (Index ((2 ^ d) + 1)) (Index ((2 ^ d) + 1))
    • In the second argument of ‘tfold’, namely ‘add’
      In the first argument of ‘(.)’, namely
        ‘tfold (resize . bv2i . pack) add’
      In the expression: tfold (resize . bv2i . pack) add . v2t . bv2v
    • Relevant bindings include
        populationCount' :: BitVector (2 ^ d) -> Index ((2 ^ d) + 1)
          (bound at ...)

because tfold expects a function of type "b -> b -> b", i.e. a function where the arguments and result all have exactly the same type.

In order to accommodate the type of our add, where the result is larger than the arguments, we must use a dependently typed fold in the form of dtfold:

{-# LANGUAGE UndecidableInstances #-}
import Data.Singletons

data IIndex (f :: TyFun Nat Type) :: Type
type instance Apply IIndex l = Index ((2^l)+1)

populationCount' :: (KnownNat k, KnownNat (2^k))
                 => BitVector (2^k) -> Index ((2^k)+1)
populationCount' bv = tdfold (Proxy @IIndex)
                             (resize . bv2i . pack)
                             (\_ x y -> add x y)
                             (v2t (bv2v bv))

And we can test that it works:

>>> :t populationCount' (7 :: BitVector 16)
populationCount' (7 :: BitVector 16) :: Index 17
>>> populationCount' (7 :: BitVector 16)
3

tscanl applied to Vec

>>> scanlPar (+) (1 :> 2 :> 3 :> 4 :> Nil)
1 :> 3 :> 6 :> 10 :> Nil

Low-depth left scan

tscanl is similar to foldl, but returns a tree of successive reduced values from the left:

tscanl f [x1, x2, x3, ...] == [x1, x1 `f` x2, x1 `f` x2 `f` x3, ...]

>>> tscanl (+) (v2t (1 :> 2 :> 3 :> 4 :> Nil))
<<1,3>,<6,10>>

tscanr applied to Vec

>>> scanrPar (+) (1 :> 2 :> 3 :> 4 :> Nil)
10 :> 9 :> 7 :> 4 :> Nil

Low-depth right scan

tscanr is similar to foldr, but returns a tree of successive reduced values from the left:

tscanr f [..., xn2, xn1, xn] == [..., xn2 `f` xn1 `f` xn, xn1 `f` xn, xn]

>>> tscanr (+) (v2t (1 :> 2 :> 3 :> 4 :> Nil))
<<10,9>,<7,4>>

Convert a vector with 2^d elements to a tree of depth d.

>>> v2t (1 :> 2 :> 3 :> 4:> Nil)
<<1,2>,<3,4>>

Convert a tree of depth d to a vector of 2^d elements

>>> (BR (BR (LR 1) (LR 2)) (BR (LR 3) (LR 4)))
<<1,2>,<3,4>>
>>> t2v (BR (BR (LR 1) (LR 2)) (BR (LR 3) (LR 4)))
1 :> 2 :> 3 :> 4 :> Nil

Given a function f that is strict in its nth RTree argument, make it lazy by applying lazyT to this argument:

f x0 x1 .. (lazyT xn) .. xn_plus_k