I previously wrote about a design pattern I called an incremental fold (or catamorphism). I described it as a design pattern, because, as written, it cannot be factored into code. That is, it is a pattern for designing part of a program.
The pattern I presented is a useful way to implement functions that can be expressed as catamorphisms such that the result is incrementally computed for each operation on a value of a datatype. Unlike a fold defined directly as a function, which traverse an entire value, the incremental fold only traverses parts that are updated. For some values, this may provide a performance benefit.
This post shows how we can adapt the above idea to a more general concept that I'm calling incremental attributes. It's more general in that incremental attributes can express the incremental fold as well as other flows of incremental computation.
Review of the incremental fold
First, let's review the implementation of the incremental fold.
[Note: This is not a literate Haskell article, because there's too much duplicated code required; however, all source files are available.]
module IncrementalAttributes1Synthesized where
data Tree a s
= Tip s
| Bin a (Tree a s) (Tree a s) s
deriving Show
data Alg a s
= Alg { stip :: s, sbin :: a -> s -> s -> s }
result :: Tree a s -> s
result (Tip s) = s
result (Bin _ _ _ s) = s
tip :: Alg a s -> Tree a s
tip alg = Tip (stip alg)
bin :: Alg a s -> a -> Tree a s -> Tree a s -> Tree a s
bin alg x lt rt = Bin x lt rt (sbin alg x (result lt) (result rt))
empty :: (Ord a) => Alg a s -> Tree a s
empty = tip
singleton :: (Ord a) => Alg a s -> a -> Tree a s
singleton alg x = bin alg x (tip alg) (tip alg)
insert :: (Ord a) => Alg a s -> a -> Tree a s -> Tree a s
insert alg x t =
case t of
Tip _ ->
singleton alg x
Bin y lt rt _ ->
case compare x y of
LT -> bin alg y (insert alg x lt) rt
GT -> bin alg y lt (insert alg x rt)
EQ -> bin alg x lt rt
fromList :: (Ord a) => Alg a s -> [a] -> Tree a s
fromList alg = foldr (insert alg) (empty alg)
heightAlg :: Alg a Integer
heightAlg = Alg 0 (\_ x y -> 1 + max x y)
t1 = fromList heightAlg "azbycx"
This will be the starting point for our discussion. We have a basic binary tree with an algebra type that gives the fold functions, stip and sbin. The application of the algebra is blended into the utility functions just as it was with the incremental fold. I have chosen to keep the representation simple, so the algebra is passed around as an argument to the functions. This could, of course, be done with type classes. Lastly, we have an example algebra that determines the height of a tree. To get the height of the example t1, simply type result t1 after loading this file into GHCi.
Incrementally inherited attributes
Suppose that, instead of height, we wanted to incrementally compute the depth of every node. Thus, we would attach a value to each node that stored its distance from the root. We can't do that with the above implementation, because attributes are only fed "upwards" or from the leaves to the root. We have no way of passing information downwards. The solution is to use inherited attributes.
Before presenting the code, here is a little side note. You may notice the use of the words synthesized and inherited here. The terminology comes from the study of attribute grammars, extending context-free grammars to support semantic operations. Wouter Swierstra wrote a great tutorial on attribute grammars in Haskell for The Monad Reader in 2005. In fact, I use an example from there at the end of this article. You can think of synthesized as "produced by the children for the parent" and inherited as "passed down from the parent to the children."
As you can now imagine, inherited attributes will allow us to bring information to the leaves. Many of the changes to the code are trivial, so we ignore them. The relevant changes are the following:
data Alg a i
= Alg { itip :: i -> i, ibin :: a -> i -> i }
tip :: Alg a i -> i -> Tree a i
tip alg i = Tip (itip alg i)
bin :: Alg a i -> i -> a -> Tree a i -> Tree a i -> Tree a i
bin alg i x lt rt = Bin x (update i lt) (update i rt) i
where
update i' t =
case t of
Tip _ ->
tip alg i'
Bin x lt rt _ ->
let s = ibin alg x i' in
Bin x (update s lt) (update s lt) s
The datatype Alg (that I'm still calling the algebra, though that may not be proper use of the category theoretical term) now has functions that take an inherited attribute from a parent and create a new inherited attribute to be stored with the node and passed on to its children. The change to bin is the more complicated of the changes, because once a Bin constructor is constructed, all of its child nodes must be updated with new inherited values.
To implement an algebra for depth, we do the following:
depthAlg :: Alg a Int
depthAlg = Alg (+1) (const (+1))
t1 = fromList depthAlg 0 "azbycx"
Load the code and check the result to see for yourself what it looks like.
One is not enough
Now that we have use cases for synthesized and inherited incremental attributes, we're going to want both. Fortunately, that's not too difficult. The new datatypes are simply a product of the two previous:
data Tree a i s
= Tip i s
| Bin a (Tree a i s) (Tree a i s) i s
deriving Show
data Alg a i s
= Alg { itip :: i -> i, ibin :: a -> i -> i,
stip :: s, sbin :: a -> s -> s -> s }
You can now see why I was using s and i to distinguish the types of the attributes. Again, most of the code modifications are trivial, and the bin function needs special attention.
bin :: Alg a i s -> i -> a -> Tree a i s -> Tree a i s -> Tree a i s
bin alg i x lt rt =
Bin x (update i lt) (update i rt) i (sbin alg x (sresult lt) (sresult rt))
where
update i' t =
case t of
Tip _ _ ->
tip alg i'
Bin y ylt yrt _ s ->
let j = ibin alg y i' in
Bin y (update j ylt) (update j yrt) j s
Defining an algebra for both depth and height is no more difficult than defining each alone.
depthAndHeightAlg :: Alg a Int Int
depthAndHeightAlg = Alg (+1) (const (+1)) 1 (\_ x y -> 1 + max x y)
Feedback
You probably know where this is going by now. There's that famous saying, "what goes down must come up." We want more than just two separate directions of information flow. We want to utilize the information flowing toward the leaves to help determine that which flows up to the root or vice versa. A simple example of this is a counter that annotates each node with its rank in an in-order traversal. This can't be done with just synthesized or inherited attributes, because it depends on a combination of input from the parent, children, and siblings for each node.
The code is similar to the previous implementation, but the differences in Alg are important.
data Alg a i s
= Alg { ftip :: i -> s, fbin :: a -> i -> s -> s -> (i, i, s) }
Each node now has a single inherited attribute, because it has a single parent. We use the synthesized attributes to store a local result, so each constructor only has one as an output. For the Bin constructor, we have a pair of incoming synthesized values and a pair of outgoing inherited values. The left component in each pair is associated with the left child, and the right with the right child. This allows us to have information flow up from the synthesized attribute of the left child and down to the inherited attribute of the right or in the opposite direction.
The bin is again tricky to write correctly.
bin :: Alg a i s -> i -> a -> Tree a i s -> Tree a i s -> Tree a i s
bin alg i x lt rt = update i (Bin x lt rt undefined undefined)
where
update j t =
case t of
Tip _ _ ->
tip alg j
Bin y ylt yrt _ _ ->
let (li, ri, s) = fbin alg y j (sresult zlt) (sresult zrt)
zlt = update li ylt
zrt = update ri yrt
in Bin y zlt zrt j s
Notice the circular programming here. The definition and uses of, for example, li and zlt show that we could easily loop infinitely. This depends on how the specific algebra functions are implemented. Here is the "counter example":
newtype CounterI = CI { cntI :: Int } deriving Show
data CounterS = CS { size :: Int, cntS :: Int } deriving Show
counterAlg :: Alg a CounterI CounterS
counterAlg = Alg ft fb
where
ft :: CounterI -> CounterS
ft i = CS { size = 1, cntS = cntI i }
fb :: a -> CounterI -> CounterS -> CounterS -> (CounterI, CounterI, CounterS)
fb _ i ls rs =
( i
, CI { cntI = 1 + cntI i + size ls }
, CS { size = 1 + size ls + size rs
, cntS = cntI i + size ls }
)
t1 = fromList counterAlg (CI { cntI = 0 }) "azbycx"
I've relied heavily on record syntax to document the flow of information. Notice in fb how the i is directly inherited by the left child and how the right child inherits the new count that depends on the size of the left subtree and the inherited count of its parent. As shown in this example, the dependency flow must be unidirectional for one desired result. But there's no reason we can't go up, down, and then up again (for example).
Revisiting the diff problem.
As I mentioned, Wouter wrote a good introduction to attribute grammars in Haskell (which I highly recommend that you read). He focuses on the use of the UUAG system to generate code for solving problems that are harder to solve with traditional functional programming techniques. He describes the problem as follows:
Suppose we want to write a function diff :: [Float] -> [Float] that given a list xs, calculates a new list where every element x is replaced with the difference between x and the average of xs. Similar problems pop up in any library for performing statistical calculations.
Great problem! And we can solve it using incremental attributes in Haskell instead of in UUAG's attribute grammar syntax.
newtype DiffI = DI { avg :: Float } deriving Show
data DiffS = DS { sumD :: Float, len :: Float, res :: Float } deriving Show
diffAlg :: Alg Float DiffI DiffS
diffAlg = Alg ft fb
where
ft :: DiffI -> DiffS
ft i =
DS { sumD = 0
, len = 0
, res = 0
}
fb :: Float -> DiffI -> DiffS -> DiffS -> (DiffI, DiffI, DiffS)
fb x i ls sr =
( i
, i
, DS { sumD = x + sumD ls + sumD sr
, len = 1 + len ls + len sr
, res = x - avg i
}
)
The implementation is not too much more difficult than the attribute grammar solution. We don't have the clean separation of concerns, but adding another attribute only means adding another field in DI or DS depending on whether it's inherited or synthesized.
Oh, but we're not done! Where's the actual average generated? Ah right, that's fed to the root inherited attribute.
t2 = let val = fromList diffAlg (DI { avg = a }) [1,4,1.5,3.5,2,3,2.5]
s = sresult val
a = sumD s / len s
in val
Here's another example of circular programming. Due to the way we implemented the application of the algebra, we can take advantage of lazy evaluation to ensure that the sum and length (and thus average) are incrementally computed and, as a result, the difference (res) is determined as needed for each node.
