User:Michiexile/MATH198/Lecture 9

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Recursion patterns

Meijer, Fokkinga & Patterson identified in the paper Functional programming with bananas, lenses, envelopes and barbed wire a number of generic patterns for recursive programming that they had observed, catalogued and systematized. The aim of that paper is to establish a number of rules for modifying and rewriting expressions involving these generic recursion patterns.

As it turns out, these patterns are instances of the same phenomenon we saw last lecture: where the recursion comes from specifying a different algebra, and then take a uniquely existing morphism induced by initiality (or, as we shall see, finality).

Before we go through the recursion patterns, we need to establish a few pieces of theoretical language, dualizing the Eilenberg-Moore algebra constructions from the last lecture.

Coalgebras for endofunctors

Definition If $P:C\to C$ is an endofunctor, then a $P$-coalgebra on $A$ is a morphism $a:A\to PA$.

A morphism of coalgebras: $f:a\to b$ is some $f:A\to B$ such that the diagram

commutes.

Just as with algebras, we get a category of coalgebras. And the interesting objects here are the final coalgebras. Just as with algebras, we have

Lemma (Lambek) If $a:A\to PA$ is a final coalgebra, it is an isomorphism.

Finally, one thing that makes us care highly about these entities: in an appropriate category (such as $\omega -CPO$), initial algebras and final coalgebras coincide, with the correspondence given by inverting the algebra/coalgebra morphism. In Haskell not quite true (specifically, the final coalgebra for the lists functor gives us streams...).

Onwards to recursion schemes!

We shall define a few specific morphisms we'll use repeatedly. This notation, introduced here, occurs all over the place in these corners of the literature, and are good to be aware of in general:

• If $a:TA\to A$ is an initial algebra for $T$, we denote $a=in_{A}$.
• If $a:A\to TA$ is a final coalgebra for $T$, we denote $a=out_{A}$.
• We write $\mu f$ for the fixed point operator

mu f = x where x = f x

• MFP write $(f\Delta g)$ for

Delta f g = \x -> (f x, g x)

• MFP write $(f\nabla g)x$ for

(Nabla f g) (Left x) = f x
(Nabla f g) (Right x) = g x

These two last constructions are directly motivated by the maps induced from the universal properties of products and coproducts.

We shall write $(f\times g)$ and $(f+g)$ for the $\Delta$ and $\nabla$ constructions, respectively.

We note that in the situation considered by MFP, inital algebras and final coalgebras coincide, and thus $in_{A},out_{A}$ are the pair of isomorphic maps induced by either the initial algebra- or the final coalgebra-structure.

Catamorphisms

A catamorphism is the uniquely existing morphism from an initial algebra to a different algebra. We have to define maps down to the return value type for each of the constructors of the complex data type we're recursing over, and the catamorphism will deconstruct the structure (trees, lists, ...) and do a generalized fold over the structure at hand before returning the final value.

The intuition is that for catamorphisms we start essentially structured, and dismantle the structure.

Example: the length function from last lecture. This is the catamorphism for the functor $P_{A}(X)=1+A\times X$ given by the maps

u :: Int
u = 0

m :: (A, Int) -> Int
m (a, n) = n+1

MFP define the catamorphism by, supposing T is initial for the functor F:

cata :: (F a b -> b) -> T a -> b
cata phi = mu (\x -> phi . fmap x . outT)

We can reframe the example above as a catamorphism by observing that here,

data F a b = Nil | Cons a b deriving (Eq, Show)
type T a = [a]

instance Functor (F a) where
  fmap _ Nil = Nil
  fmap f (Cons n a) = Cons n (f a)

outT :: T a -> F a (T a)
outT [] = Nil
outT (a:as) = Cons a as

lphi :: F a Int -> Int
lphi Nil = 0
lphi (Cons a n) = n + 1

l = cata lphi

where we observe that mu has a global definition for everything we do and out is defined once we settle on the functor F and its initial algebra. Thus, the definition of phi really is the only place that the recursion data shows up.

Anamorphisms

An anamorphism is the categorical dual to the catamorphism. It is the canonical morphism from a coalgebra to the final coalgebra for that endofunctor.

Here, we start unstructured, and erect a structure, induced by the coalgebra structures involved.

Example: we can write a recursive function

first :: Int -> [Int]
first 1 = [1]
first n = n : first (n - 1)

This is an anamorphism from the coalgebra for $P_{\mathbb {N} }(X)=1+\mathbb {N} \times X$ on $\mathbb {N}$ generated by the two maps

c 0 = Left ()
c n = Right (n, n-1)

and we observe that we can chase through the diagram

to conclude that therefore

f 0 = []
f n = n : f (n - 1)

which is exactly the recursion we wrote to begin with.

MFP define the anamorphism by a fixpoint as well, namely:

ana psi = mu (\x -> inT . fmap x . psi)

We can, again, recast our illustration above into a structural anamorphism, by:

-- Reuse mu, F, T from above
inT :: F a (T a) -> T a
inT Nil = []
inT (Cons a as) = a:as

fpsi :: Int -> F Int Int
fpsi 0 = Nil
fpsi n = Cons n (n-1)

Again, we can note that the implementation of fpsi here is exactly the c above, and the resulting function will - as we can verify by compiling and running - give us the same kind of reversed list of the n first integers as the first function above would.

Hylomorphisms

The hylomorphisms capture one of the two possible compositions of anamorphisms and catamorphisms. Parametrized over an algebra $\phi :TA\to A$ and a coalgebra $\psi :B\to TB$ the hylomorphism is a recursion pattern that computes a value in $A$ from a value in $A$ by generating some sort of intermediate structure and then collapsing it again.

It is, thus the composition of the uniquely existing morphism from a coalgebra to the final coalgebra for an endofunctor, followed by the uniquely existing morphism from the initial algebra to some other algebra.

MFP define it, again, as a fix point:

hylo phi psi = mu (\x -> phi . fmap x . psi)

First off, we can observe that by picking one or the other of $in_{A},out_{A}$ as a parameter, we can recover both the anamorphisms and the catamorphisms as hylomorphisms.

As an example, we'll compute the factorial function using a hylomorphism:

phi :: F Int Int -> Int
phi Nil = 1
phi (Cons n m) = n*m

psi :: Int -> F Int Int
psi 0 = Int
psi n = Cons n (n-1)

factorial = hylo phi psi

Metamorphisms

Paramorphisms

Apomorphisms

Further reading

Terminology in the literature: in and out, inl, inr.

Homework

1. Write a fold for the data type data T a = L a | B a a | C a a a and demonstrate how this can be written as a catamorphism by giving the algebra it maps to.
2. Write the fibonacci function as a hylomorphism.
3. * The integers have a partial order induced by the divisibility relation. We can thus take any integer and arrange all its divisors in a tree by having an edge $n\to d$ if $d|n$ and $d$ doesn't divide any other divisor of $n$. Write an anamorphic function that will generate this tree for a given starting integer. Demonstrate how this function is an anamorphism by giving the algebra it maps from.

Hint: You will be helped by having a function to generate a list of all primes. One suggestion is:

primes :: [Integer]
primes = sieve [2..]
  where
    sieve (p:xs) = p : sieve [x|x <- xs, x `mod` p > 0]

Hint: A good data structure to use is; with expected output of running the algorithm:

data Tree = Leaf Integer | Node Integer [Tree]

divisionTree 60 = 
  Node 60 [
    Node 30 [
      Node 15 [
        Leaf 5,
        Leaf 3],
      Node 10 [
        Leaf 5,
        Leaf 2],
      Node 6 [
        Leaf 3,
        Leaf 2]],
    Node 20 [
      Node 10 [
        Leaf 5,
        Leaf 2],
      Node 4 [
        Leaf 2]],
    Node 12 [
      Node 6 [
        Leaf 3,
        Leaf 2],
      Node 4 [
        Leaf 2]]]