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User:Michiexile/MATH198/Lecture 9

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Line 119: Line 119:
 
MFP define the anamorphism by a fixpoint as well, namely:
 
MFP define the anamorphism by a fixpoint as well, namely:
 
<haskell>
 
<haskell>
  +
ana :: (b -> F a b) -> b -> T a
 
ana psi = mu (\x -> inT . fmap x . psi)
 
ana psi = mu (\x -> inT . fmap x . psi)
 
</haskell>
 
</haskell>
Line 144: Line 145:
 
MFP define it, again, as a fix point:
 
MFP define it, again, as a fix point:
 
<haskell>
 
<haskell>
  +
hylo :: (F a b2 -> b2) -> (b1 -> F a b1) -> b1 -> b2
 
hylo phi psi = mu (\x -> phi . fmap x . psi)
 
hylo phi psi = mu (\x -> phi . fmap x . psi)
 
</haskell>
 
</haskell>
Line 162: Line 164:
 
factorial = hylo phi psi
 
factorial = hylo phi psi
 
</haskell>
 
</haskell>
 
   
 
====Metamorphisms====
 
====Metamorphisms====
  +
  +
The ''metamorphism'' is the ''other'' composition of an anamorphism with a catamorphism. It takes some structure, deconstructs it, and then reconstructs a new structure from it.
  +
  +
As a recursion pattern, it's kinda boring - it'll take an interesting structure, deconstruct it into a ''scalar'' value, and then reconstruct some structure from that scalar. As such, it won't even capture the richness of <math>hom(F x, G y)</math>, since any morphism expressed as a metamorphism will factor through a map <math>x\to y</math>.
   
 
====Paramorphisms====
 
====Paramorphisms====
  +
  +
''Paramorphisms'' were discussed in the MFP paper as a way to extend the catamorphisms so that the operating function can access its arguments in computation as well as in recursion. We gave the factorial above as a hylomorphism instead of a catamorphism precisely because no simple enough catamorphic structure exists.
   
 
====Apomorphisms====
 
====Apomorphisms====
  +
  +
The ''apomorphism'' is the dual of the paramorphism - it does with retention of values along the way what anamorphisms do compared to catamorphisms.
   
 
===Further reading===
 
===Further reading===
   
 
Terminology in the literature: in and out, inl, inr.
 
Terminology in the literature: in and out, inl, inr.
+
Bananas et.c.
  +
Sorting morphisms.
   
 
===Homework===
 
===Homework===
Line 179: Line 188:
 
# Write a fold for the data type <hask>data T a = L a | B a a | C a a a</hask> and demonstrate how this can be written as a catamorphism by giving the algebra it maps to.
 
# Write a fold for the data type <hask>data T a = L a | B a a | C a a a</hask> and demonstrate how this can be written as a catamorphism by giving the algebra it maps to.
 
# Write the fibonacci function as a hylomorphism.
 
# Write the fibonacci function as a hylomorphism.
  +
# Write the Towers of Hanoi as a hylomorphism. You'll probably want to use binary trees as the intermediate data structure.
  +
# Write a prime numbers generator as an anamorphism.
 
# * 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 <math>n \to d</math> if <math>d|n</math> and <math>d</math> doesn't divide any other divisor of <math>n</math>. 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.
 
# * 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 <math>n \to d</math> if <math>d|n</math> and <math>d</math> doesn't divide any other divisor of <math>n</math>. 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:
 
:''Hint'': You will be helped by having a function to generate a list of all primes. One suggestion is:

Revision as of 00:51, 17 November 2009

IMPORTANT NOTE: THESE NOTES ARE STILL UNDER DEVELOPMENT. PLEASE WAIT UNTIL AFTER THE LECTURE WITH HANDING ANYTHING IN, OR TREATING THE NOTES AS READY TO READ.


Contents

1 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.

1.1 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

CoalgebraMorphism.png

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 ω − 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 = inA.
  • If a: A\to TA is a final coalgebra for T, we denote a = outA.
  • We write μf for the fixed point operator
mu f = x where x = f x
  • MFP write (fΔg) 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 Δ and \nabla constructions, respectively.

We note that in the situation considered by MFP, inital algebras and final coalgebras coincide, and thus inA,outA are the pair of isomorphic maps induced by either the initial algebra- or the final coalgebra-structure.

1.2 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.

1.3 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

CoalgebraMorphism.png

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 :: (b -> F a b) -> b -> T a
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.

1.4 Hylomorphisms

The hylomorphisms capture one of the two possible compositions of anamorphisms and catamorphisms. Parametrized over an algebra \phi: T A\to A and a coalgebra \psi: B \to T B 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 :: (F a b2 -> b2) -> (b1 -> F a b1) -> b1 -> b2
hylo phi psi = mu (\x -> phi . fmap x . psi)

First off, we can observe that by picking one or the other of inA,outA 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

1.5 Metamorphisms

The metamorphism is the other composition of an anamorphism with a catamorphism. It takes some structure, deconstructs it, and then reconstructs a new structure from it.

As a recursion pattern, it's kinda boring - it'll take an interesting structure, deconstruct it into a scalar value, and then reconstruct some structure from that scalar. As such, it won't even capture the richness of hom(Fx,Gy), since any morphism expressed as a metamorphism will factor through a map x\to y.

1.6 Paramorphisms

Paramorphisms were discussed in the MFP paper as a way to extend the catamorphisms so that the operating function can access its arguments in computation as well as in recursion. We gave the factorial above as a hylomorphism instead of a catamorphism precisely because no simple enough catamorphic structure exists.

1.7 Apomorphisms

The apomorphism is the dual of the paramorphism - it does with retention of values along the way what anamorphisms do compared to catamorphisms.

2 Further reading

Terminology in the literature: in and out, inl, inr. Bananas et.c. Sorting morphisms.

3 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. Write the Towers of Hanoi as a hylomorphism. You'll probably want to use binary trees as the intermediate data structure.
  4. Write a prime numbers generator as an anamorphism.
  5. * 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]]]