LINQ's Aggregate() is just for IEnumerables. Catamorphisms in general refer to the pattern of folding for an arbitrary data type. So Aggregate() is to IEnumerables what FoldTree (below) is to Trees (below); both are catamorphisms for their respective data types.
I translated some of the code in part 4 of the series into C#. The code is below. Note that the equivalent F# used three less-than characters (for generic type parameter annotations), whereas this C# code uses more than 60. This is evidence why no one writes such code in C# - there are too many type annotations. I present the code in case it helps people who know C# but not F# play with this. But the code is so dense in C#, it's very hard to make sense of.
Given the following definition for a binary tree:
using System;
using System.Collections.Generic;
using System.Windows;
using System.Windows.Controls;
using System.Windows.Input;
using System.Windows.Media;
using System.Windows.Shapes;
class Tree<T> // use null for Leaf
{
public T Data { get; private set; }
public Tree<T> Left { get; private set; }
public Tree<T> Right { get; private set; }
public Tree(T data, Tree<T> left, Tree<T> rright)
{
this.Data = data;
this.Left = left;
this.Right = right;
}
public static Tree<T> Node<T>(T data, Tree<T> left, Tree<T> right)
{
return new Tree<T>(data, left, right);
}
}
One can fold trees and e.g. measure if two trees have different nodes:
class Tree
{
public static Tree<int> Tree7 =
Node(4, Node(2, Node(1, null, null), Node(3, null, null)),
Node(6, Node(5, null, null), Node(7, null, null)));
public static R XFoldTree<A, R>(Func<A, R, R, Tree<A>, R> nodeF, Func<Tree<A>, R> leafV, Tree<A> tree)
{
return Loop(nodeF, leafV, tree, x => x);
}
public static R Loop<A, R>(Func<A, R, R, Tree<A>, R> nodeF, Func<Tree<A>, R> leafV, Tree<A> t, Func<R, R> cont)
{
if (t == null)
return cont(leafV(t));
else
return Loop(nodeF, leafV, t.Left, lacc =>
Loop(nodeF, leafV, t.Right, racc =>
cont(nodeF(t.Data, lacc, racc, t))));
}
public static R FoldTree<A, R>(Func<A, R, R, R> nodeF, R leafV, Tree<A> tree)
{
return XFoldTree((x, l, r, _) => nodeF(x, l, r), _ => leafV, tree);
}
public static Func<Tree<A>, Tree<A>> XNode<A>(A x, Tree<A> l, Tree<A> r)
{
return (Tree<A> t) => x.Equals(t.Data) && l == t.Left && r == t.Right ? t : Node(x, l, r);
}
// DiffTree: Tree<'a> * Tree<'a> -> Tree<'a * bool>
// return second tree with extra bool
// the bool signifies whether the Node "ReferenceEquals" the first tree
public static Tree<KeyValuePair<A, bool>> DiffTree<A>(Tree<A> tree, Tree<A> tree2)
{
return XFoldTree((A x, Func<Tree<A>, Tree<KeyValuePair<A, bool>>> l, Func<Tree<A>, Tree<KeyValuePair<A, bool>>> r, Tree<A> t) => (Tree<A> t2) =>
Node(new KeyValuePair<A, bool>(t2.Data, object.ReferenceEquals(t, t2)),
l(t2.Left), r(t2.Right)),
x => y => null, tree)(tree2);
}
}
In this second example, another tree is reconstructed differently:
class Example
{
// original version recreates entire tree, yuck
public static Tree<int> Change5to0(Tree<int> tree)
{
return Tree.FoldTree((int x, Tree<int> l, Tree<int> r) => Tree.Node(x == 5 ? 0 : x, l, r), null, tree);
}
// here it is with XFold - same as original, only with Xs
public static Tree<int> XChange5to0(Tree<int> tree)
{
return Tree.XFoldTree((int x, Tree<int> l, Tree<int> r, Tree<int> orig) =>
Tree.XNode(x == 5 ? 0 : x, l, r)(orig), _ => null, tree);
}
}
And in this third example, folding a tree is used for drawing:
class MyWPFWindow : Window
{
void Draw(Canvas canvas, Tree<KeyValuePair<int, bool>> tree)
{
// assumes canvas is normalized to 1.0 x 1.0
Tree.FoldTree((KeyValuePair<int, bool> kvp, Func<Transform, Transform> l, Func<Transform, Transform> r) => trans =>
{
// current node in top half, centered left-to-right
var tb = new TextBox();
tb.Width = 100.0;
tb.Height = 100.0;
tb.FontSize = 70.0;
// the tree is a "diff tree" where the bool represents
// "ReferenceEquals" differences, so color diffs Red
tb.Foreground = (kvp.Value ? Brushes.Black : Brushes.Red);
tb.HorizontalContentAlignment = HorizontalAlignment.Center;
tb.VerticalContentAlignment = VerticalAlignment.Center;
tb.RenderTransform = AddT(trans, TranslateT(0.25, 0.0, ScaleT(0.005, 0.005, new TransformGroup())));
tb.Text = kvp.Key.ToString();
canvas.Children.Add(tb);
// left child in bottom-left quadrant
l(AddT(trans, TranslateT(0.0, 0.5, ScaleT(0.5, 0.5, new TransformGroup()))));
// right child in bottom-right quadrant
r(AddT(trans, TranslateT(0.5, 0.5, ScaleT(0.5, 0.5, new TransformGroup()))));
return null;
}, _ => null, tree)(new TransformGroup());
}
public MyWPFWindow(Tree<KeyValuePair<int, bool>> tree)
{
var canvas = new Canvas();
canvas.Width=1.0;
canvas.Height=1.0;
canvas.Background = Brushes.Blue;
canvas.LayoutTransform=new ScaleTransform(200.0, 200.0);
Draw(canvas, tree);
this.Content = canvas;
this.Title = "MyWPFWindow";
this.SizeToContent = SizeToContent.WidthAndHeight;
}
TransformGroup AddT(Transform t, TransformGroup tg) { tg.Children.Add(t); return tg; }
TransformGroup ScaleT(double x, double y, TransformGroup tg) { tg.Children.Add(new ScaleTransform(x,y)); return tg; }
TransformGroup TranslateT(double x, double y, TransformGroup tg) { tg.Children.Add(new TranslateTransform(x,y)); return tg; }
[STAThread]
static void Main(string[] args)
{
var app = new Application();
//app.Run(new MyWPFWindow(Tree.DiffTree(Tree.Tree7,Example.Change5to0(Tree.Tree7))));
app.Run(new MyWPFWindow(Tree.DiffTree(Tree.Tree7, Example.XChange5to0(Tree.Tree7))));
}
}
