I am new to Functional programming.
The challenge I have is regarding the mental map of how a binary search tree works in Haskell.
In other programs (C,C++) we have something called root. We store it in a variable. We insert elements into it and do balancing etc..
The program takes a break does other things (may be process user inputs, create threads) and then figures out it needs to insert a new element in the already created tree. It knows the root (stored as a variable) and invokes the insert function with the root and the new value.
So far so good in other languages. But how do I mimic such a thing in Haskell, i.e.
I see functions implementing converting a list to a Binary Tree, inserting a value etc.. That's all good
I want this functionality to be part of a bigger program and so i need to know what the root is so that i can use it to insert it again. Is that possible? If so how?
Note: Is it not possible at all because data structures are immutable and so we cannot use the root at all to insert something. in such a case how is the above situation handled in Haskell?
It all happens in the same way, really, except that instead of mutating the existing tree variable we derive a new tree from it and remember that new tree instead of the old one.
For example, a sketch in C++ of the process you describe might look like:
int main(void) {
Tree<string> root;
while (true) {
string next;
cin >> next;
if (next == "quit") exit(0);
root.insert(next);
doSomethingWith(root);
}
}
A variable, a read action, and loop with a mutate step. In haskell, we do the same thing, but using recursion for looping and a recursion variable instead of mutating a local.
main = loop Empty
where loop t = do
next <- getLine
when (next /= "quit") $ do
let t' = insert next t
doSomethingWith t'
loop t'
If you need doSomethingWith to be able to "mutate" t as well as read it, you can lift your program into State:
main = loop Empty
where loop t = do
next <- getLine
when (next /= "quit") $ do
loop (execState doSomethingWith (insert next t))
Writing an example with a BST would take too much time but I give you an analogous example using lists.
Let's invent a updateListN which updates the n-th element in a list.
updateListN :: Int -> a -> [a] -> [a]
updateListN i n l = take (i - 1) l ++ n : drop i l
Now for our program:
list = [1,2,3,4,5,6,7,8,9,10] -- The big data structure we might want to use multiple times
main = do
-- only for shows
print $ updateListN 3 30 list -- [1,2,30,4,5,6,7,8,9,10]
print $ updateListN 8 80 list -- [1,2,3,4,5,6,7,80,9,10]
-- now some illustrative complicated processing
let list' = foldr (\i l -> updateListN i (i*10) l) list list
-- list' = [10,20,30,40,50,60,70,80,90,100]
-- Our crazily complicated illustrative algorithm still needs `list`
print $ zipWith (-) list' list
-- [9,18,27,36,45,54,63,72,81,90]
See how we "updated" list but it was still available? Most data structures in Haskell are persistent, so updates are non-destructive. As long as we have a reference of the old data around we can use it.
As for your comment:
My program is trying the following a) Convert a list to a Binary Search Tree b) do some I/O operation c) Ask for a user input to insert a new value in the created Binary Search Tree d) Insert it into the already created list. This is what the program intends to do. Not sure how to get this done in Haskell (or) is am i stuck in the old mindset. Any ideas/hints welcome.
We can sketch a program:
data BST
readInt :: IO Int; readInt = undefined
toBST :: [Int] -> BST; toBST = undefined
printBST :: BST -> IO (); printBST = undefined
loop :: [Int] -> IO ()
loop list = do
int <- readInt
let newList = int : list
let bst = toBST newList
printBST bst
loop newList
main = loop []
"do balancing" ... "It knows the root" nope. After re-balancing the root is new. The function balance_bst must return the new root.
Same in Haskell, but also with insert_bst. It too will return the new root, and you will use that new root from that point forward.
Even if the new root's value is the same, in Haskell it's a new root, since one of its children has changed.
See ''How to "think functional"'' here.
Even in C++ (or other imperative languages), it would usually be considered a poor idea to have a single global variable holding the root of the binary search tree.
Instead code that needs access to a tree should normally be parameterised on the particular tree it operates on. That's a fancy way of saying: it should be a function/method/procedure that takes the tree as an argument.
So if you're doing that, then it doesn't take much imagination to figure out how several different sections of code (or one section, on several occasions) could get access to different versions of an immutable tree. Instead of passing the same tree to each of these functions (with modifications in between), you just pass a different tree each time.
It's only a little more work to imagine what your code needs to do to "modify" an immutable tree. Obviously you won't produce a new version of the tree by directly mutating it, you'll instead produce a new value (probably by calling methods on the class implementing the tree for you, but if necessary by manually assembling new nodes yourself), and then you'll return it so your caller can pass it on - by returning it to its own caller, by giving it to another function, or even calling you again.
Putting that all together, you can have your whole program manipulate (successive versions of) this binary tree without ever having it stored in a global variable that is "the" tree. An early function (possibly even main) creates the first version of the tree, passes it to the first thing that uses it, gets back a new version of the tree and passes it to the next user, and so on. And each user of the tree can call other subfunctions as needed, with possibly many of new versions of the tree produced internally before it gets returned to the top level.
Note that I haven't actually described any special features of Haskell here. You can do all of this in just about any programming language, including C++. This is what people mean when they say that learning other types of programming makes them better programmers even in imperative languages they already knew. You can see that your habits of thought are drastically more limited than they need to be; you could not imagine how you could deal with a structure "changing" over the course of your program without having a single variable holding a structure that is mutated, when in fact that is just a small part of the tools that even C++ gives you for approaching the problem. If you can only imagine this one way of dealing with it then you'll never notice when other ways would be more helpful.
Haskell also has a variety of tools it can bring to this problem that are less common in imperative languages, such as (but not limited to):
Using the State monad to automate and hide much of the boilerplate of passing around successive versions of the tree.
Function arguments allow a function to be given an unknown "tree-consumer" function, to which it can give a tree, without any one place both having the tree and knowing which function it's passing it to.
Lazy evaluation sometimes negates the need to even have successive versions of the tree; if the modifications are expanding branches of the tree as you discover they are needed (like a move-tree for a game, say), then you could alternatively generate "the whole tree" up front even if it's infinite, and rely on lazy evaluation to limit how much work is done generating the tree to exactly the amount you need to look at.
Haskell does in fact have mutable variables, it just doesn't have functions that can access mutable variables without exposing in their type that they might have side effects. So if you really want to structure your program exactly as you would in C++ you can; it just won't really "feel like" you're writing Haskell, won't help you learn Haskell properly, and won't allow you to benefit from many of the useful features of Haskell's type system.
I try to merge two signals. One is Mouse.clicks and another is Keyboard.space.
On clicks, I should get a Signal(Int,Int) from Mouse.position as return value
On space, I should get something different so I can identify different signal is triggered.
My idea is:
type Event = Click | Space
mergedSignal : Signal Event
mergedSignal =
let
clickSignal = map (\event -> Click) Mouse.clicks
timeoutSignal = map (\event -> Space) Keyboard.space
in
merge clickSignal timeoutSignal
and get position somehow:
positionOnClickSignal:Signal (Int,Int)
positionOnClickSignal = sampleOn Mouse.clicks Mouse.position
Obviously, it is wrong.
It sounds like you want the mouse position to carry over as part of the event. In that case, you could redefine Event as
type Event
= Click (Int, Int)
| Space
Inside your mergedSignal, the clickSignal is currently just checking for Mouse.clicks but based on your description and other example, I think you actually want that to be based off positionOnclickSignal, which gives you a Signal (Int, Int), and using that, you can now populate the (Int, Int) portion of the Click (Int, Int) event like this:
clickSignal =
map Click positionOnClickSignal
You'll notice that I took out the parenthesis in the above. That is more idiomatic for Elm, because Click is in essence a function that takes one (Int, Int) parameter, which will be passed in from the map function. It could have easily been written like this:
clickSignal =
map (\pos -> Click pos) positionOnClickSignal
Now, if you're just trying to see some debug text of this on screen, a quick and easy way to go about that is to use show from the Graphics.Element package.
import Graphics.Element exposing (show)
main =
map (show << toString) mergedSignal
That will give you some debug text shown as the only thing on the page, and you could easily toss it up on http://elm-lang.org/try for testing.
Is it possible to create a Signal from a List? Essentially what I want is something with the signature List a -> Signal a. I know that a Signal represents a time-varying value and so something like this doesn't actually make any sense (i.e. I can't think of a reason to use it in production code).
I could see applications of it for testing though. For example, imagine some function which depended on the past values of a Signal (via foldp for instance) and you wanted to make assertions about the state of the system given the signal had received values x, y, and z.
Note that there wouldn't have to be anything special about the Signal denoting that it only would ever receive a fixed number of values. I'm thinking of it more like: in production you have a Signal of mouse clicks, and you want to test that from a given starting position, after a given set of clicks, the system should be in some other known state. I know you could simulate this by calling the function a fixed number of times and feeding the results back in with new values, I'm just wondering if it is possible.
I guess it's possible. You use a time-based signal, and map the values from the list over it:
import Time
import Graphics.Element exposing (show)
list = [1..10]
signalFromList : List a -> Signal a
signalFromList list =
let
(Just h) =
List.head list
time =
Time.every Time.second
maybeFlatMap =
flip Maybe.andThen
lists =
Signal.foldp (always <| maybeFlatMap List.tail) (Just list) time
in
Signal.filterMap (maybeFlatMap List.head) h lists
main = Signal.map show <| signalFromList list
However!
It shouldn't be hard to do testing without signals. If you have a foldp somewhere, in a test you can use List.foldl over a list [x,y,z] instead. That should give you the ability to look at the state of your program after inputs x, y, z.
I don't think there is any way it can be done synchronously in pure elm (Apanatshka's answer illustrates well how to set up a sequence of events across time and why it's a bad idea). If we look at how most Signals are defined, we'll see they all head down into a native package at some point.
The question then becomes: can we do this natively?
f : List a -> Signal a
I often think of (Signal a) as 'an a that changes over time'. Here we provide a List of as, and want the function to make it change over time for us.
Before we go any further, I recommend a quick look at Native/Signal.js: https://github.com/elm-lang/core/blob/master/src/Native/Signal.js
Let's say we get down to native land with our List of as. We want something a bit like Signal.constant, but with some extra behaviour that 'sends' each a afterwards. When can we do the sending, though? I am guessing we can't do it during the signal construction function, as we're still building the signal graph. This leaves us a couple of other options:
something heinous with setTimeout, scheduling the sending of each 'a' at an appropriate point in the future
engineering a hook into the elm runtime so that we can run an arbitrary callback at the point when the signal graph is fully constructed
To me at least, the former sounds error prone, and I hope the latter doesn't exist (and never does)!
For testing, your suggestion of using a List fold to mimic the behaviour of foldp would be the way I would go.
I've been learning f# in the previous days, writing a small project which, at last, works (with the help of SO, of course).
I'm trying to learn to be as idiomatic as possible, which basically means that I try to not mutate my data structures. This is costing me a lot of effort :-)
In my search for idiomatic functional programming, I have been trying to use as much as possible lists, tuples and record, rather than objects. But then "praticality beats purity" and so I'm rewriting my small project using objects this time.
I thought that you could give me some advice, surely my idea of "good functional programming design" is not yet very well defined.
For instance I have to modify the nodes of a tree, modifying at the same time the states at two different levels (L and L+1). I've been able to do that without mutating data, but I needed a lot of "inner" and "helper" functions, with accumulators and so on. The nice feeling of being able to clearly express the algorithm was lost for me, due to the need to modify my data structure in an involved way. This is extremely easy in imperative languages, for instance: just dereference the pointers to the relevant nodes, modify their state and iterate over.
Surely I've not designed properly my structure, and for this reason I'm now trying the OOP approach.
I've looked at SICP, at How to design programs and have found a thesis by C. Okasaki ("Purely functional data structures") but the examples on SICP and HTDP are similar to what I did, or maybe I'm not able to understand them fully. The thesis on the other hand is a bit too hard for me at the moment :-)
What do you think about this "tension" which I am experiencing? Am I interpreting the "never mutate data" too strictly? Could you suggest me some resource?
Thanks in advance,
Francesco
When it comes to 'tree update', I think you can always do it pretty elegantly
using catamorphisms (folds over trees). I have a long blog series about this,
and most of the example code below comes from part 4 of the series.
When first learning, I find it best to focus on a particular small, concrete
problem statement. Based on your description, I invented the following problem:
You have a binary tree, where each node contains a "name" and an "amount" (can
think of it like bank accounts or some such). And I want to write a function
which can tell someone to "steal" a certain amount from each of his direct
children. Here's a picture to describe what I mean:
alt text http://oljksa.bay.livefilestore.com/y1pNWjpCPP6MbI3rMfutskkTveCWVEns5xXaOf-NZlIz2Hs_CowykUmwtlVV7bPXRwh4WHJMT-5hSuGVZEhmAIPuw/FunWithTrees.png
On the left I have an original tree. The middle example shows the result I want if node
'D' is asked to steal '10' from each of his children. And the right example
shows what the desired result is if instead I asked 'F' to steal '30' in the original example.
Note that the tree structure I use will be immutable, and the red colors in the
diagram designate "new tree nodes" relative to the original tree. That is black
nodes are shared with the original tree structure (Object.ReferenceEquals to one
another).
Now, assuming a typical tree structure like
type Tree<'T> = //'
| Node of 'T * Tree<'T> * Tree<'T> //'
| Leaf
we'd represent the original tree as
let origTree = Node(("D",1000),
Node(("B",1000),
Node(("A",1000),Leaf,Leaf),
Node(("C",1000),Leaf,Leaf)),
Node(("F",1000),
Node(("E",1000),Leaf,Leaf),
Leaf))
and the "Steal" function is really easy to write, assuming you have the usual "fold"
boilerplate:
// have 'stealerName' take 'amount' from each of its children and
// add it to its own value
let Steal stealerName amount tree =
let Subtract amount = function
| Node((name,value),l,r) -> amount, Node((name,value-amount),l,r)
| Leaf -> 0, Leaf
tree |> XFoldTree
(fun (name,value) left right ->
if name = stealerName then
let leftAmt, newLeft = Subtract amount left
let rightAmt, newRight = Subtract amount right
XNode((name,value+leftAmt+rightAmt),newLeft,newRight)
else
XNode((name,value), left, right))
XLeaf
// examples
let dSteals10 = Steal "D" 10 origTree
let fSteals30 = Steal "F" 30 origTree
That's it, you're done, you've written an algorithm that "updates" levels L and
L+1 of an immutable tree just by authoring the core logic. Rather than explain
it all here, you should go read my blog series (at least the start: parts one two three four).
Here's all the code (that drew the picture above):
// Tree boilerplate
// See http://lorgonblog.spaces.live.com/blog/cns!701679AD17B6D310!248.entry
type Tree<'T> =
| Node of 'T * Tree<'T> * Tree<'T>
| Leaf
let (===) x y = obj.ReferenceEquals(x,y)
let XFoldTree nodeF leafV tree =
let rec Loop t cont =
match t with
| Node(x,left,right) -> Loop left (fun lacc ->
Loop right (fun racc ->
cont (nodeF x lacc racc t)))
| Leaf -> cont (leafV t)
Loop tree (fun x -> x)
let XNode (x,l,r) (Node(xo,lo,ro) as orig) =
if xo = x && lo === l && ro === r then
orig
else
Node(x,l,r)
let XLeaf (Leaf as orig) =
orig
let FoldTree nodeF leafV tree =
XFoldTree (fun x l r _ -> nodeF x l r) (fun _ -> leafV) tree
// /////////////////////////////////////////
// stuff specific to this problem
let origTree = Node(("D",1000),
Node(("B",1000),
Node(("A",1000),Leaf,Leaf),
Node(("C",1000),Leaf,Leaf)),
Node(("F",1000),
Node(("E",1000),Leaf,Leaf),
Leaf))
// have 'stealerName' take 'amount' from each of its children and
// add it to its own value
let Steal stealerName amount tree =
let Subtract amount = function
| Node((name,value),l,r) -> amount, Node((name,value-amount),l,r)
| Leaf -> 0, Leaf
tree |> XFoldTree
(fun (name,value) left right ->
if name = stealerName then
let leftAmt, newLeft = Subtract amount left
let rightAmt, newRight = Subtract amount right
XNode((name,value+leftAmt+rightAmt),newLeft,newRight)
else
XNode((name,value), left, right))
XLeaf
let dSteals10 = Steal "D" 10 origTree
let fSteals30 = Steal "F" 30 origTree
// /////////////////////////////////////////
// once again,
// see http://lorgonblog.spaces.live.com/blog/cns!701679AD17B6D310!248.entry
// DiffTree: Tree<'T> * Tree<'T> -> Tree<'T * bool>
// return second tree with extra bool
// the bool signifies whether the Node "ReferenceEquals" the first tree
let rec DiffTree(tree,tree2) =
XFoldTree (fun x l r t t2 ->
let (Node(x2,l2,r2)) = t2
Node((x2,t===t2), l l2, r r2)) (fun _ _ -> Leaf) tree tree2
open System.Windows
open System.Windows.Controls
open System.Windows.Input
open System.Windows.Media
open System.Windows.Shapes
// Handy functions to make multiple transforms be a more fluent interface
let IdentT() = new TransformGroup()
let AddT t (tg : TransformGroup) = tg.Children.Add(t); tg
let ScaleT x y (tg : TransformGroup) = tg.Children.Add(new ScaleTransform(x, y)); tg
let TranslateT x y (tg : TransformGroup) = tg.Children.Add(new TranslateTransform(x, y)); tg
// Draw: Canvas -> Tree<'T * bool> -> unit
let Draw (canvas : Canvas) tree =
// assumes canvas is normalized to 1.0 x 1.0
FoldTree (fun ((name,value),b) l r trans ->
// current node in top half, centered left-to-right
let tb = new TextBox(Width=100.0, Height=100.0, FontSize=30.0, Text=sprintf "%s:%d" name value,
// the tree is a "diff tree" where the bool represents
// "ReferenceEquals" differences, so color diffs Red
Foreground=(if b then Brushes.Black else Brushes.Red),
HorizontalContentAlignment=HorizontalAlignment.Center,
VerticalContentAlignment=VerticalAlignment.Center)
tb.RenderTransform <- IdentT() |> ScaleT 0.005 0.005 |> TranslateT 0.25 0.0 |> AddT trans
canvas.Children.Add(tb) |> ignore
// left child in bottom-left quadrant
l (IdentT() |> ScaleT 0.5 0.5 |> TranslateT 0.0 0.5 |> AddT trans)
// right child in bottom-right quadrant
r (IdentT() |> ScaleT 0.5 0.5 |> TranslateT 0.5 0.5 |> AddT trans)
) (fun _ -> ()) tree (IdentT())
let TreeToCanvas tree =
let canvas = new Canvas(Width=1.0, Height=1.0, Background = Brushes.Blue,
LayoutTransform=new ScaleTransform(400.0, 400.0))
Draw canvas tree
canvas
let TitledControl title control =
let grid = new Grid()
grid.ColumnDefinitions.Add(new ColumnDefinition())
grid.RowDefinitions.Add(new RowDefinition())
grid.RowDefinitions.Add(new RowDefinition())
let text = new TextBlock(Text = title, HorizontalAlignment = HorizontalAlignment.Center)
Grid.SetRow(text, 0)
Grid.SetColumn(text, 0)
grid.Children.Add(text) |> ignore
Grid.SetRow(control, 1)
Grid.SetColumn(control, 0)
grid.Children.Add(control) |> ignore
grid
let HorizontalGrid (controls:_[]) =
let grid = new Grid()
grid.RowDefinitions.Add(new RowDefinition())
for i in 0..controls.Length-1 do
let c = controls.[i]
grid.ColumnDefinitions.Add(new ColumnDefinition())
Grid.SetRow(c, 0)
Grid.SetColumn(c, i)
grid.Children.Add(c) |> ignore
grid
type MyWPFWindow(content, title) as this =
inherit Window()
do
this.Content <- content
this.Title <- title
this.SizeToContent <- SizeToContent.WidthAndHeight
[<System.STAThread()>]
do
let app = new Application()
let controls = [|
TitledControl "Original" (TreeToCanvas(DiffTree(origTree,origTree)))
TitledControl "D steals 10" (TreeToCanvas(DiffTree(origTree,dSteals10)))
TitledControl "F steals 30" (TreeToCanvas(DiffTree(origTree,fSteals30))) |]
app.Run(new MyWPFWindow(HorizontalGrid controls, "Fun with trees")) |> ignore
I guess if you start your sentence with " I have to modify the nodes of a tree, modifying at the same time the states at two different levels" then you're not really tackling your problem in a functional way. It's like writing a paper in a foreign language by first writing it in your mother tongue, then trying to translate. Doesn't really work. I know it hurts, but in my opinion it's better to immerse yourself completely. Don't worry about comparing the approaches just yet.
One way I've found to learn "the functional way" is to look at (and implement yourself!) some functional pearls. They're basically well document uber-functional elegant progams to solve a variety of problems. Start with the older ones, and don't be afraid to stop reading and try another one if you don't get it. Just come back to it later with renewed enthousiasm and more experience. It helps :)
What do you think about this "tension"
which I am experiencing? Am I
interpreting the "never mutate data"
too strictly? Could you suggest me
some resource?
In my opinion, if you're learning functional programming for the first time, its best to start out with zero mutable state. Otherwise, you'll only end up falling back on mutable state as your first resort, and all of your F# code will be C# with a little different syntax.
Regarding data structures, some are easier to express in a functional style than others. Could you provide a description of how you're trying to modify your tree?
For now, I would recommend F# Wikibook's page on data structures to see how data structures are written in a functional style.
I've looked at SICP, at How to design
programs and have found a thesis by C.
Okasaki ("Purely functional data
structures")
I personally found Okasaki's book more readable than the thesis online.
I have to modify nodes of a tree.
No you don't. That's your problem right there.
This is costing me a lot of effort
This is typical. It's not easy to learn to program with immutable data structures. And to most beginners, it seems unnatural at first. It's doubly difficult because HTDP and SICP don't give you good models to follow (see footnote).
I thought that you could give me some advice, surely my idea of "good functional programming design" is not yet very well defined.
We can, but you have to tell us what the problem is. Then many people on this forum can tell you if it is the sort of problem whose solution can be expressed in a clear way without resorting to mutation. Most tree problems can. But with the information you've given us, there's no way for us to tell.
Am I interpreting the "never mutate data" too strictly?
Not strictly enough, I'd say.
Please post a question indicating what problem you're trying to solve.
Footnote: both HTDP and SICP are done in Scheme, which lacks pattern matching. In this setting it is much harder to understand tree-manipulation code than it is using the pattern matching provided by F#. As far as I'm concerned, pattern matching is an essential feature for writing clear code in a purely functional style. For resources, you might consider Graham Hutton's new book on Programming in Haskell.
Take a look at the Zipper data structure.
For instance I have to modify the nodes of a tree, modifying at the same time the states at two different levels (L and L+1)
Why? In a functional language, you'd create a new tree instead. It can reuse the subtrees which don't need to be modified, and just plug them into a newly created root. "Don't mutate data" doesn't mean "try to mutate data without anyone noticing, and by adding so many helper methods that no one realize that this is what you're doing".
It means "don't mutate your data. Create new copies instead, which are initialized with the new, correct, values".