I was a bit confused by the documentation for fix (although I think I understand what it's supposed to do now), so I looked at the source code. That left me more confused:
fix :: (a -> a) -> a
fix f = let x = f x in x
How exactly does this return a fixed point?
I decided to try it out at the command line:
Prelude Data.Function> fix id
...
And it hangs there. Now to be fair, this is on my old macbook which is kind of slow. However, this function can't be too computationally expensive since anything passed in to id gives that same thing back (not to mention that it's eating up no CPU time). What am I doing wrong?
You are doing nothing wrong. fix id is an infinite loop.
When we say that fix returns the least fixed point of a function, we mean that in the domain theory sense. So fix (\x -> 2*x-1) is not going to return 1, because although 1 is a fixed point of that function, it is not the least one in the domain ordering.
I can't describe the domain ordering in a mere paragraph or two, so I will refer you to the domain theory link above. It is an excellent tutorial, easy to read, and quite enlightening. I highly recommend it.
For the view from 10,000 feet, fix is a higher-order function which encodes the idea of recursion. If you have the expression:
let x = 1:x in x
Which results in the infinite list [1,1..], you could say the same thing using fix:
fix (\x -> 1:x)
(Or simply fix (1:)), which says find me a fixed point of the (1:) function, IOW a value x such that x = 1:x... just like we defined above. As you can see from the definition, fix is nothing more than this idea -- recursion encapsulated into a function.
It is a truly general concept of recursion as well -- you can write any recursive function this way, including functions that use polymorphic recursion. So for example the typical fibonacci function:
fib n = if n < 2 then n else fib (n-1) + fib (n-2)
Can be written using fix this way:
fib = fix (\f -> \n -> if n < 2 then n else f (n-1) + f (n-2))
Exercise: expand the definition of fix to show that these two definitions of fib are equivalent.
But for a full understanding, read about domain theory. It's really cool stuff.
I don't claim to understand this at all, but if this helps anyone...then yippee.
Consider the definition of fix. fix f = let x = f x in x. The mind-boggling part is that x is defined as f x. But think about it for a minute.
x = f x
Since x = f x, then we can substitute the value of x on the right hand side of that, right? So therefore...
x = f . f $ x -- or x = f (f x)
x = f . f . f $ x -- or x = f (f (f x))
x = f . f . f . f . f . f . f . f . f . f . f $ x -- etc.
So the trick is, in order to terminate, f has to generate some sort of structure, so that a later f can pattern match that structure and terminate the recursion, without actually caring about the full "value" of its parameter (?)
Unless, of course, you want to do something like create an infinite list, as luqui illustrated.
TomMD's factorial explanation is good. Fix's type signature is (a -> a) -> a. The type signature for (\recurse d -> if d > 0 then d * (recurse (d-1)) else 1) is (b -> b) -> b -> b, in other words, (b -> b) -> (b -> b). So we can say that a = (b -> b). That way, fix takes our function, which is a -> a, or really, (b -> b) -> (b -> b), and will return a result of type a, in other words, b -> b, in other words, another function!
Wait, I thought it was supposed to return a fixed point...not a function. Well it does, sort of (since functions are data). You can imagine that it gave us the definitive function for finding a factorial. We gave it a function that dind't know how to recurse (hence one of the parameters to it is a function used to recurse), and fix taught it how to recurse.
Remember how I said that f has to generate some sort of structure so that a later f can pattern match and terminate? Well that's not exactly right, I guess. TomMD illustrated how we can expand x to apply the function and step towards the base case. For his function, he used an if/then, and that is what causes termination. After repeated replacements, the in part of the whole definition of fix eventually stops being defined in terms of x and that is when it is computable and complete.
You need a way for the fixpoint to terminate. Expanding your example it's obvious it won't finish:
fix id
--> let x = id x in x
--> id x
--> id (id x)
--> id (id (id x))
--> ...
Here is a real example of me using fix (note I don't use fix often and was probably tired / not worrying about readable code when I wrote this):
(fix (\f h -> if (pred h) then f (mutate h) else h)) q
WTF, you say! Well, yes, but there are a few really useful points here. First of all, your first fix argument should usually be a function which is the 'recurse' case and the second argument is the data on which to act. Here is the same code as a named function:
getQ h
| pred h = getQ (mutate h)
| otherwise = h
If you're still confused then perhaps factorial will be an easier example:
fix (\recurse d -> if d > 0 then d * (recurse (d-1)) else 1) 5 -->* 120
Notice the evaluation:
fix (\recurse d -> if d > 0 then d * (recurse (d-1)) else 1) 3 -->
let x = (\recurse d -> if d > 0 then d * (recurse (d-1)) else 1) x in x 3 -->
let x = ... in (\recurse d -> if d > 0 then d * (recurse (d-1)) else 1) x 3 -->
let x = ... in (\d -> if d > 0 then d * (x (d-1)) else 1) 3
Oh, did you just see that? That x became a function inside our then branch.
let x = ... in if 3 > 0 then 3 * (x (3 - 1)) else 1) -->
let x = ... in 3 * x 2 -->
let x = ... in 3 * (\recurse d -> if d > 0 then d * (recurse (d-1)) else 1) x 2 -->
In the above you need to remember x = f x, hence the two arguments of x 2 at the end instead of just 2.
let x = ... in 3 * (\d -> if d > 0 then d * (x (d-1)) else 1) 2 -->
And I'll stop here!
How I understand it is, it finds a value for the function, such that it outputs the same thing you give it. The catch is, it will always choose undefined (or an infinite loop, in haskell, undefined and infinite loops are the same) or whatever has the most undefineds in it. For example, with id,
λ <*Main Data.Function>: id undefined
*** Exception: Prelude.undefined
As you can see, undefined is a fixed point, so fix will pick that. If you instead do (\x->1:x).
λ <*Main Data.Function>: undefined
*** Exception: Prelude.undefined
λ <*Main Data.Function>: (\x->1:x) undefined
[1*** Exception: Prelude.undefined
So fix can't pick undefined. To make it a bit more connected to infinite loops.
λ <*Main Data.Function>: let y=y in y
^CInterrupted.
λ <*Main Data.Function>: (\x->1:x) (let y=y in y)
[1^CInterrupted.
Again, a slight difference. So what is the fixed point? Let us try repeat 1.
λ <*Main Data.Function>: repeat 1
[1,1,1,1,1,1, and so on
λ <*Main Data.Function>: (\x->1:x) $ repeat 1
[1,1,1,1,1,1, and so on
It is the same! Since this is the only fixed point, fix must settle on it. Sorry fix, no infinite loops or undefined for you.
As others pointed out, fix somehow captures the essence of recursion. Other answers already do a great job at explaining how fix works. So let's take a look from another angle and see how fix can be derived by generalising, starting from a specific problem: we want to implement the factorial function.
Let's first define a non recursive factorial function. We will use it later to "bootstrap" our recursive implementation.
factorial n = product [1..n]
We approximate the factorial function by a sequence of functions: for each natural number n, factorial_n coincides with factorial up to and including n. Clearly factorial_n converges towards factorial for n going towards infinity.
factorial_0 n = if n == 0 then 1 else undefined
factorial_1 n = n * factorial_0(n - 1)
factorial_2 n = n * factorial_1(n - 1)
factorial_3 n = n * factorial_2(n - 1)
...
Instead of writing factorial_n out by hand for every n, we implement a factory function that creates these for us. We do this by factoring the commonalities out and abstracting over the calls to factorial_[n - 1] by making them a parameter to the factory function.
factorialMaker f n = if n == 0 then 1 else n * f(n - 1)
Using this factory, we can create the same converging sequence of functions as above. For each factorial_n we need to pass a function that calculates the factorials up to n - 1. We simply use the factorial_[n - 1] from the previous step.
factorial_0 = factorialMaker undefined
factorial_1 = factorialMaker factorial_0
factorial_2 = factorialMaker factorial_1
factorial_3 = factorialMaker factorial_2
...
If we pass our real factorial function instead, we materialize the limit of the series.
factorial_inf = factorialMaker factorial
But since that limit is the real factorial function we have factorial = factorial_inf and thus
factorial = factorialMaker factorial
Which means that factorial is a fixed-point of factorialMaker.
Finally we derive the function fix, which returns factorial given factorialMaker. We do this by abstracting over factorialMaker and make it an argument to fix. (i.e. f corresponds to factorialMaker and fix f to factorial):
fix f = f (fix f)
Now we find factorial by calculating the fixed-point of factorialMaker.
factorial = fix factorialMaker
Related
How can I write a type declaration and function in Haskell that takes either a function (that itself takes no arguments) or a value. When given a function it calls the function. When given a value it returns the value.
[edit] To give more context, I'm mostly curious how to solve this problem in Haskell without bit twiddling: Designing function f(f(n)) == -n
Sean
I'm mostly curious how to solve this problem in Haskell without bit twiddling: Designing function f(f(n)) == -n
That's actually quite easy to solve:
when :: (a -> Bool) -> (a -> a) -> a -> a
when p f x = if p x then f x else x
f :: Integer -> Integer
f = (+) <$> when even negate <*> signum
How do we derive this? Consider:
f (f n) = (-n) -- (0) - from the interview question
f x = y -- (1) - assumption
f y = (-x) -- (2) - from (0) and (1), f (f x) = (-x)
f (-x) = (-y) -- (3) - from (0) and (2), f (f y) = (-y)
f (-y) = x -- (4) - from (0) and (3), f (f (-x)) = x
Now, if you see the left hand sides of these equations then you'll notice that there are four cases:
f x.
f y.
f (-x).
f (-y).
Notice that the domain of the function f is divided into positive and negative numbers, x and (-x), and y and (-y). Let's assume that x and y together form the set of positive numbers and (-x) and (-y) together form the set of negative numbers.
The set of positive numbers is divided into two proper disjoint subsets, x and y. How do we divide the set of positive numbers into two proper disjoint subsets? Odd and even numbers are a good candidate. Hence, let's assume that x is the set of positive odd numbers and y is the set of positive even numbers.
Another advantage of using odd and even numbers is that when negated odd numbers remain odd and even numbers remain even. Hence, (-x) is the set of negative odd numbers and (-y) is the set of negative even numbers.
Now, consider the four cases again. Notice that the sign only changes when the number is even:
f x = y (sign does not change).
f y = (-x) (sign changes).
f (-x) = (-y) (sign does not change).
f (-y) = x (sign changes).
Hence, we only negate the number when it is even (i.e. when even negate).
Next, we need to convert odd numbers into even numbers and vice versa. The easiest way to do so is to add or subtract one from the number. However, care should be taken that the resulting number is not 0. Consider the special case of 0:
f 0 = z -- (a) - assumption
f z = (-0) -- (b) - from (0) and (a), f (f 0) = (-0)
f (-0) = (-z) -- (c) - from (0) and (b), f (f z) = (-z)
(-0) = 0 -- (d) - reflexivity
f (-0) = f 0 -- (e) - from (d)
(-z) = z -- (f) - from (a) and (c) and (e)
z = 0 -- (g) - from (d) and (f)
f 0 = 0 -- (h) - from (a) and (g)
Hence, f n = 0 if and only if n = 0. So let's consider the neighbors of 0, 1 and (-1). Both of them are odd numbers. Hence, they are not negated. However, they do need to be converted into an even number (except for 0). Hence, 1 is converted into 2 and (-1) is converted into (-2).
Thus, for odd numbers we simply add the sign of the number to the number itself.
Now, consider even numbers. We know:
f 1 = 2 -- (A)
f (-1) = (-2) -- (B)
Therefore:
f 2 = (-1) -- (C), from (0) and (A), f (f 1) = (-1)
f (-2) = 1 -- (D), from (0) and (B), f (f (-1)) = 1
We know that even numbers are always negated. Hence, 2 first becomes (-2) and vice versa. Let the original even number be n. Hence, first we negate n and then add signum n to it:
evenF n = negate n + signum n
evenF 2 = negate 2 + signum 2
= (-2) + 1
= (-1)
evenF (-2) = negate (-2) + signum (-2)
= 2 + (-1)
= 1
evenF 0 = negate 0 + signum 0
= 0 + 0
= 0
Thus, for both the odd case and the even case we add the sign of the original number to when even negate. Therefore, f is defined as:
f :: Integer -> Integer
f = (+) <$> when even negate <*> signum
Hope that helps.
You can't write a function with two different signatures (unless you use typeclasses, but typeclasses are not suitable for this problem). You must solve this in a way that lets you treat both functions and non-function values as the same type. There are two obvious options:
Use a sum type.
f :: Either (Int -> Char) Char -> Char
f (Left g) = g 1
f (Right c) = c
Use const to convert your non-function value into a function that ignores its argument:
f = ($ 42)
f chr --> '*'
f (const 'a') --> 'a'
However, since this is a very unHaskelly thing to ask for, I suspect that this is an XY problem.
You could do it like this:
data FunctionOrValue a
= Function (() -> a)
| Value a
getValue :: FunctionOrValue a -> a
getValue (Function f) = f ()
getValue (Value x) = x
However this is a bit silly.
It sounds like you're trying to defer values manually, but since Haskell is lazy, there's not normally a need to do this.
An answer based on the interview question you posted:
f n = if (abs fracN) > 1 then 1/fracN else - (1/fracN)
where
fracN = realToFrac n
The question specified that the input is an int; it did not specify that the result must also be an int.
Edit: if you must return an Int, note that the question allows you to specify a range of possible inputs. I use a limit of 1073741823 (half of the max value of a signed 32-bit int), which allows me to write this:
fint :: Int -> Int
fint 0 = 0
fint n = if (abs n) <= rangeVal then n+addend else -(n-addend)
where
rangeVal = 1073741823
negator = if n < 0 then -1 else 1
addend = negator*rangeVal
One of the nice thing in Haskell (IMHO) is there is no difference between a value and a function without parameter returning a value (thanks to lazyness AND purity). Or if you prefer, every value is in fact a function without parameter which will be evaluated when needed. Therefore there is no need to worry about that type of problem. There is no such thing like f(), it's just f.
For example, you can write
x = 3 :: Int
f = head [] :: Int -- should blow up but doesn't
head [x, f] -- note that f and x have the same type
> 3 -- doesn't blow up on f, because f is not called
head [f] -- blows up, when trying to print f
My homework went really well until I stumpled upon the last task.
First, I had to define a custom List structure:
data List a = Nil | Cons a (List a) deriving Show
Another task was to write a custom fold function:
foldList :: (a -> b -> b) -> b -> List a -> b
foldList f b Nil = b
foldList f b (Cons a l) = f a (foldList f b l)
The second parameter is the value that is used at the end of the list (at a Nil element).
I also had to write a function prodList that multiplies every element of the provided list with each other:
prodList :: List Int -> Int
prodList = foldList (\x y -> x * y) 1
The 1 at the end is the neutral element of multplication. It therefore has no effect on the calculation.
The last one, though, is hard for me to solve.
I have to write a function binList that calculates the decimal value of list that represents a binary number. The least significant bit is the first element of the list, the binary number is therefore reversed.
A given example is that the result of binList (Cons 1 (Cons 0 (Cons 0 (Cons 0 (Cons 1 Nil))))) should be 19 (since (10001)_2 is (19)_10). The result of the list [1,1,0,1], however, should be (1011)_2=(11)_10).
The culprit of the assignment is, that we have to use foldList.
I know how to calculate each digit, but I struggle to find a way to find out which index i I'm currently at:
binList :: List Int -> Int
binList = foldList (\x y -> 2^i*x + y)
There probably is a nice, curry way to solve this in Haskell. Could you explain to me how you would solve this assignment?
If you were to write out the calculation, it would look like this:
x0 + 2 * x1 + 4 * x2 + 8 * x3 + ...
This might suggest you need to use the index, but if you factorize this expression, you get this instead:
x0 + 2 * (x1 + 2 * (x2 + 2 * (x3 ...
Do you see how it can be written as a fold now? Notice that there is a self-similarity that looks kind of like this:
x + 2 * x'
Hopefully that's enough of a hint for you :)
I am trying to understand lambda functions (i.e. anonymous functions) in Haskell by writing a few simple functions that use them.
In the following example, I am simply trying to take in 3 parameters and add two of the three using an anonymous function and adding the third parameter after that.
I am getting an error saying that I must declare an instance first.
specialAdd x y z = (\x y -> x + y) + z
I appreciate any explanation of why my example is not working and/or any explanation that would help me better understand lambda functions.
specialAdd x y z = (\x y -> x + y) + z
In this example, what you are trying to do is add a function to a number, which is not going to work. Look at (\x y -> x + y) + z: it has the form a + b. In order for such an expression to work, the a part and the b part must be numbers of the same type.
Haskell is a bit of an unusual language, so its error messages are rarely of the form "you can't do that". So what's going on here is that Haskell sees that (\x y -> x + y) is a function, and since in an expression like a + b, b must be the same type as a, it concludes that b must also be a function. Haskell also allows you to define your own rules for adding non-built-in types; so it can't just give you an error saying "you can't add two functions," but instead the error is "you have not defined a rule that allows me to add two functions."
The following would do what you want:
specialAdd x y z = ((\x y -> x + y) x y) + z
Here you are applying the function (\x y -> x + y) to arguments x and y, then adding the result to z.
A good way to practice anonymous function is to use them with high order function as fold or map.
Using map as an entry point,
Basic definition of map,
map f [] = []
map f (x:xs) = f x : f xs
Built up an example,
>>> let list = [0..4]
>>> let f x = x + 1
Applying map we obtain,
>>> map f list
[1,2,3,4,5]
Now, we can omit the declaration of f and replace it using anonymous function,
>>> map (\x->x+1) list
[1,2,3,4,5]
then we deduce, map f list == map (\x->x+1) list, thus
f = \x-> x + 1 --- The same as f x = x + 1, but this is the equivalent lambda notation.
then starting with a simple function we see how to translate it into an anonymous function and then how an anonymous function can be rely to a lambda abstraction.
As an exercise try to translate f x = 2*x.
Now more complex, an anonymous function which take two arguments,
Again an working example,
>>> let add x y = x + y
>>> foldl' add 0 [0..4]
10
Can be rewrite using anonymous function as,
>>> foldl' (\x y -> x + y) 0 [0..4]
Again using equality we deduce that add = \x y -> x + y
Moreover as in hakell all function are function of one argument, and we can partial apply it, we can rewrite our previous anonymous function as, add = \x -> (\y -> x + y).
Then where is the trick ?? Because, I just show the use of anonymous function into high order one, and starting from that, showing how this can be exploited to rewrite function using lambda notation. I mean how can it help you to learn how to write down anonymous function ?
Simply cause I've give you (show you) an existing framework using high order function.
This framework is a huge opportunity to accommodate you with this notation.
Starting from that an infinity range of exercise can be deduce, for example try to do the following.
A - Find the corresponding anonymous function ?
1 - let f (x,y) = x + y in map f [(0,1),(2,3),(-1,1)]
2 - let f x y = x * y in foldl' f 1 [1..5]
B - Rewrite all of them using lambda notation into a declarative form (f = \x-> (\y-> ...)
And so on ....
To summarize,
A function as
(F0) f x1 x2 ... xn = {BODY of f}
can always be rewrite as,
(F1) f = \x1 x2 ... xn -> {BODY of f}
where
(F2) (\x1 x2 ... xn -> {BODY of f})
F2 form are just anonymous function, a pure translation of the function into lambda calculus form. F1 is a declarative lambda notation (because we declare f, as we define it, binding it to the anonymous F2). F0 being the usual notation of Haskeller.
A last note focusing on the fact we can put parenthesis between the argument, this create a closure. Doing that mean that a subset of the function's code can be fully evaluated using a subset of the function's argument, (mean converting to a form where no more free variable occurs), but that's another story.
Here is correct form:
specialAdd a b c = ((\x y -> x + y) a b) + c
Example from Learn You a Haskell...:
zipWith (\a b -> (a * 30 + 3) / b) [5,4,3,2,1] [1,2,3,4,5]
Great explanation:
http://learnyouahaskell.com/higher-order-functions#lambdas
From what I understand Labmbda/Anonymous functions help you declare a function "inline" without the need to give it a name. The "\" (ASCII for the Greek, λ) precedes the variable names for the expression that follows the "->". For example,
(\x y -> x + y)
is an anonymous (lambda) function similar to (+). It takes two parameters of type Num and returns their sum:
Prelude> :type (+)
(+) :: Num a => a -> a -> a
Prelude> :type (\x y -> x + y)
(\x y -> x + y) :: Num a => a -> a -> a
Your example is not working because, as others have pointed out, the right hand side of it is using a lambda function, (\x y -> x + y), as a parameter for the (+) operator, which is defined by default only for parameters of type Num. Some of the beauty of the lambda function can be in its "anonymous" use. Vladimir showed how you can use the lambda function in your declaration by passing it the variables from the left side. A more "anonymous" use could be simply calling it, with variables, without giving the function a name (hence anonymous). For example,
Prelude> (\x y z -> x + y + z) 1 2 3
6
and if you like writing parentheses:
Prelude> (((+).) . (+)) 1 2 3
6
Or in a longer expression (as in your example declaration), e.g.,
Prelude> filter (\x -> length x < 3) [[1],[1,2],[1,2,3]]
[[1],[1,2]]
You are trying to use (+) as something like (Num a) => (a -> a -> a) -> a -> ?? which is not correct.
(+) is defined in the class Num and (a -> a -> a) is not an instance of this class.
What exactly are you trying to achieve ?
The fixed point combinator doesn't always produce the right answer given the definition:
fix f = f (fix f)
The following code does not terminate:
fix (\x->x*x) 0
Of course, fix can't always produce the right answer, but I was wondering, can this be improved?
Certainly for the above example, one can implement some fix that looks like
fix f x | f x == f (f x) = f x
| otherwise = fix f (f x)
and gives the correct output.
What is the reason the above definition (or something even better, as this one only handle function with 1 parameter) is not used instead?
Fixed point combinator finds the least-defined fixed point of a function, which is ⊥ in your case (non-termination indeed is undefined value).
You can check, that in your case
(\x -> x * x) ⊥ = ⊥
i.e. ⊥ really is fixed point of \x -> x * x.
As for why is fix defined that way: the main point of fix is to allow you use anonymous recursion and for that you do not need more sophisticated definition.
Your example does not even typecheck:
Prelude> fix (\x->x*x) 0
<interactive>:1:11:
No instance for (Num (a0 -> t0))
arising from a use of `*'
Possible fix: add an instance declaration for (Num (a0 -> t0))
In the expression: x * x
In the first argument of `fix', namely `(\ x -> x * x)'
In the expression: fix (\ x -> x * x) 0
And that gives the clue as to why it doesn't work as you expect. The x in your anonymous function is supposed to be a function, not a number. The reason for this is, as Vitus suggests, that a fixpoint combinator is a way to write recursion without actually writing recursion. The general idea is that a recursive definition like
f x = if x == 0 then 1 else x * f (x-1)
can be written as
f = fix (\f' x -> if x == 0 then 1 else x * f' (x-1))
Your example
fix (\x->x*x) 0
thus corresponds to the expression
let x = x*x in x 0
which makes no sense.
I'm not entirely qualified to talk about what the "fixpoint combinator" is, or what the "least fixed point" is, but it is possible to use a fix-esque technique to approximate certain functions.
Translating Scala by Example section 4.4 to Haskell:
sqrt' :: Double -> Double
sqrt' x = sqrtIter 1.0
where sqrtIter guess | isGoodEnough guess = guess
| otherwise = sqrtIter (improve guess)
improve guess = (guess + x / guess) / 2
isGoodEnough guess = abs (guess * guess - x) < 0.001
This function works by repeatedly "improving" a guess until we determine that it is "good enough". This pattern can be abstracted:
myFix :: (a -> a) -- "improve" the guess
-> (a -> Bool) -- determine if a guess is "good enough"
-> a -- starting guess
-> a
fixApprox improve isGoodEnough startGuess = iter startGuess
where iter guess | isGoodEnough guess = guess
| otherwise = iter (improve guess)
sqrt'' :: Double -> Double
sqrt'' x = myFix improve isGoodEnough 1.0
where improve guess = (guess + x / guess) / 2
isGoodEnough guess = abs (guess * guess - x) < 0.001
See also Scala by Example section 5.3. fixApprox can be used to approximate the fixed point of the improve function passed into it. It repeatedly invokes improve on the input until the output isGoodEnough.
In fact, you can use myFix not only for approximations, but for exact answers as well.
primeAfter :: Int -> Int
primeAfter n = myFix improve isPrime (succ n)
where improve = succ
isPrime x = null [z | z <- [2..pred x], x `rem` z == 0]
This is a pretty dumb way to generate primes, but it illustrates the point. Hm...now I wonder...does something like myFix already exist? Stop...Hoogle time!
Hoogling (a -> a) -> (a -> Bool) -> a -> a, the very first hit is until.
until p f yields the result of applying f until p holds.
Well there you have it. As it turns out, myFix = flip until.
You probably meant iterate:
*Main> take 8 $ iterate (^2) (0.0 ::Float)
[0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0]
*Main> take 8 $ iterate (^2) (0.001 ::Float)
[1.0e-3,1.0000001e-6,1.0000002e-12,1.0000004e-24,0.0,0.0,0.0,0.0]
*Main> take 8 $ iterate (^2) (0.999 ::Float)
[0.999,0.99800104,0.9960061,0.9920281,0.9841198,0.96849173,0.93797624,0.8797994]
*Main> take 8 $ iterate (^2) (1.0 ::Float)
[1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0]
*Main> take 8 $ iterate (^2) (1.001 ::Float)
[1.001,1.002001,1.0040061,1.0080284,1.0161213,1.0325024,1.0660613,1.1364866]
Here you have all the execution history explicitly available for your analysis. You can attempt to detect the fixed point with
fixed f from = snd . head
. until ((< 1e-16).abs.uncurry (-).head) tail
$ _S zip tail history
where history = iterate f from
_S f g x = f x (g x)
and then
*Main> fixed (^2) (0.999 :: Float)
0.0
but trying fixed (^2) (1.001 :: Float) will loop indefinitely, so you'd need to develop separate testing for convergence, and even then detection of repellent fixed points like 1.0 will need more elaborate investigation.
You can't define fix the way you've mentioned since f x may not even be comparable. For instance, consider the example below:
myFix f x | f x == f (f x) = f x
| otherwise = myFix f (f x)
addG f a b =
if a == 0 then
b
else
f (a - 1) (b + 1)
add = fix addG -- Works as expected.
-- addM = myFix addG (Compile error)
I am not sure if this is good programming practice, but I would like to know if one can define a recursive function using the lambda expression.
This is an artificial example I made up: So one can defined the factorial function in Haskell recursively as follows
factorial :: Integer -> Integer
factorial 1 = 1
factorial (n + 1) = (n + 1) * factorial n
Now, I want a function f such that f n = (factorial n) + 1. Rather than using a name for factorial n (i.e. defining it before hand), I want to define f where factorial n is given a lambda expression within the definition of f. Can I use a recursive lambda definition in f in place of using the name factorial?
The canonical way to do recursion with pure lambda expressions is to use a fixpoint combinator, which is a function with the property
fixpoint f x = f (fixpoint f) x
If we assume that such a combinator exists, we can write the recursive function as
factorial = fixpoint (\ff n -> if n == 1 then 1 else n * ff(n-1))
The only problem is that fixpoint itself is still recursive. In the pure lambda calculus, there are ways to create fixpoint combinators that consist only of lambdas, for example the classical "Y combinator":
fixpoint f = (\x -> f (x x)) (\x -> f (x x))
But we still have problems, because this definition is not well-typed according to Haskell -- and it can be proved that there is no way to write a well-typed fixpoint combinator using only lambdas and function applications. It can be done by use of an auxiliary data type to shoehorn in some type recursion:
data Paradox a = Self (Paradox a -> a)
fixpoint f = let half (Self twin) = f (twin (Self twin))
in half (Self half)
(Note that if the injections and projections from the singleton data type are removed, this is exactly the Y combinator!)
Yes, using the fixed-point function fix:
fact :: Int -> Int
fact = fix (\f n -> if n == 0 then 1 else n * (f (n-1)))
Basically, it doesn't have a name, since it's a lambda expression, so you take the function in as an argument. Fix applies the function to itself "infinitely" many times:
fix f = f (fix f)
and is defined in Data.Function.
Why do we use lambda's instead of let in?
Prelude> (let fib n = if n == 1 then 1 else n * fib(n-1) in fib ) 4
24
Owen is spot on with the fix function
Prelude> fix f = f (fix f)
Prelude> fix (\f n -> if n == 0 then 1 else n * f (n - 1)) 6
720
The nameless Lambda function is self terminating even if fix is not.
Owen's fix function beats the fix function typically used in Haskell which is
fix :: (a -> a) -> a
fix f = let {x = f x} in x
Both allow anonymous recursive lambda functions