I have the following code which gives out [2,4,6] :
j :: [Int]
j = ((\f x -> map x) (\y -> y + 3) (\z -> 2*z)) [1,2,3]
Why? It seems that just the "z-function" is being used, what happens to the "y-function"?
And how does map work in this particular case?
Let's compute:
((\f x -> map x) (\y -> y + 3) (\z -> 2*z)) [1,2,3]
^^^ f ^^^^^^^ ^^^ x ^^^^^
=
(map x) [1,2,3]
where f = \y -> y +3
x = \z -> 2*z
=
[x 1,x 2,x 3]
where f = \y -> y +3
x = \z -> 2*z
=
[2*1,2*2,2*3]
where f = \y -> y +3
x = \z -> 2*z
=
[2,4,6]
where f = \y -> y +3
x = \z -> 2*z
As we can see, f was taken as an argument, but was never used after that. Consequently \y -> y+3 never affected the final result.
The function map x is the function that applies function x to each element of a list. Note that, above, (map x) [1,2,3] is the same as map x [1,2,3]. Indeed, each function application g x1 x2 x3 x4 can be equivalently written as (((g x1) x2) x3) x4 by left-associating the applications.
In pure functional languages like Haskell, is there an algorithm to get the inverse of a function, (edit) when it is bijective? And is there a specific way to program your function so it is?
In some cases, yes! There's a beautiful paper called Bidirectionalization for Free! which discusses a few cases -- when your function is sufficiently polymorphic -- where it is possible, completely automatically to derive an inverse function. (It also discusses what makes the problem hard when the functions are not polymorphic.)
What you get out in the case your function is invertible is the inverse (with a spurious input); in other cases, you get a function which tries to "merge" an old input value and a new output value.
No, it's not possible in general.
Proof: consider bijective functions of type
type F = [Bit] -> [Bit]
with
data Bit = B0 | B1
Assume we have an inverter inv :: F -> F such that inv f . f ≡ id. Say we have tested it for the function f = id, by confirming that
inv f (repeat B0) -> (B0 : ls)
Since this first B0 in the output must have come after some finite time, we have an upper bound n on both the depth to which inv had actually evaluated our test input to obtain this result, as well as the number of times it can have called f. Define now a family of functions
g j (B1 : B0 : ... (n+j times) ... B0 : ls)
= B0 : ... (n+j times) ... B0 : B1 : ls
g j (B0 : ... (n+j times) ... B0 : B1 : ls)
= B1 : B0 : ... (n+j times) ... B0 : ls
g j l = l
Clearly, for all 0<j≤n, g j is a bijection, in fact self-inverse. So we should be able to confirm
inv (g j) (replicate (n+j) B0 ++ B1 : repeat B0) -> (B1 : ls)
but to fulfill this, inv (g j) would have needed to either
evaluate g j (B1 : repeat B0) to a depth of n+j > n
evaluate head $ g j l for at least n different lists matching replicate (n+j) B0 ++ B1 : ls
Up to that point, at least one of the g j is indistinguishable from f, and since inv f hadn't done either of these evaluations, inv could not possibly have told it apart – short of doing some runtime-measurements on its own, which is only possible in the IO Monad.
⬜
You can look it up on wikipedia, it's called Reversible Computing.
In general you can't do it though and none of the functional languages have that option. For example:
f :: a -> Int
f _ = 1
This function does not have an inverse.
Not in most functional languages, but in logic programming or relational programming, most functions you define are in fact not functions but "relations", and these can be used in both directions. See for example prolog or kanren.
Tasks like this are almost always undecidable. You can have a solution for some specific functions, but not in general.
Here, you cannot even recognize which functions have an inverse. Quoting Barendregt, H. P. The Lambda Calculus: Its Syntax and Semantics. North Holland, Amsterdam (1984):
A set of lambda-terms is nontrivial if it is neither the empty nor the full set. If A and B are two nontrivial, disjoint sets of lambda-terms closed under (beta) equality, then A and B are recursively inseparable.
Let's take A to be the set of lambda terms that represent invertible functions and B the rest. Both are non-empty and closed under beta equality. So it's not possible to decide whether a function is invertible or not.
(This applies to the untyped lambda calculus. TBH I don't know if the argument can be directly adapted to a typed lambda calculus when we know the type of a function that we want to invert. But I'm pretty sure it will be similar.)
If you can enumerate the domain of the function and can compare elements of the range for equality, you can - in a rather straightforward way. By enumerate I mean having a list of all the elements available. I'll stick to Haskell, since I don't know Ocaml (or even how to capitalise it properly ;-)
What you want to do is run through the elements of the domain and see if they're equal to the element of the range you're trying to invert, and take the first one that works:
inv :: Eq b => [a] -> (a -> b) -> (b -> a)
inv domain f b = head [ a | a <- domain, f a == b ]
Since you've stated that f is a bijection, there's bound to be one and only one such element. The trick, of course, is to ensure that your enumeration of the domain actually reaches all the elements in a finite time. If you're trying to invert a bijection from Integer to Integer, using [0,1 ..] ++ [-1,-2 ..] won't work as you'll never get to the negative numbers. Concretely, inv ([0,1 ..] ++ [-1,-2 ..]) (+1) (-3) will never yield a value.
However, 0 : concatMap (\x -> [x,-x]) [1..] will work, as this runs through the integers in the following order [0,1,-1,2,-2,3,-3, and so on]. Indeed inv (0 : concatMap (\x -> [x,-x]) [1..]) (+1) (-3) promptly returns -4!
The Control.Monad.Omega package can help you run through lists of tuples etcetera in a good way; I'm sure there's more packages like that - but I don't know them.
Of course, this approach is rather low-brow and brute-force, not to mention ugly and inefficient! So I'll end with a few remarks on the last part of your question, on how to 'write' bijections. The type system of Haskell isn't up to proving that a function is a bijection - you really want something like Agda for that - but it is willing to trust you.
(Warning: untested code follows)
So can you define a datatype of Bijection s between types a and b:
data Bi a b = Bi {
apply :: a -> b,
invert :: b -> a
}
along with as many constants (where you can say 'I know they're bijections!') as you like, such as:
notBi :: Bi Bool Bool
notBi = Bi not not
add1Bi :: Bi Integer Integer
add1Bi = Bi (+1) (subtract 1)
and a couple of smart combinators, such as:
idBi :: Bi a a
idBi = Bi id id
invertBi :: Bi a b -> Bi b a
invertBi (Bi a i) = (Bi i a)
composeBi :: Bi a b -> Bi b c -> Bi a c
composeBi (Bi a1 i1) (Bi a2 i2) = Bi (a2 . a1) (i1 . i2)
mapBi :: Bi a b -> Bi [a] [b]
mapBi (Bi a i) = Bi (map a) (map i)
bruteForceBi :: Eq b => [a] -> (a -> b) -> Bi a b
bruteForceBi domain f = Bi f (inv domain f)
I think you could then do invert (mapBi add1Bi) [1,5,6] and get [0,4,5]. If you pick your combinators in a smart way, I think the number of times you'll have to write a Bi constant by hand could be quite limited.
After all, if you know a function is a bijection, you'll hopefully have a proof-sketch of that fact in your head, which the Curry-Howard isomorphism should be able to turn into a program :-)
I've recently been dealing with issues like this, and no, I'd say that (a) it's not difficult in many case, but (b) it's not efficient at all.
Basically, suppose you have f :: a -> b, and that f is indeed a bjiection. You can compute the inverse f' :: b -> a in a really dumb way:
import Data.List
-- | Class for types whose values are recursively enumerable.
class Enumerable a where
-- | Produce the list of all values of type #a#.
enumerate :: [a]
-- | Note, this is only guaranteed to terminate if #f# is a bijection!
invert :: (Enumerable a, Eq b) => (a -> b) -> b -> Maybe a
invert f b = find (\a -> f a == b) enumerate
If f is a bijection and enumerate truly produces all values of a, then you will eventually hit an a such that f a == b.
Types that have a Bounded and an Enum instance can be trivially made RecursivelyEnumerable. Pairs of Enumerable types can also be made Enumerable:
instance (Enumerable a, Enumerable b) => Enumerable (a, b) where
enumerate = crossWith (,) enumerate enumerate
crossWith :: (a -> b -> c) -> [a] -> [b] -> [c]
crossWith f _ [] = []
crossWith f [] _ = []
crossWith f (x0:xs) (y0:ys) =
f x0 y0 : interleave (map (f x0) ys)
(interleave (map (flip f y0) xs)
(crossWith f xs ys))
interleave :: [a] -> [a] -> [a]
interleave xs [] = xs
interleave [] ys = []
interleave (x:xs) ys = x : interleave ys xs
Same goes for disjunctions of Enumerable types:
instance (Enumerable a, Enumerable b) => Enumerable (Either a b) where
enumerate = enumerateEither enumerate enumerate
enumerateEither :: [a] -> [b] -> [Either a b]
enumerateEither [] ys = map Right ys
enumerateEither xs [] = map Left xs
enumerateEither (x:xs) (y:ys) = Left x : Right y : enumerateEither xs ys
The fact that we can do this both for (,) and Either probably means that we can do it for any algebraic data type.
Not every function has an inverse. If you limit the discussion to one-to-one functions, the ability to invert an arbitrary function grants the ability to crack any cryptosystem. We kind of have to hope this isn't feasible, even in theory!
In some cases, it is possible to find the inverse of a bijective function by converting it into a symbolic representation. Based on this example, I wrote this Haskell program to find inverses of some simple polynomial functions:
bijective_function x = x*2+1
main = do
print $ bijective_function 3
print $ inverse_function bijective_function (bijective_function 3)
data Expr = X | Const Double |
Plus Expr Expr | Subtract Expr Expr | Mult Expr Expr | Div Expr Expr |
Negate Expr | Inverse Expr |
Exp Expr | Log Expr | Sin Expr | Atanh Expr | Sinh Expr | Acosh Expr | Cosh Expr | Tan Expr | Cos Expr |Asinh Expr|Atan Expr|Acos Expr|Asin Expr|Abs Expr|Signum Expr|Integer
deriving (Show, Eq)
instance Num Expr where
(+) = Plus
(-) = Subtract
(*) = Mult
abs = Abs
signum = Signum
negate = Negate
fromInteger a = Const $ fromIntegral a
instance Fractional Expr where
recip = Inverse
fromRational a = Const $ realToFrac a
(/) = Div
instance Floating Expr where
pi = Const pi
exp = Exp
log = Log
sin = Sin
atanh = Atanh
sinh = Sinh
cosh = Cosh
acosh = Acosh
cos = Cos
tan = Tan
asin = Asin
acos = Acos
atan = Atan
asinh = Asinh
fromFunction f = f X
toFunction :: Expr -> (Double -> Double)
toFunction X = \x -> x
toFunction (Negate a) = \a -> (negate a)
toFunction (Const a) = const a
toFunction (Plus a b) = \x -> (toFunction a x) + (toFunction b x)
toFunction (Subtract a b) = \x -> (toFunction a x) - (toFunction b x)
toFunction (Mult a b) = \x -> (toFunction a x) * (toFunction b x)
toFunction (Div a b) = \x -> (toFunction a x) / (toFunction b x)
with_function func x = toFunction $ func $ fromFunction x
simplify X = X
simplify (Div (Const a) (Const b)) = Const (a/b)
simplify (Mult (Const a) (Const b)) | a == 0 || b == 0 = 0 | otherwise = Const (a*b)
simplify (Negate (Negate a)) = simplify a
simplify (Subtract a b) = simplify ( Plus (simplify a) (Negate (simplify b)) )
simplify (Div a b) | a == b = Const 1.0 | otherwise = simplify (Div (simplify a) (simplify b))
simplify (Mult a b) = simplify (Mult (simplify a) (simplify b))
simplify (Const a) = Const a
simplify (Plus (Const a) (Const b)) = Const (a+b)
simplify (Plus a (Const b)) = simplify (Plus (Const b) (simplify a))
simplify (Plus (Mult (Const a) X) (Mult (Const b) X)) = (simplify (Mult (Const (a+b)) X))
simplify (Plus (Const a) b) = simplify (Plus (simplify b) (Const a))
simplify (Plus X a) = simplify (Plus (Mult 1 X) (simplify a))
simplify (Plus a X) = simplify (Plus (Mult 1 X) (simplify a))
simplify (Plus a b) = (simplify (Plus (simplify a) (simplify b)))
simplify a = a
inverse X = X
inverse (Const a) = simplify (Const a)
inverse (Mult (Const a) (Const b)) = Const (a * b)
inverse (Mult (Const a) X) = (Div X (Const a))
inverse (Plus X (Const a)) = (Subtract X (Const a))
inverse (Negate x) = Negate (inverse x)
inverse a = inverse (simplify a)
inverse_function x = with_function inverse x
This example only works with arithmetic expressions, but it could probably be generalized to work with lists as well. There are also several implementations of computer algebra systems in Haskell that may be used to find the inverse of a bijective function.
No, not all functions even have inverses. For instance, what would the inverse of this function be?
f x = 1
Consider this trivial program:
module Study
g : Nat -> Nat -> Nat
g x y = x - y
f : Nat -> List Nat
f x = map (g x) [1, 2 .. x]
It gives an obvious error:
|
4 | g x y = x - y
| ^
When checking right hand side of g with expected type
Nat
When checking argument smaller to function Prelude.Nat.-:
Can't find a value of type
LTE y x
— Saying I should offer some proof that this subtraction is safe to perform.
Surely, in the given context, g is always invoked safely. This follows from the way enumerations behave. How can I extract a proof of that fact so that I can give it to the invocation of g?
I know that I can use isLTE to obtain the proof:
g : Nat -> Nat -> Nat
g x y = case y `isLTE` x of
(Yes prf) => x - y
(No contra) => ?s_2
This is actually the only way I know of, and it seems to me that in a situation such as we have here, where x ≥ y by construction, there should be a way to avoid a superfluous case statement. Is there?
For map (\y = x - y) [1, 2 .. x] there needs to be a proof \y => LTE y x for every element of [1, 2 .. x]. There is Data.List.Quantifiers.All for this: All (\y => LTE y x) [1, 2 .. x].
But constructing and applying this proof is not so straight-forward. You could either build a proof about the range function lteRange : (x : Nat) -> All (\y => LTE y x) (natRange x) or define a function that returns a range and its proof lteRange : (x : Nat) -> (xs : List Nat ** All (\y => LTE y x) xs). For simplicity, I'll show an example with the second type.
import Data.List.Quantifiers
(++) : All p xs -> All p ys -> All p (xs ++ ys)
(++) [] ys = ys
(++) (x :: xs) ys = x :: (xs ++ ys)
lteRange : (x : Nat) -> (xs : List Nat ** All (\y => LTE y x) xs)
lteRange Z = ([] ** [])
lteRange (S k) = let (xs ** ps) = lteRange k in
(xs ++ [S k] ** weakenRange ps ++ [lteRefl])
where
weakenRange : All (\y => LTE y x) xs -> All (\y => LTE y (S x)) xs
weakenRange [] = []
weakenRange (y :: z) = lteSuccRight y :: weakenRange z
Also, map only applies one argument, but (-) needs the proof, too. So with a little helper function …
all_map : (xs : List a) -> All p xs -> (f : (x : a) -> p x -> b) -> List b
all_map [] [] f = []
all_map (x :: xs) (p :: ps) f = f x p :: all_map xs ps f
We can roughly do what you wanted without checking for LTE during the run-time:
f : Nat -> List Nat
f x = let (xs ** prfs) = lteRange x in all_map xs prfs (\y, p => x - y)
I'm a relatively experienced Haskell programmer with a few hours of experience, so the answer might be obvious.
After watching A taste of Haskell, I got lost when Simon explained how the append (++) function really works with its arguments.
So, here's the part where he talks about this.
First, he says that (++) :: [a] -> [a] -> [a] can be understood as a function which gets two lists as arguments, and returns a list after the last arrow). However, he adds that actually, something like this happens: (++) :: [a] -> ([a] -> [a]), the function takes only one argument and returns a function.
I'm not sure to understand how the returned function closure gets the first list as it expects one argument as well.
On the next slide of the presentation, we have the following implementation:
(++) :: [a] -> [a] -> [a]
[] ++ ys = ys
(x:xs) ++ ys = x : (xs ++ ys)
If I think that (++) receives two arguments and return a list, this piece of code along with the recursion is clear enough.
If we consider that (++) receives only one argument and returns a list, where does ys come from? Where is the returned function ?
The trick to understanding this is that all haskell functions only take 1 argument at most, it's just that the implicit parentheses in the type signature and syntax sugar make it appear as if there are more arguments. To use ++ as an example, the following transformations are all equivalent
xs ++ ys = ...
(++) xs ys = ...
(++) xs = \ys -> ...
(++) = \xs -> (\ys -> ...)
(++) = \xs ys -> ...
Another quick example:
doubleList :: [Int] -> [Int]
doubleList = map (*2)
Here we have a function of one argument doubleList without any explicit arguments. It would have been equivalent to write
doubleList x = map (*2) x
Or any of the following
doubleList = \x -> map (*2) x
doubleList = \x -> map (\y -> y * 2) x
doubleList x = map (\y -> y * 2) x
doubleList = map (\y -> y * 2)
The first definition of doubleList is written in what is commonly called point-free notation, so called because in the mathematical theory backing it the arguments are referred to as "points", so point-free is "without arguments".
A more complex example:
func = \x y z -> x * y + z
func = \x -> \y z -> x * y + z
func x = \y z -> x * y + z
func x = \y -> \z -> x * y + z
func x y = \z -> x * y + z
func x y z = x * y + z
Now if we wanted to completely remove all references to the arguments we can make use of the . operator which performs function composition:
func x y z = (+) (x * y) z -- Make the + prefix
func x y = (+) (x * y) -- Now z becomes implicit
func x y = (+) ((*) x y) -- Make the * prefix
func x y = ((+) . ((*) x)) y -- Rewrite using composition
func x = (+) . ((*) x) -- Now y becomes implicit
func x = (.) (+) ((*) x) -- Make the . prefix
func x = ((.) (+)) ((*) x) -- Make implicit parens explicit
func x = (((.) (+)) . (*)) x -- Rewrite using composition
func = ((.) (+)) . (*) -- Now x becomes implicit
func = (.) ((.) (+)) (*) -- Make the . prefix
So as you can see there are lots of different ways to write a particular function with a varying number of explicit "arguments", some of which are very readable (i.e. func x y z = x * y + z) and some which are just a jumble of symbols with little meaning (i.e. func = (.) ((.) (+)) (*))
Maybe this will help. First let's write it without operator notation which might be confusing.
append :: [a] -> [a] -> [a]
append [] ys = ys
append (x:xs) ys = x : append xs ys
We can apply one argument at a time:
appendEmpty :: [a] -> [a]
appendEmpty = append []
we could equivalently could have written that
appendEmpty ys = ys
from the first equation.
If we apply a non-empty first argument:
-- Since 1 is an Int, the type gets specialized.
appendOne :: [Int] -> [Int]
appendOne = append (1:[])
we could have equivalently have written that
appendOne ys = 1 : append [] ys
from the second equation.
You are confused about how Function Currying works.
Consider the following function definitions of (++).
Takes two arguments, produces one list:
(++) :: [a] -> [a] -> [a]
[] ++ ys = ys
(x:xs) ++ ys = x : (xs ++ ys)
Takes one argument, produces a function taking one list and producing a list:
(++) :: [a] -> ([a] -> [a])
(++) [] = id
(++) (x:xs) = (x :) . (xs ++)
If you look closely, these functions will always produce the same output. By removing the second parameter, we have changed the return type from [a] to [a] -> [a].
If we supply two parameters to (++) we get a result of type [a]
If we supply only one parameter we get a result of type [a] -> [a]
This is called function currying. We don't need to provide all the arguments to a function with multiple arguments. If we supply fewer then the total number of arguments, instead of getting a "concrete" result ([a]) we get a function as a result which can take the remaining parameters ([a] -> [a]).
For example, suppose we had the functions double(x) = 2 * x, square(x) = x ^ 2 and sum(x,y) = x + y. What is a function compose such as compose(compose(sum, square), double) = x^2 + 2*x? Notice that I'm asking a function that can be used for functions of any arity. For example, you could compose f(x,y,z) with g(x), h(x), i(x) into f(g(x), h(x), i(x)).
This is a common Haskell idiom, applicative functors:
composed = f <$> g1 <*> g2 <*> ... <*> gn
(A nicer introduction can be found here).
This looks very clean because of automatic partial application, and works like this:
(<*>) f g x = f x (g x)
(<$>) f g x = f (g x) -- same as (.)
For example,
f <$> g <*> h <*> i ==>
(\x -> f (g x)) <*> h <*> i ==>
(\y -> (\x -> f (g x)) y (h y)) <*> i ==>
(\y -> f (g y) (h y)) <*> i ==>
(\z -> (\y -> f (g y) (h y)) z (i z)) ==>
(\z -> f (g z) (h z) (i z)).
Applicative functors are more general, though. They are not an "algorithm", but a concept. You could also do the same on a tree, for example (if properly defined):
(+) <$> (Node (Leaf 1) (Leaf 2)) <*> (Node (Leaf 3) (Leaf 4)) ==>
Node (Leaf 4) (Leaf 6)
But I doubt that applicatives are really usable in most other languages, due to the lack of easy partial application.