Say I have a recursive function that I want to know how many times the function has called itself per input value. Rather than putting printf expressions or changing the return type to include the number of calls, is it possible to "wrap" the function with another to achive this? I would like the wrapped function to return the number of function calls and the original functions result. It should be reusable across different functions.
Here is what I have and it doesn't work.
open System
open System.IO
open System.Collections.Generic
/// example recursive function
let rec getfilenames dir =
seq {
yield Directory.GetFiles dir
for x in Directory.GetDirectories dir do yield! getfilenames x}
/// function to count the number of calls a recursive function makes to itself
let wrapped (f: 'a -> 'b) =
let d = new Dictionary<'a, int>()
fun x ->
let ok, res = d.TryGetValue(x)
if ok then d.[x] <- d.[x] + 1
else
d.Add(x, 1)
d, f x
> let f = wrapped getfilenames
let calls, res = f "c:\\temp";;
val f : (string -> Dictionary<string,int> * seq<string []>)
val res : seq<string []>
val calls : Dictionary<string,int> = dict [("c:\temp", 1)]
This is not going to work, because getfilenames is defined as calling getfilenames, not any other function and especially not a function defined after that. So, as soon as your wrapper calls the function, the function will ignore your wrapper and start calling itself.
What you need to do is move the recursion out of the getfilenames function and into another function, by providing the function to be called recursively as a parameter.
let body recfun dir =
seq {
yield Directory.GetFiles dir
for x in Directory.GetDirectories dir do yield! recfun x}
let rec getfilenames dir = body getfilenames dir
Now, you can wrap body before plugging it into a recursive function:
let wrap f =
let d = (* ... *) in
d, fun recfun x ->
let ok, res = d.TryGetValue(x)
if ok then d.[x] <- d.[x] + 1
else d.Add(x, 1)
f recfun x
let calls, counted_body = wrap body
let getfilenames dir = counted_body getfilenames dir
Note that the wrap function returns both the wrapped function (with a signature identical to the original function) and the dictionary, for external access. The number of calls will then be found in calls.
As Victor points out, you cannot take a recursive function and "inject" some behavior into the place where the recursive call happens (because the function is already complete). You'll need to provide some extension point for that. In Victor's solution, this is done by taking a function to be called recursively as an argument, which is the most general solution.
A simpler option is to use F# value recursion which allows you to create a function value and use it in its declaration. You can use this to create a recursive function by calling another function that adds some behavior to the function (e.g. counting):
let rec factorial = counted (fun x ->
if x = 0 then 1
else x * (factorial (x - 1)) )
factorial 10
Inside the lambda function, we can directly access the function we're defining, so there is no need for passing function to be called recursively as additional parameter. The function counted simply wraps the given function f and adds some functionality:
let counted f =
let count = ref 0
(fun x ->
count := !count + 1;
printfn "call: %d" (!count)
f x)
Thanks to the value recursion, the functionality will be added to the factorial function (and so when it calls itself, it will call the version with added counting support).
Related
Anyone have a decent example, preferably practical/useful, they could post demonstrating the concept?
I came across this term somewhere that I’m unable to find, probably it has to do something with a function returning a function while enclosing on some mutable variable. So there’s no visible mutation.
Probably Haskell community has originated the idea where mutation happens in another area not visible to the scope. I maybe vague here so seeking help to understand more.
It's a good idea to hide mutation, so the consumers of the API won't inadvartently change something unexpectedly. This just means that you have to encapsulate your mutable data/state. This can be done via objects (yes, objects), but what you are referring to in your question can be done with a closure, the canonical example is a counter:
let countUp =
let mutable count = 0
(fun () -> count <- count + 1
count)
countUp() // 1
countUp() // 2
countUp() // 3
You cannot access the mutable count variable directly.
Another example would be using mutable state within a function so that you cannot observe it, and the function is, for all intents and purposes, referentially transparent. Take for example the following function that reverses a string not character-wise, but rather by taking individual text elements (which, depending on language, can be more than one character):
let reverseStringU s =
if Core.string.IsNullOrEmpty s then s else
let rec iter acc (ee : System.Globalization.TextElementEnumerator) =
if not <| ee.MoveNext () then acc else
let e = ee.GetTextElement ()
iter (e :: acc) ee
let inline append x s = (^s : (member Append : ^x -> ^s) (s, x))
let sb = System.Text.StringBuilder s.Length
System.Globalization.StringInfo.GetTextElementEnumerator s
|> iter []
|> List.fold (fun a e -> append e a) sb
|> string
It uses a StringBuilder internally but you cannot observe this externally.
This memoize function fails on any functions of type () -> 'a at runtime with a Null-Argument-Exception.
let memoize f =
let cache = System.Collections.Generic.Dictionary()
fun x ->
if cache.ContainsKey(x) then
cache.[x]
else
let res = f x
cache.[x] <- res
res
Is there a way to write a memoize function that also works for a () -> 'a ?
(My only alternative for now is using a Lazy type. calling x.Force() to get the value.)
The reason why the function fails is that F# represents unit () using null of type unit. The dictionary does not allow taking null values as keys and so it fails.
In your specific case, there is not much point in memoizing function of type unit -> 'a (because it is better to use lazy for this), but there are other cases where this would be an issue - for example None is also represented by null so this fails too:
let f : int option -> int = memoize (fun a -> defaultArg a 42)
f None
The easy way to fix this is to wrap the key in another data type to make sure it is never null:
type Key<'K> = K of 'K
Then you can just wrap the key with the K constructor and everything will work nicely:
let memoize f =
let cache = System.Collections.Generic.Dictionary()
fun x ->
if cache.ContainsKey(K x) then
cache.[K x]
else
let res = f x
cache.[K x] <- res
res
I just found that the last memoize function on the same website using Map instead of Dictionary works for 'a Option -> 'b and () -> 'a too:
let memoize1 f =
let cache = ref Map.empty
fun x ->
match cache.Value.TryFind(x) with
| Some res -> res
| None ->
let res = f x
cache.Value <- cache.Value.Add(x, res)
res
Memoization having a pure function (not just of type unit -> 'a, but any other too) as a lookup key is impossible because functions in general do not have equality comparer for the reason.
It may seem that for this specific type of function unit -> 'a it would be possible coming up with a custom equality comparer. But the only approach for implementing such comparer beyond extremes (reflection, IL, etc.) would be invoking the lookup function as f1 = f2 iff f1() = f2(), which apparently nullifies any performance improvement expected from memoization.
So, perhaps, as it was already noted, for this case optimizations should be built around lazy pattern, but not memoization one.
UPDATE: Indeed, after second look at the question all talking above about functions missing equality comparer is correct, but not applicable, because memoization happens within each function's individual cache from the closure. On the other side, for this specific kind of functions with signature unit->'a, i.e. at most single value of argument, using Dictionary with most one entry is an overkill. The following similarly stateful, but simpler implementation with just one memoized value will do:
let memoize2 f =
let notFilled = ref true
let cache = ref Unchecked.defaultof<'a>
fun () ->
if !notFilled then
cache := f ()
notFilled := false
!cache
used as let foo = memoize2(fun () -> ...heavy on time and/or space calculation...)
with first use foo() performing and storing the result of calculation and all successive foo() just reusing the stored value.
Solution with mutable dictionary and single dictionary lookup call:
let memoize1 f =
// printfn "Dictionary"
let cache = System.Collections.Generic.Dictionary()
fun x ->
let result, value = cache.TryGetValue(x)
match result with
| true -> value
| false ->
// printfn "f x"
let res = f x
cache.Add(x, res)
res
Here is what I have so far:
type Maybe<'a> = option<'a>
let succeed x = Some(x)
let fail = None
let bind rest p =
match p with
| None -> fail
| Some r -> rest r
let rec whileLoop cond body =
if cond() then
match body() with
| Some() ->
whileLoop cond body
| None ->
fail
else
succeed()
let forLoop (xs : 'T seq) f =
using (xs.GetEnumerator()) (fun it ->
whileLoop
(fun () -> it.MoveNext())
(fun () -> it.Current |> f)
)
whileLoop works fine to support for loops, but I don't see how to get while loops supported. Part of the problem is that the translation of while loops uses delay, which I could not figure out in this case. The obvious implementation below is probably wrong, as it does not delay the computation, but runs it instead!
let delay f = f()
Not having delay also hinders try...with and try...finally.
There are actually two different ways of implementing continuation builders in F#. One is to represent delayed computations using the monadic type (if it supports some way of representing delayed computations, like Async<'T> or the unit -> option<'T> type as shown by kkm.
However, you can also use the flexibility of F# computation expressions and use a different type as a return value of Delay. Then you need to modify the Combine operation accordingly and also implement Run member, but it all works out quite nicely:
type OptionBuilder() =
member x.Bind(v, f) = Option.bind f v
member x.Return(v) = Some v
member x.Zero() = Some ()
member x.Combine(v, f:unit -> _) = Option.bind f v
member x.Delay(f : unit -> 'T) = f
member x.Run(f) = f()
member x.While(cond, f) =
if cond() then x.Bind(f(), fun _ -> x.While(cond, f))
else x.Zero()
let maybe = OptionBuilder()
The trick is that F# compiler uses Delay when you have a computation that needs to be delayed - that is: 1) to wrap the whole computation, 2) when you sequentially compose computations, e.g. using if inside the computation and 3) to delay bodies of while or for.
In the above definition, the Delay member returns unit -> M<'a> instead of M<'a>, but that's perfectly fine because Combine and While take unit -> M<'a> as their second argument. Moreover, by adding Run that evaluates the function, the result of maybe { .. } block (a delayed function) is evaluated, because the whole block is passed to Run:
// As usual, the type of 'res' is 'Option<int>'
let res = maybe {
// The whole body is passed to `Delay` and then to `Run`
let! a = Some 3
let b = ref 0
while !b < 10 do
let! n = Some () // This body will be delayed & passed to While
incr b
if a = 3 then printfn "got 3"
else printfn "got something else"
// Code following `if` is delayed and passed to Combine
return a }
This is a way to define computation builder for non-delayed types that is most likely more efficient than wrapping type inside a function (as in kkm's solution) and it does not require defining a special delayed version of the type.
Note that this problem does not happen in e.g. Haskell, because that is a lazy language, so it does not need to delay computations explicitly. I think that the F# translation is quite elegant as it allows dealing with both types that are delayed (using Delay that returns M<'a>) and types that represent just an immediate result (using Delay that returns a function & Run).
According to monadic identities, your delay should always be equivalent to
let delay f = bind (return ()) f
Since
val bind : M<'T> -> ('T -> M<'R>) -> M<'R>
val return : 'T -> M<'T>
the delay has the signature of
val delay : (unit -> M<'R>) -> M<'R>
'T being type-bound to unit. Note that your bind function has its arguments reversed from the customary order bind p rest. This is technically same but does complicate reading code.
Since you are defining the monadic type as type Maybe<'a> = option<'a>, there is no delaying a computation, as the type does not wrap any computation at all, only a value. So you definition of delay as let delay f = f() is theoretically correct. But it is not adequate for a while loop: the "body" of the loop will be computed before its "test condition," really before the bind is bound. To avoid this, you redefine your monad with an extra layer of delay: instead of wrapping a value, you wrap a computation that takes a unit and computes the value.
type Maybe<'a> = unit -> option<'a>
let return x = fun () -> Some(x)
let fail = fun() -> None
let bind p rest =
match p() with
| None -> fail
| Some r -> rest r
Note that the wrapped computation is not run until inside the bind function, i. e. not run until after the arguments to bind are bound themselves.
With the above expression, delay is correctly simplified to
let delay f = fun () -> f()
Anyone have a decent example, preferably practical/useful, they could post demonstrating the concept?
(Edit: a small Ocaml FP Koan to start things off)
The Koan of Currying (A koan about food, that is not about food)
A student came to Jacques Garrigue and said, "I do not understand what currying is good for." Jacques replied, "Tell me your favorite meal and your favorite dessert". The puzzled student replied that he liked okonomiyaki and kanten, but while his favorite restaurant served great okonomiyaki, their kanten always gave him a stomach ache the following morning. So Jacques took the student to eat at a restaurant that served okonomiyaki every bit as good as the student's favorite, then took him across town to a shop that made excellent kanten where the student happily applied the remainder of his appetite. The student was sated, but he was not enlightened ... until the next morning when he woke up and his stomach felt fine.
My examples will cover using it for the reuse and encapsulation of code. This is fairly obvious once you look at these and should give you a concrete, simple example that you can think of applying in numerous situations.
We want to do a map over a tree. This function could be curried and applied to each node if it needs more then one argument -- since we'd be applying the one at the node as it's final argument. It doesn't have to be curried, but writing another function (assuming this function is being used in other instances with other variables) would be a waste.
type 'a tree = E of 'a | N of 'a * 'a tree * 'a tree
let rec tree_map f tree = match tree with
| N(x,left,right) -> N(f x, tree_map f left, tree_map f right)
| E(x) -> E(f x)
let sample_tree = N(1,E(3),E(4)
let multiply x y = x * y
let sample_tree2 = tree_map (multiply 3) sample_tree
but this is the same as:
let sample_tree2 = tree_map (fun x -> x * 3) sample_tree
So this simple case isn't convincing. It really is though, and powerful once you use the language more and naturally come across these situations. The other example with some code reuse as currying. A recurrence relation to create prime numbers. Awful lot of similarity in there:
let rec f_recurrence f a seed n =
match n with
| a -> seed
| _ -> let prev = f_recurrence f a seed (n-1) in
prev + (f n prev)
let rowland = f_recurrence gcd 1 7
let cloitre = f_recurrence lcm 1 1
let rowland_prime n = (rowland (n+1)) - (rowland n)
let cloitre_prime n = ((cloitre (n+1))/(cloitre n)) - 1
Ok, now rowland and cloitre are curried functions, since they have free variables, and we can get any index of it's sequence without knowing or worrying about f_recurrence.
While the previous examples answered the question, here are two simpler examples of how Currying can be beneficial for F# programming.
open System.IO
let appendFile (fileName : string) (text : string) =
let file = new StreamWriter(fileName, true)
file.WriteLine(text)
file.Close()
// Call it normally
appendFile #"D:\Log.txt" "Processing Event X..."
// If you curry the function, you don't need to keep specifying the
// log file name.
let curriedAppendFile = appendFile #"D:\Log.txt"
// Adds data to "Log.txt"
curriedAppendFile "Processing Event Y..."
And don't forget you can curry the Printf family of function! In the curried version, notice the distinct lack of a lambda.
// Non curried, Prints 1 2 3
List.iter (fun i -> printf "%d " i) [1 .. 3];;
// Curried, Prints 1 2 3
List.iter (printfn "%d ") [1 .. 3];;
Currying describes the process of transforming a function with multiple arguments into a chain of single-argument functions. Example in C#, for a three-argument function:
Func<T1, Func<T2, Func<T3, T4>>> Curry<T1, T2, T3, T4>(Func<T1, T2, T3, T4> f)
{
return a => b => c => f(a, b, c);
}
void UseACurriedFunction()
{
var curryCompare = Curry<string, string, bool, int>(String.Compare);
var a = "SomeString";
var b = "SOMESTRING";
Console.WriteLine(String.Compare(a, b, true));
Console.WriteLine(curryCompare(a)(b)(true));
//partial application
var compareAWithB = curryCompare(a)(b);
Console.WriteLine(compareAWithB(true));
Console.WriteLine(compareAWithB(false));
}
Now, the boolean argument is probably not the argument you'd most likely want to leave open with a partial application. This is one reason why the order of arguments in F# functions can seem a little odd at first. Let's define a different C# curry function:
Func<T3, Func<T2, Func<T1, T4>>> BackwardsCurry<T1, T2, T3, T4>(Func<T1, T2, T3, T4> f)
{
return a => b => c => f(c, b, a);
}
Now, we can do something a little more useful:
void UseADifferentlyCurriedFunction()
{
var curryCompare = BackwardsCurry<string, string, bool, int>(String.Compare);
var caseSensitiveCompare = curryCompare(false);
var caseInsensitiveCompare = curryCompare(true);
var format = Curry<string, string, string, string>(String.Format)("Results of comparing {0} with {1}:");
var strings = new[] {"Hello", "HELLO", "Greetings", "GREETINGS"};
foreach (var s in strings)
{
var caseSensitiveCompareWithS = caseSensitiveCompare(s);
var caseInsensitiveCompareWithS = caseInsensitiveCompare(s);
var formatWithS = format(s);
foreach (var t in strings)
{
Console.WriteLine(formatWithS(t));
Console.WriteLine(caseSensitiveCompareWithS(t));
Console.WriteLine(caseInsensitiveCompareWithS(t));
}
}
}
Why are these examples in C#? Because in F#, function declarations are curried by default. You don't usually need to curry functions; they're already curried. The major exception to this is framework methods and other overloaded functions, which take a tuple containing their multiple arguments. You therefore might want to curry such functions, and, in fact, I came upon this question when I was looking for a library function that would do this. I suppose it is missing (if indeed it is) because it's pretty trivial to implement:
let curry f a b c = f(a, b, c)
//overload resolution failure: there are two overloads with three arguments.
//let curryCompare = curry String.Compare
//This one might be more useful; it works because there's only one 3-argument overload
let backCurry f a b c = f(c, b, a)
let intParse = backCurry Int32.Parse
let intParseCurrentCultureAnyStyle = intParse CultureInfo.CurrentCulture NumberStyles.Any
let myInt = intParseCurrentCultureAnyStyle "23"
let myOtherInt = intParseCurrentCultureAnyStyle "42"
To get around the failure with String.Compare, since as far as I can tell there's no way to specify which 3-argument overload to pick, you can use a non-general solution:
let curryCompare s1 s2 (b:bool) = String.Compare(s1, s2, b)
let backwardsCurryCompare (b:bool) s1 s2 = String.Compare(s1, s2, b)
I won't go into detail about the uses of partial function application in F# because the other answers have covered that already.
It's a fairly simple process. Take a function, bind one of its arguments and return a new function. For example:
let concatStrings left right = left + right
let makeCommandPrompt= appendString "c:\> "
Now by currying the simple concatStrings function, you can easily add a DOS style command prompt to the front of any string! Really useful!
Okay, not really. A more useful case I find is when I want to have a make a function that returns me data in a stream like manner.
let readDWORD array i = array[i] | array[i + 1] << 8 | array[i + 2] << 16 |
array[i + 3] << 24 //I've actually used this function in Python.
The convenient part about it is that rather than creating an entire class for this sort of thing, calling the constructor, calling obj.readDWORD(), you just have a function that can't be mutated out from under you.
You know you can map a function over a list? For example, mapping a function to add one to each element of a list:
> List.map ((+) 1) [1; 2; 3];;
val it : int list = [2; 3; 4]
This is actually already using currying because the (+) operator was used to create a function to add one to its argument but you can squeeze a little more out of this example by altering it to map the same function of a list of lists:
> List.map (List.map ((+) 1)) [[1; 2]; [3]];;
val it : int list = [[2; 3]; [4]]
Without currying you could not partially apply these functions and would have to write something like this instead:
> List.map((fun xs -> List.map((fun n -> n + 1), xs)), [[1; 2]; [3]]);;
val it : int list = [[2; 3]; [4]]
I gave a good example of simulating currying in C# on my blog. The gist is that you can create a function that is closed over a parameter (in my example create a function for calculating the sales tax closed over the value of a given municipality)out of an existing multi-parameter function.
What is appealing here is instead of having to make a separate function specifically for calculating sales tax in Cook County, you can create (and reuse) the function dynamically at runtime.
Is it possible to pass a reference to a function to another function in F#? Specifically, I'd like to pass lambda functions like
foo(fun x -> x ** 3)
More specifically, I need to know how I would refer to the passed function in a function that I wrote myself.
Yes, it is possible. The manual has this example:
> List.map (fun x -> x % 2 = 0) [1 .. 5];;
val it : bool list
= [false; true; false; true; false]
Functions are first class citizens in F#. You can therefore pass them around just like you want to.
If you have a function like this:
let myFunction f =
f 1 2 3
and f is function then the return value of myFunction is f applied to 1,2 and 3.
Passing a lambda function to another function works like this:
Suppose we have a trivial function of our own as follows:
let functionThatTakesaFunctionAndAList f l = List.map f l
Now you can pass a lambda function and a list to it:
functionThatTakesaFunctionAndAList (fun x -> x ** 3.0) [1.0;2.0;3.0]
Inside our own function functionThatTakesaFunctionAndAList you can just refer to the lambda function as f because you called your first parameter f.
The result of the function call is of course:
float list = [1.0; 8.0; 27.0]