Using functional continuation in ReasonML, can I apply CPS in numeric analysis/method? e.g. euler method, finite difference method.
let factorial3 = n => {
let rec cont = (n, func) =>
if (n <= 1) {
func();
} else {
cont(n - 1, () => n * func());
};
cont(n, () => 1);
};
The question has already been answered in the question body. You can apply CPS in Reason for anything that you'd like.
You provided yourself an example of a function using CPS.
One thing I wanted to add is that on your example, it would be preferable to just use an accumulator parameter instead like this:
let factorial = n => {
let rec cont = (n, tot) =>
if (n <= 1) {
tot;
} else {
cont(n - 1, n * tot);
};
cont(n, 1);
};
as it avoids allocating a new lambda on every recursion while your function remains tail-recursive. I think that CPS would be better left for more involved examples where your have more than one recursive calls per function call but would still like your function to remain tail recursive.
This article contains examples where CPS is a more appropriate tool (written in F#): https://www.gresearch.co.uk/article/advanced-recursion-techniques-in-f/
Related
I'm looking for some way to shorten an iterator by some condition. A bit like an inverse filter but it stops iterating at the first true value. Let's call it until(f). Where:
iterator.until(f)
Would return an iterator that only runs until f is true once.
Let's use an example of finding the next prime number.
We have some structure containing known primes and a function to extend it.
// Structure for caching known prime numbers
struct PrimeGenerator {
primes:Vec<i64>
}
impl PrimeGenerator {
// Create a new prime generator
fn new()->Self{
let primes = vec![2,3];
Self {
primes,
}
}
// Extend the list of known primes by 1
fn extend_by_one(&mut self){
let mut next_option = self.primes.last().unwrap()+2;
while self.primes.iter().any(|x| next_option%x == 0) { // This is the relevant line
next_option += 2;
}
self.primes.push(next_option);
}
}
Now this snippet is a bit too exhaustive as we should only have to check until the square root of next_option, so I was looking for a some method that would shorten the iterator based on some condition, so I could write something like:
self.iter().until(|x| x*x > next_option).any(|x| next_option%x == 0)
Is there any similar pattern available?
Looks like your until is similar to inverted take_while.
self.iter().take_while(|x| x*x <= next_option).all(|x| next_option%x != 0)
I am new to recursion, and have been looking at non-tail recursion. I believe this is where the recursive call is not at the end of the function. I was looking at some recursive code examples, and I am confused about when and how the code after the recursive call is executed. To be more specific, when the current line of the program reaches the recursive call, is that last bit of code executed before actually executing the entire function again, or does the current line of execution immediately jump back to the beginning of the function, only to execute that remaining bit of code after the entire recursion bottoms out? Can anybody please clarify how this works for me?
Imagine this code here: (I'm using JS here, but it applies to all eager languages)
function add(a, b) {
return a + b;
}
function sub(a, b) {
return a - b;
}
function fib(n) {
if (n < 2) {
return n;
} else {
return add(
fib(sub(n, 2)),
fib(sub(n, 1)),
);
}
}
console.log(fib(10));
Now when n is 2 or more we we add the result of calling fib on the two predecessors to n. The order between some of the calls are JS engines choice, but sub needs to be called and return before fib can get a value and add needs to be called when both it's arguments have results. This can be rewritten as tail calls using continuation passing style:
function cadd(a, b, k) {
k(a + b);
}
function csub(a, b, k) {
k(a - b);
}
function cfib(n, k) {
if (n < 2) {
k(n);
} else {
csub(n, 2, n2 =>
cfib(n2, rn2 =>
csub(n, 1, n1 =>
cfib(n1, rn1 =>
cadd(rn2, rn1, k)))));
}
}
cfib(10, console.log);
Here you see the order which was engines choice is explicit left to right. Each call does one thing then the continuation is called with that result. Notice also that this never returns anything. "return" is only done by continuation, yet it does the exact same thing as the previous code.
In many languages they would compile to the same.
I'm trying to do some operations that would cause a StackOverflow in Kotlin just now.
Knowing that, I remembered that Kotlin has support for tailrec functions, so I tried to do:
private tailrec fun Turn.debuffPhase(): List<Turn> {
val turns = listOf(this)
if (facts.debuff == 0 || knight.damage == 0) {
return turns
}
// Recursively find all possible thresholds of debuffing
return turns + debuff(debuffsForNextThreshold()).debuffPhase()
}
Upon my surprise that IDEA didn't recognize it as a tailrec, I tried to unmake it a extension function and make it a normal function:
private tailrec fun debuffPhase(turn: Turn): List<Turn> {
val turns = listOf(turn)
if (turn.facts.debuff == 0 || turn.knight.damage == 0) {
return turns
}
// Recursively find all possible thresholds of debuffing
val newTurn = turn.debuff(turn.debuffsForNextThreshold())
return turns + debuffPhase(newTurn)
}
Even so it isn't accepted. The important isn't that the last function call is to the same function? I know that the + is a sign to the List plus function, but should it make a difference? All the examples I see on the internet for tail call for another languages allow those kind of actions.
I tried to do that with Int too, that seemed to be something more commonly used than addition to lists, but had the same result:
private tailrec fun discoverBuffsNeeded(dragon: RPGChar): Int {
val buffedDragon = dragon.buff(buff)
if (dragon.turnsToKill(initKnight) < 1 + buffedDragon.turnsToKill(initKnight)) {
return 0
}
return 1 + discoverBuffsNeeded(buffedDragon)
}
Shouldn't all those implementations allow for tail call? I thought of some other ways to solve that(Like passing the list as a MutableList on the parameters too), but when possible I try to avoid sending collections to be changed inside the function and this seems a case that this should be possible.
PS: About the question program, I'm implementing a solution to this problem.
None of your examples are tail-recursive.
A tail call is the last call in a subroutine. A recursive call is a call of a subroutine to itself. A tail-recursive call is a tail call of a subroutine to itself.
In all of your examples, the tail call is to +, not to the subroutine. So, all of those are recursive (because they call themselves), and all of those have tail calls (because every subroutine always has a "last call"), but none of them is tail-recursive (because the recursive call isn't the last call).
Infix notation can sometimes obscure what the tail call is, it is easier to see when you write every operation in prefix form or as a method call:
return plus(turns, debuff(debuffsForNextThreshold()).debuffPhase())
// or
return turns.plus(debuff(debuffsForNextThreshold()).debuffPhase())
Now it becomes much easier to see that the call to debuffPhase is not in tail position, but rather it is the call to plus (i.e. +) which is in tail position. If Kotlin had general tail calls, then that call to plus would indeed be eliminated, but AFAIK, Kotlin only has tail-recursion (like Scala), so it won't.
Without giving away an answer to your puzzle, here's a non-tail-recursive function.
fun fac(n: Int): Int =
if (n <= 1) 1 else n * fac(n - 1)
It is not tail recursive because the recursive call is not in a tail position, as noted by Jörg's answer.
It can be transformed into a tail-recursive function using CPS,
tailrec fun fac2(n: Int, k: Int = 1): Int =
if (n <= 1) k else fac2(n - 1, n * k)
although a better interface would likely hide the continuation in a private helper function.
fun fac3(n: Int): Int {
tailrec fun fac_continue(n: Int, k: Int): Int =
if (n <= 1) k else fac_continue(n - 1, n * k)
return fac_continue(n, 1)
}
I'm trying to force myself to employ functional programming in Kotlin, and wherever possible, avoid using mutable vars. Ordinarily, for an ad hoc test for a unit-returning function, I'd just println() something inside the function to see if it's working right. But for this test, I need to accumulate a string and then ultimately use assertEquals(...).
As usual, I found myself declaring a var in the enclosing scope and using += to accumulate into it. Is there a more functional way to do this by passing/chaining a function and eliminating the mutable var? Here's some simplified but illustrative code:
inline fun <T> Iterable<T>.forEachFrom(beg:Int, act:(T)->Unit) {
var i=0; if (beg>=0) for (e in this) if (i++ >= beg) act(e)
}
fun main(args:Array<String>) {
val l = listOf("zero", "one", "two", "three", "four")
// print-to-screen test
l.forEachFrom(2){print("$it-")}; println()
// output: two-three-four-
// accumulate-in-var test
var s = ""
l.forEachFrom(2){s += "$it-"}; println(s)
// output: two-three-four-
// Is there a purely functional way, without declaring a mutable var?
// val s = l.forEachFrom(2){accumulator???("$it-")}
// - OR -
// val s = l.forEachFrom(2).accumulator???("$it-")
// println(s)
}
A way to do the same with only kotlin-stdlib and retain the semantics of the code (i.e. iterate only once) is to convert the Iterable<T> to Sequence<T> and use the .drop(n) extension:
inline fun <T> Iterable<T>.forEachFrom(beg: Int, act: (T) -> Unit) =
if (beg >= 0)
asSequence().drop(beg).forEach(act) else
Unit
UPD: After discussing the overall question, we came up with another approach.
When you have a custom higher-order function that iterates over the items and only accepts a callback but does not return anything, you can wrap that custom iteration logic into a Sequence<T> by using buildSequence { ... } and passing yield(it) as the callback:
val sequenceFromCustomFunction = buildSequence {
l.forEachFrom(2) { yield(it) }
}
This allows you to work with this sequence in functional style and, in particular, fold the sequence:
val s = sequenceFromCustomFunction.fold("") { acc, it -> acc + it + "-" }
(Really strugging to title this question, so if anyone has suggestions feel free.)
Say I wanted to do an operation like:
take an array [1,2,3]
multiply each element by 2 (map): [2,4,6]
add the elements together (reduce): 12
multiply the result by 10: 120
I can do this pretty cleanly in underscore using chaining, like so:
arr = [1,2,3]
map = (el) -> 2*el
reduce = (s,n) -> s+n
out = (r) -> 10*r
reduced = _.chain(arr).map(map).reduce(reduce).value()
result = out(reduced)
However, it would be even nicer if I could chain the 'out' method too, like this:
result = _.chain(arr).map(map).reduce(reduce).out(out).value()
Now this would be a fairly simple addition to a library like underscore. But my questions are:
Does this 'out' method have a name in functional programming?
Does this already exist in underscore (tap comes close, but not quite).
This question got me quite hooked. Here are some of my thoughts.
It feels like using underscore.js in 'chain() mode' breaks away from functional programming paradigm. Basically, instead of calling functions on functions, you're calling methods of an instance of a wrapper object in an OOP way.
I am using underscore's chain() myself here and there, but this question made me think. What if it's better to simply create more meaningful functions that can then be called in a sequence without having to use chain() at all. Your example would then look something like this:
arr = [1,2,3]
double = (arr) -> _.map(arr, (el) -> 2 * el)
sum = (arr) -> _.reduce(arr, (s, n) -> s + n)
out = (r) -> 10 * r
result = out sum double arr
# probably a less ambiguous way to do it would be
result = out(sum(double arr))
Looking at real functional programming languages (as in .. much more functional than JavaScript), it seems you could do exactly the same thing there in an even simpler manner. Here is the same program written in Standard ML. Notice how calling map with only one argument returns another function. There is no need to wrap this map in another function like we did in JavaScript.
val arr = [1,2,3];
val double = map (fn x => 2*x);
val sum = foldl (fn (a,b) => a+b) 0;
val out = fn r => 10*r;
val result = out(sum(double arr))
Standard ML also lets you create operators which means we can make a little 'chain' operator that can be used to call those functions in a more intuitive order.
infix 1 |>;
fun x |> f = f x;
val result = arr |> double |> sum |> out
I also think that this underscore.js chaining has something similar to monads in functional programming, but I don't know much about those. Though, I have feeling that this kind of data manipulation pipeline is not something you would typically use monads for.
I hope someone with more functional programming experience can chip in and correct me if I'm wrong on any of the points above.
UPDATE
Getting slightly off topic, but one way to creating partial functions could be the following:
// extend underscore with partialr function
_.mixin({
partialr: function (fn, context) {
var args = Array.prototype.slice.call(arguments, 2);
return function () {
return fn.apply(context, Array.prototype.slice.call(arguments).concat(args));
};
}
});
This function can now be used to create a partial function from any underscore function, because most of them take the input data as the first argument. For example, the sum function can now be created like
var sum = _.partialr(_.reduce, this, function (s, n) { return s + n; });
sum([1,2,3]);
I still prefer arr |> double |> sum |> out over out(sum(double(arr))) though. Underscore's chain() is nice in that it reads in a more natural order.
In terms of the name you are looking for, I think what you are trying to do is just a form of function application: you have an underscore object and you want to apply a function to its value. In underscore, you can define it like this:
_.mixin({
app: function(v, f) { return f (v); }
});
then you can pretty much do what you asked for:
var arr = [1,2,3];
function m(el) { return 2*el; };
function r(s,n) { return s+n; };
function out(r) { return 10*r; };
console.log("result: " + _.chain(arr).map(m).reduce(r).app(out).value()));
Having said all that, I think using traditional typed functional languages like SML make this kind of think a lot slicker and give much lighter weight syntax for function composition. Underscore is doing a kind of jquery twist on functional programming that I'm not sure what I think of; but without static-type checking it is frustratingly easy to make errors!