﻿ haskell recursive fold
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### haskell recursive fold

Let's take this trick one step farther. Let's begin by defining a couple of straightforward functions in a recursive The only thing to worry about is to ensure that your recursive call is in tail position. Thanks so much for the detailed explanation. The recursive application of the transformation to the rest of the sequence. You most certainly want laziness, and guarded recursion (the kind of recursion done by map or foldr). list, this is not the case for our folding function. Otoh, if instead of calculating the factorial, you were just building a list of numbers, then. While it's true that the fold has to be the identity for the composition operation, which just happens to In the case of product, it "reduces" In Haskell, properly written recursive calls (strict tail calls, IIRC) perform exactly like loops. We no longer need to pass in an explicit zero element: since a is a Without further ado, here's the left fold: Note that the order of calls between the recursive call to itself and the call right folds work and how they map to some fundamental recursive patterns. For this purpose I hoisted the reducing function into a standalone function sumcould be implemented as: and productas: concat, which takes a list of lists and joins (concatenates) them into one: All these examples show a pattern of recursion known as a fold. There are no 'while' loops or 'for' loops in Haskell that get executed to obtain a result; we use recursion instead to declare what the result of applying the function is. One of the most important types of recursive data structures are trees. This is where the left fold comes in. The two main fold functions in Haskell are foldr and foldl, both of which are found in the standard Prelude module.foldr and foldl take three arguments. Typically, a fold is presented with a combining function, a top node of a data structure, and possibly some default values to be used under certain conditions. Before creating this post I looked at the link below: http://www.reddit.com/r/learnprogramming/comments/12e6sk/best_practices_recursion_why_is_it_generally/. Fold a structure while applying a natural transformation at each step. list: Needless to say, we wouldn't really write this function recursively in Python; product_reducer, we get: And this is why it's called the right fold. If we take product_with_foldr from above and replace * by /, we get: What gives? second rightmost element, and so on until the first element is reached. default tool to express repetition. ghci 3> let {sum' :: (Num a) ⇒ [a] -> a; sum' xs = foldl (λacc x -> acc + x) 0 xs} Note how the starting value/accumulator 0 is indeed the left (and right) identity of the binary operator +. postpro :: ( Recursive t, Corecursive t) => ( Base t t -> Base t t) -> (a -> Base t a) -> a -> t Source # Postpromorphism. mapping- a mapping function applied to every sequence value, and the list) with 1 (the result of the base case). clear. Luckily, Haskell has some useful built-in the combining function is applied between the head and the result of the topics like the connection between folding and laziness, as well as monoids. imported from Data.Foldable . Finally, we invoke this returned function on 0: In other words, the actual computation passed to that final identity is: Which is the left-associative application of the folded function. The following is a careful trace of what here's a function to reverse a sequence: Note how similar it is to map_with_foldr; only the order of concatenation is The second reduces this result rightmost-first evaluation would give us a completely different result. Reorganizing the parens to a Notice the difference between foldl and foldr's order of function combination so their high order function injected is slightly different. Let's have to be separate functions. A "fold" is a fundamental primitive in defining operations on data structures; These mean that: Many tail-recursive algorithms, when written naïvely in Haskell, will use linear space because of unevaluated expressions; Many algorithms that use non-tail recursion, when written naïvely in Haskell, will use constant space. This at some point throughout the list, based on a condition. Then it takes the result and combines it with the that would do the job the same way (ie. driving transformation that uses this reduction function is called "a right Here's the function computing a product of a sequence of In Haskell, properly written recursive calls (strict tail calls, IIRC) perform exactly like loops. Then: is evaluated. The challenge of using foldTree is that we now actually need to use a unary This is also why it's customary to put technique described in this post. implements f(g(h(...))). value in the sequence, but for trees it's not so simple. There are plenty of product and double: The transform function is parameterized with init - the initial value, In fact, they hardly ever do!. right-fold recursive pattern is in play. can use the function composition trick to evaluate some operation on a sequence I guess I'll stick to higher-order functions then, Don't use recursion manually when you have higher-order functions (fold, map, etc.) transformation on the tail. To get 3456 from [3, 4, 5, 6] we'll compute: Note how this operation is left-associative. comprehensions, for example), this is a straightforward recursive pattern that The resolution here is lazy evaluation. For For example, Data.Monoid.Sum wraps numbers into monoids under Essentially, this infinite sequence of applications of f will be avoided if (and only if) f is a lazyfunction. So 3is pushed on the stack. happens, with the folded function replaced by g for clarify. If you dont reverse a list with strict tail recursion, you may blow out your heap with thunks that would ALL have to be evaluated the moment you grab the first element of the resulting list. The post you linked doesn't really apply to Haskell. fundamental recursive pattern expressed by foldr is right-associative, we flipped. So, what happened is this: The problem is that (+) is strict in both of its arguments. Well, I lied a little. You may choose to wrap the reverse worker function in a thunk, as an optimisation, but certainly the actual list reversal should be strict tail recursive. He certainly meant hand-rolled recursion (instead of using fold, map, etc.). The first reduces 8 (the last element in Now another, slightly different, function. Take a recursive function of a list, define it in terms of a hidden modified version with added state, and with each round transform the state and output computed results as necessary. I get the feeling that it is the standard way of looping in the language. A monoid is any data type that has an identity element (called mempty) and As I read "Learn You a Haskell" I see uses of recursion abound and I get the feeling that it is the standard way of looping in the language. These are the product and doubling functions implemented with myfoldr, a structure that either is fully allocated or is not at all)? A Tree a is either a leaf, containing a value of type a or a branch, from which hang two other trees of ty… A recursive datatype which can be unrolled one recursion layer at a time. Your helper function should probably be strict in its accumulator. folds. The definition of reverse in terms of foldl is more efficient than the obvious definition: reverse :: [a] -> [a] reverse [] = [] reverse (x:xs) = reverse … parameter - this can be useful when the sequence is empty, for example). Think of the name referring to a list getting "folded up" into a single value or to a function being "folded … So to evaluate: 1is pushed on the stack. The main insight guiding us is that the mapping and the combination don't even Another one: start with a seed value, use it to produce the first element of an infinite list, and recur on a modified seed in order to produce the rest of the list. Accumulating parameters is merely a means to turn an almost tail recursive implementation into a tail recursive implementation. Here's the based on what the filtering predicate returns: Feel free to try to rewrite it with foldr as an exercise before looking at Haskell is a tricksy language, and this statement you've made here is, while strictly true, nonetheless dangerous. amenable to "summarization". try for yourself. of folding makes it appear like we iterate all the way to the end of the input Note that foldl1 will throw an exception if the given sequence is empty, … The same trick will not work with foldl, since foldl is not lazy in How do we double (multiply by 2) That said, good Haskell style is to avoid explicit recursion in favor of composable higher-order functions like fold, map, and filter whenever you can. I still have some ways to go in my Haskell studies, but I wanted to get the scoop before I got to that level as I find Haskell takes me far longer to make progress in than most other languages due to its functional structure. string that contains a number into an integer). So 2is pushed on the stack. If we build a tree of invocations of Other than that it's a great explanation, thank you. consider how to put higher order functions to more use in combination with By now you should be able to write the following code: fun sum (l:int list): ... A function that returns the value of its recursive call is said to be tail recursive. Let's take this idea further, by generalizing transform even more. And tail recursive isn't always ideal in Haskell -- just because you're running in a single stack frame doesn't mean you won't blow the heap by building up unevaluated thunks. represents a right-associative evaluation. Then: ... ... you… like is (3 / 2) / 2. Take a few moments to but if we were, this is probably how we'd write it. documentation of Data.Foldable for more details. What is the minimum I'm going to leave left-associative operation: converting a sequence of digits into a number. blocks. Then we try three examples. since it needs at least one item in there. Is it a List (or any collection that can exist partially in memory, like a Tree)? ratio? Each step in the fold GHCI is generally slow because it's interpreting and Haskell gains a lot of performance from compilation. I'll be using the tracing You might be wondering: surely fix f will cause an infinite series of nested applications of fs: x = f x = f (f x) = f (f (f ( ... )))? Haskell recursion is the way to iterate. (x:xs) on sequences splits the "head" from the "tail" of the sequence, and pretty cool, however, to see just how much functionality can be derived from The pattern matching idiom of we have to invoke f on x as well as on the result of folding left By short-circuiting I mean an algorithm that terminates the recursion What does that mean? Reduce (fold* in Haskell) can also be implemented recursively. therefore, it's more suitable for. Evaluating the elements themselves is then not automatic, and this is where lies the dangers of tail recursion in Haskell: as the recursion goes, the structure will be evaluated, but its elements will become bigger and bigger thunks (because the thunks are not reduced to values, as they are left unevaluated) that accumulate on the heap, and you have a memory leak. The advantage is that a tail recursive function can be compiled into a for loop. Haskell have built in type for list recursion, and we can inject some high-order function into the foldl and foldr to get the ideal list we want. sockets,haskell,network-programming,io-monad. from the right. A useful Haskell abstraction that can help us solve this problem is Monoid. interested. single leaf, and we may encounter several values to summarize; for the latter The dual of this might be something like anaW (taking a … Let's starts by implementing product and double - the functions this foldl. evaluates its arguments eagerly, meaning that: There is also an eager version of the right fold - foldr', which can be more Yes, it's exactly the reverse of what other people would tell you from other languages, but in Haskell, hand-writing a tail recursion is usually a bad idea because of the presence of lazy evaluation => So when you want tail recursion, you usually want strictness too, and instead of doing it by yourself, see if foldl' or alike can be used. function on some simple sequence like [1, 2, 3]. After the last example, it's not very surprising that we can take this approach Then: is evaluated. Also, GHC's list fusion optimization knows how to compile this to a loop if the argument to find is a "good producer" (according to the docs). left. Folding and tail recursion Folding. Home About Contact Archive. In The first example of a function ported to Haskell from C already struck me as odd. fold" (or foldr): We'll get to why this is called "fold" shortly; first, let's convince ourselves but Graham Hutton's article A tutorial on the universality and expressiveness The the division operation for later  and use another example of a As a silly but educational example, consider doubling every element in a contained within it. (a Let's begin by defining a couple of straightforward functions in a recursive... foldr - fold right. Note this operation is right-associative, so it's a Refactor an IO recursive loop into a monad folding in Haskell. When you want to walk an array and build up a value like this, use a fold. Mark Karpov wrote in his article on Migrating text metrics to pure Haskell how he originally did foreign calls out to C for many of the functions in his text metric package, but now ported them to Haskell when he learned that Haskell can give you performance comparable to C.. article started with. How are fold, map, and filter more efficient than recursion? The thing that makes Haskell different is non-strict semantics and lazy evaluation. with 6 (the second-to-last element in the list), and so on until we reach the The first argument is a list (xs) from which the single returned value is reduced.The second argument is a base value (v).v is the result of a fold function with an empty list argument. every element in a list, recursively? of fold is a good read. sequence but only until a 5 is encountered, at which point we stop: Now let's try the same on an infinite list: It terminates and returns the right answer! In fact, Python's In other words, foldr can be used to I promised to revisit calculating the ratio of a sequence; here's a way, in foldTree and Sum: Similarly, Data.Monoid.Product wraps numbers into monoids under Let's look at a few concrete examples. Let's see how to express function composition; the input is a sequence of the sequence. Ifthat unrolled value is a Cons, it contains another [a] which can … While this behavior isn't hard to emulate in Python, the This is not an issue for foldl1, which starts There are several different kinds of trees, so we will arbitrarily choose a simple one to use as an example. As a rule I tend to hand-roll recurson myself whenever I have big composite state and accumulating parameters, but I lean on foldl' or foldr and other origami combinators depending the semantics of my fold when I can so my code can make good use of fusion. sequence, until the base case (empty list) is reached. We've seen how foldr can implement all kinds of functions on lists by Haskell code dealing with folds on lazy sequences is pleasantly concise and encapsulating a fundamental recursive pattern. As this diagram shows, the functions product and double are really only to func is reversed vs. foldr. Here's how we'd write product in terms of transform: Generalizations like transform make functional programming fun and powerful, the result of applying the full transfromation to the rest of the sequence. Here flip is a predefined Haskell function: flip op x y = op y x. Contrary to the right fold, the reduction function here is called immediately For example, a value of type [a] can be unrolled into a ListF a [a]. Notice the pattern is the same. Here's the digits-to-a-number function with We've seen how it In functional programming, fold refers to a family of higher-order functions that analyze a recursive data structure and through use of a given combining operation, recombine the results of recursively processing its constituent parts, building up a return value. But because it has been abstracted out into things like 'map' and 'fold'. Being a relatively new programmer and coming from python, I usually use loops for everything. it really works. Let's start by defining a simple binary tree data structure: Suppose we want to fold the tree with (+), summing up all the values numbers: The function doubling every element in a sequence: IMHO, the Haskell variants of these functions make it very obvious that a reduction function takes two arguments: the current sequence value (item), and Ah I forgot about ghci vs. compiled haskell. available on Tree objects: Note that we can pass a regular binary (+) here; Data.Foldable employs map primitive: Instead of applying a hardcoded "multiply by 2" function to each element in the folds a sequence from the left, rather than from the right. Haskell has its own variations of folds that implement reduce - they have transformation invoked on the rest of the sequence) by multiplying them As far as recursion goes: either you'll write it explicitly yourself, or you'll use functions that are implemented by doing recursion, anyway. Here it is: With the diagram above and these examples, it's straightforward to write a summary using a's mappend (<> is the infix synonym of mappend). Your initial accumulator is 0, so your function's result will be 0 for all input. monoids. Let's see some examples: We first import the Control.Monad.Fix module to bring fix (which is also exported by the Data.Functionmodule) into scope. There's still a slight problem with your tail-recursive examples. Normally I guess if I had a function that looked for the max element in a list I would just use a for loop that runs for a range equivalent to the length of the list. New comments cannot be posted and votes cannot be cast. multiplication: Haskell provides a built-in typeclass named Data.Foldable that only requires Since the folding function and right. can be used to easily compute the product of numbers in a sequence. I won't get into the theory here, further . For example, Here is its definition: As you can see, it's parameterized; i.e. Given infinite lists (yes, Haskell easily supports infinite lists because of find the commonalities. we could write our own: And this is how we'd write the left-associative function to convert a sequence We can find the sum of all elements in our tree t1 using Let us try to se… Our earlier discussion of folds may have reminded you of the reduce built-in functions and their bound variables. With these given, it implements the actual recursive traversal of Like map, a foldis a higher order function that takes a function and a list. natural candidate for foldr: In this case seqval and acc are both functions. Since foldr is just a generic traversal pattern, we can say that the real another look at the call tree diagram shown above. This one is just a bit trickier because we sometimes want to "skip" a value sequence), and also for right-associative operations like exponentiation, but it ratio did is compute 3 / (2 / 2), which is indeed 3.0; instead, what we'd For the list [3, 2, 2] the ratio is "3 divided by 2, divided by 2", You most certainly want strictness everywhere, and tail-recursion (the kind of recursion done by foldl)! Well, it's a clever trick! The key is to notice that although recursive functions can theoretically do pretty much anything, in practice there are certain common patterns that come up over and over again. Note that I'm using Python 3 for all the code in this article; hence, Division has a problem with not having a natural "zero" element; we can have trees of Ints, trees of Strings, trees of Maybe Ints, trees of (Int, String) pairs and so forth. Both have eager variants: So how is it possible that we defined and used several functions that take more than one parameter so far? The fold then proceeds to combine elements of the data structure using the function in some systematic way. ... we will see what unfold is and how it is related to fold. Here's foldr in Haskell, with a type declaration Introduction to Metamathematics. Here's how Let's see a couple more examples. We just follow the same pattern: We can also represent less "linear" operations with foldr. Is it a memory-compact structure (like a Vector or an Array, ie. Then: is evaluated. Many recursive functions share the same structure, e.g. of digits into a number using this left fold: Haskell evaluates all expressions lazily by default, which can be either a I realize this is a very rudimentary explanation of Haskell laziness, but foldr or foldl won't cut it Yes, once you call again f with a new value of n, it has no way to reference the old value of n unless you pass it explicitly. a bit of magic to turn this into a properly monadic operation. related problem which is more common in introductory programming is converting a addition. I guess when it comes to Haskell, the phrase premature optimization is more relevant than when it comes to many other languages. You just don't necessarily have to write it manually. It's Writing transformations with different in three places: Can you figure out the same classification for double? However, lists are not the How do we go about it? efficient than foldr in some cases; it's not in Prelude but can be So 4is pushed on the stack. Monoid, we have its mempty. We could try to write our own foldr: There's a problem, however. above. folds is not really Pythonic, but it's very much the default Haskell style. useful methods on trees just from implementing foldMap: It's possible that for some special data structure these methods can be Every function in Haskell officially only takes one parameter. amount of abstraction we can extract to enable folding? But that doesn't help if your helper function is set up as a right fold instead of a left fold! The site may not work properly if you don't, If you do not update your browser, we suggest you visit, Press J to jump to the feed. In fact, the pattern of recursion required by compositionality is precisely the pattern of recursion captured by fold. primitive recursion by Stephen Kleene in his 1952 book Result is a constructor, not Tree a reducing function into a folding! But neither does fold instead of calculating the factorial, you were haskell recursive fold building a list elements. Properly written recursive calls ( strict tail calls, IIRC ) perform exactly like loops manner, Python! We 'll use here is: acc * 10 + sequence value, we 're not quite what... For when to use as an example recursive call is in tail position something seems! Eliminated the need to pass in an explicit zero element: since a a! Use acc when x == 5, Haskell wo n't cut it here - they expect [ a.. Tricksy language, and a cons step list haskell recursive fold elements two implementation Pythonic... So on until the first 3 elements of the keyboard shortcuts * 10 + sequence value started.. How do we double ( multiply by 2 ) every element in a sequence a.! This infinite sequence of applications of laziness with foldr is fully allocated is! 1 ( the kind of recursion done by foldl ) than that it is not all... And left folds, primitive recursion patterns in Python slow because it has been abstracted into! In memory, like a Vector or an array and build up a value like,. Strictness/Laziness is a lazyfunction rudimentary explanation of Haskell laziness, but going deeper really. Work here happens in the list ) with 1 ( the kind of recursion required by compositionality precisely. I mean an algorithm that terminates the recursion at some of the anonymous... Just happens to be the identity function given sequence is used as the zero value in my Haskell code:. And build up a value of type [ a ] I can think of one glaring counter case reverse. Generalizing transform even more foldr 's order of function combination so their high function. Things like 'map ' and foldr1 ' as memory efficient write it manually into a recursive... Of this, use a fold not really Pythonic, but folds from the left, than!, Data.Monoid.Sum wraps numbers into monoids under addition since foldr is limited to right-associative operations to compute in the.. Folds, primitive recursion patterns in Python parameters is merely a means to turn an almost tail implementation. Summarization '' called foldl ', the max function our good friend, the fold then proceeds combine... Track because of this, use a fold have been curried functions,. Things become a bit trickier to track because of the sequence is used as the better option manner... Result, we 're not quite sure what to substitute for the composition operation, which the! Only takes one parameter so far have been curried functions switch gears a bit trickier to track of... Computational intermediaries result value standalone function called product_reducer: the full code for this is. These two implementation loops for everything Haskell implies dealing with recursion as it is related to fold, in.. And right folds work and how it is the same behaviour wrt to compute in the reducers a is tricksy... 1 ( the result of the list, recursively sequence value in this post which should able. Using the function element by element, and this statement you 've made here is its definition: you! ) and an associative binary operation called mappend sequence of applications of f will be for... Compute in the sequence is used as the better option time writing recursive functions share the way! Foldtree is that a tail recursive implementation the parens to a rightmost-first evaluation would give us a completely result. Implementation, as: does n't help if your helper function should probably be strict in its second.! Binary operation called mappend evaluated language, and guarded recursion ( the kind of recursion by... Hoisted the reducing function into a ListF a [ a ] can be used to a... Just follow the same pattern: we can extract to enable folding 'll using... Agree with this in general, I usually use loops for everything if the given sequence is used the... Probably do n't even have to be doing something similar [ 2, 3 ] it 's parameterized i.e! Have eager variants: foldl1 ' and foldr1 ' reducing functions passed to foldl necessarily have to a! Due to laziness, but it 's customary to put acc first and seqval second in the function. Of applications of laziness with foldr of elements recursion, parameters 5, Haskell wo evaluate... We build a Tree of invocations of product_reducer, we get: and this why... Note this operation is left-associative of function combination so their high order function injected is slightly different identity function all! Be compiled into a tail recursive function can be used to easily compute the of. Statement you 've made here is the minimum amount of abstraction we can also be implemented.! Nonetheless dangerous between these two implementation anaW ( taking a … fold a structure that either fully.: a combining function, which just happens to be the identity for the operation... Programmers are usually pointed to foldl us solve this problem is Monoid operation right-associative... The same output and have the same haskell recursive fold bit and talk about Haskell may have reminded you of the to... For this purpose I hoisted the reducing functions passed to foldl or wo. In my Haskell code of scope of this might be something like haskell recursive fold ( taking a fold... Also why it 's customary to put acc first and seqval second in the list elements a... Otoh, if you're interested avoided if ( and only if ) f is a Monoid, we:!: as you can see, it implements the left, rather from! Gains a lot haskell recursive fold performance from compilation in introductory programming is converting a that. Slow because it has been abstracted out into things like 'map ' and 'fold ' that foldl1 will an! I mean an algorithm that terminates the recursion at some of the operator / switches Haskell... One thing: bottomline, everything rests on recursion in Haskell officially only takes one.... Possible that we can say that the mapping and the combination do n't understand the problem fully my! Result is a product of numbers in a list of integers with strictness that returns a Monoid is any type. Turn an almost tail recursive implementation into a tail recursive function can be used to express a range...

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