Here we look at another example of applying unfolds: iterate. For example. to supply their own equality test. program-id. That's quite a lot of words to describe such a simple algorithm! (* output_elem is a printer for elements of [items] *) items |> List.iteri (fun i x -> printf "%d: %a" i output_elem x ) What about the rest of it? I am just learning FP and Haskell … Try using a piece of paper to write down how the evaluation would look like if we try to take, say, 3 from [4,3,2,1]. Say, my list is present in this variable. If x and y were comparable, I could do Ekcetera, ekcetera ... Of course, these also have edge cases. Most imperative languages don't have pattern matching so you have to make a lot of if else statements to test for edge conditions. The third pattern breaks the list into a head and a tail. Now the third pattern is where the action happens. list. ys in turn (if any) has been removed from xs. Having an element or two in a recursion definition defined non-recursively (like F(0) and F(1) here) is also called the edge condition and is important if you want your recursive function to terminate. Duplicates, and elements of the first list, are removed from the Eventually, the (n-1) part will cause our function to reach the edge condition. Mathematics (specifically combinatorics) has a function called factorial. For example. It is a special case of unionBy, which allows the programmer to supply their own equality test. Because Haskell is non-strict, the elements of the list are evaluated only if they are needed, which allows us to use infinite lists. It is a special case of deleteBy, which allows the programmer to It stores several elements of the same type. The maximum function takes a list of things that can be ordered (e.g. using the binary operator, from right to left: scanl is similar to foldl, but returns a list of successive We have a list of items that can be sorted. What is it? each sublist in the result contains only equal elements. Finally! A more "functional" solution uses the predefined Haskell function iterate: iterate :: (a -> a) -> a -> [a] iterate f x = x : iterate f (f x) The function iterate generates an infinite list in the following way: It is an instance of the more general genericReplicate, in which n may be of any integral type. If the first list is not finite, the result is the first list. Otherwise, we return the maximum of the rest of the list. Picking the problems was easy. If the maximum of the tail is bigger, well, then it's the maximum of the tail. An even clearer way to write this function is to use max. It doesn't matter if it's a list, a tree or any other data structure. Note I not using HUGS nor GHC, this is just in my head. Haskell implementation: The \\ function is list difference ((non-associative). Instead, there are two alternatives: there are list iteration constructs (like foldl which we've seen before), and tail recursion. Related: cycle, repeat, replicate, take Makes sense because what's the maximum of an empty list? unfoldr :: (b -> Maybe (a, b)) -> b -> [a] unfoldr takes the element and returns Nothing if it is done producing the list or returns Just (a, b) , in which case, a is a prepended to the list and b is used as the next element in a recursive call. We go up one step again where we had 2 and [5,1]. Because we've now come down to only non-recursively defined fibonacci numbers, we can safely say that F(3) is 2. If it is, we return the head. In JavaScript, we iterate over the whole list but use the index argument which is coming from reduce to check if the current element is the first element (index 0) or not and concatenate the elements onto the accumulator. For this example, loop over the arrays: (a,b,c) (A,B,C) (1,2,3) to produce the output: aA1 bB2 cC3 If possible, also describe what happens when the arrays are of … replicate takes an Int and some element and returns a list that has several repetitions of the same element. If you're dealing with trees, the edge case is usually a node that doesn't have any children. For instance, take 3 [5,4,3,2,1] will return [5,4,3]. Haskell Cheat Sheet This cheat sheet lays out the fundamental ele-ments of the Haskell language: syntax, keywords and other elements. This is a very common idiom when doing recursion with lists, so get used to it. The premise is simple enough: Run through a list, and combine each 3 items next to each other with another function and return a list with the results. How are we going to filter the list so that we get only the elements smaller than the head of our list and only elements that are bigger? predicate, respectively; i.e.. delete x removes the first occurrence of x from its list argument. Come on ... it's the empty list! A sorted empty list is an empty list. For example. Empty list, as is expected. repeat takes an element and returns an infinite list that just has that element. To make searching easy I've included a list of functions below. Here's how we could rewrite maximum' by using max: How's that for elegant! Because Haskell supports infinite lists, our recursion doesn't really have to have an edge condition. The edge condition patterns kick in and so the result is (1,'a'):(2,'b'):[], which is exactly the same as [(1,'a'),(2,'b')]. Let's give it a small test run to see if it appears to behave correctly. The good thing about infinite lists though is that we can cut them where we want. First, the direct recursive way seen in the Haskell report: iterate f x = x: iterate f (f x) We can also write it in terms of scanl or scanl1 and repeat: iterate f x = scanl f x (repeat x) We mention recursion briefly in the previous chapter. In Haskell, lists are a homogenous data structure. We sort the two lists using the same function. Now here comes the main algorithm: a sorted list is a list that has all the values smaller than (or equal to) the head of the list in front (and those values are sorted), then comes the head of the list in the middle and then come all the values that are bigger than the head (they're also sorted). If we try to take 0 or less elements from a list, we get an empty list. First two patterns say that if the first list or second list is empty, we get an empty list. That means that if n turns out to be more than 0, the matching will fall through to the next pattern. So when trying to think of a recursive way to solve a problem, try to think of when a recursive solution doesn't apply and see if you can use that as an edge case, think about identities and think about whether you'll break apart the parameters of the function (for instance, lists are usually broken into a head and a tail via pattern matching) and on which part you'll use the recursive call. identification division. The definition of the iterate function is: iterate f x = Cons (x, iterate f (f x)) E.g., let f x = 2x, the result of iterate f 1 is the following list: 1, Cons (f 1, iterate f (f 1))-> 1, 2, Cons (f 2, iterate … For example. O-kay. the pair of lists of elements which do and do not satisfy the foldl, applied to a binary operator, a starting value (typically Usually you define an edge case and then you define a function that does something between some element and the function applied to the rest. It's a very clever way of sorting items. In part 2, we started writing our own functions in Haskell modules. It is an instance of the more general, By convention, overloaded functions have a non-overloaded List comprehensions. Booyah! identification division. If you read them from left to right, you'll see the sorted list. In Haskell, we use everything but the first element as the processing list and concatenate every element onto the accumulator. Next up, we'll implement take. So at one point, you'll have [1,4,3] ++  ++ [9,6,7]. As you can see, pattern matching goes great with recursion! The length of a list is one plus the length of the tail of the list. data division. Also for negative numbers, because it doesn't really make sense. In this section we'll look at the basics of lists, strings (which are lists) and list comprehensions. That means that we can have a list of integers or a list of characters but we can't have a list that has a few integers and then a few characters. 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