I'm struggling with the syntax with Haskell. It's very simple, but i'm stuck.
I'm trying to write a findMin function that takes a list and finds the minimum. Here's my code, I have tried so many syntactical things that I'm up for any help I can get.
findMin [] = [0]
findMin list = if any < head list then findMin(tail) else take 1
And i get all sorts of type errors. What is going wrong??
(if it helps at all i have a background in object oriented programming)
I see that you've got things figured out in the comments but I'll add some things here hopefully to help. I also feel like I should quickly mention that Haskell already has a minimum function just in case anyone stumbles upon this who isn't just trying to learn the language and actually needs the function for something.
First of all let's talk about types. I would normally expect a findMin style of function to return the minimum value rather than that value inside a list so the type will be:
findMin :: (Num a, Ord a) => [a] -> a
The things before => add context to the function type. This restricts all the things that a can be to only be things that have order (otherwise how can we find a minimum). Secondly Num a forces a to be a number, this is necessary because you specified that the case for the empty list should be 0.
I'll explain 2 other ways to write the findMin function, trying to make them more concise than your definition (one of the benefits of Haskell is how concise it can be and I also find it helps when learning to see the multiple possibilities). The first will be using recursion and guards the second will be using a list comprehension.
We can't do much with findMin [] = 0 so we'll move onto the lists with stuff in them.
We need to be careful with a recursive definition because eventually we will evaluate findMin [] and always get 0 so we need to stop the recursion before that by defining a case for a single value:
findMin [x] = x
When passing a list as an argument to a function you can separate out its elements and give them each a name so (x:xs) means a value x is the first element followed by a list of elements xs.
For this definition we will define the first two elements on their own followed by the rest of the elements:
findMin (x:y:xs)
| x < y = findMin (x:xs)
| otherwise = findMin (y:xs)
The guards allow us to have multiple definitions for the function depending on a condition. If x < y we want to get rid of y as it cannot be the minimum so we find the minimum of x and the remaining elements, xs. If x is not smaller than y then the minimum value is either y or one of the values in xs.
The second way to define this function is using a list comprehension (this is my favourite as it is particularly concise).
We aren't using recursion so we don't need the case for one element we can keep our definition for an empty list and go straight to any list with elements:
findMin xs = head [x | x <- xs, all (>= x) xs]
So what's going on here? [x | x <- xs] creates a list of x values where x is all the elements from xs. We then add a condition to say we only want those values if all (>= x) xs meaning if all elements of xs are greater than or equal to xs.
This results in a list of the minimum elements. This might have one element if the minimum occurs once or it may have several if it occurs multiple times. Either way they are all the same so we just take the first using head.
Hope this helps and hope you have fun learning Haskell. Feel free to ask if you have any questions :)
In ghci this function seems to do your trick:
let findMin x = if length x > 1 then min (head x) (findMin (tail x)) else head x
I'm learning as I try to answer some questions here, so any feedback would be appreciated.
Related
I have to write my own List.map function using 'for elem in list' and tail/non-tail recursive. I have been looking all around Google for some tips, but didn't find much. I am used to Python and it's pretty hard to not think about using its methods, but of course, these languages are very different from each other.
For the first one I started with something like:
let myMapFun funcx list =
for elem in list do
funcx elem::[]
Tail recursive:
let rec myMapFun2 f list =
let cons head tail = head :: tail
But anyway, I know it's wrong, it feels wrong. I think I am not used yet to F# strcture. Can anyone give me a hand?
Thanks.
As a general rule, when you're working through a list in F#, you want to write a recursive function that does something to the head of the list, then calls itself on the tail of the list. Like this:
// NON-tail-recursive version
let rec myListFun list =
match list with
| [] -> valueForEmptyList // Decision point 1
| head :: tail ->
let newHead = doSomethingWith head // Decision point 2
newHead :: (myListFun tail) // Return value might be different, too
There are two decisions you need to make: What do I do if the list is empty? And what do I do with each item in the list? For example, if the thing you're wanting to do is to count the number of items in the list, then your "value for empty list" is probably 0, and the thing you'll do with each item is to add 1 to the length. I.e.,
// NON-tail-recursive version of List.length
let rec myListLength list =
match list with
| [] -> 0 // Empty lists have length 0
| head :: tail ->
let headLength = 1 // The head is one item, so its "length" is 1
headLength + (myListLength tail)
But this function has a problem, because it will add a new recursive call to the stack for each item in the list. If the list is too long, the stack will overflow. The general pattern, when you're faced with recursive calls that aren't tail-recursive (like this one), is to change your recursive function around so that it takes an extra parameter that will be an "accumulator". So instead of passing a result back from the recursive function and THEN doing a calculation, you perform the calculation on the "accumulator" value, and then pass the new accumulator value to the recursive function in a truly tail-recursive call. Here's what that looks like for the myListLength function:
let rec myListLength acc list =
match list with
| [] -> acc // Empty list means I've finished, so return the accumulated number
| head :: tail ->
let headLength = 1 // The head is one item, so its "length" is 1
myListLength (acc + headLength) tail
Now you'd call this as myListLength 0 list. And since that's a bit annoying, you can "hide" the accumulator by making it a parameter in an "inner" function, whose definition is hidden inside myListLength. Like this:
let myListLength list =
let rec innerFun acc list =
match list with
| [] -> acc // Empty list means I've finished, so return the accumulated number
| head :: tail ->
let headLength = 1 // The head is one item, so its "length" is 1
innerFun (acc + headLength) tail
innerFun 0 list
Notice how myListLength is no longer recursive, and it only takes one parameter, the list whose length you want to count.
Now go back and look at the generic, NON-tail-recursive myListFun that I presented in the first part of my answer. See how it corresponds to the myListLength function? Well, its tail-recursive version also corresponds well to the tail-recursive version of myListLength:
let myListFun list =
let rec innerFun acc list =
match list with
| [] -> acc // Decision point 1: return accumulated value, or do something more?
| head :: tail ->
let newHead = doSomethingWith head
innerFun (newHead :: acc) tail
innerFun [] list
... Except that if you write your map function this way, you'll notice that it actually comes out reversed. The solution is to change innerFun [] list in the last line to innerFun [] list |> List.rev, but the reason why it comes out reversed is something that you'll benefit from working out for yourself, so I won't tell you unless you ask for help.
And now, by the way, you have the general pattern for doing all sorts of things with lists, recursively. Writing List.map should be easy. For an extra challenge, try writing List.filter next: it will use the same pattern.
let myMapFun funcx list =
[for elem in list -> funcx elem]
myMapFun ((+)1) [1;2;3]
let rec myMapFun2 f = function // [1]
| [] -> [] // [2]
| h::t -> (f h)::myMapFun f t // [3]
myMapFun2 ((+)1) [1;2;3] // [4]
let myMapFun3 f xs = // [6]
let rec g f xs= // [7]
match xs with // [1]
| [] -> [] // [2]
| h::t -> (f h)::g f t // [3]
g f xs
myMapFun3 ((+)1) [1;2;3] // [4]
// [5] see 6 for a comment on value Vs variable.
// [8] see 8 for a comment on the top down out-of-scopeness of F#
(*
Reference:
convention: I've used a,b,c,etc refer to distinct aspects of the numbered reference
[1] roughly function is equivalent to the use of match. It's the way they do it in
OCaml. There is no "match" in OCaml. So this is a more compatible way
of writing functions. With function, and the style that is used here, we can shave
off a whole two lines from our definitions(!) Therefore, readability is increased(!)
If you end up writing many functions scrolling less to be on top
of the breadth of what is happening is more desirable than the
niceties of using match. "Match" can be
a more "rounded" form. Sometimes I've found a glitch with function.
I tend to change to match, when readability is better served.
It's a style thing.
[1b] when I discovered "function" in the F# compiler source code + it's prevalence in OCaml,
I was a little annoyed that it took so long to discover it + that it is deemed such an
underground, confusing and divisive tool by our esteemed F# brethren.
[1c] "function" is arguably more flexible. You can also slot it into pipelines really
quickly. Whereas match requires assignment or a variable name (perhaps an argument).
If you are into pipelines |> and <| (and cousins such as ||> etc), then you should
check it out.
[1d] on style, typically, (fun x->x) is the standard way, however, if you've ever
appreciated the way you can slot in functions from Seq, List, and Module, then it's
nice to skip the extra baggage. For me, function falls into this category.
[2a] "[]" is used in two ways, here. How annoying. Once it grows on you, it's cool.
Firstly [] is an empty list. Visually, it's a list without the stuff in it
(like [1;2;3], etc). Left of the "->" we're in the "pattern" part of the partern
matching expression. So, when the input to the function (lets call it "x" to stay
in tune with our earliest memories of maths or "math" classes) is an empty list,
follow the arrow and do the statement on the right.
Incidentally, sometimes it's really nice to skip the definition of x altogether.
Behold, the built in "id" identity function (essentially fun (x)->x -- ie. do nothing).
It's more useful than you realise, at first. I digress.
[2b] "[]" on the right of [] means return an empty list from this code block. Match or
function symantics being the expression "block" in this case. Block being the same
meaning as you'll have come across in other languages. The difference in F#, being
that there's *always* a return from any expression unless you return unit which is
defined as (). I digress, again.
[3a] "::" is the "cons" operator. Its history goes back a long way. F# really only
implements two such operators (the other being append #). These operators are
list specific.
[3b] on the lhs of "->" we have a pattern match on a list. So the first element
on the lhs of :: goes into the value (h)ead, and the rest of the list, the tail,
goes into the (t)ail value.
[3c] Head/tail use is very specific in F#. Another language that I like a lot, has
a nicer terminology for obviously interesting parts of a list, but, you know, it's
nice to go with an opinionated simplification, sometimes.
[3d] on the rhs of the "->", the "::", surprisingly, means join a single element
to a list. In this case, the result of the function f or funcx.
[3e] when we are talking about lists, specifically, we're talking about a linked
structure with pointers behind the scenes. All we have the power to do is to
follow the cotton thread of pointers from structure to structure. So, with a
simple "match" based device, we abstract away from the messy .Value and .Next()
operations you may have to use in other languages (or which get hidden inside
an enumerator -- it'd be nice to have these operators for Seq, too, but
a Sequence could be an infinite sequence, on purpose, so these decisions for
List make sense). It's all about increasing readability.
[3f] A list of "what". What it is is typically encoded into 't (or <T> in C#).
or also <T> in F#. Idiomatically, you tend to see 'someLowerCaseLetter in
F# a lot more. What can be nice is to pair such definitions (x:'x).
i.e. the value x which is of type 'x.
[4a] move verbosely, ((+)1) is equivilent to (fun x->x+1). We rely on partial
composition, here. Although "+" is an operator, it is firstmost, also a
function... and functions... you get the picture.
[4b] partial composition is a topic that is more useful than it sounds, too.
[5] value Vs variable. As an often stated goal, we aim to have values that
never ever change, because, when a value doesn't change, it's easier to
think and reason about. There are nice side-effects that flow from that
choice, that mean that threading and locking are a lot simpler. Now we
get into that "stateless" topic. More often than not, a value is all you
need. So, "value" it is for our cannon regarding sensible defaults.
A variable, implies, that it can be changed. Not strictly true, but in
the programming world this is the additional meaning that has been strapped
on to the notion of variable. Upon hearing the word variable, ones mind might
start jumping through the different kinds of variable "hoops". It's more stuff
that you need to hold in the context of your mind. Apparently, western people
are only able to hold about 7 things in their minds at once. Introduce mutability
and value in the same context, and there goes two slots. I'm told that more uniform
languages like Chinese allow you to hold up to 10 things in your mind at once.
I can't verify the latter. I have a language with warlike Saxon and elegant
French blended together to use (which I love for other reasons).
Anyway, when I hear "value", I feel peace. That can only mean one thing.
[6] this variation really only achieves hiding of the recursive function. Perhaps
it's nice to be a little terser inside the function, and more descriptive to
the outside world. Long names lead to bloat. Sometimes, it's just simpler.
[7a] type inference and recursion. F# is one of the nicest
languages that I've come across for elegantly dealing with recursive algorithms.
Initially, it's confusing, but once you get past that
[7b] If you are interested in solving real problems, forget about "tail"
recursion, for now. It's a cool compiler trick. When you get performance conscious,
or on a rainy day, it
might be a useful thing to look up.
Look it up by all means if you are curious, though. If you are writing recursive
stuff, just be aware that the compiler geeks have you covered (sometimes), and
that horrible "recursive" performance hole (that is often associated with
recursive techniques -- ie. perhaps avoid at all costs in ancient programming
history) may just be turned into a regular loop for you, gratis. This auto-to-loop
conversion has always been a compiler geek promise. You can rely on it more though.
It's more predictable in F# as to when "tail recursion" kicks in. I digress.
Step 1 correctly and elegantly solve useful problems.
Step 2 (or 3, etc) work out why the silicon is getting hot.
NB. depending on the context, performance may be an equally important thing
to think about. Many don't have that problem. Bear in mind that by writing
functionally, you are structuring solutions in such a way that they are
more easily streamlineable (in the cycling sense). So... it's okay not to
get caught in the weeds. Probably best for another discussion.
[8] on the way the file system is top down and the way code is top down.
From day one we are encouraged in an opinionated (some might say coerced) into
writing code that has flow + code that is readable and easier to navigate.
There are some nice side-effects from this friendly coercion.
I am trying to understand what's useful and how to actually use lambda expression in Haskell.
I don't really understand the advantage of using lambda expression over the convention way of defining functions.
For example, I usually do the following:
let add x y = x+y
and I can simply call
add 5 6
and get the result of 11
I know I can also do the following:
let add = \x->(\y-> x+y)
and get the same result.
But like I mentioned before, I don't understand the purpose of using lambda expression.
Also, I typed the following code (a nameless function?) into the prelude and it gave me an error message.
let \x -> (\y->x+y)
parse error (possibly incorrect indentation or mismatched backets)
Thank you in advance!
Many Haskell functions are "higher-order functions", i.e., they expect other functions as parameters. Often, the functions we want to pass to such a higher-order function are used only once in the program, at that particular point. It's simply more convenient then to use a lambda expression than to define a new local function for that purpose.
Here's an example that filters all even numbers that are greater than ten from a given list:
ghci> filter (\ x -> even x && x > 10) [1..20]
[12,14,16,18,20]
Here's another example that traverses a list and for every element x computes the term x^2 + x:
ghci> map (\ x -> x^2 + x) [1..10]
[2,6,12,20,30,42,56,72,90,110]
I know that "a powerset is simply any number between 0 and 2^N-1 where N is number of set members and one in binary presentation denotes presence of corresponding member".
(Hynek -Pichi- Vychodil)
I would like to generate a powerset using this mapping from the binary representation to the actual set elements.
How can I do this with Erlang?
I have tried to modify this, but with no success.
UPD: My goal is to write an iterative algorithm that generates a powerset of a set without keeping a stack. I tend to think that binary representation could help me with that.
Here is the successful solution in Ruby, but I need to write it in Erlang.
UPD2: Here is the solution in pseudocode, I would like to make something similar in Erlang.
First of all, I would note that with Erlang a recursive solution does not necessarily imply it will consume extra stack. When a method is tail-recursive (i.e., the last thing it does is the recursive call), the compiler will re-write it into modifying the parameters followed by a jump to the beginning of the method. This is fairly standard for functional languages.
To generate a list of all the numbers A to B, use the library method lists:seq(A, B).
To translate a list of values (such as the list from 0 to 2^N-1) into another list of values (such as the set generated from its binary representation), use lists:map or a list comprehension.
Instead of splitting a number into its binary representation, you might want to consider turning that around and checking whether the corresponding bit is set in each M value (in 0 to 2^N-1) by generating a list of power-of-2-bitmasks. Then, you can do a binary AND to see if the bit is set.
Putting all of that together, you get a solution such as:
generate_powerset(List) ->
% Do some pre-processing of the list to help with checks later.
% This involves modifying the list to combine the element with
% the bitmask it will need later on, such as:
% [a, b, c, d, e] ==> [{1,a}, {2,b}, {4,c}, {8,d}, {16,e}]
PowersOf2 = [1 bsl (X-1) || X <- lists:seq(1, length(List))],
ListWithMasks = lists:zip(PowersOf2, List),
% Generate the list from 0 to 1^N - 1
AllMs = lists:seq(0, (1 bsl length(List)) - 1),
% For each value, generate the corresponding subset
lists:map(fun (M) -> generate_subset(M, ListWithMasks) end, AllMs).
% or, using a list comprehension:
% [generate_subset(M, ListWithMasks) || M <- AllMs].
generate_subset(M, ListWithMasks) ->
% List comprehension: choose each element where the Mask value has
% the corresponding bit set in M.
[Element || {Mask, Element} <- ListWithMasks, M band Mask =/= 0].
However, you can also achieve the same thing using tail recursion without consuming stack space. It also doesn't need to generate or keep around the list from 0 to 2^N-1.
generate_powerset(List) ->
% same preliminary steps as above...
PowersOf2 = [1 bsl (X-1) || X <- lists:seq(1, length(List))],
ListWithMasks = lists:zip(PowersOf2, List),
% call tail-recursive helper method -- it can have the same name
% as long as it has different arity.
generate_powerset(ListWithMasks, (1 bsl length(List)) - 1, []).
generate_powerset(_ListWithMasks, -1, Acc) -> Acc;
generate_powerset(ListWithMasks, M, Acc) ->
generate_powerset(ListWithMasks, M-1,
[generate_subset(M, ListWithMasks) | Acc]).
% same as above...
generate_subset(M, ListWithMasks) ->
[Element || {Mask, Element} <- ListWithMasks, M band Mask =/= 0].
Note that when generating the list of subsets, you'll want to put new elements at the head of the list. Lists are singly-linked and immutable, so if you want to put an element anywhere but the beginning, it has to update the "next" pointers, which causes the list to be copied. That's why the helper function puts the Acc list at the tail instead of doing Acc ++ [generate_subset(...)]. In this case, since we're counting down instead of up, we're already going backwards, so it ends up coming out in the same order.
So, in conclusion,
Looping in Erlang is idiomatically done via a tail recursive function or using a variation of lists:map.
In many (most?) functional languages, including Erlang, tail recursion does not consume extra stack space since it is implemented using jumps.
List construction is typically done backwards (i.e., [NewElement | ExistingList]) for efficiency reasons.
You generally don't want to find the Nth item in a list (using lists:nth) since lists are singly-linked: it would have to iterate the list over and over again. Instead, find a way to iterate the list once, such as how I pre-processed the bit masks above.
I am having trouble understanding what my lecturer want me to do from this question. Can anyone help explain to me what he wants me to do?
Define a higher order version of the insertion sort algorithm. That is define
functions
insertBy :: Ord b => (a->b) -> a -> [a] -> [a]
inssortBy :: Ord b => (a->b) -> [a] -> [a]
and this bit is where it got me confused:
such that inssort f l sorts the list l such that an element x comes before an elementyif f x < f y.
If you were sorting numbers, then it's clear what x < y means. But what if you were sorting letters? Or customers? Or anything else without a clear (to the computer) ordering?
So you are supposed to create a function f() that defines that ordering for the sorting procedure. That f() will take the letters or customers or whatever and will return an integer for each one that the computer can actually sort on.
At least, that's how the problem is described. I personally would have designed a predicate that accepted two items, x and y and returned a boolean if x < y. But whichever is fine.
The code wants you to rewrite the insertion sort algorithm, but using a function as a parameter - thus a higher order function.
I would like to point out that this code, typo included, seems to stem from a piece of work currently due at a certain university - I found this page while searching for "insertion sort algortihm", as I copy pasted the term out of the document as well, typo included.
Seeking code from the internet is a risky business. Might I recommend the insertion sort algorithm wikipedia entry, or the Haskell code provided in your lecture slides (you are looking for the 'insertion sort algorithm' and for 'higher-order functions), as opposed to the several queries you have placed on Stack Overflow?
I'm trying to use scheme to write a function f that takes a number n and a function g and returns a list of lists of length n, but according with booleans according to the pattern indicated by g. For example, the function f should take n say 3 and the function g which makes every 3rd item on the list a true. It should return this:
(list (list true true false)
(list true true false)
(list true true false))
I have no idea where to start with this, so any help or tips would be greatly appreciated. THanks!
Key to this looks like using map.
In short, map takes a function and a list, and applies the function to that list, returning the modified list.
To do this, as I'm assuming you don't want a direct solution, I'd first construct your list of lists (just use an accumulator -- count down -- to make the list as long as needed, and use cons and list and quote for everything else.
Then simply apply map (probably twice, depending on how you implement your solution) to the list of lists. (eg list l and function fn: (map fn (car l)) and recur with the cdr of l and cons the results back together).
Good luck! Hope this helps!
I don't quite follow what you're trying to do, but in addition to map, you might find build-list to be useful. build-list takes a number and a procedure, takes the range from 0 to that number less 1, and maps your procedure over that range. e.g.
> (build-list 5 (lambda (x) (* x x)))
(0 1 4 9 16)