using ML as a programming language we have list and tuple, in the case of lists we can form a list from another list by removing or appending elements from and to the original list, for example if we have:
val x = [7,8,9] : int list
in REPL we can do some operations like the following:
- hd x;
val it = 7 : int
- tl x;
val it = [8,9] : int list
now if we have a tuple lets say:
val y = (7,8,9) :int*int*int
now the question is that , can we have a smaller tuple by removing the first element from the original tuple ? in other words , how to remove (#1 y) and have new tuple (8,9) in a similar way that we do it in the case of list.
Thanks.
Tuples are very different from lists. With lists, size need not be known at compile time, but with tuples, not only should the number of elements be known at compile time, the type of each element is independent of the others.
Take the type signature of tl:
- tl;
val it = fn : 'a list -> 'a list
It is 'a list -> 'a list - in other words tl takes a list of 'a and returns another one. Why don't we have one for tuples as well? Assume we wanted something like
y = (1,2,3);
tail y; (* returns (2,3) *)
Why does this not make sense? Think of the type signature of tail. What would it be?
In this case, it would clearly be
'a * 'b * 'c -> 'b * 'c
Takes product of an 'a, a 'b and a 'c and returns a product of
a 'b and a 'c. In ML, all functions defined must have a statically determined
type signature. It would be impossible to have a tail function for tuples that
handles all possible tuple sizes, because each tuple size is essentially a different type.
'a list
Can be the type of many kinds of lists: [1,2,3,4], or ["A", "short", "sentence"], or
[true, false, false, true, false]. In all these cases, the value of the type
variable 'a is bound to a different type. (int, string, and bool). And 'a list can be a list of any size.
But take tuples:
(1, true, "yes"); (* (int * bool * string) *)
("two", 2) (* (string, int) *)
("ok", "two", 2) (* (string, string, int) *)
Unlike list, these are all of different types. So while the type signature of all lists is simple ('a list), there is no 'common type' for all tuples - a 2-tuple has a different type from a 3-tuple.
So you'll have to do this instead:
y = (7, 8, 9);
(a, b, c) = y;
and a is your head and you can re-create the tail with (b,c).
Or create your own tail:
fun tail (a,b,c) = (b, c)
This also gives us an intuitive understanding as to why such a function would not make sense: If is impossible to define a single tail for use across all tuple types:
fun tail (a,b) = (b)
| tail (a,b,c) = (b, c) (* won't compile *)
You can also use the # shorthand to get at certain elements of the tuple:
#1 y; (* returns 7 *)
But note that #1 is not a function but a compile time shorthand.
Lists and tuples are immutable so there is no such thing like removing elements from them.
You can construct a new tuple by decomposing the original tuple. In SML, the preferred way is to use pattern matching:
fun getLastTwo (x, y, z) = (y, z)
If you like #n functions, you can use them as well:
val xyz = (7, 8, 9)
val yz = (#2 xyz, #3 xyz) (* (8, 9) *)
Related
Suppose I have the following list of tuple [("a", "b"); ("c", "d"); ("e", "f"); ("g", "h"); ("i", "j")]. I would like to create a sub list from that list by taking the second element of each tuple of that list. In other words, I want to obtain ["a"; "d"; "f", "h"].
I tried iter (fun (x,y) -> liste#[y]) old_list;; but I got
Error: This expression has type (string * string) list
but an expression was expected of type unit
Do I have to use a recursive function? Is there a workaround?
The type of List.iter is ('a -> unit) -> 'a list -> unit. In other words, it returns unit, which is the way in OCaml to return nothing (or nothing interesting anyway).
Since you want your function to return a result, you can't use List.iter. Instead, you want to transform each element of a list in some way. The function that does this is List.map.
Here's a function that changes a list of ints into a list of bools according to whether each int is odd or even:
let is_even list = List.map (fun x -> x mod 2 = 0) list
It works like this:
# is_even [3;4;5;6;7];;
- : bool list = [false; true; false; true; false]
You want something like this with a different fun expression.
As a side comment, your result isn't what would usually be called a sublist of the original list. A sublist is a new list containing some of the elements of the old list. You could call it a projection more accurately I think.
I got a problem that needs to turn a list of tuples into a flattened list for example:
[(1,2), (3,4), (5,6)] can be turned into [1,2,3,4,5,6]
I have tried to write a function like this:
fun helper2(nil,b) = []
| helper2(a,nil) = []
| helper2(a::l1,b::l2) =l1::l2
fun flatten2 [] = []
| flatten2 ((a,b)::tl) = helper2(a,b)
It shows:
val flatten2 = fn : ('a list * 'a list list) list -> 'a list list
And when I tried to run it using command flatten2[(1,2),(3,4),(5,6)];
It will give me the following error message:
stdIn:1.2-1.29 Error: operator and operand do not agree [overload conflict]
operator domain: ('Z list * 'Z list list) list
operand: ([int ty] * [int ty]) list
in expression:
flatten2 ((1,2) :: (3,4) :: (<exp>,<exp>) :: nil)
My questions are:
Why SML see the a and b values as lists, not just simply a and b
How can I revise my code so SML can see a and b as 'a and 'b not lists
How to make this code work the way it should be?
Thanks
First question: As to why the type comes out as ('a list * 'a list list) it's because type inference is looking at this part of the code:
| helper2(a::l1,b::l2) =l1::l2
^^
here
Keep in mind that the type of the "cons" (::) operator is 'a -> 'a list -> 'a list, it is gluing a single element onto a list of that same type of element. So SML has concluded that whatever l1 and l2 are, the relationship is that l2 is a list of whatever l1 is.
fun helper2(nil,b) = []
Says that a must be a list because nil has type 'a list. Therefore, l2 has to be a list of lists (of some type 'a).
Question 2 and 3: I'm not quite sure how to correct the code as it is written. I'd probably write something like this:
fun helper2 [] accum = List.rev accum
| helper2 ((a,b)::tl) accum = helper2 tl (b :: a :: accum);
fun flatten2 list = helper2 list [];
helper2 does all of the dirty work. If the input list is empty then we're all done and we can return the reversed accumulator that we've been building up. The second case is where we actually add things to the accumulator. We pattern match on the head and the tail of the list. This pattern match means that the input has type ('a * 'a) list (a list of tuples where both elements are the same type). In the head, we have a tuple and we name the first and second element a and b, respectively. We prepend a then b onto the accumulator and recursively call helper2 on the tail of the list. Eventually, we'll chew through all the elements in the list and then we'll be left with just the accumulator -- which, recall, has all the elements but in the reverse order. Calling List.rev reverses the accumulator and that's our answer.
And when I load and run it I get this:
- flatten2 [(1,2), (3,4), (5,6)];
val it = [1,2,3,4,5,6] : int list
Why SML see the a and b values as lists, not just simply a and b
Chris already answered this in-depth.
You're passing a as the first argument to helper2, which expects a list as its first argument. And you're passing b as the second argument to helper2, which uses its second argument, b::l2, also a list, as the tail of a list where a is the head. So b must be a list of those lists.
This doesn't make any sense, and is most likely a consequence of confusing syntax: You are passing in what you think of single elements a and b in flatten2, but when you deal with them in helper2 they're now lists where the heads are called a and b. Those are not the same a and b.
How can I revise my code so SML can see a and b as 'a and 'b not lists
You could ditch the helper function to begin with:
fun flatten2 [] = []
| flatten2 ((a,b)::pairs) = a :: b :: flatten2 pairs
The purpose of having a helper function is so that it can accumulate the result during recursion, because this version of flatten2 uses a lot of stack space. It can do this with an extra argument so that flatten2 doesn't need to mention it:
This is the version Chris made.
How to make this code work the way it should be?
You can make this code in a lot of ways. Two ways using explicit recursion were mentioned.
Here are some alternatives using higher-order functions:
(* Equivalent to my first version *)
fun flatten2 pairs =
foldr (fn ((a,b), acc) => a :: b :: acc) [] pairs
(* Equivalent to Chris'es version *)
fun flatten2 pairs =
rev (foldl (fn ((a,b), acc) => b :: a :: acc) [] pairs)
(* Yet another alternative *)
fun concatMap f xs =
List.concat (List.map f xs)
fun flatten2 pairs =
concatMap (fn (a,b) => [a,b]) pairs
I've defined functions:
fun concaten(x,y) =
if (x = [])
then y
else hd(x) :: concaten(tl(x),y);
as well as:
fun existsin(x,L) =
if (L=[])
then false
else if (x = hd(L))
then true
else existsin(x,tl(L));
and am now trying to define a function of type (((list * list) -> list) -> list) that looks vaguely like the following:
fun strongunion(x,y) =
val xy = concaten(x,y);
if xy=[]
then []
if (existsin(hd(xy),tl(xy)) andalso x!= [])
then strongunion(tl(x),y)
else if (existsin(hd(xy),tl(xy)) andalso x = [])
then strongunion(x,tl(y))
else if (x != [])
then hd(xy) :: strongunion(tl(x),y)
else hd(xy) :: strongunion(x,tl(y));
which takes the "strong" union of two lists, i.e. it combats faulty inputs (lists with element duplicates). This code is, of course, syntactically invalid, but the reason I included it was to show what such a function would look like in an imperative language.
The way I started going about doing this was to first concatenate the lists, then remove duplicated elements from that concatenation (well, technically I am adding non-duplicates to an empty list, but these two operations are consequentially equivalent). To do this, I figured I would design the function to take two lists (type list*list), transform them into their concatenation (type list), then do the duplicate removal (type list), which would be of type (((list*list) -> list) -> list).
My issue is that I have no idea how to do this in SML. I'm required to use SML for a class for which I'm a TA, otherwise I wouldn't bother with it, and instead use something like Haskell. If someone can show me how to construct such higher-order functions, I should be able to take care of the rest, but I just haven't come across such constructions in my reading of SML literature.
I'm a bit unsure if strong union means anything other than just union. If you assume that a function union : ''a list * ''a list -> ''a list takes two lists of elements without duplicates as inputs, then you can make it produce the unions without duplicates by conditionally inserting each element from the one list into the other:
(* insert a single element into a list *)
fun insert (x, []) = [x]
| insert (x, xs as (y::ys)) =
if x = y
then xs
else y::insert(x, ys)
(* using manual recursion *)
fun union ([], ys) = ys
| union (x::xs, ys) = union (xs, insert (x, ys))
(* using higher-order list-combinator *)
fun union (xs, ys) = foldl insert ys xs
Trying this:
- val demo = union ([1,2,3,4], [3,4,5,6]);
> val demo = [3, 4, 5, 6, 1, 2] : int list
Note, however, that union wouldn't be a higher-order function, since it doesn't take functions as input or return functions. You could use a slightly stretched definition and make it curried, i.e. union : ''a list -> ''a list -> ''a list, and say that it's higher-order when partially applying it to only one list, e.g. like union [1,2,3]. It wouldn't even be fully polymorphic since it accepts only lists of types that can be compared (e.g. you can't take the union of two sets of functions).
I would like to iterate over all combinations of elements from a list of lists which have the same length but not necessarily the same type. This is like the cartesian product of two lists (which is easy to do in OCaml), but for an arbitrary number of lists.
First I tried to write a general cartesian (outer) product function which takes a list of lists and returns a list of tuples, but that can't work because the input list of lists would not have elements of the same type.
Now I'm down to a function of the type
'a list * 'b list * 'c list -> ('a * 'b * 'c) list
which unfortunately fixes the number of inputs to three (for example). It's
let outer3 (l1, l2, l3) =
let open List in
l1 |> map (fun e1 ->
l2 |> map (fun e2 ->
l3 |> map (fun e3 ->
(e1,e2,e3))))
|> concat |> concat
This works but it's cumbersome since it has to be redone for each number of inputs. Is there a better way to do this?
Background: I want to feed the resulting flat list to Parmap.pariter.
To solve your task for arbitrary ntuple we need to use existential types. We can use GADT, but they are close by default. Of course we can use open variants, but I prefer a little more syntactically heavy but more portable solution with first class modules (and it works because GADT can be expressed via first class modules). But enough theory, first of all we need a function that will produce the n_cartesian_product for us, with type 'a list list -> 'a list list
let rec n_cartesian_product = function
| [] -> [[]]
| x :: xs ->
let rest = n_cartesian_product xs in
List.concat (List.map (fun i -> List.map (fun rs -> i :: rs) rest) x)
Now we need to fit different types into one type 'a, and here comes existential types, let's define a signature:
module type T = sig
type t
val x : t
end
Now let's try to write a lifter to this existential:
let int x = (module struct type t = int let x = x end : T)
it has type:
int -> (module T)
Let's extend the example with few more cases:
let string x = (module struct type t = string let x = x end : T)
let char x = (module struct type t = char let x = x end : T)
let xxs = [
List.map int [1;2;3;4];
List.map string ["1"; "2"; "3"; "4"];
List.map char ['1'; '2'; '3'; '4']
]
# n_cartesian_product xxs;;
- : (module T) list list =
[[<module>; <module>; <module>]; [<module>; <module>; <module>];
[<module>; <module>; <module>]; [<module>; <module>; <module>];
...
Instead of first class modules you can use other abstractions, like objects or functions, if your type requirements allow this (e.g., if you do not need to expose the type t). Of course, our existential is very terse, and maybe you will need to extend the signature.
I used #ivg 's answer but in a version with a GADT. I reproduce it here for reference. In a simple case where only the types float and int can appear in the input lists, first set
type wrapped = Int : int -> wrapped | Float : float -> wrapped
this is a GADT without type parameter. Then
let wrap_f f = Float f
let wrap_i i = Int f
wrap types into the sum type. On wrapped value lists we can call n_cartesian_product from #ivg 's answer. The result is a list combinations: wrapped list list which is flat (for the present purposes).
Now to use Parmap, i have e.g. a worker function work : float * int * float * float -> float. To get the arguments out of the wrappers, I pattern match:
combinations |> List.map (function
| [Float f1; Int i; Float f2; Float f3] -> (f1, i, f2, f3)
| _ -> raise Invalid_argument "wrong parameter number or types")
to construct the flat list of tuples. This can be finally fed to Parmap.pariter with the worker function work.
This setup is almost the same as using a regular sum type type wrapsum = F of float | I of int instead of wrapped. The pattern matching would be the same; the only difference seems to be that getting a wrong input, e.g. (F 1, I 1, F 2.0, F, 3.0) would be detected only at runtime, not compile time as here.
I have a question about the way SML of New Jersey interprets lists:
Suppose I have a function f(x : 'a, n : int) : 'a list such that f returns a list of n copies of x, e.g. f(2,5) = [2,2,2,2,2], f(9,0) = [].
So then I go into the REPL, and I check f(9,0) = nil, and it returns true. From this, I assumed that you could use list = nil to check whether a list is the empty list. I used this in a function, and it wouldn't run. I ended up learning that the type definitions are different:
sml:121.2-123.10 Error: operator and operand don't agree [equality type required]
operator domain: ''Z * ''Z
operand: 'a list * 'Y list
in expression:
xs = nil
(Where xs was my list). I then learned that the way to check if a list is the empty list is with null list. Why is this so? What's going on with nil? Can someone explain this behavior to me?
I also note that apparently (case xs = of nil is the same as checking null xs. Does this mean nil is a type?
This is an error related to polymorphism. By default, when an empty list is evaluated, it has the type 'a list. This means that the list can contain elements of any type. If you try to evaluate 1::[], you won't get a type error because of that. This is called polymorphism, it is a feature that allows your functions to take arguments of any type. This can be useful in functions like null, because you don't care about the contents of the list in that case, you only care about its length (in fact, you only care if it's empty or not).
However, you can also have empty lists with different types. You can make your function return an empty int list. In fact, you are doing so in your function.
This is the result in a trivial implementation of your function:
- fun f(x : 'a, n : int) : 'a list =
case n of
0 => []
| _ => x::f(x, n-1);
val f = fn : 'a * int -> 'a list
- f(4,5);
val it = [4,4,4,4,4] : int list
- f(4,0);
val it = [] : int list
As you can see, even if the second argument is 0, your function returns an int list. You should be able to compare it directly with an list of type 'a list.
- it = [];
val it = true : bool
However, if you try to compare two empty lists that have different types and are not type of 'a list, you should get an error. You can see an example of it below:
- [];
val it = [] : 'a list
- val list1 : int list = [];
val list1 = [] : int list
- val list2 : char list = [];
val list2 = [] : char list
- list1 = [];
val it = true : bool
- list2 = [];
val it = true : bool
- list1 = list2;
stdIn:6.1-6.14 Error: operator and operand don't agree [tycon mismatch]
operator domain: int list * int list
operand: int list * char list
in expression:
list1 = list2
Also, case xs of nil is a way of checking if a list is empty, but this is because nil (which is just a way to write []) has the type 'a list by default. (Note that case expressions don't directly return a boolean value.) Therefore, nil is not a type, but 'a list is a polymorphic type that you can compare with lists of any type, but if your empty lists don't have polymorphic type, you will get a type error, which I think what is happening in your case.