Can everyone explain to me this piece of code ?
let safe_division n = function
| 0 -> failwith "divide by 0"
| m -> n / m
When I excute safeDiv 3 0 , what is the m and n in this case ?
In general case, when does the function match the first and second pattern ?
It is easy to see what this means once you realise that
let f x y z = e
is just a short-hand for
let f = function x -> function y -> function z -> e
That is, a function of n arguments actually is n nested functions of 1 argument. That representation is called "currying". It is what allows you to apply a function partially, e.g.
let g = f 3
returns a function of 2 arguments.
Of course, the short-hand above can be mixed freely with the explicit form on the right, and that's what your example does. You can desugar it into:
let safe_division = function n -> function
| 0 -> failwith "divide by 0"
| m -> n / m
When you execute safe_division 3 0, first, 3 is bound to the name n and the right-hand side of the declaration is then evaluated.
This is a function, so the next argument, 0, is matched against the different cases, in order. Here, it matches the first case, so the right-hand side is evaluated and an exception is thrown. In this case, the name m is never bound to anything.
If the second argument was, for example, 1, then it would have matched the second case (this case matches every possible value anyway, it's a default case), binding the name m to the value 1 and then returning the result of n / m.
let safe_division n
define a function which type is int -> ...
function
| 0 -> failwith "divide by 0"
| m -> n / m
define a function which type is int -> int
So the resulting type of the whole is int -> int -> int where n is the first argument, and m the second. The last int is the result.
let safe_division n = function
| 0 -> failwith "divide by 0"
| m -> n / m
is just equivalent to:
let safe_division n = fun x -> match x with
| 0 -> failwith "divide by 0"
| m -> n / m
Note fun and function are slightly different. See: Function definition.
Related
needing some help (if possible) in how to count the amount of times a recursive function executes itself.
I don't know how to make some sort of counter in OCaml.
Thanks!
Let's consider a very simple recursive function (not Schroder as I don't want to do homework for you) to calculate Fibonacci numbers.
let rec fib n =
match n with
| 0 | 1 -> 1
| _ when n > 0 -> fib (n - 2) + fib (n - 1)
| _ -> raise (Invalid_argument "Negative values not supported")
Now, if we want to know how many times it's been passed in, we can have it take a call number and return a tuple with that call number updated.
To get each updated call count and pass it along, we explicitly call fib in let bindings. Each time c shadows its previous binding, as we don't need that information.
let rec fib n c =
match n with
| 0 | 1 -> (1, c + 1)
| _ when n > 0 ->
let n', c = fib (n - 1) (c + 1) in
let n'', c = fib (n - 2) (c + 1) in
(n' + n'', c)
| _ -> raise (Invalid_argument "Negative values not supported")
And we can shadow that to not have to explicitly pass 0 on the first call.
let fib n = fib n 0
Now:
utop # fib 5;;
- : int * int = (8, 22)
The same pattern can be applied to the Schroder function you're trying to write.
You can create a reference in any higher scope like so
let counter = ref 0 in
let rec f ... =
counter := !counter + 1;
... (* Function body *)
If the higher scope happens to be the module scope (or file top-level scope) you should omit the in
You can return a tuple (x,y) where y you increment by one for each recursive call. It can be useful if your doing for example a Schroder sequence ;)
I am new to F# and haven't done functional programming since I was an undergraduate, but I've been trying teach myself. I wrote a naive recursive Extended Euclidean implementation, which works just fine, and am now trying again but with continuations.
I walked through the code by hand twice with a small example and got the correct answer, but when I run it through the interpreter I am not getting the same result, so I'm clearly misunderstanding something I am trying to do.
I ran eea 7 3 by hand, and the (correct) result I computed was (1, 1, -2)
But when I run it in the interpreter I get
eea 7 3;;
val it : int * int * int = (1, 0, 1)
Here's my implementation:
let eea a b =
let rec contEEA a b f =
match b with
| 0 -> f () (a,1,0)
| _ ->
contEEA b (a%b) (fun () t ->
let (d,x',y') = t
(d, y', x'-(y'*(a/b)))
)
contEEA a b (fun () t -> t)
For reference the naive approach, straight from a textbook, is
let rec eea_gcd a b =
match b with
| 0 -> (a, 1, 0)
| _ ->
let d, x', y' = eea_gcd b (a % b)
(d, y', x'-(y'*(a/b)))
Your continuation-based version is always doing exactly one iteration (the last one). When you make the recursive call, your continuation just straight up returns the result instead of "returning" it to the previous call by passing to the previous continuation.
So the call sequence goes like this:
eea 7 3
contEEA 7 3 (fun () t -> t)
b <> 0 ==> second case matches
contEEA 3 1 (fun () t -> ... (d, y', ...))
b <> 0 ==> second case matches
contEEA 1 0 (fun () t -> ... (d, y', ...))
b = 0 ==> first case matches
The continuation is called f () (1, 1, 0)
The continuation calculates result (1, 0, 1 - (0*(3/1)) = (1, 0, 1) and immediately returns it
What you want to do instead is when the first continuation calculates the result of (1, 0, 1) it should pass it to the previous continuation, so that it may carry on the calculations from there, ultimately passing the result to the very first continuation fun () t -> t, which returns it back to the consumer.
To do that, replace this line:
(d, y', x'-(y'*(a/b)))
With this:
f (d, y', x'-(y'*(a/b)))
Also, a few notes on some other aspects.
The first parameter of the continuation (the unit, ()) is not necessary, since it's never actually used (and how can it be?). You can lose it.
After removing the unit parameter, the first continuation becomes fun t -> t, which has a special name id (aka "the identity function")
Rather than destructure the triple with a let, you can do it right in the parameter declaration. Parameters can be patterns!
Applying all of the above, as well as the actual problem fix, here's a better version:
let eea a b =
let rec contEEA a b f =
match b with
| 0 -> f (a,1,0)
| _ ->
contEEA b (a%b) (fun (d,x',y') ->
f (d, y', x'-(y'*(a/b)))
)
contEEA a b id
I am doing practice with F#. I am trying to create a simple program capable to find me out a couple of prime numbers that, summed together, equal a natural number input. It is the Goldbach conjecture. A single couple of primes will be enough. We will assume the input to be a even number.
I first created a function to check if a number is prime:
let rec isPrime (x: int) (i: int) :bool =
match x % i with
| _ when float i > sqrt (float x) -> true
| 0 -> false
| _ -> isPrime x (i + 1)
Then, I am trying to develop a function that (a) looks for prime numbers, (b) compare their sum with the input 'z' and (c) returns a tuple when it finds the two numbers. The function should not be correct yet, but I would get the reason behind this problem:
let rec sumPrime (z: int) (j: int) (k: int) :int * int =
match isPrime j, isPrime k with
| 0, 0 when j + k > z -> (0, 0)
| 0, 0 -> sumPrime (j + 1) (k + 1)
| _, 0 -> sumPrime j (k + 1)
| 0, _ -> sumPrime (j + 1) k
| _, _ -> if j + k < z then
sumPrime (j + 1) k
elif j + k = z then
(j, k)
The problem: even if I specified that the output should be a tuple :int * int the compiler protests, claiming that the expected output should be of type bool. When in trouble, I usually refer to F# for fun and profit, that i love, but this time I cannot find out the problem. Any suggestion is greatly appreciated.
Your code has three problems that I've spotted:
Your isPrime returns a bool (as you've specified), but your match expression in sumPrime is matching against integers (in F#, the Boolean value false is not the same as the integer value 0). Your match expression should look like:
match isPrime j, isPrime k with
| false, false when j + k > z -> (0, 0)
| false, false -> ...
| true, false -> ...
| false, true -> ...
| true, true -> ...
You have an if...elif expression in your true, true case, but there's no final else. By default, the final else of an if expression returns (), the unit type. So once you fix your first problem, you'll find that F# is complaining about a type mismatch between int * int and unit. You'll need to add an else condition to your final match case to say what to do if j + k > z.
You are repeatedly calling your sumPrime function, which takes three parameters, with just two parameters. That is perfectly legal in F#, since it's a curried language: calling sumPrime with two parameters produces the type int -> int * int: a function that takes a single int and returns a tuple of ints. But that's not what you're actually trying to do. Make sure you specify a value for z in all your recursive calls.
With those three changes, you should probably see your compiler errors go away.
Is it possible to have nested if without else statements. I wrote the following useless program to demonstrate nested ifs. How do I fix this so it's correct in terms of syntax. lines 5 and 6 gives errors.
let rec move_helper b sz r = match b with
[] -> r
|(h :: t) ->
if h = 0 then
if h - 1 = sz then h - 1 ::r
if h + 1 = sz then h + 1 ::r
else move_helper t sz r
;;
let move_pos b =
move_helper b 3 r
;;
let g = move_pos [0;8;7;6;5;4;3;2;1]
You can't have if without else unless the result of the expression is of type unit. This isn't the case for your code, so it's not possible.
Here's an example where the result is unit:
let f x =
if x land 1 <> 0 then print_string "1";
if x land 2 <> 0 then print_string "2";
if x land 4 <> 0 then print_string "4"
You must understand that if ... then is an expression like any other. If no else is present, it must be understood as if ... then ... else () and thus has type unit. To emphasize the fact that it is an expression, suppose you have two functions f and g of type, say, int → int. You can write
(if test then f else g) 1
You must also understand that x :: r does not change r at all, it constructs a new list putting x in front of r (the tail of this list is shared with the list r). In your case, the logic is not clear: what is the result when h=0 but the two if fail?
let rec move_helper b sz r = match b with
| [] -> r
| h :: t ->
if h = 0 then
if h - 1 = sz then (h - 1) :: r
else if h + 1 = sz then (h + 1) :: r
else (* What do you want to return here? *)
else move_helper t sz r
When you have a if, always put an else. Because when you don't put an else, Java will not know if the case is true or false.
I'm doing some homework but I've been stuck for hours on something.
I'm sure it's really trivial but I still can't wrap my head around it after digging through the all documentation available.
Can anybody give me a hand?
Basically, the exercise in OCaml programming asks to define the function x^n with the exponentiation by squaring algorithm.
I've looked at the solution:
let rec exp x = function
0 -> 1
| n when n mod 2 = 0 -> let y = exp x (n/2) in y*y
| n when n mod 2 <> 0 -> let y = exp x ((n-1)/2) in y*y*x
;;
What I don't understand in particular is how the parameter n can be omitted from the fun statement and why should it be used as a variable for a match with x, which has no apparent link with the definition of exponentiation by squaring.
Here's how I would do it:
let rec exp x n = match n with
0 -> 1
| n when (n mod 2) = 1 -> (exp x ((n-1)/2)) * (exp x ((n-1)/2)) * x
| n when (n mod 2) = 0 -> (exp x (n/2)) * (exp x (n/2))
;;
Your version is syntaxically correct, yields a good answer, but is long to execute.
In your code, exp is called recursively twice, thus yielding twice as much computation, each call yielding itself twice as much computation, etc. down to n=0. In the solution, exp is called only once, the result is storred in the variable y, then y is squared.
Now, about the syntax,
let f n = match n with
| 0 -> 0
| foo -> foo-1
is equivalent to:
let f = function
| 0 -> 0
| foo -> foo-1
The line let rec exp x = function is the begging of a function that takes two arguments: x, and an unnammed argument used in the pattern matching. In the pattern matching, the line
| n when n mod 2 = 0 ->
names this argument n. Not that a different name could be used in each case of the pattern matching (even if that would be less clear):
| n when n mod 2 = 0 -> let y = exp x (n/2) in y*y
| p when p mod 2 <> 0 -> let y = exp x ((p-1)/2) in y*y*x
The keyword "function" is not a syntaxic sugar for
match x with
but for
fun x -> match x with
thus
let rec exp x = function
could be replaced by
let rec exp x = fun y -> match y with
which is of course equivalent with your solution
let rec exp x y = match y with
Note that i wrote "y" and not "n" to avoid confusion. The n variable introduced after the match is a new variable, which is only related to the function parameter because it match it. For instance, instead of
let y = x in ...
you could write :
match x with y -> ...
In this match expression, the "y" expression is the "pattern" matched. And like any pattern, it binds its variables (here y) with the value matched. (here the value of x) And like any pattern, the variables in the pattern are new variables, which may shadow previously defined variables. In your code :
let rec exp x n = match n with
0 -> 1
| n when (n mod 2) = 1 -> (exp x ((n-1)/2)) * (exp x ((n-1)/2)) * x
| n when (n mod 2) = 0 -> (exp x (n/2)) * (exp x (n/2))
;;
the variable n in the two cases shadow the parameter n. This isn't a problem, though, since the two variable with the same name have the same value.