Implicitly, primesBelow shouldn't ever in fact call itself, not as it is articulated here **at the very least**, not without wasting a lot of calculus.

As it is, and maybe no matter what (i'm not sure, don't have the knowledge to certify that), when primesBelow checks if a value "v" is prime or not, well no matter what it'll already have calculated and stored all primes below this value n (this, according to how primesBelow is articulated, aka filtering of Naturals bottom-top).

Thus, if for each potential element "v" of the result (in my version, "list") of primesBelow, you call once again primesBelow, asking it to generate again all primes below sqrt(v), you'll do nothing more than doing again what you already did, because all those previous primes have already been generated, stored away, and especially very accessible, in the list-result in-construction of the **current** call to primesBelow, so if you don't use it but call again primesBelow to get a copy of what you already have, you'll multiply immensely the work without any gain.
That's why I named the very result of primesBelow, to get a way to use "list" (the previously generated items of the future result-list) in "checker".

2016-05-05 15:44 GMT+02:00 Dushyant Juneja <juneja.dushyant@gmail.com>:
Hi Akash,

Thanks for the response. A very simple and lucid explanation. Looks interesting.

So, here's the big picture now, for which I need this. I intend to implement a lookalike Sieve of Eratosthenes algorithm in haskell. For this, I intend to use the earlier function recursively, as follows:

primesBelowN :: Integer -> [Integer]
primesBelowN n = 2:3:filter f [6*k+i | k <- [1..(n-1)`div`6], i <- [-1, 1]]
                     where f x = foldr g True xs
                                 where g t ac = (x `rem` t /= 0) && ac
                                       xs = [ m | m <- primesBelowN n, m <= (truncate (sqrt (fromInteger x)))]

Of course, I could do something like this:

primesBelowN :: Integer -> [Integer]
primesBelowN n = 2:3:filter f [6*k+i | k <- [1..(n-1)`div`6], i <- [-1, 1]]
                     where f x = foldr g True xs
                                 where g t ac = (x `rem` t /= 0) && ac
                                       xs = [ m | m <- primesBelowN (truncate (sqrt (fromInteger x)))]

However, this calls primesBelowN function with a new argument everytime. I suppose that is not optimal (correct me if I am wrong).

Point number 2: both fail. Grrh.

Any ideas how I could go recursive with this function?

Dushyant


On Thu, May 5, 2016 at 6:31 PM akash g <akaberto@gmail.com> wrote:
Hi Dushyant,

The problem most likely is
[m | m <- [5,7..], m <= (truncate (sqrt (fromInteger x)))]

 This is because, the filter condition (the last part) does a very simple thing:  It filters out any element that does not fulfil the criteria.  You are operating on a list that is monotonically increasing.  However, the filter isn't aware of this property.  Hence, this list comprehension never ends because it doesn't know that once the condition fails, it will always fail.

Thus, the solution would be to generate a finite set (or take a part of the infinite set using takeWhile or something like that), instead of using an infinite one.

Regards,
G Akash.

On Thu, May 5, 2016 at 6:13 PM, Dushyant Juneja <juneja.dushyant@gmail.com> wrote:
Hi,

I seem to be landing into infinite recursion when using higher order functions with list comprehension. Take this for an example. The following works well, and gives answers for numbers like 2000000 as well:

primesBelowN :: Integer -> [Integer]
primesBelowN n = 2:3:filter f [6*k+i | k <- [1..(n-1)`div`6], i <- [-1, 1]]
                     where f x = foldr g True xs
                                 where g t ac = (x `rem` t /= 0) && ac
                                       xs = [5, 7..(truncate (sqrt (fromInteger x)))]


However, the following never returns anything for the same number, probably due to some kind of loop malfunction:

primesBelowN :: Integer -> [Integer]
primesBelowN n = 2:3:filter f [6*k+i | k <- [1..(n-1)`div`6], i <- [-1, 1]]
                     where f x = foldr g True xs
                                 where g t ac = (x `rem` t /= 0) && ac
                                       xs = [ m | m <- [5, 7, ..], m <= (truncate (sqrt (fromInteger x)))]

Any ideas what might be going wrong?

Thanks in advance!

DJ

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