[Haskell-beginners] Informal proof of bijective mapping between Naturals and Natural pairs
Lyndon Maydwell
maydwell at gmail.com
Fri Feb 4 01:06:59 CET 2011
Thanks for your feedback Brent!
The diagonal approach seems like it is much cleaner to me. There are
certainly many less concepts required to understand the proof.
I agree completely about the variable names, and the removal of 'y' by
currying. I should really take the time to tidy things like this up
:-)
One advantage I can think of for my system is that you could split the
primes not just into even/odd, but any number of lists. This would
mean that it is then trivial to apply the same approach to tuples of
any size. I guess that the equivalent progression in the diagonal
approach would be to use higher-dimensional spaces, and take 'plane'
intersections. Although with this approach the proof could probably be
expressed as succinctly, it would probably get very messy in terms of
implementation.
Michael Katelman also linked me to
http://en.wikipedia.org/wiki/Pairing_function, which seems very
relevant.
Probably the most trivial higher-arity aproach would just be a
'exponentiation' of any proven pairing function, with the right or
left or both tuple element(s) representing another tuple, rather than
just a number.
I was also wondering if the prime approach could be related to
cryptography in some way.
Anyway. Thanks again!
On Fri, Feb 4, 2011 at 5:14 AM, Brent Yorgey <byorgey at seas.upenn.edu> wrote:
> Hi Lyndon,
>
> On Tue, Feb 01, 2011 at 12:43:14AM +0800, Lyndon Maydwell wrote:
>> Hi Beginners.
>>
>> I was musing over weather Naturals were isomorphic to Natural pairs
>> and wrote the following "proof".
>
> Indeed they are. This is a classic result first due to Cantor, who
> came at it from the point of view of discovering that the natural
> numbers have the same cardinality (size) as the positive rational
> numbers (which can be thought of as pairs of naturals).
>
> Your proof is great -- it has the advantage of working on rather
> elementary principles.
>
> However, there are many such bijections. Here's another one, shorter
> than yours although somewhat harder to understand just by looking at
> the code:
>
> f :: (Integer, Integer) -> Integer
> f (x,y) = (r*r + r) `div` 2 + x
> where r = x + y
>
> f' :: Integer -> (Integer, Integer)
> f' n = (n - t, r - (n - t))
> where r = floor $ (sqrt(1 + 8*(fromIntegral n)) - 1) / 2
> t = (r*r + r) `div` 2
>
> *Main Test.QuickCheck> quickCheck (\(NonNegative x) (NonNegative y) -> f' (f (x,y)) == (x,y))
> +++ OK, passed 100 tests.
> *Main Test.QuickCheck> quickCheck (\(NonNegative x) -> f (f' x) == x)
> +++ OK, passed 100 tests.
>
> This one looks a bit complicated but at heart it's rather simple: we
> think of the pairs of naturals as residing in a grid, like so:
>
> . . .
> : : :
> (0,2) (1,2) (2,2) ...
> (0,1) (1,1) (2,1) ...
> (0,0) (1,0) (2,0) ...
>
> and we imagine listing them by diagonals, like
>
> (0,0) (0,1) (1,0) (0,2) (1,1) (2,0) (0,3) ...
>
> Now, the math to go back and forth between a pair and its index in
> this listing by diagonals isn't quite as nice-looking as one might
> hope (essentially we have to use the formula for triangular numbers
> and solve a quadratic equation) but it's not too bad. (Actually, what
> I wrote above won't work once you get to integers big enough that the
> sqrt starts losing precision, but you could write an accurate integer
> square root operation to get around that.)
>
> Some comments on your code style are interspersed below.
>
>> import Control.Monad.Logic
>> import Data.Maybe
>> import Data.List
>> import Data.Numbers.Primes
>> import Test.QuickCheck
>>
>> -- This program demonstrates a mapping between the pairs of natural
>> numbers and a subset of co-primes.
>> -- The property should hold for all sized lists, not just pairs.
>> Ordering is preserved.
>> -- Question: Are we able to compress the range to create a bijection?
>> -- Answer: Yes! We can use the breadth-wise indces of the products,
>> rather than the products themselves.
>>
>> -- Pair to Number
>>
>> x_p = (evens !!)
>> y_p = (odds !!)
>>
>> xy_p x y = x_p x * y_p y
>>
>> xy_c x = ns_nc . xy_p x
>
> The definition of xy_c strikes me as a gratuitous use of points-free
> style just for its own sake. It's hard to read because of the
> asymmetric treatment of x and the (implicit) second argument. I would
> find it much clearer to just write
>
> xy_c x y = ns_nc (xy_p x y)
>
> If you really want to do it points-free you could use a 'double
> composition' operator,
>
> oo = (.) . (.)
> xy_c = ns_nc `oo` xy_p
>
> but I don't recommend it.
>
> Also, most of the difficulty I had reading your code was due to your
> choice of function names, which were too telegraphic and didn't give
> me any clues as to what they did. Of course naming is rather an
> idiomatic thing, but if you want others to read your code I suggest
> trying to use names that are a bit more descriptive/suggestive.
>
>>
>> -- Number to Pair
>>
>> p_xy n = (x,y)
>> where
>> x = fromJust $ findIndex (`divides` n) evens
>> y = fromJust $ findIndex (`divides` n) odds
>>
>> c_xy = p_xy . nc_ns
>>
>> -- Sparse Number to Compact Number
>>
>> numbers = odds >>- \x -> evens >>- \y -> return (x*y) -- I don't
>> really understand LogicT... :-(
>>
>> ns_nc n = fromJust $ findIndex (==n) numbers
>>
>> -- Compact Numner to Sparse Number
>>
>> nc_ns = (numbers !!)
>>
>> -- Helpers
>>
>> evens = map (primes !!) [0,2..]
>> odds = map (primes !!) [1,3..]
>
> This strikes me as a rather inefficient way to define evens and odds.
> It would be better to do something like
>
> deinterlace :: [a] -> ([a],[a])
> deinterlace (x1:x2:xs) = (x1:l1, x2:l2)
> where (l1,l2) = deinterlace xs
>
> (evens, odds) = deinterlace primes
>
>>
>> x `divides` y = y `mod` x == 0
>>
>> -- Id referencing properties
>>
>> prop_p1 = forAll (elements [(x,y) | x <- [1..6], y <- [1..7]]) f
>> where
>> f (x,y) = (x,y) == c_xy (xy_c x y)
>>
>> prop_p2 = forAll (elements [1..100]) f
>> where
>> f c = c == uncurry xy_c (c_xy c)
>
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