[Haskell-cafe] natural mergesort in Data.List
Christian Sternagel
c.sternagel at gmail.com
Tue Sep 18 13:14:31 UTC 2018
Dear David,
I am guaranteeing (since I proved it in Isabelle/HOL) that the following
version of "sequences" does not contain empty lists in its result (I am
copying from my Isabelle formalization, in order to avoid typos) ;)
fun sequences :: "'a list ⇒ 'a list list"
and asc :: "'a ⇒ ('a list ⇒ 'a list) ⇒ 'a list ⇒ 'a list list"
and desc :: "'a ⇒ 'a list ⇒ 'a list ⇒ 'a list list"
where
"sequences (a # b # xs) =
(if key a > key b then desc b [a] xs else asc b ((#) a) xs)"
| "sequences [x] = [[x]]"
| "sequences [] = []"
| "asc a as (b # bs) =
(if ¬ key a > key b then asc b (λys. as (a # ys)) bs
else as [a] # sequences (b # bs))"
| "asc a as [] = [as [a]]"
| "desc a as (b # bs) =
(if key a > key b then desc b (a # as) bs
else (a # as) # sequences (b # bs))"
| "desc a as [] = [a # as]"
The "key" function is an implicit first parameter for each of
"sequences", "asc", and "desc" above. The fact that I am using a "key"
function instead of a comparator is due to Isabelle/HOL's standard
library. Also, there are no pattern guards in Isabelle/HOL. Anyway, it
should be relatively straight-forward to translate these functions into
Haskell.
Another thing: I just realized that "merge_pairs" in my formalization
also differs from "mergePairs", since with the changed "sequences" it
might actually get an empty list as input, in which case the current
"mergePairs" wouldn't terminate at all.
So for those who are interested, the full definition of mergesort can be
found here
https://devel.isa-afp.org/browser_info/current/AFP/Efficient-Mergesort/Efficient_Mergesort.html
where mergesort is called "msort_key".
cheers
chris
Btw: What is "NonEmpty x"?
On 09/18/2018 11:55 AM, David Feuer wrote:
> If you're guaranteeing that the result won't contain empty lists, it
> would be worth benchmarking the effect of using NonEmpty x instead of
> [x] in that spot.
>
> On Tue, Sep 18, 2018, 4:21 AM Christian Sternagel <c.sternagel at gmail.com
> <mailto:c.sternagel at gmail.com>> wrote:
>
> Dear Cafe,
>
> some years ago I formalized the mergesort implementation [1] from
>
>
> http://hackage.haskell.org/package/base-4.11.1.0/docs/src/Data.OldList.html#sort
>
> (as far as I can tell it didn't change since 2012) in the proof
> assistant Isabelle/HOL [2].
>
> More specifically, I proved its functional correctness (the result is
> sorted and contains all elements of the input with exactly the same
> multiplicities) and that it is a stable sorting algorithm.
>
> Very recently I also formalized a complexity result in Isabelle/HOL,
> namely that the number of comparisons is bounded from above by
>
> n + n * ⌈log 2 n⌉
>
> for lists of length n.
>
> For this proof I had to change the definition of "sequences",
> "ascending", and "descending" slightly.
>
> Now here is my question: Does anyone now of reasons why the current
> implementation of "sequences" is allowed to produce spurious empty lists
> in its result? The version I used in my formalization only differs in
> the following three spots:
>
> sequences [x] = [[x]] -- this is the only important change, since
> sequences [] = [] -- then the result does not contain empty lists
>
> instead of
>
> sequences xs = [xs]
>
> and
>
> ascending a as [] = let !x = as [a] in [x]
>
> instead of
>
> ascending a as bs = let !x = as [a] in x : sequences bs
>
> and
>
> descending a as [] = [a:as]
>
> instead of
>
> descending a as bs = (a:as) : sequences bs
>
> [1] https://www.isa-afp.org/entries/Efficient-Mergesort.html
> [2] http://isabelle.in.tum.de/
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