For a list of items I and an operator LEQ which returns bool for any pair of items in I, SORT() returns a list S such that:
1. Every item in I is present exactly once in S
2. For each consecutive pair of items (S_i, S_j) in S, LEQ(S_i, S_j) is true.
aka, one item in I is missing in your output.
Also, SORT(1,2,3,4) = 1,2,3,4,7
but I do take the view that ( 1, 2, 3, 4, 5, 5, 6 ) is a list of seven values (perhaps the number of dollars in the pockets of seven distinct unique people) and when sorted the output should also have seven items that correspond to the seven input items.
> Also ...
Yeah, that needs tightening up by pastel8739
L = [(5,foo), (2,bar), (2,baz),...]
SORT(L, key=first) # or however it'd be specified
> You need a way to differentiate the two 5s
As there's no unique filtering or other reduction going on here, there's a permutation chain from input to output.
SORT ( 3, 2, 5, 5 ) ->> ( 2, 3, 5, 5 ) and
SORT ( 3, 2, 5, 5 ) ->> ( 2, 3, 5, 5 )
( I'm teasing, perhaps )
Every item in I is present exactly once in S
Are we talking about Values? Then inigyou is correct.
Memory locations? Then it is trivially true, but that does not prevent me from writing 0 into every memory location.
Value + Memory location? Then it does not work for arrays since we are modifying the memory locations by moving the values between them.
The abstract notion of manipulable things in a indexable order? You need to show how that correlates to reality in a way where you can not put in a hole even larger than this one you are trying to close.
To loop back, this very discussion shows how non-trivial it really is and how much thought actually needs to be put into handling even "trivial" problems. Almost everybody who talks about how we can replace these complex implementations with simpler, understandable specifications has little to no experience with the difficulties of actually creating correct specifications. Anybody who would bring up sorting as "trivial" either has no idea what they are talking about or is so far ahead that they have weird ideas as to what constitutes as "trivial". In both cases, their opinion is highly divorced from practical reality.
That is not to say that it is not worthwhile or even that the specifications are "more complex". It is quite possible the specification is still simpler despite the difficulty, but it is also likely the complex implementation was already totally incomprehensible and a simplified specification is also incomprehensible, it is now just formally incomprehensible.
You have that /s tag, but this is actually the problem with pastel8739's spec as written.
>> 1. Every item in I is present exactly once in S
This actually does require inigyou's example to be the result of calling SORT when you cannot distinguish repeated items from each other.
SORT([1,1]) => [1,1]
pastel8739's spec also doesn't exclude the possibility of inserting new values (so long as they aren't duplicates of items in the source list).
2. If the comparison implements a strict total order, the output is sorted according to it.
However, that specification is not trivial. Almost nobody correctly articulates property 1 when first encountering the problem if they do not already know the answer or are already aware it is a trick question (and even then most software developers still fail).
Furthermore, that also sidesteps the problem of formally specifying what a permutation is. Unless you have a grab bag of already proven powerful theorems, the author is most likely also going to make a error doing that as well even if we start at a proof abstraction level comparable to normal programming.
Reality is that trivial problems admit trivially wrong specifications exceedingly easily. There is little reason to assume that much more complicated problems that are hard to even articulate will magically support obviously correct specifications that are simpler and more understandable than the code.