I wonder: if we have a decidable partial ordering: like leq
method in PartialOrd
type class in lattices
package.
Can we sort a list using leq
so we get kind of a topological sorting?
Note, topological sorting is used for graphs: there we don’t have leq
-like luxury. We’d first need to compute a transitive closure of a graph. On the other hand, given a list of "node", it may not include all nodes; and may include duplicates.
Insertion sort is a simple sorting algorithm, trivial to implement for lists (i.e. not in-place). I continue wondering: maybe it will produce topological ordering. Even insertion sort is simple, I still had an uneasy feeling: does it really work. Insertion sort terminates despite what function you pass as a comparator (even impure random one!), so what kind of output it produces when given lawful leq
?
I must admit that exposure to Haskell and Agda made me suspect all pen and paper proofs, especially made by myself. So let’s certify insertion sort. The plan is to show how to certify insertion order first for total order and then for partial order.
Module definitions and imports:
module Topo where
open import Data.List
open import Relation.Nullary
open import Relation.Binary.PropositionalEquality
Next, the definition of insert
and sort
. These are straight-forward and are familiar. Note that _≤?_
decides whether ≤
-relation holds. Returning Dec
doesn’t forget what we compute (it returns the evidence), compared with simply returning Bool
.
module Sort {A : Set} (_≤_ : A → A → Set)
(_≤?_ : (x y : A) → Dec (x ≤ y))
where
insert : A → List A → List A
insert x [] = x ∷ []
insert x (y ∷ ys) with x ≤? y
... | yes x≤y = x ∷ y ∷ ys
... | no ¬x≤y = y ∷ insert x ys
sort : List A → List A
sort [] = []
sort (x ∷ xs) = insert x (sort xs)
Twan van Laarhoven proved complete correctness of various sort algorithms (code gist). We’ll only do a sortedness of insertion sort.
Twan uses _≤?_ : (x y : A) → (x ≤ y ⊎ y ≤ x)
comparator (⊎
is Either
), that implies that ordering have to be total. Our ... → Dec (x ≤ y)
version is less powerful, therefore we’ll need to assume ≤-flip
.
Twan also defines insert
to operate on Sorted xs
, our proof is completely external.
There are few auxiliary lemmas, culminating with lemma that insert
preserves sortedness, and the theorem that sort
produces a sorted list.
One could also show that sort
produces a permutation of input list, but that’s something I’m quite confident about already. Note: the proof of that fact won’t need any additional assumptions about ≤
.
By the way, these proofs show how dependently typed programming is full with various lists.^{1} Luckily (or not) Agda allows reuse of constructor names.
module Total (A : Set) (_≤_ : A → A → Set)
(_≤?_ : (x y : A) → Dec (x ≤ y))
(≤-trans : ∀ {x y z} → x ≤ y → y ≤ z → x ≤ z)
(≤-flip : ∀ {x y} → ¬ (x ≤ y) → y ≤ x) -- precise enough!
where
infix 4 _≤*_
open Sort _≤_ _≤?_
data _≤*_ (x : A) : List A → Set where
[] : x ≤* []
_∷_ : ∀ {y ys} → x ≤ y → x ≤* ys → x ≤* y ∷ ys
data Sorted : List A → Set where
[] : Sorted []
_∷_ : ∀ {x xs} → x ≤* xs → Sorted xs → Sorted (x ∷ xs)
≤*-trans : ∀ x y ys → x ≤ y → y ≤* ys → x ≤* ys
≤*-trans x y [] x≤y [] = []
≤*-trans x y (y' ∷ ys) x≤y (y≤y' ∷ y'≤ys) =
≤-trans x≤y y≤y' ∷ (≤*-trans x y ys x≤y y'≤ys)
lem-cons-all≤ : ∀ x y ys → x ≤ y → y ≤* ys → x ≤* y ∷ ys
lem-cons-all≤ x y ys x≤y y≤ys = x≤y ∷ ≤*-trans x y ys x≤y y≤ys
lem-skip : ∀ x y ys → y ≤ x → y ≤* ys → y ≤* insert x ys
lem-skip x y [] y≤x y≤ys = y≤x ∷ y≤ys
lem-skip x y (y' ∷ ys) y≤x (y≤y' ∷ y'≤ys ) with x ≤? y'
... | yes x≤y' = y≤x ∷ y≤y' ∷ y'≤ys
... | no ¬x≤y' = y≤y' ∷ (lem-skip x y ys y≤x y'≤ys)
lem-insert-sorted : ∀ x xs → Sorted xs → Sorted (insert x xs)
lem-insert-sorted x [] s = [] ∷ s
lem-insert-sorted x (y ∷ ys) (y≤ys ∷ sys) with x ≤? y
... | yes x≤y = lem-cons-all≤ x y ys x≤y y≤ys ∷ y≤ys ∷ sys
... | no ¬x≤y = lem-skip x y ys (≤-flip ¬x≤y) y≤ys ∷ sorted-insert-x-ys
where sorted-insert-x-ys = lem-insert-sorted x ys sys
thm-sort-sorted : ∀ xs → Sorted (sort xs)
thm-sort-sorted [] = []
thm-sort-sorted (x ∷ xs) =
lem-insert-sorted x (sort xs) (thm-sort-sorted xs)
But what about partial order? Wikipedia says following about the topological sort:
In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge from vertex to vertex , comes before in the ordering.
We massage that into simple "there aren’t edges pointing backwards". So instead of saying: "for all sublists x ∷ ys
, x
is less-than-or-equal than any of y
ys
" we say "for all subsets x ∷ ys
, x is not greater-than any of y
ys
".
After that, the proof structure is quite similar. I needed to use antisymmetry of ≤
, which shows that this Sorted
predicate won’t hold for preorder. I’m not sure whether insertion order would work for preorder, I’m not sure it won’t work either.
module Partial (A : Set) (_≤_ : A → A → Set)
(_≤?_ : (x y : A) → Dec (x ≤ y))
(≤-trans : ∀ {x y z} → x ≤ y → y ≤ z → x ≤ z)
(≤-antisym : ∀ {x y} → x ≤ y → y ≤ x → x ≡ y)
where
open Sort _≤_ _≤?_
record _<_ (x y : A) : Set where
constructor le
field
is-le : x ≤ y
not-eq : ¬ (x ≡ y)
open _<_
<-trans₁ : ∀ {x y z} → x < y → y ≤ z → x < z
<-trans₁ (le x≤y ¬x≡y) y≤z = le
(≤-trans x≤y y≤z)
(λ x≡z → ¬x≡y (≤-antisym x≤y (subst (λ i → _ ≤ i) (sym x≡z) y≤z)))
infix 4 _¬>*_
-- x ¬>* ys = x is not larger than any in y ∈ ys
data _¬>*_ (x : A) : List A → Set where
[] : x ¬>* []
_∷_ : ∀ {y ys} → ¬ y < x → x ¬>* ys → x ¬>* y ∷ ys
data Sorted : List A → Set where
[] : Sorted []
_∷_ : ∀ {x xs} → x ¬>* xs → Sorted xs → Sorted (x ∷ xs)
lem-trans-none> : ∀ x y ys → x ≤ y → y ¬>* ys → x ¬>* ys
lem-trans-none> x y [] x≤y [] = []
lem-trans-none> x y (z ∷ zs) x≤y (¬z<y ∷ y≤zs) =
(λ z<y → ¬z<y (<-trans₁ z<y x≤y)) ∷ lem-trans-none> x y zs x≤y y≤zs
lem-flip : ∀ {x y} → x ≤ y → ¬(y < x)
lem-flip {x} {y} x≤y with y ≤? x
... | yes y≤x = λ y<x → not-eq y<x (≤-antisym y≤x x≤y)
... | no ¬y≤x = λ y<x → ¬y≤x (is-le y<x)
lem-cons-none> : ∀ x y ys → x ≤ y → y ¬>* ys → x ¬>* y ∷ ys
lem-cons-none> x y ys x≤y y≤ys =
lem-flip x≤y ∷ lem-trans-none> _ _ _ x≤y y≤ys
lem-skip : ∀ x y ys → ¬ (x ≤ y) → y ¬>* ys → y ¬>* insert x ys
lem-skip x y [] ¬x≤y [] = (λ p → ¬x≤y (is-le p)) ∷ []
lem-skip x y (z ∷ zs) ¬x≤y (¬z<y ∷ y≤zs) with x ≤? z
... | yes x≤z = (λ x<y → ¬x≤y (is-le x<y)) ∷ ¬z<y ∷ y≤zs
... | no ¬x≤z = ¬z<y ∷ (lem-skip x y zs ¬x≤y y≤zs)
lem-insert-sorted : ∀ x xs → Sorted xs → Sorted (insert x xs)
lem-insert-sorted x [] s = [] ∷ s
lem-insert-sorted x (y ∷ ys) (y≤ys ∷ sys) with x ≤? y
... | yes x≤y = lem-cons-none> x y ys x≤y y≤ys ∷ y≤ys ∷ sys
... | no ¬x≤y = lem-skip x y ys ¬x≤y y≤ys ∷ sorted-insert-x-ys
where sorted-insert-x-ys = lem-insert-sorted x ys sys
thm-sort-sorted : ∀ xs → Sorted (sort xs)
thm-sort-sorted [] = []
thm-sort-sorted (x ∷ xs) =
lem-insert-sorted x (sort xs) (thm-sort-sorted xs)
Would merge sort work with partial order? I don’t know yet!
Let’s try in Haskell. You should try your hypothesis first, before trying to formally prove them. Proving false statements can take a lot of time!
After a little of imports, we’ll defined isSorted
check, and try it on a N5
lattice. insertionSort
works with leq
, but a mergeSort
only with <=
.
import Algebra.PartialOrd
import Algebra.Lattice.N5
import Algebra.Lattice.Ordered
import Data.List (sortBy)
import Test.QuickCheck
lt :: PartialOrd a => a -> a -> Bool
lt x y = x /= y && leq x y
notGt :: PartialOrd a => a -> a -> Bool
notGt x y = not (lt y x)
-- | This checks that list is sorted in PartialOrd sense.
isSorted :: PartialOrd a => [a] -> Bool
isSorted [] = True
isSorted (x : ys) = isSorted ys && all (notGt x) ys
-- | Sorted holds when list is sorted using @Ord@
--
-- +++ OK, passed 100 tests.
totalProp :: [Ordered Int] -> Bool
totalProp = isSorted . sortBy compare
-- | Next, let's define insertion sort.
insertionSort :: (a -> a -> Bool) -> [a] -> [a]
insertionSort le = go where
go [] = []
go (x:xs) = insert x (go xs)
insert x [] = [x]
insert x (y : ys) | le x y = x : y : ys
| otherwise = y : insert x ys
-- | And try with a partially ordered set.
--
-- +++ OK, passed 100 tests.
--
-- Works!
m5Prop :: [N5] -> Bool
m5Prop = isSorted . insertionSort leq
-- Then, naive mergesort.
mergeSort :: (a -> a -> Bool) -> [a] -> [a]
mergeSort f = go where
go [] = []
go [x] = [x]
go xs = let (ys, zs) = split xs
in merge (go ys) (go zs)
merge [] ys = ys
merge xs [] = xs
merge (x:xs) (y:ys) | f x y = x : merge xs (y:ys)
| otherwise = y : merge (x:xs) ys
-- | >>> split [1..10]
-- ([1,3,5,7,9],[2,4,6,8,10])
split :: [a] -> ([a],[a])
split [] = ([], [])
split (x:xs) = case split xs of
~(ys,zs) -> (x:zs,ys)
-- | Our 'mergeSort' returns correct results. Works like 'sort'.
--
-- +++ OK, passed 100 tests
mergeProp :: [Int] -> Property
mergeProp xs = mergeSort (<=) xs === sortBy compare xs
-- >>> quickCheck m5Prop2
-- *** Failed! Falsified (after 18 tests and 4 shrinks):
-- [N5b,N5a,N5c]
-- sorted [N5a,N5c,N5b]
--
-- >>> insertionSort leq [N5b,N5a,N5c]
-- [N5c,N5b,N5a]
--
-- Sort is not unique: insertionSort finds an ordering:
-- See picture at https://hackage.haskell.org/package/lattices-2/docs/Algebra-Lattice-N5.html
--
-- >>> leq N5b N5a
-- True
--
-- >>> isSorted [N5c,N5b,N5a]
-- True
--
-- >>> isSorted [N5b,N5a,N5c]
-- True
--
-- >>> isSorted [N5b,N5c,N5a]
-- True
--
m5Prop2 :: [N5] -> Property
m5Prop2 xs =
let xs' = mergeSort leq xs
in counterexample ("sorted " ++ show xs') $ isSorted xs'
Note: Sorted
is somewhat lax:
-- >>> isSorted [N5b,N5c,N5b]
-- True
But that’s not an issue to me, as I’ll be sorting lists with distinct elements. More on that later.
In Haskell there’s a variety of string types. In Dependent Haskell there will be a variety of various lists and naturals numbers... ... and strings types. https://twitter.com/phadej/status/1147829454187761664↩︎