Correctness of short cut fusion
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1 Short cut fusion
Short cut fusion allows elimination of intermediate data structures using rewrite rules that can also be performed automatically during compilation.
The two most popular instances are the 1.1 foldr/build
The foldr :: (a > b > b) > b > [a] > b foldr c n [] = n foldr c n (x:xs) = c x (foldr c n xs) build :: (forall b. (a > b > b) > b > b) > [a] build g = g (:) []
foldr c n (build g) → g c n
1.2 destroy/unfoldr
The destroy :: (forall b. (b > Maybe (a,b)) > b > c) > [a] > c destroy g = g step step :: [a] > Maybe (a,[a]) step [] = Nothing step (x:xs) = Just (x,xs) unfoldr :: (b > Maybe (a,b)) > b > [a] unfoldr p e = case p e of Nothing > [] Just (x,e') > x:unfoldr p e'
destroy g (unfoldr p e) → g p e
2 Correctness
If theThat is, the left and righthand sides should be semantically the same for each instance of either rule. Unfortunately, this is not so in Haskell.
We can distinguish two situations, depending on whether 2.1 In the absence of seq
2.1.1 foldr/build
If foldr c n (build g) = g c n
The two sides are interchangeable in any program without affecting semantics.
2.1.2 destroy/unfoldr
The To see this, consider the following instance:
g = \x y > case x y of Just z > 0 p = \x > if x==0 then Just undefined else Nothing e = 0
destroy g (unfoldr p e) = g step (unfoldr p e) = case step (unfoldr p e) of Just z > 0 = case step (case p e of Nothing > [] Just (x,e') > x:unfoldr p e') of Just z > 0 = case step (case Just undefined of Nothing > [] Just (x,e') > x:unfoldr p e') of Just z > 0 = undefined
while its righthand side "evaluates" as follows:
g p e = case p e of Just z > 0 = case Just undefined of Just z > 0 = 0
The obvious questions now are:
 Can the converse also happen, that is, can a safely terminating program be transformed into a failing one?
 Can a safely terminating program be transformed into another safely terminating one that gives a different value as result?
There is no formal proof yet, but strong evidence supporting the conjecture that the answer to both questions is "No!".
The conjecture goes that ifdestroy g (unfoldr p e) ⊑ g p e
What is known is that semantic equivalence can be recovered here by putting moderate restrictions on p.
More precisely, ifdestroy g (unfoldr p e) = g p e
2.2 In the presence of seq
This is the more interesting setting, given that in Haskell there is no way to restrict the use of Unsurprisingly, it is also the setting in which more can go wrong than above.
2.2.1 foldr/build
In the presence of The instance
g = seq c = undefined n = 0
The converse cannot happen, because the following always holds:
foldr c n (build g) ⊒ g c n
Moreover, semantic equivalence can again be recovered by putting restrictions on the involved functions.
More precisely, iffoldr c n (build g) = g c n
2.2.2 destroy/unfoldr
Contrary to the situation without This is witnessed by the following instance:
g = \x y > seq x 0 p = undefined e = 0
Conditions for semantic approximation in either direction can be given as follows.
Ifdestroy g (unfoldr p e) ⊑ g p e
destroy g (unfoldr p e) ⊒ g p e
Of course, conditions for semantic equivalence can be obtained by combining the two laws above.
3 Discussion
Correctness of short cut fusion is not just an academic issue.
All recent versions of GHC (at least 6.0  6.6) automatically perform transformations likeThere has been at least one occasion where, as a result, a safely terminating program was turned into a failing one "in the wild", with a less artificial example than the ones given above.
3.1 foldr/build
As pointed out above, everything is fine with 3.2 destroy/unfoldr
As above, the compiler cannot figure out automatically whether (and how) a given instance of data Step a b = Done  Yield a b destroy' :: (forall b. (b > Step a b) > b > c) > [a] > c destroy' g = g step' step' :: [a] > Step a [a] step' [] = Done step' (x:xs) = Yield x xs unfoldr' :: (b > Step a b) > b > [a] unfoldr' p e = case p e of Done > [] Yield x e' > x:unfoldr' p e'
But it allows some of the laws above to be simplified a bit.
We would still have that ifdestroy g' (unfoldr' p e) ⊑ g p e
destroy' g (unfoldr' p e) = g p e
destroy' g (unfoldr' p e) ⊑ g p e
destroy' g (unfoldr' p e) ⊒ g p e
The worst change in program behavior from a complier user's point of view is when, through application of "optimization" rules, a safely terminating program is transformed into a failing one or one delivering a different result.
This can happen in the presence ofrepeat x = unfoldr (\y > Just (x,y)) undefined
or
repeat x = unfoldr' (\y > Yield x y) undefined
A lefttoright approximation as in
destroy g (unfoldr p e) ⊑ g p e
under suitable preconditions might be acceptable in practice. After all, it only means that the transformed program may be "more terminating" than the original one, but not less so.
If one insists on semantic equivalence rather than approximation, then the conditions imposed on the producer of the intermediate list become quite severe, in particular in the potential presence ofFor example, the following producer has to be outlawed then:
enumFromTo n m = unfoldr (\i > if i>m then Nothing else Just (i,i+1)) n
4 Literature
Various parts of the above story, and elaborations thereof, are also told in the following papers:
 A. Gill, J. Launchbury, and S.L. Peyton Jones. A short cut to deforestation. Functional Programming Languages and Computer Architecture, Proceedings, pages 223232, ACM Press, 1993.
 J. Svenningsson. Shortcut fusion for accumulating parameters & ziplike functions. International Conference on Functional Programming, Proceedings, pages 124132, ACM Press, 2002.
 P. Johann. On proving the correctness of program transformations based on free theorems for higherorder polymorphic calculi. Mathematical Structures in Computer Science, 15:201229, 2005.
 P. Johann and J. Voigtländer. The impact of seq on free theoremsbased program transformations. Fundamenta Informaticae, 69:63102, 2006.
 J. Voigtländer and P. Johann. Selective strictness and parametricity in structural operational semantics, inequationally. Theoretical Computer Science, 388:290318, 2007.
 J. Voigtländer. Proving Correctness via Free Theorems: The Case of the destroy/buildRule. Partial Evaluation and SemanticsBased Program Manipulation, Proceedings, pages 1320, ACM Press, 2008.
 J. Voigtländer. Semantics and Pragmatics of New Shortcut Fusion Rules. Functional and Logic Programming, Proceedings, LNCS 4989:163179, SpringerVerlag, 2008.
 P. Johann and J. Voigtländer. A family of syntactic logical relations for the semantics of Haskelllike languages. Information and Computation, 207:341368, 2009.