# Correctness of short cut fusion

## 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 `foldr`

/`build`

- and the `destroy`

/`unfoldr`

-rule for Haskell lists.

`foldr`

/`build`

`foldr`

/`build`

The `foldr`

/`build`

-rule eliminates intermediate lists produced by `build`

and consumed by `foldr`

, where these functions are defined as follows:

```
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 (:) []
```

Note the *rank-2 polymorphic* type of `build`

.

The `foldr`

/`build`

-rule now means the following replacement for appropriately typed `g`

, `c`

, and `n`

:

```
foldr c n (build g) → g c n
```

`destroy`

/`unfoldr`

`destroy`

/`unfoldr`

The `destroy`

/`unfoldr`

-rule eliminates intermediate lists produced by `unfoldr`

and consumed by `destroy`

, where these functions are defined as follows:

```
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'
```

Note the *rank-2 polymorphic* type of `destroy`

.

The `destroy`

/`unfoldr`

-rule now means the following replacement for appropriately typed `g`

, `p`

, and `e`

:

```
destroy g (unfoldr p e) → g p e
```

## Correctness

If the `foldr`

/`build`

- and the `destroy`

/`unfoldr`

-rule are to be automatically performed during compilation, as is possible using GHC's *RULES pragmas*, we clearly want them to be equivalences.
That is, the left- and right-hand 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 `g`

is defined using `seq`

or not.

### In the absence of `seq`

`seq`

`foldr`

/`build`

`foldr`

/`build`

If `g`

does not use `seq`

, then the `foldr`

/`build`

-rule really is a semantic equivalence, that is, it holds that

```
foldr c n (build g) = g c n
```

The two sides are interchangeable in any program without affecting semantics.

`destroy`

/`unfoldr`

`destroy`

/`unfoldr`

The `destroy`

/`unfoldr`

-rule, however, is not a semantic equivalence.
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
```

These values have appropriate types for being used in the `destroy`

/`unfoldr`

-rule. But with them, that rule's left-hand side "evaluates" as follows:

```
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 right-hand side "evaluates" as follows:

```
g p e = case p e of Just z -> 0
= case Just undefined of Just z -> 0
= 0
```

Thus, by applying the `destroy`

/`unfoldr`

-rule, a nonterminating (or otherwise failing) program can be transformed into a safely terminating one.
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 if `g`

does not use `seq`

, then the `destroy`

/`unfoldr`

-rule is a semantic approximation from left to right, that is, it holds that

```
destroy 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, if `g`

does not use `seq`

and `p`

is a strict function that never returns `Just ⊥`

(where ⊥ denotes any kind of failure or nontermination), then indeed:

```
destroy g (unfoldr p e) = g p e
```

### In the presence of `seq`

`seq`

This is the more interesting setting, given that in Haskell there is no way to restrict the use of `seq`

, so in any given program we must be prepared for the possibility that the `g`

appearing in the `foldr`

/`build`

- or the `destroy`

/`unfoldr`

-rule is defined using `seq`

.
Unsurprisingly, it is also the setting in which more can go wrong than above.

`foldr`

/`build`

`foldr`

/`build`

In the presence of `seq`

, the `foldr`

/`build`

-rule is not anymore a semantic equivalence.
The instance

```
g = seq
c = undefined
n = 0
```

shows, via similar "evaluations" as above, that the right-hand side (`g c n`

) can be strictly less defined than the left-hand side (`foldr c n (build g)`

).

The converse cannot happen, because the following always holds:

```
foldr c n (build g) <math>\sqsupseteq</math> g c n
```

Moreover, semantic equivalence can again be recovered by putting restrictions on the involved functions.
More precisely, if `(c <math>\bot~\bot)\neq\bot</math>`

and `n <math>\neq\bot</math>`

, then even in the presence of `seq`

:

```
foldr c n (build g) = g c n
```

`destroy`

/`unfoldr`

`destroy`

/`unfoldr`

Contrary to the situation without `seq`

, now also the `destroy`

/`unfoldr`

-rule may decrease the definedness of a program.
This is witnessed by the following instance:

```
g = \x y -> seq x 0
p = undefined
e = 0
```

Here the left-hand side of the rule (`destroy g (unfoldr p e)`

) yields `0`

, while the right-hand side (`g p e`

) yields `undefined`

.

Conditions for semantic approximation in either direction can be given as follows.

If `p <math>\neq\bot</math>`

and `(p <math>\bot</math>)<math>\in\{\bot,</math>Just <math>\bot\}</math>`

, then:

```
destroy g (unfoldr p e) <math>\sqsubseteq</math> g p e
```

If `p`

is strict and total and never returns `Just <math>\bot</math>`

, then:

```
destroy g (unfoldr p e) <math>\sqsupseteq</math> g p e
```

Of course, conditions for semantic equivalence can be obtained by combining the two laws above.

## 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 like `foldr`

/`build`

during their optimization pass (also in the disguise of more specialized rules such as `head`

/`build`

). The rules are specified in the GHC.Base and GHC.List modules.
There 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.

`foldr`

/`build`

`foldr`

/`build`

As pointed out above, everything is fine with `foldr`

/`build`

in the absence of `seq`

.
If the producer (`build g`

) of the intermediate list may be defined using `seq`

, then the conditions `(c <math>\bot~\bot)\neq\bot</math>`

and `n <math>\neq\bot</math>`

better be satisified, lest the compiler transform a perfectly fine program into a failing one.

The mentioned conditions are equivalent to requiring that the consumer (`foldr c n`

) is a total function, that is, maps non- lists to a non- value.
It is thus relatively easy to identify whether a list consumer defined in terms of `foldr`

is eligible for `foldr`

/`build`

-fusion in the presence of `seq`

or not.
For example, the Prelude functions `head`

and `sum`

are generally not, while `map`

is.

There is, however, currently no way to detect automatically, inside the compiler, whether a particular instance of `foldr`

/`build`

-fusion is safe, i.e., whether the producer avoids `seq`

or the consumer is total.

`destroy`

/`unfoldr`

`destroy`

/`unfoldr`

As above, the compiler cannot figure out automatically whether (and how) a given instance of `destroy`

/`unfoldr`

-fusion will change the semantics of a program.

An easy way to get rid of the condition regarding `p`

never returning `Just <math>\bot</math>`

is to slightly change the definitions of the functions involved:

```
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'
```

The new type `Step a b`

is almost isomorphic to `Maybe (a,b)`

, but avoids the "junk value" `Just <math>\bot</math>`

. This change does not affect the expressiveness of `unfoldr`

or `unfoldr'`

with respect to list producers.
But it allows some of the laws above to be simplified a bit.

We would still have that if `g`

does not use `seq`

, then:

```
destroy g' (unfoldr' p e) <math>\sqsubseteq</math> g p e
```

Moreover, if `g`

does not use `seq`

and `p`

is strict, then even:

```
destroy' g (unfoldr' p e) = g p e
```

In the potential presence of `seq`

, if `p <math>\neq\bot</math>`

and `p`

is strict, then:

```
destroy' g (unfoldr' p e) <math>\sqsubseteq</math> g p e
```

Also without restriction regarding `seq`

, if `p`

is strict and total, then:

```
destroy' g (unfoldr' p e) <math>\sqsupseteq</math> 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 of `seq`

, for example with a producer of the form

```
repeat x = unfoldr (\y -> Just (x,y)) undefined
```

or

```
repeat x = unfoldr' (\y -> Yield x y) undefined
```

Fortunately, it cannot happen for any producer of a nonempty, spine-total list (i.e., one that contains at least one element and ends with `[]`

).
The reason is that for any such producer expressed via `unfoldr`

or `unfoldr'`

the conditions imposed on `p`

in the left-to-right approximation laws above are necessarily fulfilled.

A left-to-right approximation as in

```
destroy g (unfoldr p e) <math>\sqsubseteq</math> 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 of `seq`

.
For 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
```

## 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 223-232, ACM Press, 1993.
- J. Svenningsson. Shortcut fusion for accumulating parameters & zip-like functions. International Conference on Functional Programming, Proceedings, pages 124-132, ACM Press, 2002.
- P. Johann. On proving the correctness of program transformations based on free theorems for higher-order polymorphic calculi. Mathematical Structures in Computer Science, 15:201-229, 2005.
- P. Johann and J. Voigtländer. The impact of seq on free theorems-based program transformations. Fundamenta Informaticae, 69:63-102, 2006.
- J. Voigtländer and P. Johann. Selective strictness and parametricity in structural operational semantics, inequationally. Theoretical Computer Science, 388:290-318, 2007.
- J. Voigtländer. Proving Correctness via Free Theorems: The Case of the destroy/build-Rule. Partial Evaluation and Semantics-Based Program Manipulation, Proceedings, pages 13-20, ACM Press, 2008.
- J. Voigtländer. Semantics and Pragmatics of New Shortcut Fusion Rules. Functional and Logic Programming, Proceedings, LNCS 4989:163-179, Springer-Verlag, 2008.
- P. Johann and J. Voigtländer. A family of syntactic logical relations for the semantics of Haskell-like languages. Information and Computation, 207:341-368, 2009.