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- 1. Real World Haskell: Lecture 6 Bryan O’Sullivan 2009-11-11
- 2. Models of evaluation Coming from a C++ or Python background, you’re surely used to the the || and or operators in those languages. If the operator’s left argument evaluates to true, it “short circuits” the right (i.e. doesn’t evaluate it). We see the same behaviour in Haskell. Prelude> undefined *** Exception: Prelude.undefined Prelude> True || undefined True
- 3. Eager, or strict, evaluation In most languages, the usual model of is of strict evaluation: a function’s arguments are fully evaluated before the evaluation of the function’s body begins. def inc(x): return x + 1 def bar(a): b = 5 if a % 2 else 8 return inc(a + 3) * inc(inc(b)) For example, if we run bar(1) then: The local variable b gets the value 5. The two calls to inc are fully evaluated before we invoke * on 5 and 7, respectively.
- 4. Are things diﬀerent in Haskell? Let’s think about our old friend the list constructor. Prelude> 1:[] [1] Prelude> 1:undefined [1*** Exception: Prelude.undefined And a function that operates on a list: Prelude> head (1:[]) 1 What do you think will happen now?
- 5. Are things diﬀerent in Haskell? Let’s think about our old friend the list constructor. Prelude> 1:[] [1] Prelude> 1:undefined [1*** Exception: Prelude.undefined And a function that operates on a list: Prelude> head (1:[]) 1 What do you think will happen now? Prelude> head (1:undefined) 1
- 6. Was that a special case? Well, uh, perhaps lists are special in Haskell. Riiight? −− d e f i n e d i n Data . Maybe i s J u s t ( Just ) = True isJust = False Let’s see how the Maybe type fares. Prelude Data.Maybe> Just undefined Just *** Exception: Prelude.undefined Prelude Data.Maybe> isJust (Just undefined) True
- 7. What else should we expect? Here’s a slightly diﬀerent function: i s J u s t O n e ( Just a ) | a == 1 = True isJustOne = False How will it behave? *Main> isJustOne (Just 2) False Okay, we expected that. What if we follow the same pattern as before, and package up an undeﬁned value?
- 8. What else should we expect? Here’s a slightly diﬀerent function: i s J u s t O n e ( Just a ) | a == 1 = True isJustOne = False How will it behave? *Main> isJustOne (Just 2) False Okay, we expected that. What if we follow the same pattern as before, and package up an undeﬁned value? *Main> isJustOne (Just undefined) *** Exception: Prelude.undefined
- 9. Haskell’s evaluation model Haskell follows a semantic model called non-strict evaluation: expressions are not evaluated unless (and usually until) their values are used. Perhaps you’ve heard of lazy evaluation: this is a speciﬁc kind of non-strict semantics. Haskell compilers go further, using call by need as an implementation strategy. Call by need: evaluate an expression when needed, then overwrite the location of the expression with the evaluated result (i.e. memoize it), in case it is needed again.
- 10. What does this mean in practice? Consider the isJust function again. i s J u s t ( Just ) = True isJust = False It only evaluates its argument to the point of seeing whether it was constructed with a Just or Nothing constructor. Notably, the function does not inspect the argument of the Just constructor. When we try isJust (Just undeﬁned) in ghci, the value undeﬁned is never evaluated.
- 11. And our other example, revisited Who can explain why this code crashes when presented with Just undeﬁned? i s J u s t O n e ( Just a ) | a == 1 = True isJustOne = False
- 12. A classic example Here is the inﬁnite list of Fibonacci numbers in Haskell: f i b s = 0 : 1 : zipWith (+) f i b s ( t a i l f i b s ) Even though this list is conceptually inﬁnite, its components only get generated on demand. *Main> head (drop 256 fibs) 141693817714056513234709965875411919657707794958199867
- 13. Traversing lists What do these functions have in common? map f [ ] = [] map f ( x : x s ) = f x : map f x s bunzip [ ] = ([] ,[]) b u n z i p ( ( a , b ) : x s ) = l e t ( as , b s ) = b u n z i p x s i n ( a : as , b : b s ) *Main> map succ "button" "cvuupo" *Main> bunzip [(1,’a’),(3,’b’),(5,’c’)] ([1,3,5],"abc")
- 14. What does map do? If you think about what map does to the structure of a list, it replaces every (:) constructor with a new (:) constructor that has a transformed version of its arguments. map succ ( 1 : 2 : []) == ( succ 1 : succ 2 : [ ] )
- 15. And bunzip? Thinking structurally, bunzip performs the same kind of operation as map. bunzip ((1 , ’ a ’ ) : (2 , ’ b ’ ) : [ ] ) == ((1 : 2 : [ ] ) , ( ’ a ’ : ’b ’ : [ ] ) ) This time, the pattern is much harder to see, but it’s still there: Every time we see a (:) constructor, we replace it with a transformed piece of data. In this case, the transformed data is the head pair pulled apart and grafted onto the heads of a pair of lists.
- 16. Abstraction! Abstraction! Abstraction! If we have two functions that do essentially the same thing, don’t we have a design pattern? In Haskell, we can usually do better than waﬄe about design patterns: we can reify them into code! To wit: foldr f z [ ] = z fol dr f z ( x : xs ) = f x ( fold r f z xs )
- 17. The right fold (no, really) The foldr function is called a right fold, because it associates to the right. What do I mean by this? f o l d r (+) 1 [ 2 , 3 , 4 ] == f o l d r (+) 1 ( 2 : 3 : 4 : [ ] ) == f o l d r (+) 1 ( 2 : ( 3 : ( 4 : [ ] ) ) ) == 2 + (3 + (4 + 1)) Notice a few things: We replaced the empty list with the “empty” value. We replaced each non-empty constructor with an addition operator. That’s our structural transformation in a nutshell!
- 18. map as a right fold Because map follows the same pattern as foldr, we can actually write map in terms of foldr! bmap : : ( a −> b ) −> [ a ] −> [ b ] bmap f x s = f o l d r g [ ] x s where g y y s = f y : y s Since we can write a map as a fold, this implies that a fold is somehow more primitive than a map.
- 19. unzip as a right fold And here’s our unzip-like function as a fold: b u n z i p : : [ ( a , b ) ] −> ( [ a ] , [ b ] ) bunzip xs = fold r g ( [ ] , [ ] ) xs where g ( x , y ) ( xs , y s ) = ( x : xs , y : y s ) In fact, I’d suggest that bunzip-in-terms-of-foldr is actually easier to understand than the original deﬁnition. Many other common list functions can be expressed as right folds, too!
- 20. Function composition (I) Remember your high school algebra? f ◦ g (x) ≡ f (g (x)) f ◦ g ≡ λx → f (g x) Taking the above notation and writing it in Haskell, we get this: f . g = x −> f ( g x ) The backslash is Haskell ASCII art for λ, and introduces an anonymous function. Between the backslash and the ASCII arrow are the function’s arguments, and following the arrow is its body.
- 21. Function composition (II) f . g = x −> f ( g x ) The result of f . g is a function that accepts one argument, applied f to it, then applies g to the result. So the expression succ . succ adds two to a number, for instance. Why care about this? Well, here’s our old deﬁnition of map-as-foldr. bmap f x s = f o l d r g [ ] x s where g y y s = f y : y s And here’s a more succinct version. bmap f x s = f o l d r ( ( : ) . f ) [ ] x s
- 22. The left fold So we’ve seen folds that associate to the right: 1 + (2 + (3 + 4)) What about folds that associate to the left? ((1 + 2) + 3) + 4 Not surprisingly, the left fold does indeed exist, and is named foldl . foldl f z [] = z f o l d l f z ( x : xs ) = f o l d l f ( f z x ) xs
- 23. The right fold in pictures
- 24. The left fold in pictures
- 25. Folds and laziness Which of these deﬁnitions for adding up the elements of a list is better? sum1 x s = f o l d r (+) 0 x s sum2 x s = f o l d l (+) 0 x s That’s a hard question to approach without a sense of what lazy evaluation will cause to happen. Suppose an oracle generates the list [1..1000] for us at a rate of one element per second.
- 26. Sum as right fold In the ﬁrst second, we see the partial expression 1 : ( . . . ) {− can ’ t s e e a n y t h i n g more y e t −} But we want to know the result as soon as possible, so we generate a partial result: 1 + ( . . . ) {− can ’ t make any more p r o g r e s s y e t −}
- 27. Second number two In the second second, we now have the partial expression 1 : ( 2 : . . . ) {− can ’ t s e e a n y t h i n g more y e t −} We thus construct a little more of our eventual result: 1 + ( 2 + ( . . . ) ) {− s t i l l no f u r t h e r p r o g r e s s −} Because we’re constructing a right-associative expression (that’s what a right fold is for), we can’t create an intermediate result of the-sum-so-far at any point. In other words, we’re creating a big expression containing 1000 nested applications of (+), which we’ll only be able to fully evaluate at the end of the list!
- 28. What happens in practice? On casual inspection, it’s not clear that this right fold business really matters. Prelude> foldr (+) 0 [0..100000] 5000050000 But if we try to sum a longer list, we get a problem: Prelude> foldr (+) 0 [0..1000000] *** Exception: stack overflow The GHC runtime imposes a limit on the size of a deferred expression to reduce the likelihood of us shooting ourselves in the foot. Or at least to make the foot-shooting happen early enough that it won’t be a serious problem.
- 29. Left folds are better . . . uh, right? Obviously, a left fold can’t tell us the sum before we reach the end of a list, but it has a promising property. Given a list like this: 1 : 2 : 3 : ... Then our sum-via- foldl will produce a result like this: ((1 + 2) + 3) + . . . This is left associative, so we could potentially evaluate the leftmost portion on the ﬂy: add 1 + 2 to give 3, then add 3 to give 6, and so on, keeping a single Int as the rolling sum-so-far.
- 30. Are we out of the woods? So we know that this will fail: Prelude> foldr (+) 0 [0..1000000] *** Exception: stack overflow But what about this? Prelude> foldl (+) 0 [0..1000000] *** Exception: stack overflow Hey! Shouldn’t the left fold have saved our bacon?
- 31. Why did foldl not help? Alas, consider the deﬁnition of foldl : f o l d l : : ( a −> b −> a ) −> a −> [ b ] −> a foldl f z [] = z f o l d l f z ( x : xs ) = f o l d l f ( f z x ) xs Because foldl is polymorphic, there is no way it can inspect the result of f z x. And since the intermediate result of each f z x can’t be evaluated, a huge unevaluated expression piles up until we reach the end of the list, just as with foldr!
- 32. Tracing evaluation How can we even get a sense of what pure Haskell code is actually doing? import Debug . T r a c e f o l d l T : : (Show a ) = > ( a −> b −> a ) −> a −> [ b ] −> a foldlT f z [] = z f o l d l T f z ( x : xs ) = let i = f z x i n t r a c e ( ”now ” ++ show i ) f o l d l T f i x s
- 33. What does trace do? The trace function is a magical function of sin: it prints its ﬁrst argument on stderr, then returns its second argument. The expression trace (”now ”++ show i) foldlT prints something, then returns foldT. If you have the patience, run this in ghci: Prelude> foldlT (+) 0 [0..1000000] now 0 now 1 now 3 ...blah blah blah... 500000500000 Whoa! It eventually prints a result, where plain foldl failed!
- 34. What’s going on? In order to print an intermediate result, trace must evaluate it ﬁrst. Haskell’s call-by-need evaluation ensures that an unevaluated expression will be overwritten with the evaluated result. Instead of a constantly-growing expression, we thus have a single primitive value as the running sum at each iteration through the loop.
- 35. Is a debugging hack the answer to our problem? Clearly, using trace seems like an incredibly lame solution to the problem of evaluating intermediate results, although it’s very useful for debugging. (Actually, it’s the only Haskell debugger I use.) The real solution to our problem lies in a function named seq. seq : : a −> t −> t This function is a rather magical hack; all it does is evaluate its ﬁrst argument until it reaches a constructor, then return its second argument.
- 36. Folding with seq The Data.List module deﬁnes the following function for us: f o l d l ’ : : ( a −> b −> a ) −> a −> [ b ] −> a foldl ’ f z [ ] = z foldl ’ f z ( x : xs ) = l e t i = f z x in i ‘ seq ‘ f o l d l ’ f i x s Let’s compare the two in practice: Prelude> foldl (+) 0 [0..1000000] *** Exception: stack overflow Prelude> import Data.List Prelude> foldl’ (+) 0 [0..1000000] 500000500000
- 37. Rules of thumb for folds If you can generate your result lazily and incrementally, e.g. as map does, use foldr. If you are generating what is conceptually a single result (e.g. one number), use foldl ’, because it will evaluate those tricky intermediate results strictly. Never use plain old foldl without the little tick at the end.
- 38. Homework For each of the following, choose the appropriate fold for the kind of result you are returning. You can ﬁnd the type signature for each function using ghci. Write concat using a fold. Write length using a fold. Write (++) using a fold. Write and using a fold. Write unwords using a fold. Write () from Data.List using a fold. For super duper bonus points: Write either foldr or foldl in terms of the other. (Hint 1: only one is actually possible. Hint 2: the answer is highly non-obvious, and involves higher order functions.)

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