I'm just starting out with Haskell, so I'm trying to wrap my head around the "Haskell way of thinking." Is there a reason to use pattern matching to solve Problem 1 here basically by unwrapping the whole list and calling the function recursively, instead of just retrieving the last element directly like myLast lst = lst !! ((length lst) - 1)? It seems almost brute-force, but I assume it's just my lack of familiarity here.
A few things I can think of:
(!!) and length are ultimately implemented using recursion over the structure of the list. That being so, it can be a worthwhile learning exercise to implement those basic functions using explicit recursion.
Keep in mind that, under the hood, the retrieval of the last element is not direct. Since we are dealing with linked lists, length has to go through all elements of the lists, and (!!) has to go through all elements up to the desired index. That being so, lst !! (length lst - 1) runs through the whole list twice, rather than once. (This is one of the reasons why, as a rule of thumb, length is better avoided unless you actually need to know the number of elements in and of itself, and not just as a proxy to something else.)
Pattern matching is a neat way of stating facts about the structure of data types. If, while consuming a list recursively, you match a [x] pattern (or, equivalently, x : [] -- an element consed to the empty list), you know that x is the last element. In a way, matching [x] involves one less level of indirection than accessing the list element at index length lst - 1, as it only deals with the structure of the list, without requiring an indexing scheme to be bolted on the top of it.
With all that said, there is something fundamentally right about your feeling that explicit recursion feels "almost brute-force". In time, you'll find out about folds, mapping functions, and other ways to capture and abstract common recursive patterns, making it possible to write in a more fluent manner.
Related
From my understanding, lazy evaluation is the arguments are not evaluated before they are passed to a function, but only when their values are actually used.
But in a haskell tutorial, I see an example.
xs = [1,2,3,4,5,6,7,8]
doubleMe(doubleMe(doubleMe(xs)))
The author said an imperative language would probably pass through the list once and make a copy and then return it. Then it would pass through the list another two times and return the result.
But in a lazy language, it would first compute
doubleMe(doubleMe(doubleMe(1)))
This will give back a doubleMe(1), which is 2. Then 4, and finally 8.
So it only does one pass through the list and only when you really need it.
This makes me confused. Why don't lazy language take the list as a whole, but split it? I mean we can ignore what the list or the expression is before we use it. But we need to evaluate the whole thing when we use it, isn't it?
A list like [1,2,3,4,5,6,7,8] is just syntactic sugar for this: 1:2:3:4:5:6:7:8:[].
In this case, all the values in the list are numeric constants, but we could define another, smaller list like this:
1:1+1:[]
All Haskell lists are linked lists, which means that they have a head and a tail. In the above example, the head is 1, and the tail is 1+1:[].
If you only want the head of the list, there's no reason to evaluate the rest of the list:
(h:_) = 1:1+1:[]
Here, h refers to 1. There's no reason to evaluate the rest of the list (1+1:[]) if h is all you need.
That's how lists are lazily evaluated. 1+1 remains a thunk (an unevaluated expression) until the value is required.
I am wondering exactly how list-comprehensions are evaluated in Haskell. After reading this Removing syntactic sugar: List comprehension in Haskell and this: Haskell Lazy Evaluation and Reuse I still don't really understand if
[<function> x|x <- <someList>, <somePredicate>]
is actually exactly equivalent (not just in outcome but in evaluation) to
filter (<somePredicate>) . map (<function>) $ <someList>
and if so, does this mean it can potentially reduce time complexity drastically to build up the desired list with only desired elements?
Also, how does this work in terms of infinite lists? To be specific: I assume something like:
[x|x <- [1..], x < 20]
will be evaluated in finite time, but how "obvious" does the fact that there are no more elements above some value which satisfy the predicate need to be, for the compiler to consider it? Would
[x|x <- [1..], (sum.map factorial $ digits x) == x]
work (see project Euler problem 34 https://projecteuler.net/problem=34). There is obviously an upper bound because from some x on x*9! < 10^n -1 always holds, but do I need to supply that bound or will the compiler find it?
There's nothing obvious about the fact that a particular infinite sequence has no more elements matching a predicate. When you pass a list to filter, it has no way of knowing any other properties of the elements than that an element can be passed to the predicate.
You can write your own version of Ord a => List a which can describe a sequence as ascending or not, and a version of filter that can use this information to stop looking at elements past a particular threshold. Unfortunately, list comprehensions won't use either of them.
Instead, I'd use a combination of takeWhile and a comprehension without a predicate / a plain map. Somewhere in the takeWhile arguments, you will supply the compiler the information about the expected upper bound; for a number of n decimal digits, it would be 10^n.
[<function> x|x <- <someList>, <somePredicate>]
should always evaluate to the same result as
filter (<somePredicate>) . map (<function>) $ <someList>
However, there is no guarantee that this is how the compiler will actually do it. The section on list comprehensions in the Haskell Report only mentions what list comprehensions should do, not how they should work. So each compiler is free to do as its developers find best. Therefore, you should not assume anything about the performance of list comprehensions or that the compiler will do any optimizations.
If it's linked list, why doesn't it support push_back?
If it's simply array, why does it need linear time when accessed by subscript?
Appreciated for your help.
Edit: We can append element in front of a list like this 1:[2,3], that's push_front; but we can't do it this way: [2,3]:4, that's push_back.
ps. actually I borrow push_front/back from C++'s STL
Haskell list are singly linked list. It supports appending to the end of the list, but it has to traverse the entire list for this operation.:
λ> let x = [1,2,3]
λ> x ++ [4]
[1,2,3,4]
If by push_back you mean adding an element to the end, of course it "supports" it. To add an element x to a list list (and get a new list so constructed), use the expression list ++ [x].
Beware though, that this is an O(n) operation, n being the length of list.
That's because it is a singly linked list. It also answers your other question about subscript.
Since the question asks about the "underlying implementation":
The list type is (conceptually) defined like this:
data [a] = [] | a:[a]
You can't actually write this declaration, because lists have funky syntax. But you can make exactly the same kind of thing yourself
data List a = Nil | a :* List a
infixr 5 :*
When you write something like [1,2,3], that's just special syntax for 1:2:3:[]. Since : associates to the right, this really means 1:(2:(3:[])). The functions on lists are all, fundamentally, based on pattern matching, which deals with just one constructor at a time. You may have to open up a lot of : constructors to get to the end of a list, and then build a bunch of : constructors to put together a new version with an extra element at the end. Any functions (like ++) that you may use will end up going through this sort of process internally.
This has been a question I've been wondering for a while. if statements are staples in most programming languages (at least then ones I've worked with), but in Haskell it seems like it is quite frowned upon. I understand that for complex situations, Haskell's pattern matching is much cleaner than a bunch of ifs, but is there any real difference?
For a simple example, take a homemade version of sum (yes, I know it could just be foldr (+) 0):
sum :: [Int] -> Int
-- separate all the cases out
sum [] = 0
sum (x:xs) = x + sum xs
-- guards
sum xs
| null xs = 0
| otherwise = (head xs) + sum (tail xs)
-- case
sum xs = case xs of
[] -> 0
_ -> (head xs) + sum (tail xs)
-- if statement
sum xs = if null xs then 0 else (head xs) + sum (tail xs)
As a second question, which one of these options is considered "best practice" and why? My professor way back when always used the first method whenever possible, and I'm wondering if that's just his personal preference or if there was something behind it.
The problem with your examples is not the if expressions, it's the use of partial functions like head and tail. If you try to call either of these with an empty list, it throws an exception.
> head []
*** Exception: Prelude.head: empty list
> tail []
*** Exception: Prelude.tail: empty list
If you make a mistake when writing code using these functions, the error will not be detected until run time. If you make a mistake with pattern matching, your program will not compile.
For example, let's say you accidentally switched the then and else parts of your function.
-- Compiles, throws error at run time.
sum xs = if null xs then (head xs) + sum (tail xs) else 0
-- Doesn't compile. Also stands out more visually.
sum [] = x + sum xs
sum (x:xs) = 0
Note that your example with guards has the same problem.
I think the Boolean Blindness article answers this question very well. The problem is that boolean values have lost all their semantic meaning as soon as you construct them. That makes them a great source for bugs and also makes the code more difficult to understand.
Your first version, the one preferred by your prof, has the following advantages compared to the others:
no mention of null
list components are named in the pattern, so no mention of head and tail.
I do think that this one is considered "best practice".
What's the big deal? Why would we want to avoid especially head and tail? Well, everybody knows that those functions are not total, so one automatically tries to make sure that all cases are covered. A pattern match on [] not only stands out more than null xs, a series of pattern matches can be checked by the compiler for completeness. Hence, the idiomatic version with complete pattern match is easier to grasp (for the trained Haskell reader) and to proof exhaustive by the compiler.
The second version is slightly better than the last one because one sees at once that all cases are handled. Still, in the general case the RHS of the second equation could be longer and there could be a where clauses with a couple of definitions, the last of them could be something like:
where
... many definitions here ...
head xs = ... alternative redefnition of head ...
To be absolutly sure to understand what the RHS does, one has to make sure common names have not been redefined.
The 3rd version is the worst one IMHO: a) The 2nd match fails to deconstruct the list and still uses head and tail. b) The case is slightly more verbose than the equivalent notation with 2 equations.
In many programming languages, if-statements are fundamental primitives, and things like switch-blocks are just syntax sugar to make deeply-nested if-statements easier to write.
Haskell does it the other way around. Pattern matching is the fundamental primitive, and an if-expression is literally just syntax sugar for pattern matching. Similarly, constructs like null and head are simply user-defined functions, which are all ultimately implemented using pattern matching. So pattern matching is the thing at the bottom of it all. (And therefore potentially more efficient than calling user-defined functions.)
In many cases - such as the ones you list above - it's simply a matter of style. The compiler can almost certainly optimise things to the point where all versions are roughly equal in performance. But generally [not always!] pattern matching makes it clearer exactly what you're trying to achieve.
(It's annoyingly easy to write an if-expression where you get the two alternatives the wrong way around. You'd think it would be a rare mistake, but it's surprisingly common. With a pattern match, there's little chance of making that specific mistake, although there's still plenty of other things to screw up.)
Each call to null, head and tail entails a pattern match. But the 1st version in your answer does just one pattern match, and reuses its results through named components of the pattern.
Just for that, it is better. But it is also more visually apparent, more readable.
Pattern matching is better than a string of if-then-else statements for (at least) the following reasons:
it is more declarative
it interacts well with sum-types
Pattern matching helps to reduce the amount of "accidental complexity" in your code - that is, code that is really more about implementation details rather than the essential logic of your program.
In most other languages when the compier/run-time sees a string of if-then-else statements it has no choice but to test the conditions in exactly the order the programmer specified them. But pattern matching encourages the programmer to focus more on describing what should happen versus how things should be performed. Due to purity and immutability of values in Haskell the compiler can consider the collection of patterns as a whole and decide the how best to implement them.
An analogy would be C's switch statement. If you dump the assembly code for various switch statements you will see that sometimes the compiler will generate a chain/tree of comparisons and in other cases it will generate a jump table. The programmer uses the same syntax in both cases - the compiler chooses the implementation based on what the comparison values are. If they form a contiguous block of values the jump table method is used, otherwise a comparison tree is used. And this separation of concerns allows the compiler to implement even more strategies in the future if other patterns among the comparison values are detected.
Note to other potential contributors: Please don't hesitate to use abstract or mathematical notations to make your point. If I find your answer unclear, I will ask for elucidation, but otherwise feel free to express yourself in a comfortable fashion.
To be clear: I am not looking for a "safe" head, nor is the choice of head in particular exceptionally meaningful. The meat of the question follows the discussion of head and head', which serve to provide context.
I've been hacking away with Haskell for a few months now (to the point that it has become my main language), but I am admittedly not well-informed about some of the more advanced concepts nor the details of the language's philosophy (though I am more than willing to learn). My question then is not so much a technical one (unless it is and I just don't realize it) as it is one of philosophy.
For this example, I am speaking of head.
As I imagine you'll know,
Prelude> head []
*** Exception: Prelude.head: empty list
This follows from head :: [a] -> a. Fair enough. Obviously one cannot return an element of (hand-wavingly) no type. But at the same time, it is simple (if not trivial) to define
head' :: [a] -> Maybe a
head' [] = Nothing
head' (x:xs) = Just x
I've seen some little discussion of this here in the comment section of certain statements. Notably, one Alex Stangl says
'There are good reasons not to make everything "safe" and to throw exceptions when preconditions are violated.'
I do not necessarily question this assertion, but I am curious as to what these "good reasons" are.
Additionally, a Paul Johnson says,
'For instance you could define "safeHead :: [a] -> Maybe a", but now instead of either handling an empty list or proving it can't happen, you have to handle "Nothing" or prove it can't happen.'
The tone that I read from that comment suggests that this is a notable increase in difficulty/complexity/something, but I am not sure that I grasp what he's putting out there.
One Steven Pruzina says (in 2011, no less),
"There's a deeper reason why e.g 'head' can't be crash-proof. To be polymorphic yet handle an empty list, 'head' must always return a variable of the type which is absent from any particular empty list. It would be Delphic if Haskell could do that...".
Is polymorphism lost by allowing empty list handling? If so, how so, and why? Are there particular cases which would make this obvious? This section amply answered by #Russell O'Connor. Any further thoughts are, of course, appreciated.
I'll edit this as clarity and suggestion dictates. Any thoughts, papers, etc., you can provide will be most appreciated.
Is polymorphism lost by allowing empty
list handling? If so, how so, and why?
Are there particular cases which would
make this obvious?
The free theorem for head states that
f . head = head . $map f
Applying this theorem to [] implies that
f (head []) = head (map f []) = head []
This theorem must hold for every f, so in particular it must hold for const True and const False. This implies
True = const True (head []) = head [] = const False (head []) = False
Thus if head is properly polymorphic and head [] were a total value, then True would equal False.
PS. I have some other comments about the background to your question to the effect of if you have a precondition that your list is non-empty then you should enforce it by using a non-empty list type in your function signature instead of using a list.
Why does anyone use head :: [a] -> a instead of pattern matching? One of the reasons is because you know that the argument cannot be empty and do not want to write the code to handle the case where the argument is empty.
Of course, your head' of type [a] -> Maybe a is defined in the standard library as Data.Maybe.listToMaybe. But if you replace a use of head with listToMaybe, you have to write the code to handle the empty case, which defeats this purpose of using head.
I am not saying that using head is a good style. It hides the fact that it can result in an exception, and in this sense it is not good. But it is sometimes convenient. The point is that head serves some purposes which cannot be served by listToMaybe.
The last quotation in the question (about polymorphism) simply means that it is impossible to define a function of type [a] -> a which returns a value on the empty list (as Russell O'Connor explained in his answer).
It's only natural to expect the following to hold: xs === head xs : tail xs - a list is identical to its first element, followed by the rest. Seems logical, right?
Now, let's count the number of conses (applications of :), disregarding the actual elements, when applying the purported 'law' to []: [] should be identical to foo : bar, but the former has 0 conses, while the latter has (at least) one. Uh oh, something's not right here!
Haskell's type system, for all its strengths, is not up to expressing the fact that you should only call head on a non-empty list (and that the 'law' is only valid for non-empty lists). Using head shifts the burden of proof to the programmer, who should make sure it's not used on empty lists. I believe dependently typed languages like Agda can help here.
Finally, a slightly more operational-philosophical description: how should head ([] :: [a]) :: a be implemented? Conjuring a value of type a out of thin air is impossible (think of uninhabited types such as data Falsum), and would amount to proving anything (via the Curry-Howard isomorphism).
There are a number of different ways to think about this. So I am going to argue both for and against head':
Against head':
There is no need to have head': Since lists are a concrete data type, everything that you can do with head' you can do by pattern matching.
Furthermore, with head' you're just trading off one functor for another. At some point you want to get down to brass tacks and get some work done on the underlying list element.
In defense of head':
But pattern matching obscures what's going on. In Haskell we are interested in calculating functions, which is better accomplished by writing them in point-free style using compositions and combinators.
Furthermore, thinking about the [] and Maybe functors, head' allows you to move back and forth between them (In particular the Applicative instance of [] with pure = replicate.)
If in your use case an empty list makes no sense at all, you can always opt to use NonEmpty instead, where neHead is safe to use. If you see it from that angle, it's not the head function that is unsafe, it's the whole list data-structure (again, for that use case).
I think this is a matter of simplicity and beauty. Which is, of course, in the eye of the beholder.
If coming from a Lisp background, you may be aware that lists are built of cons cells, each cell having a data element and a pointer to next cell. The empty list is not a list per se, but a special symbol. And Haskell goes with this reasoning.
In my view, it is both cleaner, simpler to reason about, and more traditional, if empty list and list are two different things.
...I may add - if you are worried about head being unsafe - don't use it, use pattern matching instead:
sum [] = 0
sum (x:xs) = x + sum xs