The meaning of "pure", and the meaning of "effect" are closely intertwined, because essentially, "pure" (in this usage) means "not having any effects", and "effect" means "the part of the function result that isn't pure".  If what you have in your mind is a function `div :: Int -> Int -> Int`, and instead have to settle for `div :: Int -> Int -> Maybe Int`, they you could consider the first the type it should have "if it were pure", and call the Maybe type "an effect".  The words are relative to your starting assumption of what type div should have, though.  As you mention, you could also quite reasonably admit that `Maybe Int` is a perfectly good type on its own, and consider `safeDiv` to be a pure function with this type as a codomain.

You can even pull the same trick with more powerful effects.  The type `IO Int` is a (more or less) defined type, and its values are ACTIONS that your computer could take, which if they don't fail return an Int.  From this perspective, even a function like `readFile :: FilePath -> IO ByteString` is a "pure" function, which maps file paths to actions.  But if you consider it as a map from file paths to bytestrings, then it is effectful.  Again, these words are defined relative to what you consider the result to be.  (I'm ignoring, here, some questions about what the correct semantics for IO types even is...)

If you want a more formal (but less intuitive) way to think about this, then you can turn to category theory.  In category theory, a monad (say, F) is an endofunctor in some category -- for us, typically the category of Haskell types and functions.  But F also defines a SECOND category, called the Kleisli category of F: the set of types here the same, but a "Kleisli arrow" between two objects A and B is a function A -> F B in the base category.  Notice that any Kleisli arrow IS an arrow in the base category, so in that sense you could claim that it's "pure".  But IF you choose to think about it as an arrow from A to B, THEN you must be talking about the Kleisli category, and it has an effect captured by F.  If that wasn't what you were looking for, though, feel free to ignore it.

On Mon, Oct 30, 2017 at 7:11 PM, Steven Leiva <leiva.steven@gmail.com> wrote:
Hello Again Brandon,

Thank you for the explanation. I'll have to mull it over a bit to let it sink in. I am finding the overloading of purity to be easier to grasp than the meaning of effect. I think the reason for that is precisely because it depends on the context (generally speaking) in which it is being used. For example, in the case of Maybe, the effect is possible failure. In the case of lists, the effect is non-determinism, etc. 



On Mon, Oct 30, 2017 10:02 PM, Brandon Allbery allbery.b@gmail.com wrote:
In this specific case it is actually pure, because Maybe is pure, but in the general case it behaves with respect to Applicative (and Monad, which this appears to be leading up to) as effectful. In this context, an effect is just whatever behavior is captured by the Applicative/Monad.

"purity" is a bit overloaded:

- purity with respect to an effect of some unspecified kind, as here;

- purity with respect to IO which encapsulates behavior not contained specifically within your program, the most common meaning in Haskell;

- purity with respect to cross-thread effects in IO/STM;

- purity with respect to mutability in ST;

....


On Mon, Oct 30, 2017 at 9:49 PM, Steven Leiva <leiva.steven@gmail.com> wrote:
Hi Everyone, 

I am reading the 2nd edition of Graham Hutton's Programming in Haskell. I'm not reading the entire book, just the parts of Haskell that I am still iffy on.

Anyway, in Chapter 12, Section 3, Hutton introduces monads. 

He start off with the following code:

first 
                  
module Expr where
data Expr = Val Int | Div Expr Expr
eval :: Expr -> Int
eval (Val n) = n
eval (Div el er) = eval el `div` eval er
Mixmax Not using Mixmax yet?


And then he points out that the second clause of eval will raise an error if eval er evaluates to 0. 

One solution is that, instead of using the div function, we use a safeDiv :: Int -> Int -> Maybe Int function, which evaluate to Nothing if the divisor is 0. This means that expr's type changes from eval :: Eval -> Int to eval :: Eval -> Maybe Int, and this means that implementing eval becomes very verbose:


second 
                  
module Expr where
data Expr = Val Int | Div Expr Expr
eval :: Expr -> Maybe Int
eval (Val n) = Just n
eval (Div el er) = case eval el of
                    Nothing -> Nothing
                    Just y -> case eval er of
                                Nothing -> Nothing
                                Just x -> y `safeDiv` x
safeDiv :: Int -> Int -> Maybe Int
safeDiv x y
    | y == 0 = Nothing
    | otherwise = Just (x `div` y)
Mixmax Not using Mixmax yet?


In order to make eval more concise, we can try the applicative style, where the second clause of the eval function becomes pure safeDiv <*> eval el <*> eval er. Of course, that doesn't work because pure safeDiv has the type Int -> Int -> Maybe Int, and what we need is a function of type Int -> Int -> Int. 

Anyways, this is all setup / context to what Hutton says next:

The conclusion is that the function eval does not fit the pattern of effectful programming that is capture by applicative functors. The applicative style restricts us to applying pure functions to effectful arguments: eval does not fit this pattern because the function safeDiv that is used to process the resulting values is not a pure function, but may itself fail. 

I am confused by Hutton's use of the word effectful and by his description of safeDiv as "not a pure function". I tried skimming the other sections of the book to see if he provided a definition of this somewhere, but if he did, I couldn't find it. So my question is, in what way does Hutton mean for the reader to understand the words effect / effectful, and why does he describe the function safeDiv as not a pure function? 

Thank you!

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--
brandon s allbery kf8nh                               sine nomine associates
allbery.b@gmail.com                                  ballbery@sinenomine.net
unix, openafs, kerberos, infrastructure, xmonad        http://sinenomine.net


Steven Leiva
305.528.6038
leiva.steven@gmail.com
http://www.linkedin.com/in/stevenleiva

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