In functional optics, a well-behaved prism (called a partial lens in scala, I believe) is supposed to have a set function of type 'subpart -> 'parent -> 'parent, where if the prism "succeeds" and is structurally compatible with the 'parent argument given, then it returns the 'parent given with the appropriate subpart modified to have the 'subpart value given. If the prism "fails" and is structurally incompatible with the 'parent argument, then it returns the 'parent given unmodified.
I'm wondering why the prism doesn't return a 'parent option (Maybe for Haskellers) to represent the pass/fail nature of the set function? Shouldn't the programmer be able to tell from the return type whether the set was "successful" or not?
I know there's been a lot of research and thought put into the realm of functional optics, so I'm sure there must be a definitive answer that I just can't seem to find.
(I'm from an F# background, so I apologize if the syntax I've used is a bit opaque for Haskell or Scala programmers).
I doubt there's one definitive answer, so I'll give you two here.
Origin
I believe prisms were first imagined (by Dan Doel, if my vague recollection is correct) as "co-lenses". Whereas a lens from s to a offers
get :: s -> a
set :: (s, a) -> s
a prism from s to a offers
coget :: a -> s
coset :: s -> Either s a
All the arrows are reversed, and the product, (,), is replaced by a coproduct, Either. So a prism in the category of types and functions is a lens in the dual category.
For simple prisms, that s -> Either s a seems a bit weird. Why would you want the original value back? But the lens package also offers type-changing optics. So we end up with
get :: s -> a
set :: (s, b) -> t
coget :: a -> s
coset :: t -> Either s b
Suddenly what we're getting back in the non-matching case may actually be a bit different! What's that about? Here's an example:
cogetLeft :: a -> Either a x
cogetLeft = Left
cosetLeft :: Either b x -> Either (Either a x) b
cosetLeft (Left b) = Right b
cosetLeft (Right x) = Left (Right x)
In the second (non-matching) case, the value we get back is the same, but its type has been changed.
Nice hierarchy
For both Van Laarhoven (as in lens) and profunctor style frameworks, both lenses and prisms can also stand in for traversals. To do that, they need to have similar forms, and this design accomplishes that. leftaroundabout's answer gives more detail on this aspect.
To answer the “why” – lenses etc. are pretty rigidly derived from category theory, so this is actually quite clear-cut – the behaviour you describe just drops out of the maths, it's not something anybody defined for any purpose but follows from far more general ideas.
Ok, that's not really satisfying.
Not sure if other languages' type systems are powerful enough to express this, but in principle and in Haskell, a prism is a special case of a traversal.
A traversal is a way to “visit” all occurences of “elements” within some “container”. The classical example is
mapM :: Monad m => (a -> m b) -> [a] -> m [b]
This is typically used like
Prelude> mapM print [1..4]
1
2
3
4
[(),(),(),()]
The focus here is on: sequencing the actions/side-effects, and gathering back the result in a container with the same structure as the one we started with.
What's special about a prism is simply that the containers are restricted to contain either one or zero elements† (whereas a general traversal can go over any number of elements). But the set operator doesn't know about that because it's strictly more general. The nice thing is that you can therefore use this on a lens, or a prism, or on mapM, and always get a sensible behaviour. But it's not the behaviour of “insert exactly once into the structure or else tell me if it failed”.
Not that this isn't a sensible operation, just it's not what lens libraries call “setting”. You can do it by explicitly matching and re-building:
set₁ :: Prism s a -> a -> s -> Maybe s
set₁ p x = case matching p x of
Left _ -> Nothing
Right a -> Just $ a ^. re p
†More precisely: a prism seperates the cases: a container may either contain one element, and nothing else apart from that, or it may have no element but possibly something unrelated.
what I am looking for
Let T be an OCaml data type, (example: type t = A | B of int), and x be a value of type T, is there a function f that satisfies the following requirements:
f maps x to a string, i.e. f(x) is a string representation of x
for all u, v in T, u = v if and only if f(u) = f(v)
f can be derived automatically, like type t = ... [##deriving yojson]
the string representation of a value of a relatively simple type, like the one defined above, should be human editable
(not essential, but nice to have) locality, i.e., if you extend the type t above to type t = A | B of int | C of something, then f("A the one before the extending") should be equal to f("A the one after the extending"), in another word, it should make upgrading an old version of a type to the new version easy
why I want this
Store an OCaml data into Postgres column. I have a small web app that uses PGOCaml to fetch data from Postgres, and PGOCaml type checks the SQL statement at compile time, so if you create domain some_type as text in Postgres, and change the source code of PGOCaml a bit (to use the above f to convert a Postgres text into an OCaml type), you can store ADT into a Postgres table, while maintain type safety.
The second point in the requirements is important, for on the Postgres side, you likely need to test for equality on that column, and such tests are done on Postgres text type.
I looked into Sexp, didn't find information about the second point.
PS, new to OCaml, does this kind of thing already have a mature solution?
Update
I end up using yojson, since my type is very simple, just nullary variants, I can get away with it, it's a galaxy away from perfect though.
Update 2
For those who has the similar problem, I think the current best solution is to use yojson, and instead of storing it in a text column, storing it in a jsonb, this way, you get white space and order insensitive comparison, (though I cannot find pg's documentation on the equality of jsonb type).
Upgrading RichouHunters comment to an answere as I think Sexplib is a good module for this:
I think the way s-expressions may (or may not) satisfy your second requirement will depend a lot on your type T and on the way you define the serializer. You'll find more about them (and Sexplib) here.
Let's say I have a pair of conversion functions
string2int :: String -> Maybe Int
int2string :: Int -> String
I could represent these fairly easily using Optics.
stringIntPrism :: Prism String Int
However if I want to represent failure reason, I'd need to keep these as two separate functions.
string2int :: String -> Validation [ParseError] Int
int2string :: Int -> String`
For this simple example Maybe is perfectly fine, since we can always assume that a failure is a parse failure, thus we don't actually have to encode this using an Either or Validation type.
However imagine, in addition to my parsing Prism, I want to perform some validation
isOver18 :: Int -> Validation [AgeError] Int
isUnder55 :: Int -> Validation [AgeError] Int
It would be ideal to be able compose these things together, such that I could have
ageField = isUnder55 . isOver18 . string2Int :: ValidationPrism [e] String Int
This is fairly trivial to build by hand, however it seems like a common enough concept that there might be something lurking in the field of Lenses/Optics that does this already. Is there an existing abstraction that handles this?
tl;dr
Is there a standard way of implementing a partial lens / prism / iso that can be parameterised over an arbitrary functor instead of being tied directly to Maybe?.
I've used Haskell notation above since it's more straight forward, however I'm actually using Monocle in Scala to implement this. I would, however, be perfectly happy with an answer specific to i.e. ekmett's Lens library.
I have recently written a blog post about indexed optics; which explores a bit how we can do coindexed optics as well.
In short: Coindexed-optics are possible, but we have yet to do some further research there. Especially, because if we try to translate that approach into lens encoding of lenses (from Profunctor to VL) it gets even more hairy (but I think we can get away with only 7 type-variables).
And we cannot really do this without altering how indexed optics are currently encoded in lens. So for now, you'll better to use validation specific libraries.
To give a hint of the difficulties: When we try to compose with Traversals, should we have
-- like `over` but also return an errors for elements not matched
validatedOver :: CoindexedOptic' s a -> (a -> a) -> s -> (ValidationErrors, s)
or something else? If we could only compose Coindexed Prisms their value won't justify their complexity; they won't "fit" into optics framework.
I am just starting to learn Purescript so I hope that this is not a stupid question.
Suppose that we have an object
a = {x:1,y:2}
an we want to change x to equal 2. As far as I can see if we use the ST monad we will have to copy the whole object in order to change the value. If the initial object is big this would be very inefficient. What is the right way to mutate objects in place?
The ST monad is a fine approach, but depending on your use case, there may or may not be standard library functions for this.
The Data.StrMap module in purescript-maps defines a foreign type for homogeneous records with string keys, so if your values all have the same type, you could use Data.StrMap.ST to mutate your record in place.
If not, you should be easily able to define a function to update a record in place using ST and the FFI. The tricky bit is picking the right type. If you want to do something for a specific key, you could write a function
setFoo :: forall r a h eff. STRef h { foo :: a | r } -> a -> Eff (st :: ST h | eff) Unit
for example. Defining a generic setter would be more difficult without losing type safety. This is the trade-off made by Data.StrMap: you restrict yourself to a single value type, but get to use arbitrary keys.
I am a Scala programmer, learning Haskell now. It's easy to find practical use cases and real world examples for OO concepts, such as decorators, strategy pattern etc. Books and interwebs are filled with it.
I came to the realization that this somehow is not the case for functional concepts. Case in point: applicatives.
I am struggling to find practical use cases for applicatives. Almost all of the tutorials and books I have come across so far provide the examples of [] and Maybe. I expected applicatives to be more applicable than that, seeing all the attention they get in the FP community.
I think I understand the conceptual basis for applicatives (maybe I am wrong), and I have waited long for my moment of enlightenment. But it doesn't seem to be happening. Never while programming, have I had a moment when I would shout with a joy, "Eureka! I can use applicative here!" (except again, for [] and Maybe).
Can someone please guide me how applicatives can be used in a day-to-day programming? How do I start spotting the pattern? Thanks!
Applicatives are great when you've got a plain old function of several variables, and you have the arguments but they're wrapped up in some kind of context. For instance, you have the plain old concatenate function (++) but you want to apply it to 2 strings which were acquired through I/O. Then the fact that IO is an applicative functor comes to the rescue:
Prelude Control.Applicative> (++) <$> getLine <*> getLine
hi
there
"hithere"
Even though you explicitly asked for non-Maybe examples, it seems like a great use case to me, so I'll give an example. You have a regular function of several variables, but you don't know if you have all the values you need (some of them may have failed to compute, yielding Nothing). So essentially because you have "partial values", you want to turn your function into a partial function, which is undefined if any of its inputs is undefined. Then
Prelude Control.Applicative> (+) <$> Just 3 <*> Just 5
Just 8
but
Prelude Control.Applicative> (+) <$> Just 3 <*> Nothing
Nothing
which is exactly what you want.
The basic idea is that you're "lifting" a regular function into a context where it can be applied to as many arguments as you like. The extra power of Applicative over just a basic Functor is that it can lift functions of arbitrary arity, whereas fmap can only lift a unary function.
Since many applicatives are also monads, I feel there's really two sides to this question.
Why would I want to use the applicative interface instead of the monadic one when both are available?
This is mostly a matter of style. Although monads have the syntactic sugar of do-notation, using applicative style frequently leads to more compact code.
In this example, we have a type Foo and we want to construct random values of this type. Using the monad instance for IO, we might write
data Foo = Foo Int Double
randomFoo = do
x <- randomIO
y <- randomIO
return $ Foo x y
The applicative variant is quite a bit shorter.
randomFoo = Foo <$> randomIO <*> randomIO
Of course, we could use liftM2 to get similar brevity, however the applicative style is neater than having to rely on arity-specific lifting functions.
In practice, I mostly find myself using applicatives much in the same way like I use point-free style: To avoid naming intermediate values when an operation is more clearly expressed as a composition of other operations.
Why would I want to use an applicative that is not a monad?
Since applicatives are more restricted than monads, this means that you can extract more useful static information about them.
An example of this is applicative parsers. Whereas monadic parsers support sequential composition using (>>=) :: Monad m => m a -> (a -> m b) -> m b, applicative parsers only use (<*>) :: Applicative f => f (a -> b) -> f a -> f b. The types make the difference obvious: In monadic parsers the grammar can change depending on the input, whereas in an applicative parser the grammar is fixed.
By limiting the interface in this way, we can for example determine whether a parser will accept the empty string without running it. We can also determine the first and follow sets, which can be used for optimization, or, as I've been playing with recently, constructing parsers that support better error recovery.
I think of Functor, Applicative and Monad as design patterns.
Imagine you want to write a Future[T] class. That is, a class that holds values that are to be calculated.
In a Java mindset, you might create it like
trait Future[T] {
def get: T
}
Where 'get' blocks until the value is available.
You might realize this, and rewrite it to take a callback:
trait Future[T] {
def foreach(f: T => Unit): Unit
}
But then what happens if there are two uses for the future? It means you need to keep a list of callbacks. Also, what happens if a method receives a Future[Int] and needs to return a calculation based on the Int inside? Or what do you do if you have two futures and you need to calculate something based on the values they will provide?
But if you know of FP concepts, you know that instead of working directly on T, you can manipulate the Future instance.
trait Future[T] {
def map[U](f: T => U): Future[U]
}
Now your application changes so that each time you need to work on the contained value, you just return a new Future.
Once you start in this path, you can't stop there. You realize that in order to manipulate two futures, you just need to model as an applicative, in order to create futures, you need a monad definition for future, etc.
UPDATE: As suggested by #Eric, I've written a blog post: http://www.tikalk.com/incubator/blog/functional-programming-scala-rest-us
I finally understood how applicatives can help in day-to-day programming with that presentation:
https://web.archive.org/web/20100818221025/http://applicative-errors-scala.googlecode.com/svn/artifacts/0.6/chunk-html/index.html
The autor shows how applicatives can help for combining validations and handling failures.
The presentation is in Scala, but the author also provides the full code example for Haskell, Java and C#.
Warning: my answer is rather preachy/apologetic. So sue me.
Well, how often in your day-to-day Haskell programming do you create new data types? Sounds like you want to know when to make your own Applicative instance, and in all honesty unless you are rolling your own parser, you probably won't need to do it very much. Using applicative instances, on the other hand, you should learn to do frequently.
Applicative is not a "design pattern" like decorators or strategies. It is an abstraction, which makes it much more pervasive and generally useful, but much less tangible. The reason you have a hard time finding "practical uses" is because the example uses for it are almost too simple. You use decorators to put scrollbars on windows. You use strategies to unify the interface for both aggressive and defensive moves for your chess bot. But what are applicatives for? Well, they're a lot more generalized, so it's hard to say what they are for, and that's OK. Applicatives are handy as parsing combinators; the Yesod web framework uses Applicative to help set up and extract information from forms. If you look, you'll find a million and one uses for Applicative; it's all over the place. But since it's so abstract, you just need to get the feel for it in order to recognize the many places where it can help make your life easier.
I think Applicatives ease the general usage of monadic code. How many times have you had the situation that you wanted to apply a function but the function was not monadic and the value you want to apply it to is monadic? For me: quite a lot of times!
Here is an example that I just wrote yesterday:
ghci> import Data.Time.Clock
ghci> import Data.Time.Calendar
ghci> getCurrentTime >>= return . toGregorian . utctDay
in comparison to this using Applicative:
ghci> import Control.Applicative
ghci> toGregorian . utctDay <$> getCurrentTime
This form looks "more natural" (at least to my eyes :)
Coming at Applicative from "Functor" it generalizes "fmap" to easily express acting on several arguments (liftA2) or a sequence of arguments (using <*>).
Coming at Applicative from "Monad" it does not let the computation depend on the value that is computed. Specifically you cannot pattern match and branch on a returned value, typically all you can do is pass it to another constructor or function.
Thus I see Applicative as sandwiched in between Functor and Monad. Recognizing when you are not branching on the values from a monadic computation is one way to see when to switch to Applicative.
Here is an example taken from the aeson package:
data Coord = Coord { x :: Double, y :: Double }
instance FromJSON Coord where
parseJSON (Object v) =
Coord <$>
v .: "x" <*>
v .: "y"
There are some ADTs like ZipList that can have applicative instances, but not monadic instances. This was a very helpful example for me when understanding the difference between applicatives and monads. Since so many applicatives are also monads, it's easy to not see the difference between the two without a concrete example like ZipList.
I think it might be worthwhile to browse the sources of packages on Hackage, and see first-handedly how applicative functors and the like are used in existing Haskell code.
I described an example of practical use of the applicative functor in a discussion, which I quote below.
Note the code examples are pseudo-code for my hypothetical language which would hide the type classes in a conceptual form of subtyping, so if you see a method call for apply just translate into your type class model, e.g. <*> in Scalaz or Haskell.
If we mark elements of an array or hashmap with null or none to
indicate their index or key is valid yet valueless, the Applicative
enables without any boilerplate skipping the valueless elements while
applying operations to the elements that have a value. And more
importantly it can automatically handle any Wrapped semantics that
are unknown a priori, i.e. operations on T over
Hashmap[Wrapped[T]] (any over any level of composition, e.g. Hashmap[Wrapped[Wrapped2[T]]] because applicative is composable but monad is not).
I can already picture how it will make my code easier to
understand. I can focus on the semantics, not on all the
cruft to get me there and my semantics will be open under extension of
Wrapped whereas all your example code isn’t.
Significantly, I forgot to point out before that your prior examples
do not emulate the return value of the Applicative, which will be a
List, not a Nullable, Option, or Maybe. So even my attempts to
repair your examples were not emulating Applicative.apply.
Remember the functionToApply is the input to the
Applicative.apply, so the container maintains control.
list1.apply( list2.apply( ... listN.apply( List.lift(functionToApply) ) ... ) )
Equivalently.
list1.apply( list2.apply( ... listN.map(functionToApply) ... ) )
And my proposed syntactical sugar which the compiler would translate
to the above.
funcToApply(list1, list2, ... list N)
It is useful to read that interactive discussion, because I can't copy it all here. I expect that url to not break, given who the owner of that blog is. For example, I quote from further down the discussion.
the conflation of out-of-statement control flow with assignment is probably not desired by most programmers
Applicative.apply is for generalizing the partial application of functions to parameterized types (a.k.a. generics) at any level of nesting (composition) of the type parameter. This is all about making more generalized composition possible. The generality can’t be accomplished by pulling it outside the completed evaluation (i.e. return value) of the function, analogous to the onion can’t be peeled from the inside-out.
Thus it isn’t conflation, it is a new degree-of-freedom that is not currently available to you. Per our discussion up thread, this is why you must throw exceptions or stored them in a global variable, because your language doesn’t have this degree-of-freedom. And that is not the only application of these category theory functors (expounded in my comment in moderator queue).
I provided a link to an example abstracting validation in Scala, F#, and C#, which is currently stuck in moderator queue. Compare the obnoxious C# version of the code. And the reason is because the C# is not generalized. I intuitively expect that C# case-specific boilerplate will explode geometrically as the program grows.