In the previous two posts, we've built up a whole range of applicatives, out of Const, Identity, Reader, Compose, Product, Sum, and Fix (and some higher-order analogues). Sum has given us the most trouble, but in some sense has been the most powerful, letting us write things like possibly eventually terminating lists, or trees, or in fact any sort of structure with branching alternatives. In this post, I want to think a bit more about why it is that Sum is the trickiest of the bunch, and more generally, what we can say about when two applicative structures are the "same". In the process of doing so, we'll invent something a lot like Traversable en passant.

Let's do some counting exercises. Product Identity Identity holds exactly two things. It is therefore isomorphic to ((->) Bool), or if we prefer, ((->) Either () ()). That is to say that a pair that holds two values of type a is the same as a function that takes a two-valued type and yields a value of type a. A product of more functors in turn is isomorphic to the reader of the sum of each of the datatypes that "represent" them. E.g. Product (Product Identity Identity) (Product (Const ()) Identity) is iso to ((->) (Either (Either () ()) ()), i.e. a data type with three possible inhabitants. In making this move we took Product to Either -- multiplication to sum. We can pull a similar trick with Compose. Compose (Product Identity Identity) (Product Identity Identity) goes to ((->) (Either () (),Either () ())). So again we took Product to a sum type, but now we took Compose to a pair -- a product type! The intuition is that composition multiplies the possibilities of spaces in each nested functor.

Hmm.. products go to sums, composition goes to multiplication, etc. This should remind us of something -- these rules are exactly the rules for working with exponentials. x^n * x^m = x^(n + m). (x^n)^m = x^(n*m). x^0 = 1, x^1 = x.

Seen from the right standpoint, this isn't surprising at all, but almost inevitable. The functors we're describing are known as "representable," a term which derives from category theory. (See appendix on representable functors below).

In Haskell-land, a "representable functor" is just any functor isomorphic to the reader functor ((->) a) for some appropriate a. Now if we think back to our algebraic representations of data types, we call the arrow type constructor an exponential. We can "count" a -> x as x^a, since e.g. there are 3^2 distinct functions that inhabit the type 2 -> 3. The intuition for this is that for each input we pick one of the possible results, so as the number of inputs goes up by one, the number of functions goes up by multiplying through by the set of possible results. 1 -> 3 = 3, 2 -> 3 = 3 * 3, (n + 1) -> 3 = 3 * (n -> 3).

Hence, if we "represent" our functors by exponentials, then we can work with them directly as exponentials as well, with all the usual rules. Edward Kmett has a library encoding representable functors in Haskell.

Meanwhile, Peter Hancock prefers to call such functors "Naperian" after John Napier, inventor of the logarithm (See also here). Why Naperian? Because if our functors are isomorphic to exponentials, then we can take their logs! And that brings us back to the initial discussion of type mathematics. We have some functor F, and claim that it is isomorphic to -^R for some concrete data type R. Well, this means that R is the logarithm of F. E.g. (R -> a, S -> a) =~ Either R S -> a, which is to say that if log F = R and log G =~ S, then log (F * G) = log F + log G. Similarly, for any other data type n, again with log F = R, we have n -> F a =~ n -> R -> a =~ (n * R) -> a, which is to say that log (F^n) =~ n * log F.

This gives us one intuition for why the sum functor is not generally representable -- it is very difficult to decompose log (F + G) into some simpler compound expression of logs.

So what functors are Representable? Anything that can be seen as a fixed shape with some index. Pairs, fixed-size vectors, fixed-size matrices, any nesting of fixed vectors and matricies. But also infinite structures of regular shape! However, not things whose shape can vary -- not lists, not sums. Trees of fixed depth or infinite binary trees therefore, but not trees of arbitrary depth or with ragged structure, etc.

Representable functors turn out to be extremely powerful tools. Once we know a functor is representable, we know exactly what its applicative instance must be, and that its applicative instance will be "zippy" -- i.e. acting pointwise across the structure. We also know that it has a monad instance! And, unfortunately, that this monad instance is typically fairly useless (in that it is also "zippy" -- i.e. the monad instance on a pair just acts on the two elements pointwise, without ever allowing anything in the first slot to affect anything in the second slot, etc.). But we know more than that. We know that a representable functor, by virtue of being a reader in disguise, cannot have effects that migrate outwards. So any two actions in a representable functor are commutative. And more than that, they are entirely independent.

This means that all representable functors are "distributive"! Given any functor f, and any data type r, then we have


distributeReader :: Functor f => f (r -> a) -> (r -> f a)
distributeReader fra = \r -> fmap ($r) fra  That is to say, given an arrow "inside" a functor, we can always pull the arrow out, and "distribute" application across the contents of the functor. A list of functions from Int -> Int becomes a single function from Int to a list of Int, etc. More generally, since all representable functors are isomorphic to reader, given g representable, and f any functor, then we have: distribute :: (Functor f, Representable g) => f (g a) -> g (f a). This is pretty powerful sauce! And if f and g are both representable, then we get the transposition isomorphism, witnessed by flip! That's just the beginning of the good stuff. If we take functions and "unrepresent" them back to functors (i.e. take their logs), then we can do things like move from ((->) Bool) to pairs, etc. Since we're in a pervasively lazy language, we've just created a library for memoization! This is because we've gone from a function to a data structure we can index into, representing each possible argument to this function as a "slot" in the structure. And the laziness pays off because we only need to evaluate the contents of each slot on demand (otherwise we'd have a precomputed lookup table rather than a dynamically-evaluated memo table). And now suppose we take our representable functor in the form s -> a and paired it with an "index" into that function, in the form of a concrete s. Then we'd be able to step that s forward or backwards and navigate around our structure of as. And this is precisely the Store Comonad! And this in turn gives a characterization of the lens laws. What this all gives us a tiny taste of, in fact, is the tremendous power of the Yoneda lemma, which, in Haskell, is all about going between values and functions, and in fact captures the important universality and uniqueness properties that make working with representable functors tractable. A further tiny taste of Yoneda comes from a nice blog post by Conal Elliott on memoization. Extra Credit on Sum Functors There in fact is a log identity on sums. It goes like this: log(a + c) = log a + log (1 + c/a)  Do you have a useful computational interpretation of this? I've got the inklings of one, but not much else. Appendix: Notes on Representable Functors in Hask. The way to think about this is to take some arbitrary category C, and some category that's basically Set (in our case, Hask. In fact, in our case, C is Hask too, and we're just talking about endofunctors on Hask). Now, we take some functor F : C -> Set, and some A which is an element of C. The set of morphisms originating at A (denoted by Hom(A,-)) constitutes a functor called the "hom functor." For any object X in C, we can "plug it in" to Hom(A,-), to then get the set of all arrows from A to X. And for any morphism X -> Y in C, we can derive a morphism from Hom(A,X) to Hom(A,Y), by composition. This is equivalent to, in Haskell-land, using a function f :: x -> y to send g :: a -> x to a -> y by writing "functionAToY = f . g". So, for any A in C, we have a hom functor on C, which is C -> Set, where the elements of the resultant Set are homomorphisms in C. Now, we have this other arbitrary functor F, which is also C -> Set. Now, if there is an isomorphism of functors between F, and Hom(A,_), then we say F is "representable". A representable functor is thus one that can be worked with entirely as an appropriate hom-functor. A couple of weeks back one of my coworkers brought to my attention a several hour long workshop in Japan to go over and describe a number of my libraries, hosted by TANAKA Hideyuki — not the voice actor, I checked! I was incredibly honored and I figured that if that many people (they had 30 or so registered attendees and 10 presentations) were going to spend that much time going over software that I had written, I should at least offer to show up! I'd like to apologize for any errors in the romanization of people's names or misunderstandings I may have in the following text. My grasp of Japanese is very poor! Please feel free to send me corrections or additions! ## Surprise! Sadly, my boss's immediate reaction to hearing that there was a workshop in Japan about my work was to quip that "You're saying you're huge in Japan?" With him conspicuously not offering to fly me out here, I had to settle for surprising the organizers and attending via Google Hangout. ## Commentary and Logs @nushio was very helpful in getting me connected, and while the speakers gave their talks I sat on the irc.freenode.net #haskell-lens channel and Google Hangout and answered questions and provided a running commentary with more details and references. Per freenode policy the fact that we were logging the channel was announced -- well, at least before things got too far underway. Here is the IRC session log as a gist. IKEGAMI Daisuke @ikegami__ (ikeg in the IRC log) tried to keep up a high-level running commentary about what was happening in the video to the log, which may be helpful if you are trying to follow along through each retroactively. Other background chatter and material is strewn across twitter under the #ekmett_conf hash tag and on a japanese twitter aggregator named togetter While the previous post in this series was relatively immediately applicable, this one has constructions I definitely wouldn't recommend in production code. However, they do take us further in exploring the universe of applicative functors, and, more broadly, exploring which data types provide which properties by construcion. It's well known that if you have any Functor F a, you can take its "fixpoint", creating a structure of infinitely nested Fs, like so. F (F (F (...) ) ) Since we can't have infinite types directly in Haskell, we introduce the Fix newtype:  newtype Fix f = Fix (f (Fix f)) This "wraps up" the recursion so that GHC accepts the type. Fix f is a Fix constructor, containing an "f" of Fix f inside. Each in turn expands out, and soforth. Fixpoints of functors have fixedpoints of functors inside 'em. And so on, and so on, ad infinitum. (Digression: We speak of "algebraic data types" in Haskell. The "algebra" in question is an "F-algebra", and we can build up structures with fixpoints of functors, taking those functors as initial or terminal objects and generating either initial algebras or terminal coalgebras. These latter two concepts coincide in Haskell in the Fix description given above, as greatest and least fixpoints of data types in Haskell turn out to be the same thing. For more background, one can go to Wadler's "Recursive Types for Free," or Jacobs and Rutten's "Tutorial on (Co)Algebras and (Co)Induction" for starters.) The family of functors built from our friends Const, Sum, Product, and Reader (exponentiation) are known as Polynomial Functors. If we take closure of these with a proper fixpoint construct (that lets us build infinite structures), we get things that are variously known as Containers, Shapely Types, and Strictly Positive types. One irritating thing is that the fixpoint of a functor as we've written it is no longer itself a functor. The type constructor Fix is of kind (* -> *) -> *, which says it takes an "f" which takes one argument (e.g. "Maybe" or "Identity" or etc.) and returns a proper type (i.e. a value at the type level of kind *). We want a fixpoint construction that gives back something of kind * -> * — i.e. something that is a type constructor representing a functor, and not just a plain old type. The following does the trick.  newtype FixF f a = FixF (f (FixF f) a) deriving instance (Show (f (FixF f) a)) => Show (FixF f a) (I learned about FixF from a paper by Ralf Hinze, but I'm sure the origins go back much further). FixF is of kind ((* -> *) -> * -> *) -> * -> *. It takes the fixpoint of a "second-order Functor" (a Functor that sends a Functor to another Functor, i.e. an endofunctor on the functor category of hask), to recover a standard "first order Functor" back out. This sounds scary, but it isn't once you load it up in ghci and start playing with it. In fact, we've encountered second order functors just recently. Product, Sum, and Compose are all of kind (* -> *) -> (* -> *) -> * -> *. So they all send two functors to a third functor. That means that Product Const, Sum Identity and Compose Maybe are all second-order functors, and things appropriate to take our "second-order fixpoint" of. Conceptually, "Fix f" took a value with one hole, and we filled that hole with "Fix f" so there was no room for a type parameter. Now we've got an "f" with two holes, the first of which takes a functor, and the second of which is the hole of the resulting functor. Unlike boring old "Fix", we can write Functor and Applicative instances for "FixF", and they're about as simple and compositional as we could possibly hope. instance Functor (f (FixF f)) => Functor (FixF f) where fmap f (FixF x) = FixF$ fmap f x

instance Applicative (f (FixF f)) => Applicative (FixF f) where
pure x = FixF $pure x (FixF f) < *> (FixF x) = FixF (f < *> x) But now we run into a new problem! It seems like this "a" parameter is just hanging out there, doing basically nothing. We take our classic functors and embed them in there, and they still only have "one hole" at the value level, so don't actually have any place to put the "a" type we now introduced. For example, we can write the following: -- FixF . Compose . Just . FixF . Compose$ Nothing
-- > FixF (Compose (Just (FixF (Compose Nothing))))
-- :t FixF (Compose (Just (FixF (Compose Nothing))))
-- > FixF (Compose Maybe) a

We now introduce one new member of our basic constructions — a second order functor that acts like "const" on the type level, taking any functor and returning Identity.


newtype Embed (f :: * -> *) a = Embed a deriving (Show)

instance Functor (Embed f) where
fmap f (Embed x) = Embed $f x instance Applicative (Embed f) where pure x = Embed x (Embed f) < *> (Embed x) = Embed (f x) Now we can actually stick functorial values into our fixpoints: -- FixF$ Embed "hi"
-- > FixF (Embed "hi")

-- fmap (++ " world") $FixF (Embed "hi") -- > FixF (Embed "hi world") -- FixF . Product (Embed "hi") . -- FixF . Product (Embed "there") . FixF$ undefined
-- > FixF (Product (Embed "hi")
--   (FixF (Product (Embed "there")
--   (FixF *** Exception: Prelude.undefined

You may have noticed that we seem to be able to use "product" to begin a chain of nested fixpoints, but we don't seem able to stick a "Maybe" in there to stop the chain. And it seems like we're not even "fixing" where we intend to be:

-- :t FixF . Product (Embed "hi") . FixF . Product (Embed "there") . FixF $undefined -- > FixF (Product (Embed f)) [Char] That's because we're applying Product, takes and yields arguments of kind * -> * in a context where we really want to take and yield second-order functors as arguments — things of kind (* -> *) -> * -> *. If we had proper kind polymorphism, "Product" and "ProductF" would be able to collapse (and maybe, soon, they will). But at least in the ghc 7.4.1 that I'm working with, we have to write the same darn thing, but "up a kind". data ProductF f g (b :: * -> *) a = ProductF (f b a) (g b a) deriving Show instance (Functor (f b), Functor (g b)) => Functor (ProductF f g b) where fmap f (ProductF x y) = ProductF (fmap f x) (fmap f y) instance (Applicative (f b), Applicative (g b)) => Applicative (ProductF f g b) where pure x = ProductF (pure x) (pure x) (ProductF f g) < *> (ProductF x y) = ProductF (f < *> x) (g < *> y) We can now do the following properly. yy = FixF . ProductF (Embed "foo")$ InL $Const () xx = FixF . ProductF (Embed "bar") . InR . FixF . ProductF (Embed "baz") . InL$ Const ()

So we've recovered proper lists in Haskell, as the "second-order fixpoint" of polynomial functors. And the types look right too:

-- :t yy
-- > FixF (ProductF Embed (Sum (Const ()))) [Char]

Because we've built our Applicative instances compositionally, we have an applicative for our list list construction automatically:

-- (++) < $> yy < *> xx -- > FixF (ProductF (Embed "foobar") (InL (Const ()))) This is precisely the "ZipList" applicative instance. In fact, our applicative instances from this "functor toolkit" are all "zippy" — matching up structure where possible, and "smashing it down" where not. This is because Sum, with its associated natural transformation logic, is the only way to introduce a disjoint choice of shape. Here are some simple examples with Sum to demonstrate: -- liftA2 (++) (InL (Identity "hi"))$
--      InR (Product (Identity " there") (Const ([12::Int])))
-- > InL (Identity "hi there")
-- liftA2 (++) (InR (Identity "hi")) $InL (Product (Identity " there") (Const ([12::Int]))) -- > InL (Product (Identity "hi there") (Const [12])) We're always "smashing" towards the left. So in the first case, that means throwing away half of the pair. In the second case, it means injecting Const mempty into a pair, and then operating with that. In any case, we now have infinite and possibly infinite branching structures. And not only are they Functors, but they're also Applicatives, and in a way that's uniform and straightforward to reason about. In the next post, we'll stop building out our vocabulary of "base" Functors (though it's not quite complete) and instead push forward on what else these functors can provide "beyond" Applicative. Consider the humble Applicative. More than a functor, less than a monad. It gives us such lovely syntax. Who among us still prefers to write liftM2 foo a b when we could instead write foo <$> a <*> b? But we seldom use the Applicative as such — when Functor is too little, Monad is too much, but a lax monoidal functor is just right. I noticed lately a spate of proper uses of Applicative —Formlets (and their later incarnation in the reform library), OptParse-Applicative (and its competitor library CmdTheLine), and a post by Gergo Erdi on applicatives for declaring dependencies of computations. I also ran into a very similar genuine use for applicatives in working on the Panels library (part of jmacro-rpc), where I wanted to determine dependencies of a dynamically generated dataflow computation. And then, again, I stumbled into an applicative while cooking up a form validation library, which turned out to be a reinvention of the same ideas as formlets.

Given all this, It seems post on thinking with applicatives is in order, showing how to build them up and reason about them. One nice thing about the approach we'll be taking is that it uses a "final" encoding of applicatives, rather than building up and then later interpreting a structure. This is in fact how we typically write monads (pace operational, free, etc.), but since we more often only determine our data structures are applicative after the fact, we often get some extra junk lying around (OptParse-Applicative, for example, has a GADT that I think is entirely extraneous).

So the usual throat clearing:

{-# LANGUAGE TypeOperators, MultiParamTypeClasses, FlexibleInstances,
StandaloneDeriving, FlexibleContexts, UndecidableInstances,

module Main where
import Control.Applicative hiding (Const)
import Data.Monoid hiding (Sum, Product)
instance Show a => Show (Identity a) where
show (Identity x) = "(Identity " ++ show x ++ ")"

And now, let's start with a classic applicative, going back to the Applicative Programming With Effects paper:

data Const mo a = Const mo deriving Show

instance Functor (Const mo) where
fmap _ (Const mo) = Const mo

instance Monoid mo => Applicative (Const mo) where
pure _ = Const mempty
(Const f) < *> (Const x) = Const (f <> x)

(Const lives in transformers as the Constant functor, or in base as Const)

Note that Const is not a monad. We've defined it so that its structure is independent of the a type. Hence if we try to write (>>=) of type Const mo a -> (a -> Const mo b) -> Const mo b, we'll have no way to "get out" the first a and feed it to our second argument.

One great thing about Applicatives is that there is no distinction between applicative transformers and applicatives themselves. This is to say that the composition of two applicatives is cleanly and naturally always also an applicative. We can capture this like so:


newtype Compose f g a = Compose (f (g a)) deriving Show

instance (Functor f, Functor g) => Functor (Compose f g) where
fmap f (Compose x) = Compose $(fmap . fmap) f x instance (Applicative f, Applicative g) => Applicative (Compose f g) where pure = Compose . pure . pure (Compose f) < *> (Compose x) = Compose$ (< *>) < $> f < *> x (Compose also lives in transformers) Note that Applicatives compose two ways. We can also write: data Product f g a = Product (f a) (g a) deriving Show instance (Functor f, Functor g) => Functor (Product f g) where fmap f (Product x y) = Product (fmap f x) (fmap f y) instance (Applicative f, Applicative g) => Applicative (Product f g) where pure x = Product (pure x) (pure x) (Product f g) < *> (Product x y) = Product (f < *> x) (g < *> y) (Product lives in transformers as well) This lets us now construct an extremely rich set of applicative structures from humble beginnings. For example, we can reconstruct the Writer Applicative. type Writer mo = Product (Const mo) Identity tell :: mo -> Writer mo () tell x = Product (Const x) (pure ()) -- tell [1] *> tell [2] -- > Product (Const [1,2]) (Identity ()) Note that if we strip away the newtype noise, Writer turns into (mo,a) which is isomorphic to the Writer monad. However, we've learned something along the way, which is that the monoidal component of Writer (as long as we stay within the rules of applicative) is entirely independent from the "identity" component. However, if we went on to write the Monad instance for our writer (by defining >>=), we'd have to "reach in" to the identity component to grab a value to hand back to the function yielding our monoidal component. Which is to say we would destroy this nice seperation of "trace" and "computational content" afforded by simply taking the product of two Applicatives. Now let's make things more interesting. It turns out that just as the composition of two applicatives may be a monad, so too the composition of two monads may be no stronger than an applicative! We'll see this by introducing Maybe into the picture, for possibly failing computations. type FailingWriter mo = Compose (Writer mo) Maybe tellFW :: Monoid mo => mo -> FailingWriter mo () tellFW x = Compose (tell x *> pure (Just ())) failFW :: Monoid mo => FailingWriter mo a failFW = Compose (pure Nothing) -- tellFW [1] *> tellFW [2] -- > Compose (Product (Const [1,2]) (Identity Just ())) -- tellFW [1] *> failFW *> tellFW [2] -- > Compose (Product (Const [1,2]) (Identity Nothing)) Maybe over Writer gives us the same effects we'd get in a Monad — either the entire computation fails, or we get the result and the trace. But Writer over Maybe gives us new behavior. We get the entire trace, even if some computations have failed! This structure, just like Const, cannot be given a proper Monad instance. (In fact if we take Writer over Maybe as a Monad, we get only the trace until the first point of failure). This seperation of a monoidal trace from computational effects (either entirely independent of a computation [via a product] or independent between parts of a computation [via Compose]) is the key to lots of neat tricks with applicative functors. Next, let's look at Gergo Erdi's "Static Analysis with Applicatives" that is built using free applicatives. We can get essentially the same behavior directly from the product of a constant monad with an arbitrary effectful monad representing our ambient environment of information. As long as we constrain ourselves to only querying it with the takeEnv function, then we can either read the left side of our product to statically read dependencies, or the right side to actually utilize them. type HasEnv k m = Product (Const [k]) m takeEnv :: (k -> m a) -> k -> HasEnv k m a takeEnv f x = Product (Const [x]) (f x) If we prefer, we can capture queries of a static environment directly with the standard Reader applicative, which is just a newtype over the function arrow. There are other varients of this that perhaps come closer to exactly how Erdi's post does things, but I think this is enough to demonstrate the general idea. data Reader r a = Reader (r -> a) instance Functor (Reader r) where fmap f (Reader x) = Reader (f . x) instance Applicative (Reader r) where pure x = Reader$ pure x

takeEnvNew :: (env -> k -> a) -> k -> HasEnv k (Reader env) a
takeEnvNew f x = Product (Const [x]) (Reader $flip f x) So, what then is a full formlet? It's something that can be executed in one context as a monoid that builds a form, and in another as a parser. so the top level must be a product. type FormletOne mo a = Product (Const mo) Identity a Below the product, we read from an environment and perhaps get an answer. So that's reader with a maybe. type FormletTwo mo env a = Product (Const mo) (Compose (Reader env) Maybe) a Now if we fail, we want to have a trace of errors. So we expand out the Maybe into a product as well to get the following, which adds monoidal errors: type FormletThree mo err env a = Product (Const mo) (Compose (Reader env) (Product (Const err) Maybe)) a But now we get errors whether or not the parse succeeds. We want to say either the parse succeeds or we get errors. For this, we can turn to the typical Sum functor, which currently lives as Coproduct in comonad-transformers, but will hopefully be moving as Sum to the transformers library in short order. data Sum f g a = InL (f a) | InR (g a) deriving Show instance (Functor f, Functor g) => Functor (Sum f g) where fmap f (InL x) = InL (fmap f x) fmap f (InR y) = InR (fmap f y) The Functor instance is straightforward for Sum, but the applicative instance is puzzling. What should "pure" do? It needs to inject into either the left or the right, so clearly we need some form of "bias" in the instance. What we really need is the capacity to "work in" one side of the sum until compelled to switch over to the other, at which point we're stuck there. If two functors, F and G are in a relationship such that we can always send f x -> g x in a way that "respects" fmap (that is to say, such that (fmap f . fToG == ftoG . fmap f) then we call this a natural transformation. The action that sends f to g is typically called "eta". (We actually want something slightly stronger called a "monoidal natural transformation" that respects not only the functorial action fmap but the applicative action <*>, but we can ignore that for now). Now we can assert that as long as there is a natural transformation between g and f, then Sum f g can be made an Applicative, like so: class Natural f g where eta :: f a -> g a instance (Applicative f, Applicative g, Natural g f) => Applicative (Sum f g) where pure x = InR$ pure x
(InL f) < *> (InL x) = InL (f < *> x)
(InR g) < *> (InR y) = InR (g < *> y)
(InL f) < *> (InR x) = InL (f < *> eta x)
(InR g) < *> (InL x) = InL (eta g < *> x)

The natural transformation we'll tend to use simply sends any functor to Const.

instance Monoid mo => Natural f (Const mo) where
eta = const (Const mempty)

However, there are plenty of other natural transformations that we could potentially make use of, like so:

instance Applicative f =>
Natural g (Compose f g) where
eta = Compose . pure

instance (Applicative g, Functor f) => Natural f (Compose f g) where
eta = Compose . fmap pure

instance (Natural f g) => Natural f (Product f g) where
eta x = Product x (eta x)

instance (Natural g f) => Natural g (Product f g) where
eta x = Product (eta x) x

instance Natural (Product f g) f where
eta (Product x _ ) = x

instance Natural (Product f g) g where
eta (Product _ x) = x

instance Natural g f => Natural (Sum f g) f where
eta (InL x) = x
eta (InR y) = eta y

instance Natural Identity (Reader r) where
eta (Identity x) = pure x

In theory, there should also be a natural transformation that can be built magically from the product of any other two natural transformations, but that will just confuse the Haskell typechecker hopelessly. This is because we know that often different "paths" of typeclass choices will often be isomorphic, but the compiler has to actually pick one "canonical" composition of natural transformations to compute with, although multiple paths will typically be possible.

For similar reasons of avoiding overlap, we can't both have the terminal homomorphism that sends everything to "Const" and the initial homomorphism that sends "Identity" to anything like so:

-- instance Applicative g => Natural Identity g where
--     eta (Identity x) = pure x


We choose to keep the terminal transformation around because it is more generally useful for our purposes. As the comments below point out, it turns out that a version of "Sum" with the initial transformation baked in now lives in transformers as Lift.

In any case we can now write a proper Validation applicative:

type Validation mo = Sum (Const mo) Identity

validationError :: Monoid mo => mo -> Validation mo a
validationError x = InL (Const x)

This applicative will yield either a single result, or an accumulation of monoidal errors. It exists on hackage in the Validation package.

Now, based on the same principles, we can produce a full Formlet.

type Formlet mo err env a =
Product (Const mo)
(Sum (Const err) Identity))
a

All the type and newtype noise looks a bit ugly, I'll grant. But the idea is to think with structures built with applicatives, which gives guarantees that we're building applicative structures, and furthermore, structures with certain guarantees in terms of which components can be interpreted independently of which others. So, for example, we can strip away the newtype noise and find the following:

type FormletClean mo err env a = (mo, env -> Either err a)

Because we built this up from our basic library of applicatives, we also know how to write its applicative instance directly.

Now that we've gotten a basic algebraic vocabulary of applicatives, and especially now that we've produced this nifty Sum applicative (which I haven't seen presented before), we've gotten to where I intended to stop.

But lots of other questions arise, on two axes. First, what other typeclasses beyond applicative do our constructions satisfy? Second, what basic pieces of vocabulary are missing from our constructions — what do we need to add to flesh out our universe of discourse? (Fixpoints come to mind).

Also, what statements can we make about "completeness" — what portion of the space of all applicatives can we enumerate and construct in this way? Finally, why is it that monoids seem to crop up so much in the course of working with Applicatives? I plan to tackle at least some of these questions in future blog posts.

By and large, there are two sorts of proof systems that people use (these days) when studying logic: natural deduction, and sequent calculus. I know of at least one other---Hilbert style---but it is older, and the above systems were invented due to dissatisfaction with Hilbert systems (for a programming analogy, Hilbert systems are like programming entirely with combinators (S, K, etc.), rather than a lambda calculus).

## Natural Deduction

Probably the best way to categorize the difference, for the purpose of where we're eventually going, is that natural deduction focuses on the ways to build proof terms up from their constituent parts. This comes in the form of introduction and elimination rules for the various propositions. For instance, the rules for conjunction are:

This spartan style gets a bit annoying (in my opinion) for the hypothetical premises of the implication introduction, but this can be solved by adding contexts:

This is the style most commonly adopted for presenting type theories, except we reason about terms with a type, rather than just propositions. The context we added for convenience above also becomes fairly essential for keeping track of variables:

As can be seen, all the rules involve taking terms from the premise and building on them in the conclusion.

## Sequent Calculi

The other type of system in question, sequent calculus, looks very similar, but represents a subtle shift in focus for our purposes (sequent calculi are a lot more obviously different when presenting classical logics). First, the inference rules relate sequents, which look a lot like our contextual judgments above, and I'll write them the same way. The difference is that not all rules operate on the conclusion side; some operate just on the context. Generally, introduction rules stay similar to natural deduction (and are called right rules), while elimination rules are replaced by manipulations of the context, and are called left rules. For pairs, we can use the rules:

We could also have two separate left rules:

But these two different sets are equivalent as long as we're not considering substructural logics. Do note, however, that we're moving from on the top left to on the bottom left, using the fact that is sufficient to imply . That is, projections apply contravariantly to the left.

It turns out that almost no type theory is done in this style; natural deduction is far and away more popular. There are, I think, a few reasons for this. The first is: how do we even extend the left rules to type theory (eliminations are obvious, by contrast)? I know of two ways. The first is to introduce pattern matching into the contexts, so our left rule becomes:

This is an acceptable choice (and may avoid some of the pitfalls in the next option), but it doesn't gel with your typical lambda calculus. It's probably more suited to a pattern calculus of some sort (although, even then, if you want to bend your brain, go look at the left rule for implication and try to figure out how it translates into such a theory; I think you probably need higher-order contexts of some sort). Anyhow, I'm not going to explore this further.

The other option (and one that I've seen in the literature) is that left rules actually involve a variable substitution. So we come up with the following rule:

And with this rule, it becomes (I think) more obvious why natural deduction is preferred over sequent calculus, as implementing this rule in a type checker seems significantly harder. Checking the rules of natural deduction involves examining some outer-most structure of the term, and then checking the constituents of the term, possibly in an augmented context, and which rule we're dealing with is always syntax directed. But this left rule has no syntactic correspondent, so it seems as though we must nondeterministically try all left rules at each step, which is unlikely to result in a good algorithm. This is the same kind of problem that plagues extensional type theory, and ultimately results in only derivations being checkable, not terms.

## The Type Class Connection

However, there are certain problems that I believe are well modeled by such a sequent calculus, and one of them is type class checking and associated dictionary translations. This is due mainly to the fact that the process is mainly context-directed term building, rather than term-directed type checking. As far as the type class algorithm goes, there are two interesting cases, having to do with the following two varieties of declaration:


class Eq a => Ord a where ...
instance (Eq a, Eq b) => Eq (a, b) where ...


It turns out that each of these leads to a left rule in a kind of type class sequent calculus:

That is:

1. if Eq a is a sufficient constraint for M : T, then the stronger constraint Ord a is also sufficient, so we can discharge the Eq a constraint and use Ord a instead.
2. We can discharge an Eq (a, b) constraint using two constraints, Eq a, Eq b together with an instance telling us how to do so. This also works for instances without contexts, giving us rules like:

Importantly, the type inference algorithm for type classes specifies when we should use these rules based only on the contexts we're dealing with. Now, these look more like the logical sequent rules, but it turns out that they have corresponding type theory-like versions when we consider dictionary passing:

And this kind of substituting into dictionary variables produces exactly the evidence passing translation we want.

Another way to look at the difference in feasibility is that type checking involves moving bottom-to-top across the rules; in natural deduction, this is always easy, and we need look only at the terms to figure out which we should do. Type class checking and dictionary translation moves from top-to-bottom, directed by the left hand context, and produces terms on the right via complex operations, and that is a perfect fit for the sequent calculus rules.

I believe this corresponds to the general opinion on those who have studied sequent calculi with regard to type theory. A quick search revealed mostly papers on proof search, rather than type checking, and type classes rather fall into that realm (they're a very limited form of proof search).

Recently, a fellow in category land discovered a fact that we in Haskell land have actually known for a while (in addition to things most of us probably don't). Specifically, given two categories and , a functor , and provided some conditions in hold, there exists a monad , the codensity monad of .

In category theory, the codensity monad is given by the rather frightening expression:

Luite Stegeman has a mirror of the packages from Hackage.

He uses it to power his incredibly useful hdiff website.

During a Hackage outage, you can set up your local cabal configuration to point to it instead by (temporarily) replacing the remote-repo in your ~/.cabal/config file with:

 remote-repo: hdiff.luite.com:http://hdiff.luite.com/packages/archive 

and then running cabal update.

I have a ~/.cabal/config that I use whenever hackage goes down in my lens package.

If you use travis-ci, you can avoid build failures during hackage outages by first copying that config to ~/.cabal/config during before_install. -- You'll still be stuck waiting while it first tries to refresh from the real hackage server, but it only adds a few minutes to buildbot times.

Lenses are a great way to deal with functional references, but there are two common issues that arise from their use.

1. There is a long-standing folklore position that lenses do not support polymorphic updates. This has actually caused a fair bit of embarrassment for the folks who'd like to incorporate lenses in any Haskell record system improvement.
2. Access control. It'd be nice to have read-only or write-only properties -- "one-way" or "mirrored" lenses, as it were. Moreover, lenses are commonly viewed as an all or nothing proposition, in that it is hard to mix them with arbitrary user functions.
3. Finally there is a bit of a cult around trying to generalize lenses by smashing a monad in the middle of them somewhere, it would be nice to be able to get into a list and work with each individual element in it without worrying about someone mucking up our lens laws, and perhaps avoid the whole generalized lens issue entirely.

We'll take a whack at each of these concerns in turn today.
(more...)

No, I don't mean like this, but rather, If you spent any time trying to figure out xkcd's Umwelt April Fool comic this year, you may be interested in the Haskell source code. They used all sorts of information about you, the browser you were using, the resolution of your screen, to the geocoding of the network address you came from, etc. to serve up a custom web comic.

Today, davean posted to github the code for waldo, the engine he wrote to drive that comic.

Alas, he was not kind enough to actually supply the code for the umwelt comic strip itself, so you'll still be left wondering if the internet managed to find all of the Easter eggs. (Are they still Easter eggs when you release something a week before Easter?) You may find the list of links below useful if you want to get a feel for the different responses it gave people.

[Update: Jun 10, 9:09pm] davean just posted a rather insightful post mortem of the development of waldo that talks a bit about why xkcd uses Haskell internally.

I was contacted by someone who wanted to read my old catamorphism knol, despite the fact that Google Knol is no more.

Fortunately, while it was rather inconvenient that they shut down Google Knol completely, and I'll forever remember a knol as a "unit of abandonment", Google did provide a nice way to download at least your own user content and for that I am grateful.

I have fixed up the internal linkage as much as possible and have placed a copy of the original article below.

Catamorphisms: A Knol

Sadly, as I am not "Dark Magus", I am unable to download the Russian translation. If anyone knows how to contact him, I would love to obtain and preserve a copy of the translation as well.

In light of the burgeoning length of the ongoing record discussion sparked off by Simon Peyton-Jones in October, I would like to propose that we recognize an extension to Wadler's law (supplied in bold), which I'll refer to as the "Weak Record Conjecture" below.

In any language design, the total time spent discussing a feature in this list is proportional to two raised to the power of its position.

• 0. Semantics
• 1. Syntax
• 2. Lexical syntax
• 3. Lexical syntax of comments
• 4. Semantics of records

Andrej Bauer recently gave a really nice talk on how you can exploit side-effects to make a faster version of Martin Escardo's pseudo-paradoxical combinators.

A video of his talk is available over on his blog, and his presentation is remarkably clear, and would serve as a good preamble to the code I'm going to present below.

Andrej gave a related invited talk back at MSFP 2008 in Iceland, and afterwards over lunch I cornered him (with Dan Piponi) and explained how you could use parametricity to close over the side-effects of monads (or arrows, etc) but I think that trick was lost in the chaos of the weekend, so I've chosen to resurrect it here, and improve it to handle some of his more recent performance enhancements, and show that you don't need side-effects to speed up the search after all!

Last time we derived an entailment relation for constraints, now let's get some use out of it.

## Reflecting Classes and Instances

Most of the implications we use on a day to day basis come from our class and instance declarations, but last time we only really dealt with constraint products.

Max Bolingbroke has done a wonderful job on adding Constraint kinds to GHC.

Constraint Kinds adds a new kind Constraint, such that Eq :: * -> Constraint, Monad :: (* -> *) -> Constraint, but since it is a kind, we can make type families for constraints, and even parameterize constraints on constraints.

So, let's play with them and see what we can come up with!

As requested, here are the slides from Dan Doel's excellent presentation on Homotopy and Directed Type Theory from this past Monday's Boston Haskell.

I am very pleased to officially announce Hac Boston, a Haskell hackathon to be held January 20-22, 2012 at MIT in Cambridge, MA. The hackathon will officially kick off at 2:30 Friday afternoon, and go until 5pm on Sunday with the occasional break for sleep.

Everyone is welcome -- you do not have to be a Haskell guru to attend! Helping hack on someone else's project could be a great way to increase your Haskell skills.

If you plan on coming, please officially register, even if you already put your name on the wiki. Registration, travel, some information about lodging and many other details can now be found on the Hac Boston wiki.

We have confirmed space for about 30 people, so please register early! Beyond that we'll have to either seek additional space or close registration.

We're also looking for a few people interested in giving short (15-20 min.) talks, probably on Saturday afternoon. Anything of interest to the Haskell community is fair game---a project you've been working on, a paper, a quick tutorial. If you'd like to give a talk, add it on the wiki.

We look forward to seeing you at MIT!

Last time, I showed that we can build a small parsec clone with packrat support.

This time I intend to implement packrat directly on top of Parsec 3.

One of the main topics of discussion when it comes to packrat parsing since Bryan Ford's initial release of Pappy has been the fact that in general you shouldn't use packrat to memoize every rule, and that instead you should apply Amdahl's law to look for the cases where the lookup time is paid back in terms of repetitive evaluation, computation time and the hit rate. This is great news for us, since, we only want to memoize a handful of expensive combinators.

You never heard of the Millenium Falcon? It's the ship that made the Kessel Run in 12 parsecs.

I've been working on a parser combinator library called trifecta, and so I decided I'd share some thoughts on parsing.

Packrat parsing (as provided by frisby, pappy, rats! and the Scala parsing combinators) and more traditional recursive descent parsers (like Parsec) are often held up as somehow different.

Today I'll show that you can add monadic parsing to a packrat parser, sacrificing asymptotic guarantees in exchange for the convenient context sensitivity, and conversely how you can easily add packrat parsing to a traditional monadic parser combinator library.

Today I hope to start a new series of posts exploring constructive abstract algebra in Haskell.

In particular, I want to talk about a novel encoding of linear functionals, polynomials and linear maps in Haskell, but first we're going to have to build up some common terminology.

Having obtained the blessing of Wolfgang Jeltsch, I replaced the algebra package on hackage with something... bigger, although still very much a work in progress.

The question I hope to answer this time, is whether or not we turn any Haskell Comonad into a comonad transformer.