Wednesday, January 28, 2015

Haxl anti-tutorial

It's time for another anti-tutorial! Whereas a tutorial is an advanced user giving step-by-step instructions to help newbies, an anti-tutorial is a new user describing their path to enlightenment. My approach is usually to follow the types, so my anti-tutorials are also examples of how to do that.

Previously in the series:

  1. pipes anti-tutorial
  2. reactive-banana anti-tutorial
  3. netwire anti-tutorial

Today, inspired by a question from Syncopat3d, I'll try to learn how to use Simon Marlow's Haxl library. I think Haxl is supposed to improve the performance of complicated queries which use multiple data sources, such as databases and web services, by somehow figuring out which parts of the query should be executed in parallel and which ones should be batched together in one request. Since Syncopat3d is looking for a way to schedule the execution of a large computation which involves running several external processes in parallel, caching the results which are used more than once, and batching together the processes which use the same input, Haxl seemed like a good fit!

Black triangle

To understand the basics of the library, I'd like to create a black triangle, that is, a trivial program which nevertheless goes through the whole pipeline. So as a first step, I need to figure out what the stages of Haxl's pipeline are.

Since I'm using a type-directed approach, I need some type signature from which to begin my exploration. Hunting around Haxl's hackage page for something important-looking, I find GenHaxl, "the Haxl monad". Despite the recent complaints about the phrase "the <something> monad", finding that phrase here is quite reassuring, as it gives me a good idea of what to expect in this package: a bunch of commands which I can string together into a computation, and some function to run that computation.

Thus, to a first approximation, the Haxl pipeline has two stages: constructing a computation, and then running it.

A trivial computation

Since GenHaxl is a monad, I already know that return 42 is a suitably trivial and valid computation, so all I need now is a function to run a GenHaxl computation.

That function is typically right after the definition of the datatype, and indeed, that's where I find runHaxl. I see that in addition to my trivial GenHaxl computation, I'll need a value of type Env u. How do I make one?

Clicking through to the definition of Env, I see that emptyEnv can make an Env u out of a u. Since there are no constraints on u so far, I'll simply use (). I fully expect to revisit that decision once I figure out what the type u represents in the type GenHaxl u a.

>>> myEnv <- emptyEnv ()
>>> runHaxl myEnv (return 42)
42

(source)

Good, we now have a base on which to build! Let's now make our computation slightly less trivial.

What's a data source?

There are a bunch of GenHaxl commands listed after runHaxl, but most of them seem to be concerned with auxiliary matters such as exceptions and caching. Except for one:

dataFetch :: (DataSource u r, Request r a) => r a -> GenHaxl u a

That seems to be our link to another stage of Haxl's pipeline: data sources. So the first stage is a data source, then we describe a computation which fetches from the data source, then finally, we run the computation.

So, I want an r a satisfying DataSource u r. Is there something simple I could use for r? The documentation for DataSource doesn't list any instances, so I guess I'll have to define one myself. Let's see, there is only one method to implement, fetch, and it uses both u and r. The way in which they're used should give me a hint as to what those type variables represent.

fetch :: State r
      -> Flags
      -> u
      -> [BlockedFetch r]
      -> PerformFetch

I find it surprising that neither u nor r seem to constrain the output type. In particular, u is again completely unconstrained, so I'll keep using (). The description of the u parameter, "User environment", makes me think that indeed, I can probably get away with any concrete type of my choosing. As for r, which seems to be the interesting part here, we'll have to look at the definitions for State and BlockedFetch to figure out what it's about.

class Typeable1 r => StateKey r Source
    data State r

data BlockedFetch r
  = forall a . BlockedFetch (r a) (ResultVar a)

Okay, so State r is an associated type in an otherwise-empty typeclass, so I can again pick whatever I want. BlockedFetch r is much more interesting: it has an existential type a, which ties the r a to its ResultVar a. The documentation for BlockedFetch explains this link very clearly: r a is a request with result a, whose result must be placed inside the ResultVar a. This explains why r wasn't constraining fetch's output type: this ResultVar is the Haskell equivalent of an output parameter. So instead of being a pure function returning something related to r, this fetch method must be an imperative computation which fills in its output parameters before returning to the caller. The type of fetch's return type, PerformFetch, is probably some monad which has commands for filling in ResultVars.

data PerformFetch = SyncFetch (IO ()) | ...

At least in the simple case, PerformFetch is a simple wrapper around IO (), so I guess ResultVar must be a simple wrapper around MVar or IORef.

A trivial data source

Anyway, we now have a clear idea of what r a is: a request whose result has type a. Let's create a simple data source, Deep Thought, which only knows how to answer a single request.

data DeepThought a where
    AnswerToLifeTheUniverseAndEverything :: DeepThought Int

I'm using a GADT so that each request can specify the type of its answer. For example, I could easily add a request whose answer is a string instead of a number:

data DeepThought a where
    AnswerToLifeTheUniverseAndEverything :: DeepThought Int
    QuestionOfLifeTheUniverseAndEverything :: DeepThought String

But of course, Deep Thought isn't powerful enough to answer that request.

We also know that fullfilling a request isn't done by returning an answer, but by assigning the answer to a ResultVar.

runDeepThought :: DeepThought a -> ResultVar a -> IO ()
runDeepThought AnswerToLifeTheUniverseAndEverything var
  = putSuccess var 42

Alright, let's try to make DeepThought an official data source by implementing the DataSource typeclass:

instance DataSource () DeepThought where
    fetch _ _ _ reqs = SyncFetch $
        forM_ reqs $ \(BlockedFetch req var) ->
          runDeepThought req var

There's also a bunch of other easy typeclasses to implement, see the next source link for details.

A trivial state

I now have everything I need for my dataFetch to compile...

>>> runHaxl myEnv (dataFetch AnswerToLifeTheUniverseAndEverything)
*** DataSourceError "data source not initialized: DeepThought"

...but the execution fails at runtime. Now that I think about it, it makes a lot of sense: even though I don't use it, fetch receives a value of type State DeepThought, but since this is a custom type and I haven't given any of its inhabitants to anything, there is no way for Haxl to conjure one up from thin air. There must be a way to initialize the state somehow.

I must say that I'm a bit disappointed by how imperative Haxl's API has been so far. Whether we're assigning values to result variables or initializing a state, correctness requires us to perform actions which aren't required by the types and thus can't be caught until runtime. This is unusual for a Haskell library, and if the rest of the API is like this, I'm afraid following the types won't be a very useful exploration technique.

Anyway, I couldn't find any function with "init" in the name, but by looking for occurences of State in the types, I figured out how to perform the initialization: via the environment u which I had left empty until now.

instance StateKey DeepThought where
    data State DeepThought = NoState

initialState :: StateStore
initialState = stateSet NoState stateEmpty

>>> myEnv <- initEnv initialState ()
>>> runHaxl myEnv (dataFetch AnswerToLifeTheUniverseAndEverything)
42

(source)

It worked! We have a trivial data source, we have a trivial expression which queries it, we can run our expression, and we obtain the right answer. That's our black triangle!

Multiple data sources, multiple requests

Next, I'd like to try a slightly more complicated computation. Syncopat3d gives the following example:

F_0(x, y, z) = E(F_1(x, y), F_2(y, z))

Here we clearly have two different data sources, E and F. Syncopat3d insists that E is computed by an external program, which is certainly possible since our data sources can run any IO code, but I don't think this implementation detail is particularly relevant to our exploration of Haxl, so I'll create two more trivial data sources.

data E a where
    E :: String -> String -> E String
  deriving Typeable

data F a where
    F_1 :: String -> String -> F String
    F_2 :: String -> String -> F String
  deriving Typeable

runE :: E a -> ResultVar a -> IO ()
runE (E x y) var = putSuccess var (printf "E(%s,%s)" x y)

runF :: F a -> ResultVar a -> IO ()
runF (F_1 x y) var = putSuccess var (printf "F_1(%s,%s)" x y)
runF (F_2 x y) var = putSuccess var (printf "F_2(%s,%s)" x y)

Since GenHaxl is a monad, assembling those three requests should be quite straightforward...

>>> runHaxl myEnv $ do
...     f1 <- dataFetch (F_1 "x" "y")
...     f2 <- dataFetch (F_2 "y" "z")
...     dataFetch (E f1 f2)
"E(F_1(x,y),F_2(y,z))"

(source)

Batching

...but if I add a bit of tracing to my DataSource instances, I see that this computation is performed in three phases: F_1, F_2, then E.

>>> runHaxl myEnv ...
Computing ["F_1(x,y)"]
Computing ["F_2(y,z)"]
Computing ["E(F_1(x,y),F_2(y,z))"]
"E(F_1(x,y),F_2(y,z))"

(source)

This is not the trace I was hoping to see. Since fetch is receiving a list of request/var pairs, I expected Haxl to send me multiple requests at once, in case my data source knows how to exploit commonalities in the requests. But it doesn't look like Haxl figured out that the F_1 and F_2 requests could be performed at the same time.

It turns out that this is a well-known problem with Haxl's monadic interface. I remember about it now, it was described in a presentation about Haxl (slide 45) when it came out. The solution is to use the Applicative syntax to group the parts which are independent of each other:

>>> runHaxl myEnv $ do
...     (f1,f2) <- liftA2 (,) (dataFetch (F_1 "x" "y"))
...                           (dataFetch (F_2 "y" "z"))
...     dataFetch (E f1 f2)
Computing ["F_2(y,z)","F_1(x,y)"]
Computing ["E(F_1(x,y),F_2(y,z))"]
"E(F_1(x,y),F_2(y,z))"

(source)

Good, the F_1 and F_2 requests are now being performed together.

Style

I don't like the way in which we have to write our computations. Consider a slightly more complicated example:

E(
  E(F_1(x,y), F_2(y,z)),
  E(F_1(x',y'), F_2(y',z'))
)

Since the four F_1 and F_2 requests at the leaves are all independent, it would make sense for Haxl to batch them all together. But in order to obtain this behaviour, I have to list their four subcomputations together.

>>> runHaxl myEnv $ do
...     (f1,f2,f1',f2') <- (,,,) <$> (dataFetch (F_1 "x" "y"))
...                              <*> (dataFetch (F_2 "y" "z"))
...                              <*> (dataFetch (F_1 "x'" "y'"))
...                              <*> (dataFetch (F_2 "y'" "z'"))
...     (e1,e2) <- (,) <$> (dataFetch (E f1 f2))
...                    <*> (dataFetch (E f1' f2'))
...     dataFetch (E e1 e2)
Computing ["F_2(y',z')","F_1(x',y')","F_2(y,z)","F_1(x,y)"]
Computing ["E(F_1(x',y'),F_2(y',z'))","E(F_1(x,y),F_2(y,z))"]
Computing ["E(E(F_1(x,y),F_2(y,z)),E(F_1(x',y'),F_2(y',z')))"]
"E(E(F_1(x,y),F_2(y,z)),E(F_1(x',y'),F_2(y',z')))"

(source)

I feel like I'm doing the compiler's job, manually converting from the nested calls I want to write to the leaves-to-root, layered style I have to write if I want batching to work.

So I stopped working on my anti-tutorial and wrote a toy library which converts from one style to the other :)

...and when I came back here to show it off, I discovered that GenHaxl already behaved exactly like my library did! You just have to know how to define your intermediate functions:

f_1 :: GenHaxl () String -> GenHaxl () String -> GenHaxl () String
f_1 x y = join (dataFetch <$> (F_1 <$> x <*> y))

f_2 :: GenHaxl () String -> GenHaxl () String -> GenHaxl () String
f_2 x y = join (dataFetch <$> (F_1 <$> x <*> y))

e :: GenHaxl () String -> GenHaxl () String -> GenHaxl () String
e x y = join (dataFetch <$> (E <$> x <*> y))

And with those, we can now describe the computation as nested function calls, as desired.

>>> x = pure "x"
>>> y = pure "y"
>>> z = pure "z"
>>> x' = pure "x'"
>>> y' = pure "y'"
>>> z' = pure "z'"
>>> runHaxl myEnv $ e (e (f_1 x y) (f_2 y z))
...                   (e (f_1 x' y') (f_2 y' z'))
Computing ["F_1(y',z')","F_1(x',y')","F_1(y,z)","F_1(x,y)"]
Computing ["E(F_1(x',y'),F_1(y',z'))","E(F_1(x,y),F_1(y,z))"]
Computing ["E(E(F_1(x,y),F_1(y,z)),E(F_1(x',y'),F_1(y',z')))"]
"E(E(F_1(x,y),F_1(y,z)),E(F_1(x',y'),F_1(y',z')))"

(source)

Conclusion

I now understand Haxl's purpose much better. With the appropriate intermediate functions, Haxl allows us to describe a computation very concisely, as nested function calls. Haxl executes this computation one layer at a time: all of the leaves, then all the requests which only depend on the leaves, and so on. Within a single layer, the requests are subdivided again, this time according to their respective data sources. Finally, for a given data source, it is fetch's responsibility to find and exploit opportunities for reusing work across the different requests belonging to the same batch. There are also some features related to caching and parallelism which I didn't explore.

I also understand Haxl's implementation much better, having reimplemented part of it myself. In fact, I'd be interested in writing a follow-up post named "Homemade Haxl", in the same vein as my "Homemade FRP" series. What do you think? Are you more interested in watching me learn some new libraries, watching me reimplement libraries, or watching me implement new stuff? I'll be doing all three anyway, I just want to know which of those activities I should blog about :)

Really, your feedback would be greatly appreciated, as the only reason I started this anti-tutorial series in the first place is that my first write-up on understanding Pipes was so surprisingly popular. I've streamlined the format a lot since that first post, and I want to make sure I haven't lost any of the magic in the process!

Sunday, December 21, 2014

The "99 Bottles of Beers" of Type Systems

"Hello World" is a good first example program because it is small, but also because it encourages the reader to get into the driver's seat and take control of the program. Copy-paste the "Hello World" listing from a website, and you're just blindly following instructions. Change it to print "Hello Mom", and you're boldly taking your first step towards the unknown, into a world where it is now you who is giving the instructions.

New programmers need to take that step, because programming anything non-trivial requires taking your own decisions about how things are implemented. If your boss was taking all the decisions for you, you wouldn't be a programmer, you'd be a typist.

The "Hello World" of Type Systems

Once you become an experienced programmer, "Hello World" examples are still useful as an introduction to new languages and new systems. Once you have a working base, you can experiment by making small changes and verifying whether they work as you expect, or if you need to read more tutorials.

For type systems, I guess a "Hello World" program would be a small datatype/structure/class containing a few simple fields. The standard here isn't as well-established as with "Hello World", but describing a person is quite common:

data Person = Person
  { name :: String
  , age :: Int
  }

I've used Haskell here, but regardless of the language in which I would have written this, you could still easily infer how to add a field for the person's height.

99 Bottles of Beer

A little down the road, another introductory program is "99 bottles of beer on a wall". This one teaches budding programmers another important lesson: it's possible to write a program which prints out more text than what you've written in its source code. More specifically, the program shows how to use a variable to abstract over the part of the text which varies from one iteration to the other, and how to use a loop to determine how many iterations to make and which value the variable should take in each one.

For type systems, a "99 bottles of beer" program would teach the same lesson: it's possible to write a program which uses larger types than those you've written in the source code. This is rarely needed, but it's possible! Even in a large, complicated application, you might have a manager of pools of worker threads processing lists of person values, but Manager (Pool (WorkerThread (List Person))) is still a fixed type which you write down explicitly in your program. It's as if you had abstracted out the number of beers to print, but then wrote explicit calls with n = 99, n = 98 and so on, instead of using a loop to generate the calls at runtime. Our "99 bottles of beer" example should generate types at runtime.

The "99 Bottles of Beer" of Type Systems

The simplest such example I could think of is as follows:

  1. Parse a non-negative integer n from standard input or from a command-line argument.
  2. If n is 0, print 42.
  3. Otherwise, print the pair (x,x), where x is the text which would have been printed if n was one unit smaller. For example, the output for n = 3 should be "(((42,42),(42,42)),((42,42),(42,42)))".

With the important restriction that the pair (x, x) must first be constructed before being printed, and its representation must not have the same type as x.

An incorrect solution

The reason the restriction is important is that otherwise, it would be possible to implement the program using a single type, that of integer trees:

-- *not* a valid solution
data Tree a = Leaf a | Branch (Tree a) (Tree a)

showTree :: Show a => Tree a -> String
showTree (Leaf x)       = show x
showTree (Branch t1 t2) = printf "(%s,%s)" (showTree t1)
                                           (showTree t2)

printTree :: Tree Int -> Int -> IO ()
printTree v 0 = putStrLn (showTree v)
printTree v n = printTree (Branch v v) (n-1)

main :: IO ()
main = readLn >>= printTree (Leaf 42)

That program does not demonstrate that it's possible to write a program which uses larger types than those you've written in the source code.

Haskell solution

Instead of using the same type Tree Int at every iteration, we want to construct a sequence of larger and larger types:

  1. Int
  2. (Int,Int)
  3. ((Int,Int),(Int,Int))
  4. (((Int,Int),(Int,Int)),((Int,Int),(Int,Int)))
  5. ...

In Haskell, this can be achieved via polymorphic recursion, meaning that we recur at a different type than the one which the current call is being instantiated at. For example, the call printTree 42 1 instantiates the type variable a = Int, while the recursive call printTree (42,42) 0 instantiates the type variable a = (Int,Int).

printTree :: Show a => a -> Int -> IO ()
printTree v 0 = print v
printTree v n = printTree (v,v) (n-1)

main :: IO ()
main = readLn >>= printTree 42

Polymorphic recursion is often used to recur on a smaller type, but since in this function it is the Int argument which is getting smaller, we can recur on a larger type without risking an infinite loop.

C++ solution

Speaking of infinite loops, C++ uses compile-time templates to handle polymorphic recursion, and this implementation strategy causes the compiler to instantiate more and more templates when we recur on a larger type. Eventually, gcc gives up with "template instantiation depth exceeds maximum of 900".

We can work around the problem by specializing the template at one of the types encountered before that limit, and printing an error message instead of recurring further.

#include <stdio.h>

template<typename A>
struct Pair {
  A fst;
  A snd;
  
  Pair(A fst, A snd)
  : fst(fst), snd(snd)
  {}
};

void print(int n) {
  printf("%d", n);
}

template<typename A>
void print(Pair<A> pair) {
  printf("(");
  print(pair.fst);
  printf(",");
  print(pair.snd);
  printf(")");
}

template<typename A>
void println(A value) {
  print(value);
  printf("\n");
}


template<typename A>
struct PrintTree {
  static void call(int depth, A value) {
    if (depth == 0) {
      println(value);
    } else {
      PrintTree<Pair<A> >::call(depth - 1, Pair<A>(value, value));
    }
  }
};

template<>
struct PrintTree<
  Pair<Pair<Pair<Pair<Pair<Pair<Pair<Pair<int>
> > > > > > > >
{
  static void call(int, Pair<
                          Pair<Pair<Pair<Pair<Pair<Pair<Pair<int>
                        > > > > > > >
  ) {
    fprintf(stderr, "maximum depth exceeded.\n");
  }
};

int main() {
  int depth;
  scanf("%d", &depth);
  
  PrintTree<int>::call(depth, 42);
  
  return 0;
}
Java solution

Other implementation strategies, such as Java's type erasure, need no such artificial bounds.

class Pair<A> {
  private A fst;
  private A snd;
  
  public Pair(A fst, A snd) {
    this.fst = fst;
    this.snd = snd;
  }
  
  public String toString() {
    return "(" + fst.toString() + "," + snd.toString() + ")";
  }
}

public class Main {
  public static <A> void printTree(int depth, A value) {
    if (depth == 0) {
      System.out.println(value.toString());
    } else {
      printTree(depth - 1, new Pair<A>(value, value));
    }
  }
  
  public static void main(String[] args) {
    Integer n = Integer.valueOf(args[0]);
    Integer m = 42;
    printTree(n, m);
  }
}
Conclusion

Many programming languages have the ability to work with larger types than those which are known at compile time, but for some reason, the feature is rarely used.

Perhaps one of the reasons is that the feature is rarely covered in tutorials. I have presented a small example demonstrating the feature, and I have demonstrated that the example isn't specific to one particular type system by implementing it in a few different languages. If you're writing a tutorial for a language and you have already covered "Hello World", "99 bottles of beer" and the "Hello World" of type systems, please consider also covering the "99 bottles of beer" of type systems.

Although, if I want this example to catch on, I should probably give it a better name. Maybe "Complete trees whose leaves are 42", or simply "Complete 42" for short?

Monday, December 08, 2014

How to package up binaries for distribution

This weekend, I wrote a game (in Haskell of course!) for Ludum Dare, an event in which you have 48h or 72h to create a game matching an imposed theme. It was really challenging!

Once the event was over, it was time to package my game in a form which others could play. Since the packaging procedure wasn't obvious, I'm documenting it here for future reference. The procedure isn't specific to Haskell, but I'll mention that linking Haskell programs statically, as advised around the web, didn't work for me on any platform.

Windows


While your program is running, use Process Explorer to list the .dll files your program is currently using (There is also Dependency Walker, but on my program it missed glut32.dll). Copy those DLLs to the same folder as your executable, zip the folder, and ship it.

OS X


Use otool -L to list the .dylib files on which your executable depends, and copy them to the same folder as your executable (or a libs subfolder). Use install_name_tool to change all the dylib paths embedded in your executable to @executable_path/foo.dylib (or @executable_path/libs/foo.dylib). Zip the folder, and ship it.

Linux


Use ldd to list the .so files on which your executable depends, and copy all of them except libc.so.X to the same folder as your executable (or a libs subfolder). Add ld-options: -Wl,-rpath -Wl,$ORIGIN (or ld-options: -Wl,-rpath -Wl,$ORIGIN/libs) to your cabal file, pass those flags directly to gcc, or use chrpath to change the existing RPATH if there is one. Zip the folder, and ship it.

Tuesday, October 28, 2014

Understanding "Strongly-typed Bound", part 1

First, giving credits where credit is due. The Bound library is written by Edward Kmett, and so is the strongly-typed variant I want to explore in this series. I learned about the strongly-typed version via a comment by geezusfreeek, in response to a question by _skp.

I have a lot to say about this script, and since the first thing I want to say about it involves writing down some typing rules, I thought I'd write them on the whiteboard and publish a video! Please let me know what you think of this new format.


Saturday, September 06, 2014

Prisms lead to typeclasses for subtractive types

In my last post, I identified some issues with subtractive types, namely that math identities such as ∀ a. a + -a = 0, once they are translated into Haskell, would not be valid for all a. More precisely, the existence of a function of type

cancelSub :: forall a. (a :+: Negative a) -> Zero

would make it easy to implement a contradiction, regardless of the way in which we represent Negative a:

type a :+: b = Either a b
type Zero = Void

contradiction :: Void
contradiction = cancelSub (Left ())

I concluded by blaming the unconstrained forall. That is, I was hoping that the identity could be saved by finding some typeclass C such that C a => (a :+: Negative a) -> Void would be inhabited, or something along those lines. But what should C look like?

Prisms

Earlier today, I was re-listening to Edward Kmett on Lenses, in the first Haskell Cast episode. While explaining Prisms at 38:30, Kmett explained that a Lens' s a splits an s into a product consisting of an a and of something else, and that correspondingly, a Prism' s a splits an s into a sum consisting of an a and of something else. It occurred to me that the first "something else" should be s :/: a, while the second "something else" should be s :-: a.

Since Prism' s a is only inhabited for some combinations of s and a but not others, I thought a good choice for my C typeclass might be a proof that there exists a prism from s to a.

cancelSub :: HasPrism a (Negative a)
          => (a :+: Negative a) -> Void

That is, instead of restricting which types can be negated, let's restrict which types are allowed to appear together on the left- and right-hand sides of a subtractive type.

Four typeclasses

All right, so what should the HasPrism typeclass look like? In the podcast, Kmett explains that we can "construct the whole thing out of the target of a prism", and that we can pattern-match on the whole thing to see if it contains the target. In other words:

class HasPrism s a where
    construct :: a -> s
    match :: s -> Maybe a

This Maybe a discards the case I am interested in, the s :-: a. Let's ask the typeclass to provide a representation for this type, so we can give a more precise type to match.

class HasPrism s a where
    type s :-: a
    constructLeft :: a -> s
    constructRight :: (s :-: a) -> s
    match :: s -> Either a (s :-: a)

Our typeclass now has three methods, for converting back and forth between s and its two alternatives. We can combine those three methods into a single bijection, and with this final transformation, we obtain a form which is easily transferable to the other inverse types:

class Subtractable a b where
    type a :-: b
    asSub :: Iso a ((a :-: b) :+: b)

class Divisible a b where
    type a :/: b
    asDiv :: Iso a ((a :/: b) :*: b)

class Naperian b a where
    type Log b a
    asLog :: Iso a (Log b a -> b)

class Rootable n a where
    type Root n a
    asRoot :: Iso a (n -> Root n a)
Routing around the contradiction

The real test for these new definitions is whether they allow us to define constructive versions of the math identities for subtraction, division, logarithms and roots. Once annotated with the proper type class constraint, does cancelSub still lead to a contradiction? If not, can it be implemented?

It can!

type Negative a = Zero :-: a

cancelSub :: forall a. Subtractable Zero a
          => Iso (a :+: Negative a) Zero
         -- a :+: Negative a
cancelSub = swap
         -- Negative a :+: a
        >>> inverse iso
         -- Zero
  where
    iso :: Iso Zero (Negative a :+: a)
    iso = asSub

The math version of the constrained type is still ∀ a. a + -a = 0, but with a new proviso "whenever -a exists". It's still the same identity, it's just that with real numbers, -a always exists, so the proviso does not usually need to be spelled out.

In the world of types, Negative a does not always exist. If fact, there's only one possible instance of the form Subtractable Zero a:

instance Subtractable Zero Zero where
    type Zero :-: Zero = Zero
    
    asSub :: Iso Zero ((Zero :-: Zero) :+: Zero)
    asSub = Iso Right (either id id)

In other words, in the world of types, the proviso "whenever -a exists" simply means "when a = 0".

Other identities

I wish I could say that all the other identities become straightforward to implement once we add the appropriate typeclass constraints, but alas, this is not the case. I plan to discuss the remaining issues in a subsequent post.

For now, I am content to celebrate the fact that at least one contradiction has been slain :)

Friday, August 29, 2014

Edward Kmett likes my library :)

I have be re-listening to old episodes of the Haskell Cast, and it turns out I missed something really, shall we say, relevant to my interests.



In the very first episode, Edward Kmett talks about lens and a few of his other libraries. Then, near the end, he is asked about interesting Haskell stuff aside from his libraries. His answer, at 59:45:

"There was a really cool Commutativity monad [...] that really struck me as an interesting approach to things, I thought it was particularly neat toy."
— Edward Kmett


Yay, that's my library! And here are Wren's blog posts he mentions, about generalizing my approach.

Monday, August 18, 2014

Issues with subtractive types

I tried to expand on my earlier attempt at understanding root types, but I hit several obstacles. Here is a summary of those obstacles, in case future-me or someone else feels like trying again.

Goals
The conversation began with Naperian types, and the way in which they satisfy the same laws as mathematical logarithms, except at the type level. For example, the law logb xy = logb x + logb y would translate to a function of type

logProduct :: Log b (a1, a2) -> Either (Log b a1) (Log b a2)

where, as usual, pairs represent multiplication and Either represents a sum. We would also expect a function with the converse type, and we would expect the two functions to form an isomorphism.

My goal would be to find implementations for Log and the other inverse types such that the corresponding isomorphisms exist and are useful. As the rest of the post will demonstrate, "exists" is already quite a strong requirement.

I should mention before moving on that yes, I am familiar with "The Two Dualities of Computation: Negative and Fractional Types", and I am intentionally using a different approach. Their non-deterministic invertible language in quite interesting, but ultimately too weird for me.

I would prefer to find a way to implement negative and fractional stuff as Haskell datatypes, or failing that, to understand why it can't be done. Today's post is about the latter: if negative and fractional types can exist in Haskell, then identities such as 0 ↔ x + -x wouldn't be valid for all types x, like they are in that paper.

Naïve definitions
Since subtraction is the inverse of addition, I tried to define a subtractive type and the other inverses types in term of the types we already have.

type a :+: b = Either a b
type a :*: b = (a, b)

data ab :-: b where
  MkSub :: a -> (a :+: b) :-: b

data ab :/: b where
  MkDiv :: a -> (a :*: b) :/: b

data Log b a where
  MkLog :: p -> Log b (p -> b)

data Root n a where
  MkRoot :: b -> Root n (n -> b)

One obvious problem with those definitions is that they don't support any of the math identities, except for the ones used in the definitions themselves: x + y - y = x, etc. For most types a, the type a :-: b isn't even inhabited, so while we might be able to implement absurd-style isomorphisms, the result would be completely useless.

Isomorphisms to the rescue
In the reddit comment linked at the top of this post, I worked around the problem via the hypothesis that maybe we shouldn't expect math identities to correspond to type isomorphisms so directly. Instead, I postulated an extra operator which would lift an isomorphism between a and a' to an isomorphism between a :-: b and a' :-: b, and similarly for the right-hand side and for the other type operators. It worked well enough, but since this transformation isn't constructive, we still don't get useful isomorphic functions at the end.

So, can we make the transformation constructive? Something like this:

data a :-: b where
  MkSub :: r -> Iso a (r :+: b) -> a :-: b

data a :/: b where
  MkDiv :: r -> Iso a (r :*: b) -> a :/: b

data Log b a where
  MkLog :: p -> Iso a (p -> b) -> Log b a

data Root n a where
  MkRoot :: b -> Iso a (n -> b) -> Root n a

By using id as the Iso, we can construct the same inhabitants as with the previous definitions. In addition, we can now constructively lift an isomorphism on the left- or right-hand side to an isomorphism on the whole type. The code for doing this is a bit longer than I'd like, but the idea is that since isomorphisms can already be lifted to either side of a (:+:), (:*:), or (->), we should therefore be able to concatenate an isomorphism for the left- or right-hand side with the existing Iso. For example:

liftRightAdd :: forall a b b'
              . Iso b b'
             -> Iso (a :+: b) (a :+: b')
liftRightAdd isoB = ...

liftRightSub :: forall a b b'
              . Iso b b'
             -> Iso (a :-: b) (a :-: b')
liftRightSub isoB = Iso fwdS bwdS
  where
    fwdS :: a :-: b -> a :-: b'
    fwdS (MkSub r iso) = MkSub r (liftIso iso)
      where
        liftIso :: Iso a (r :+: b) -> Iso a (r :+: b')
        liftIso iso = liftRightAdd isoB . iso
    
    bwdS :: a :-: b' -> a :-: b
    bwdS (MkSub r iso) = MkSub r (liftIso iso)
      where
        liftIso :: Iso a (r :+: b') -> Iso a (r :+: b)
        liftIso iso = liftRightAdd (inverse isoB) . iso)

With those tools, we should be able to take a non-constructive transformation like the one I wrote in my reddit comment:

log_b(a1) + log_b(a2) ~ Either (Log b a1) (Log b a2)
                     ≈≈ Either (Log b (p1 -> b))
                               (Log b (p2 -> b))
                      ≈ Either p1 p2
                      ≈ Log b (Either p1 p2 -> b)
                     ≈≈ Log b (p1 -> b, p2 -> b)
                     ≈≈ Log b (a1, a2)
                      ~ log_b(a1*a2)

And translate it into a constructive version:

addLogs :: forall a1 a2 b. Iso (Log b a1 :+: Log b a2)
                               (Log b (a1 :*: a2))
          -- Log b (a1 :*: a2)
addLogs = liftRightLog (liftBothMul (inverse asExp1)
                                    (inverse asExp2))
          -- Log b ((p1 -> b) :*: (p2 -> b))
        . liftRightLog expSum
          -- Log b (p1 :+: p2 -> b)
        . inverse logExp
          -- p1 :+: p2
        . liftBothAdd logExp logExp
          -- Log b (p1 -> b) :+: Log b (p2 -> b)
        . liftBothAdd (liftRightLog asExp1)
                      (liftRightLog asExp2)
          -- Log b a1 :+: Log b a2
  where
    asExp1 :: Iso a1 (P1 -> b)
    asExp2 :: Iso a2 (P2 -> b)

expSum :: Iso (a1 :+: a2 -> b)
              ((a1 -> b) :*: (a2 -> b))
logExp:: Iso (Log b (p -> b))
             p

Success? Not so fast.

Impossible isomorphisms
In order to execute the above constructive proof, we must of course implement all the smaller isomorphisms on which it is based. Two of them, asExp1 and asExp2, seem pretty silly: can we really expect any type a1 to be isomorphic to a function of the form p1 -> b, for any type b of our choosing?

I had originally postulated that I could do this because in my original definition for Log b a1, log types were only inhabited when a1 had the required form p1 -> b. With the new Iso-based definition, I'm no longer sure I can do this, and even with the old definition, it was only ever justified to transform Log b a1 into Log b (p1 -> b), not to transform a naked a1 into p1 -> b.

However, if we simply pick p1 = Log b a1, then the math identity blogb x = x justifies the principle. Can we write a constructive version of this identity?

One direction is easy:

mkExpLog :: a -> (Log b a -> b)
mkExpLog x (MkLog p (Iso fwd _)) = fwd x p

But the other direction direction is impossible. It would allow us to implement unsafeCoerce!

unExpLog :: (Log b a -> b) -> a

unsafeCoerce' :: b -> a
unsafeCoerce' = unExpLog . const

Similar issues occur with identities from the other inverse types. With a constructive version of the identity x - x = 0, for example, we can construct an inhabitant for the empty type:

unSubSelf :: a :-: a -> Void

subSelf :: [()] :-: [()]
subSelf = MkSub () (Iso fwdL bwdL)
  where
    fwdL :: [()] -> () :+: [()]
    fwdL []      = Left ()
    fwdL (():xs) = Right xs
    
    bwdL :: () :+: [()] -> [()]
    bwdL (Left  ()) = []
    bwdL (Right xs) = ():xs

bottom :: Void
bottom = unSubSelf subSelf

The fact that a recursive type is used in this counter-example hints at an explanation of the issue. The identity x - x = 0 is certainly true in math, for any number x. So at least for types with exactly x inhabitants, we would expect the isomorphism to hold. But in this case, the type [()] has infinitely-many inhabitants, and as we know from math, ∞ - ∞ is not zero, it's an indeterminate form.

Here is another implementation of the empty type, based on the identities x / x = 1 and x / y = x * (1/y):

mkDivSelf :: () -> a :/: a
mkTimesReciprocal :: a :/: b
                  -> a :*: (() :/: b)

divZero :: Void :/: Void
divZero = mkDivSelf ()

bottom' :: Void
bottom' = fst (mkTimesReciprocal divZero)

I haven't managed to derive a contradiction from any of the root identities, but they seem just as impossible to implement.

Restricted isomorphisms
Okay, so the indeterminate forms ∞ - ∞ and 0 / 0 both led to contradictions. In math, we work around the problem by saying that the division identities are only valid when the denominator is non-zero, and we don't even bother mentioning infinity because it's usually not a member of our universe of discourse, the real numbers for example.

In the world of types, instead of forbidding particular elements such as zero, it's much more common to require a witness proving that the types under consideration are valid for the current operation. In Agda, this would be a predicate on Set, while in Haskell, it would be a typeclass constraint. So, if future me is reading this and wants to continue exploring the world of inverse types, I leave him with the following recommendation: try looking for typeclass constraints under which the identities don't lead to contradictions, or even better, under which they can actually be implemented.