Using Lawvere theories to combine effects

> {-# LANGUAGE MultiParamTypeClasses, ExplicitForAll, RankNTypes, FlexibleInstances, FlexibleContexts, TypeSynonymInstances #-}

> import Data.Monoid
> import Data.Functor.Identity
> import Control.Monad.Writer

In an earlier post I talked about how monads arise from free algebras. Let me recap a bit.

In Part 1 I described algebras. They're sets with operations on them satisfying some laws. We can build new elements of an algebra from old ones by using its operations. Eg. if x and y are in an algebra then x `mappend` y must be in it too. Starting with a bunch of symbols, thought of as leaves, we can consider the set of all expressions trees we can build from them. If we consider pairs of trees to be equivalent if the laws say the corresponding expressions are equal, then the set of trees itself forms an algebra known as a free algebra (for the given theory).

Let's start with some code. This type class says that the type b has leaves of type a:

> class Free a b where
>   leaf :: a -> b

Effects from monoids
Now we can make the type of all trees built from Monoid operations and including all leaves of type a:

> data FreeMonoid a = FreeMonoid (forall b. (Monoid b, Free a b) => b)

And we have:

> instance Monoid (FreeMonoid a) where
>   mempty = FreeMonoid mempty
>   FreeMonoid a `mappend` FreeMonoid b = FreeMonoid (a `mappend` b)

Unfortunately elements like e1 and e2 two ought to be equal but Haskell doesn't know this:

> e1, e2 :: FreeMonoid Char
> e1 = FreeMonoid (leaf 'a' `mappend` (leaf 'b' `mappend` leaf 'c'))
> e2 = FreeMonoid ((leaf 'a' `mappend` leaf 'b') `mappend` leaf 'c')

Instead we can manually construct a type that does respect equality in monoids. Elements of FreeMonoid are binary trees with a `mappend` at each node. Associativity means that we can always replace a tree with an equivalent one where the left branch is a leaf. We can also use the laws to eliminate any occurrence of mempty. So every element of FreeMonoid a is equivalent to one of the form:

Leaf x1 `mappend` (Leaf x2 `mappend` (... mempty))

In other words, free monoids are lists. We can make this explicit. The standard prelude already makes [] an instance of Monoid so we just need:

> instance Free a [a] where
>       leaf x = [x]

Here's the isomorphism (modulo tree equivalence):

> iso1 :: FreeMonoid a -> [a]
> iso1 (FreeMonoid x) = x

> iso1' :: [a] -> FreeMonoid a
> iso1' [] = FreeMonoid mempty
> iso1' (a : as) = let FreeMonoid r = iso1' as
>                  in FreeMonoid (leaf a `mappend` r)

As I talked about in that earlier article, free algebras give monads and the trees representing expressions in the algebra can be thought of as abstract syntax trees for domain specific languages. In this case it's the usual list monad. So the Monoid type class gives us a language for talking about non-determinism. The operation mappend gives us a way to "fork" a process and mempty gives as a way to "kill a thread". Here's an example using non-determinism to search for some Pythagorean triples:

> test1 :: [(Int, Int, Int)]
> test1 = do
>   a <- return 3 `mappend` return 4
>   b <- return 4 `mappend` return 5
>   c <- return 5 `mappend` return 6
>   if a*a+b*b==c*c then return (a, b, c) else mempty

Effects form M-sets
We can do exactly the same for -sets.

> class Monoid m => MSet m s where
>       act :: m -> s -> s

> data FreeMSet w a = FreeMSet (forall b. (MSet w b, Free a b) => b)

> instance Monoid w => MSet w (FreeMSet w a) where
>   m `act` FreeMSet b = FreeMSet (m `act` b)

Again we have the problem that FreeMSet doesn't automatically make equivalent elements equal. But it's not hard to see that every element of FreeMSet is equivalent to one of the form:

m `act` (leaf x)

So the free -set on the set of variables is simply the set of pairs . This is the basis of Haskell's writer monad:

> instance Monoid w => MSet w (Writer w a) where
>   act w1 m = let (a, w2) = runWriter m in WriterT (Identity (a, w1 `mappend` w2))

> instance Monoid w => Free a (Writer w a) where
>   leaf x = return x

Here's the isomorphism (again treating equivalent elements of FreeMSet as equal):

> iso2 :: Monoid w => FreeMSet w a -> Writer w a
> iso2 (FreeMSet x) = x

> iso2' :: Writer w a -> FreeMSet w a
> iso2' m = let (a, w) = runWriter m in FreeMSet (act w (leaf a))

And now the -set operation gives us an interface to an effect. This time the side effect of accumulating in a monoid:

> test2 :: Writer String Int
> test2 = do
>   act "foo" (return ())
>   a <- return 2
>   act "bar" (return ())
>   b <- return (10*a)
>   return b

Combining effects

And now we can finally combine the two effects of non-determinism and accumulation. We make the free algebra that is both a monoid and an -set:

> data FreeMMonoid w a = FreeMMonoid (forall b. (Monoid b, MSet w b, Free a b) => b)

> instance Monoid w => Monoid (FreeMMonoid w a) where
>   mempty = FreeMMonoid mempty
>   FreeMMonoid a `mappend` FreeMMonoid b = FreeMMonoid (a `mappend` b)

> instance Monoid w => MSet w (FreeMMonoid w a) where
>   m `act` FreeMMonoid b = FreeMMonoid (m `act` b)

Again we have the problem that equivalent elements aren't recognised as equal so we have to manually find a suitable type. For this we need to use the compatibility notion I introduced in Part 1. We can take 2 variables and and write them in a 1 by 2 array:

Apply mappend horizontally and act vertically to get:

m `act` (x `mappend` y)

Now apply act vertically and then mappend horizontally to get:

(m `act` x) `mappend` (m `act` y)

The law we want is:

m `act` (x `mappend` y) == (m `act` x) `mappend` (m `act` y)

Given an arbitrary tree in FreeMMonoid we can use this law to "push" all occurrences of act inwards. Ultimately every element can be written uniquely in the form:

act m1 (leaf x1) `mappend` (act m2 (leaf x2) `mappend` (... mempty)

We can then use the same argument as above to show that we end up with a list of pairs of elements of . This is exactly what we get if we apply the WriterT monad transformer to []. Here are the relevant instances:

> instance Monoid w => Monoid (WriterT w [] a) where
>   mempty = WriterT []
>   WriterT xs `mappend` WriterT ys = WriterT (xs ++ ys)

> instance Monoid w => MSet w (WriterT w [] a) where
>   m `act` WriterT xs = WriterT $ map (\(x, w) -> (x, m `mappend` w)) xs

> instance Monoid w => Free a (WriterT w [] a) where
>   leaf x = return x

Here's the isomorphism though we won't use it:

> iso3 :: Monoid w => FreeMMonoid w a -> WriterT w [] a
> iso3 (FreeMMonoid x) = x

> iso3' :: Monoid w => WriterT w [] a -> FreeMMonoid w a
> iso3' m = let xws = runWriterT m in FreeMMonoid $
>     foldr mappend mempty $ map (\(x, w) -> act w (leaf x)) xws

The monad WriterT (Product Float) [] is in fact the probability monad. Here's an example of its use:

> coin :: (Monoid a, MSet (Product Float) a, Free Bool a) => a
> coin = act (Product 0.5 :: Product Float) (leaf False)
>            `mappend`
>        act (Product 0.5 :: Product Float) (leaf True)

Compute unnormalised conditional probability distribution on a pair of coin tosses given that first coin can't be True unless second one is:

> test3 :: WriterT (Product Float) [] (Bool, Bool)
> test3 = do
>   coin1 <- coin
>   coin2 <- coin
>   if coin1>coin2 then mempty else return (coin1, coin2)

(Compare with Eric Kidd's article that also 'refactors' probability theory.)

What just happened?
Something miraculous just happened though it may have been lost in the details. We combined the list monad and the writer monad to get a new monad. We did it without using monad transformers and without specifying an order for the two monads. It just so happens in this case that the result was the same as using a monad transformer.

M-set with M-set
We can try other products of theories. It's tricky to deal with a theory combined with itself because repeating a type class in a context doesn't do anything. We need to make another type class that looks exactly like MSet but with different names. The result is that the product of the theory of -sets and the theory of -sets is the theory of -sets. This agrees with what we'd get from using monad transformers. It also agrees with intuition. -sets correspond to the effect of accumulating data in a monoid. The product theory corresponds to using two accumulators simultaneously.

(This makes me think type classes should take as arguments the name of the operations within them. That way a type can be an instance of the same type class in multiple ways. Compare with Agda modules.)

Monoid with monoid
This example illustrates why we can't expect a programming language to use the above method to combine theories. If an algebra has two multiplication operators with identities on it, and the two operators are compatible, then something surprising happens. The multiplications turn out to be the same operation. What's more, the operation is commutative. So the product of the theory of monoids with itself is the theory of commutative monoids. A free commutative monoid is a multiset. Multisets require a very different implementation to lists and I doubt any automatic algebra combiner in the near future could discover one. (The Eckmann-Hilton argument also appears here.)

The compatibility condition
To form the product of two theories we add in extra laws to ensure commutativity. If we don't add in such laws we get the sum of two theories. For the example theories I used here these theories can lead to quite complex types. For example the sum of the theory of -sets and -sets is, I think, the theory of -sets where is the "free product" of monoids. I this is a bit of a messy object from the perspective of types. Other effects, however, may behave nicely with respect to . I haven't yet investigated.

Conclusion
If you don't mind computing the relevant types by hand there are perfectly good alternative to monad transformers for combining effects. But it seems very difficult to automatically combine theories. In fact, I expect finding canonical forms for the elements of free algebras for a product theory isn't even computable. So this approach isn't going to replace monad transformers any time soon.

Exercise
Make a multiplication table showing the result of forming the product of algebras for lots of useful effects.

Lawvere theories made a bit easier

Introduction
I still don't think anyone has found a completely satisfactory way to combine effects in Haskell. (That's computational effects, not visual effects.) Monads are great for one kind of effect at a time, but it's not clear how to combine two arbitrary monads. Instead of monads we can work with monad transformers, but they are tricky both to implement and to use.

I want to sketch a different, though incomplete approach to combining effects. There are a bunch of papers that describe this approach, and even some code that implements part of it. Almost everything I say is from a paper by Hyland and Powers that I read a few years ago though I recently ran into this helpful answer by Andrej Bauer on mathoverflow. Even if we don't get code we can immediately use, we still get a good way to think about and analyse effects.

I'll get onto the effects part in another post. This one concentrates on what are known as Lawvere theories.

Monoids

> {-# LANGUAGE MultiParamTypeClasses, FlexibleInstances, FlexibleContexts, ExplicitForAll #-}
> import Data.Monoid

The (finite) lists of integers form a monoid. Here are some functions that operate on such lists:

> f1 xs = map (+1) $ reverse xs
> f2 xs = xs `mappend` xs

They can both be given signature [Integer] -> [Integer]. But there's one very important difference between these functions. f2 has been written using only operations from the type class Monoid. It's a sort of universal function that can be applied to any monoid. On the other hand, the function f1 can only applied to very specific monoids. In fact, the type signature of f2 can be written as:

> f2 :: forall a. Monoid a => a -> a

That type signature essentially says that we're not going to do anything with elements of a except use the interface defined by the Monoid type class.

(Although Haskell type classes can't enforce them, we also assume that any instance of Monoid satisfies the axioms for a monoid.)

We can also define functions on tuples of monoids. Here are some examples:

> g1 :: forall a. Monoid a => (a, a, a) -> (a, a)
> g1 (xs, ys, zs) = (mempty, ys `mappend` xs)

> g2 :: forall a. Monoid a => (a, a) -> (a, a)
> g2 (xs, ys) = (xs `mappend` ys, ys `mappend` xs)

Notice that we can compose these functions. So we have

g2 . g1 :: forall a. Monoid a => (a, a, a) -> (a, a)

We also have have identity functions for tuples. Armed with functions, identities and compositions can now form a category that I'll call . I'll call the (distinct) objects of this category , , and so on. The arrows from to are the total functions of type . So, for example, g1 is an arrow from to . Note that it doesn't matter what the objects are (as long as they're distinct). They're just placeholders between which we can string our arrows. Note how because of the universal quantifier, the functions we use have a type not of the form A -> B. So we can't represent the objects of our category as types in the usual way. We can think of mempty as a 0-ary operator, ie. an element of forall a. Monoid a => () -> a.

But there's one more detail I want. I'll consider two arrows to be equal if they can be proved equal using the axioms for monoids. For example, these two Haskell functions represent the same arrow in because of the associativity law:

> h1 (x, y, z) = (x `mappend` y) `mappend` z
> h2 (x, y, z) = x `mappend` (y `mappend` z)

We now have a bona fide category. Note that contains lots of arrows. Anything you can build using mempty and mappend as well as all of the projections and permutations between tuples. completely captures everything that can be deduced using the axioms for monoids. For example, the associativity law is contained in the fact that h1 and h2 represent the same arrow.

The category also has products. In fact it is given by with the projections back to the factors being represented by the obvious projection functions. serves as the product of no factors.

So captures the properties shared by all monoids. But what is its relationship to actual monoids? It's pretty nice. A monoid is a functor that preserves products.

Let's unpack that. First must take objects in to sets. But we've stipulated that so is completely determined on objects once we know . In fact will be the carrier for our monoid and is its th power. takes arrows in to functions on sets. So, for example, it gives concrete realisations of mempty and mappend. Because, for example, h1 and h2 represent the same arrow in , and must be equal. So associativity must hold for these realisations. The same goes for all of the other laws, and everything we can deduce from them. So the requirement of functoriality makes into a monoid with identity F(mempty) and product F(mappend).

Given an instance of the Monoid type class we can immediately apply an arrow from to it. The functor is applied implicitly by the Haskell compiler. For example h1 can be applied to ("a", "b", "c") :: (String, String, String).

The object in is weird. It's a lot like the universal monoid sharing all of the properties you expect to hold simultaneously in all monoids. Except for one important one: it's not a monoid itself.

Note that in a sense the category isn't anything new. It's just a convenient 'datastructure' into which we can pack everything we can deduce about monoids.

M-sets
Before generalising, let's try a similar treatment for another algebraic structure, the horribly named -set. An -set is a structure on a set that assumes we already have some choice of monoid . It defines an action of on the set. An action is simply a function defined for each element of and which is compatible with the operations on . Here's a suitable type class:

> class Monoid m => MSet m s where
> act :: m -> s -> s

and the laws are

act mempty x == x
act a (act b x) = act (a `mappend` b) x

We're thinking of act not as an operation on two arguments but instead thinking of act a being an operation on s for each element a. Note how that second law is really a lot of laws, one for every pair (a, b) in our monoid.

A simple example is the way scalars act on vectors.

> data Vector a = V a a a deriving Show
> instance Monoid a => MSet a (Vector a) where
>    act w (V x y z) = V (w `mappend` x) (w `mappend` y) (w `mappend` z)

Given any particular monoid (eg. String) we can define functions like:

> j1 :: forall a. MSet String a => (a, a) -> (a, a)
> j1 (x, y) = ("x" `act` x, "y" `act` y)
> j2 :: forall a. MSet String a => (a, a) -> a
> j2 (x, y) = "abc" `act` x

We can form a category in exactly the same way as for monoids. As the objects were just placeholders we may as well reuse them. The arrows from to are the functions on tuples we can make from repeated applications of act, ie. . For each choice of we get a new category encoding everything we want to know about -sets. In much the same way, any functor gives an -set. We have and maps act a to the actual action on the set.

Lawvere theories
So what do these two categories have in common?

Let's start with the simplest possible algebraic structure: a plain old set with no operations on it. The corresponding category will have objects . The arrows from to will be represented by functions of the form . That includes functions like the projections onto elements of tuples or permutations of tuples. This category is called .

Both and defined earlier contain all of the objects and arrows of . But both and also contain arrows corresponding to all of the operations you can perform in their respective algebraic structures. So we can define a Lawvere theory as nothing more than a category that is with extra arrows. The category defined earlier is the "theory of monoids" and is the "theory of -sets". A 'model' of , or a -algebra, is a product-preserving functor .

Product theories
Suppose we have two theories and . There's a straightforward way to combine them into one theory. Form the category that (like all Lawvere theories) shares the same objects as and but has all of the arrows from both. (Obviously we'll have to keep just one identity, not an identity from and another from .) The catch is that if is an arrow in and is an arrow in we'll need the composition too. We simply take the smallest set of arrows that contains all of the arrows from and and contains all of their compositions modulo all of the laws in and .

Thus far we already have a new Lawvere theory built from and . But sometimes it's useful to ensure the operations from are 'compatible', in some sense, with those from . We want them to commute with each other. I'll describe what that commutativity means now:

Suppose we have an arrows and on and respectively. If we have a set that is both a -algebra and a -algebra, then gives an operation and gives an operation . Write out an array of variables:

We can apply across the rows to get the array:

and now we can apply down the columns to get an array.

We could also have done this by applying down the columns first and then across the rows. We now throw in extra commutativity laws stating that these two operations give equal results, whatever the operations and .

For example, if the theory has a binary multiplication operator and the theory has binary multiplication operator then commutativity requires . This is the usual interchange law. In this example the commutativity law is therefore an assertion about the equality of two arrows in .

The result of combining the two theories and , additionally throwing in these commutativity laws, is known as the product theory .

Product theories in Haskell
Although we can't get Haskell to enforce the commutativity rules we can get part way to defining product theories using type classes. We make our type an instance of both classes. For and defined as the theories of monoids and -sets above, a -algebra is given by a type that is an instance of both Monoid and MSet. Unfortunately we can't make Haskell automatically enforce the rule that elements be equal if the laws say they should be.

Coming soon
We've defined Lawvere theories. I've explained previously how many kinds of effects can be modelled by algebraic theories. I've defined a product on Lawvere theories. Which means we now have a way to combine effects. But for that you'll have to wait for the sequel.