Conal Elliott has been working for several years now on using categories, specifically cartesian closed category, as a way to abstract Haskell functions at compile-time, so you can render the resulting “categorical term” into other categories.

Here’s an example Haskell function:

`\x -> f x (g x)`

And here’s its categorical rendering, just to give the flavor of the idea:

`eval ∘ (f' △ g')`

Where `eval`

means `uncurry ($)`

, and `f'`

and `g'`

are the renderings of those two functions; and the `△`

operator is `(&&&)`

. I’m not using the typical Haskell names for these, by the way, in order to convince myself not to “think in Haskell” when working with these terms, but rather I’m choosing whatever symbols I find most often using in the literature on catgeory theory.

There are a few things to notice about these categorical terms:

They must be point-free. There is no such thing as naming a term, only morphisms that use or produce objects. Hence Awodey calls category theory “the algebra of functions”.

They quickly become very large and unreadable. All but the simplest terms are nearly impossible to understand just by looking at them. Think of it as the binary code for categories.

Because they are just, in effect, chains of composition, without any name binding or scoping issue to consider, the nature of the computation is laid out in a very direct (albeit verbose) way, making rewrite rules available throughout the abstract term.

Although it seems a bit technical at first, the idea is quite simple: Discern the abstract, categorical meaning of a Haskell function, then realize that term in any other category that is cartesian (has products) and closed (has functions as objects, i.e., higher-order constructions). Nothing else needs to be known about the target category for the abstract term to have meaning there. That’s the beauty of using category theory as a universal language for expressing ideas: the meaning transports everywhere.

Here’s an equation meant for the solver, written in plain Haskell:

```
equation :: (Num a, Ord a) => a -> a -> Bool
equation x y =
x < y &&
y < 100 &&
0 <= x - 3 + 7 * y &&
(x == y || y + 20 == x + 30)
```

Here’s how I run the solver, using z3cat, which is built on top of Conal’s concat library:

```
mres <- liftIO $ runZ3 (ccc (uncurry (equation @Int))) $ do
x <- mkFreshIntVar "x"
y <- mkFreshIntVar "y"
return $ PairE (PrimE x) (PrimE y)
case mres of
Nothing -> error "No solution found."
Just sol -> putStrLn $ "Solution: " ++ show sol
```

And the result, also showing the equation submitted to Z3:

```
(let ((a!1 (ite (<= 0 (+ (- x!0 3) (* 7 y!1)))
(ite (= x!0 y!1) true (= (+ y!1 20) (+ x!0 30)))
false)))
(ite (< x!0 y!1) (ite (< y!1 100) a!1 false) false))
Solution: [-8,2]
```

Now with one function, I have either a predicate function I can use in Haskell, or an input for Z3 to find arguments for which it is true!

Note that the typical approach of deeply embedding a DSL, and writing evaluators, requires those evaluators to know the intimate details of both semantic domains. What the categorical compilation approach does is to introduce the *lingua franca* of category theory, so that both domains only have to agree on a common abstract language. This means that every time you write a new set of instances, you’ve automatically created an interpreter for every input domain renderable into CCCs. It’s like a toolbox whose power multiplies with each new set of instances.

In addition to using Conal’s work in Haskell, I’m also working on a Coq rendering of his idea, which I hope will give me a more principled way to extract Coq programs into Haskell, by way of their categorical representation.