# Proving Equivalence of Polysemy Interpreters

Let’s talk more about polysemy-check. Last week we looked at how to do property-testing for a polysemy effects’ laws. Today, we’ll investigate how to show that two interpretations are equivalent.

To continue with last week’s example, let’s say we have an effect that corresponds to having a Stack that we can push and pop:

data Stack s m a where
Push      :: s -> Stack s m ()
Pop       :: Stack s m (Maybe s)
RemoveAll :: Stack s m ()
Size      :: Stack s m Int

deriving instance Show s => Show (Stack s m a)
deriveGenericK ''Stack

makeSem ''Stack

Since we’d like to prove the equivalence of two interpretations, we’ll need to first write two interpretations. But, to illustrate, we’re going simulate multiple interpreters via a single interpretation, parameterized by which bugs should be present in it.

purposes of brevity, we’ll write a single interpretation of Stack s in terms of State [s], and then interpret that in two different ways. In essence, what we’re really testing here is the equivalence of two State interpretations, but it’s good enough for an example.

data Bug
= PushTwice
| DontRemove
deriving stock (Eq, Ord, Show, Enum, Bounded)

instance Arbitrary Bug where
arbitrary = elements [minBound..maxBound]

hasBug :: [Bug] -> Bug -> Bool
hasBug = flip elem

The PushTwice bug, as you might expect, dispatched a Push command so that it pushes twice onto the stack. The DontRemove bug causes RemoveAll to be a no-op. Armed with our bugs, we can write a little interpreter for Stack that translates Stack s commands into State [s] commands, and then immediately runs the State effect:

runStack
:: [Bug]
-> Sem (Stack s ': r) a
-> Sem r ([s], a)
runStack bugs =
(runState [] .) $reinterpret$ \case
Push s -> do
modify (s :)
when (hasBug bugs PushTwice) $modify (s :) Pop -> do r <- gets listToMaybe modify (drop 1) pure r RemoveAll -> unless (hasBug bugs DontRemove)$
put []

Size ->
gets length

For our efforts we are rewarded: runState gives rise to four interpreters for the price of one. We can now ask whether or not these interpreters are equivalent. Enter propEquivalent:

With these interpreters out of the way, it’s time to answer our original question: are pureStack and ioStack equivalent? Which is to say, do they get the same answer for every possible program? Enter propEquivalent:

prepropEquivalent
:: forall effs r1 r2 f
. ( forall a. Show a => Show (f a)
, forall a. Eq a => Eq (f a)
)
=> ( Inject effs r1
, Inject effs r2
, Arbitrary (Sem effs Int)
)
=> (forall a. Sem r1 a -> IO (f a))
-> (forall a. Sem r2 a -> IO (f a))
-> Property

All of the functions in polysemy-check have fun type signatures like this one. But despite the preponderance of foralls, it’s not as terrible as you might think. The first ten lines here are just constraints. There are only two arguments to prepropEquivalent, and they are the two interpreters you’d like to test.

This type is crazy, and it will be beneficial to understand it. There are four type variables, three of which are effect rows. We can distinguish between them:

• effs: The effect(s) you’re interested in testing. In our case, our interpreter handles Stack s, so we let effs ~ Stack s.
• r1: The effects handled by interpreter 1. Imagine we had an interpreter for Stack s that ran it via IO instead. In that case, r1 ~ '[State s, Embed IO].
• r2 The effects handled by interpreter 2.

The relationships that must between effs, r1 and r2 are $effs \subset r1$ and $effs \subset r2$. When running prepropEquivalent, you must type-apply effs, because Haskell isn’t smart enough to figure it out for itself.

The other type variable to prepropEquivalent is f, which allows us to capture the “resulting state” of an interpreter. In runStack :: [Bug] -> Sem (Stack s ': r) a -> Sem r ([s], a), you’ll notice we transform a program returning a into one returning ([s], a), and thus f ~ (,) [s]. If your interpreter doesn’t produce any resulting state, feel free to let f ~ Identity.

We’re finally ready to test our interpreters! For any equivalence relationship, we should expect something to be equivalent to itself. And this is true regardless of which bugs we enable:

prop_reflexive :: Property
prop_reflexive = do
bugs <- arbitrary
pure $prepropEquivalent @'[Stack Int] (pure . run . runStack bugs) -- pure is getting us into IO (pure . run . runStack bugs) So what’s happening here? Internally, prepropEquivalent is generating random programs of type Sem '[Stack Int] Int, and lifting that into Sem r1 Int and Sem r2 Int, and then running both interpreters and ensuring the result is the same for every program. Note that this means any fundamental non-determinism in your interpretation will break the test! Make sure to use appropriate interpreters for things like clocks and random values! To strengthen our belief in prepropEquivalent, we can also check that runStack is not equivalent to itself if different bugs are enabled: prop_bugsNotEquivalent :: Property prop_bugsNotEquivalent = expectFailure$
prepropEquivalent @'[Stack Int]
(pure . run . runStack [PushTwice])
(pure . run . runStack [])

Running this test will give us output like:

+++ OK, failed as expected. Falsified (after 3 tests):
([0,0],1) /= (,1)

The counterexample here isn’t particularly helpful (I haven’t yet figured out how to show the generated program that fails,) but you can get a hint here by noticing that the stack (the [0,0]) is twice as big in the first result as in the second.

Importantly, by specifying @'[Stack Int] when calling prepropEquivalent, we are guaranteed that the generated program will only use actions from Stack Int, so it’s not too hard to track down. This is another win for polysemy in my book — that we can isolate bugs with this level of granularity, even if we can’t yet perfectly point to them.

All of today’s code (and more!) is available as a test in polysemy-check, if you’d like to play around with it. But that’s all for now. Next week we’ll investigate how to use polysemy-check to ensure that the composition of your effects themselves is meaningful. Until then!