Proposal: Laws for mtl classes

[Härmel Nestra]

Hi people interested in mtl laws,

I have been studying axiomatization of MonadError for some time and 
recently published this:

Laws of Monadic Error Handling,
https://link.springer.com/chapter/10.1007%2F978-3-030-32505-3_21

Compared to the proposal posted here, my set of axioms also contains an 
asymmetry introduction law that enables one to switch the order of bind 
and catch (named Bnd-UnitCch or Bnd-Cch in the paper). It is somewhat 
similar to the (erroneous) law included in the initially posted proposal 
but removed later, however it is preserved also by the State transformer.

And of course the dual homomorphism law

bimap h id (m >>= k) = bimap h id m >>= bimap h id . k

(named Bnd-FunHom in the paper). In fact, the other homomorphism law 
turns out to be unnecessary (provable) in the presence of Bnd-UnitCch.

My approach is cheating in the sense that it allows catch to change the 
error type which is not the case in Haskell (Malakhovski does the same). 
Otherwise, laws such as Bnd-UnitCch would be impossible. Anyway, I 
consider this useful because this enables one to prove more facts also 
for the restricted case with fixed error type.

Regards,
--
Härmel Nestra
Lecturer, Institute of Computer Science (http://www.cs.ut.ee/eng)
University of Tartu
Phone +372 7375428
Mobile +372 53401682

On 20.10.19 22:25, Li-yao Xia wrote:
> Hello Libraries,
> 
> Some time ago I proposed some laws for the most common mtl classes 
> (State, Reader, Error): 
> https://mail.haskell.org/pipermail/libraries/2019-April/029549.html
> 
> To address some concerns about completeness I reorganized the laws into 
> groups describing more high-level properties, and formalized them in Coq 
> to ensure the laws are strong enough to lift themselves through common 
> transformers (StateT, ExceptT, ContT; ReaderT and WriterT are similar to 
> StateT, but formalizing that relationship also makes the laws interact 
> in interesting ways).
> 
> I'm open to suggestions for better ways to verify the "completeness" of 
> the laws. However, as long there aren't any objections to the existing 
> laws themselves, it still seems worth having some documentation earlier 
> rather than never.
> 
> In the end the actual changes to the initial proposal were quite minor. 
> Feel free to weigh in on the following pull requests:
> 
> https://github.com/haskell/mtl/pull/61 (MonadReader, MonadState)
> https://github.com/haskell/mtl/pull/62 (MonadError)
> 
> ---
> 
> # Changes
> 
> 1. Three laws were added,
> 
> In MonadReader,
> 
>  > local id = id
> 
> to complete the property that local is a monoid homomorphism (rather 
> than only semigroup); thanks to Andreas Abel for pointing out its 
> absence in the initial proposal.
> 
> 
>  > local f u     =   ask >>= \s -> local (\_ -> f s) u
> 
> This law was necessary to verify ContT's MonadReader instance. ContT is 
> not actually lawful, but a certain subset of it seems to be (the 
> elements of ContT that satisfy the commutativity of ask).
> 
> I did not manage to find out whether it's implied by the other laws.
> 
> 
> In MonadError, a naturality law
> 
>  > fmap f (catchError u h) = catchError (fmap f u) (fmap f . h)
> 
> whose need arose when describing the relationship between StateT's and 
> ReaderT's instances.
> 
> 
> 2. Another finding is the fact that, much like the laws of MonadState 
> are equivalent to saying that 'state' is a monad morphism, the 'ask' 
> fragment of MonadReader (which to many *is* MonadReader) can also be 
> characterized by a monad morphism, which cannot be 'reader', as it only 
> yields two of the 'ask' laws.
> 
> 
> 3. There were two mistakes in the original proposal.
> 
> One MonadError law was too strong:
> 
>  > catchError (m >>= k) h = tryError m >>= either h k   -- broken by StateT
> 
> And local should flip the order of composition
> 
>  > local g . local f = local (g . f)  -- wrong
>  > local g . local f = local (f . g)  -- right
> 
> ---
> 
> # Updated proposal
> 
> ## MonadState
> 
>  > get    >>= put    = pure ()
>  > put s  >>  get    = put s >> pure s
>  > put s1 >>  put s2 = put s2
> 
> Those three laws imply the following equations expressing that get has 
> no side effects:
> 
>  > get >> m   =   m
>  > get >>= \s1 -> get >>= \s2 -> k s1 s2   =   get >>= \s -> k s s
> 
> state must be equivalent to its default definition in terms of get and 
> put, and conversely. Under that last condition, a property which is 
> equivalent to the laws above is that state must be a monad morphism, 
> from State s to m.
> 
> ---
> 
> ## MonadReader
> 
> ask has no side effects, and produces the same result at any time.
> 
>  > ask >> m    =   m
>  > ask >>= \s1 -> ask >>= \s2 -> k s1 s2   =   ask >>= \s -> k s s
>  >
>  > m <*> ask   =   ask <**> m
> 
> local f applies f to the environment produced by ask.
> 
>  > local f ask   =   f <$> ask
>  > local f u     =   ask >>= \s -> local (\_ -> f s) u
> 
> local is a monoid morphism from (r -> r) to (reversed) (m a -> m a) 
> (i.e., (Endo r -> Dual (Endo (m a)))).
> 
>  > local id          = id
>  > local g . local f = local (f . g)
> 
> local is a monad morphism from m to m.
> 
>  > local f (pure x)   =  pure x
>  > local f (a >>= k)  =  local f a >>= \x -> local f (k x)
> 
> reader must be equivalent to its default definition in terms of ask, and 
> conversely.
> 
> Under that last condition, a property which is equivalent to the first 
> two laws is that reader must be a monad morphism from Reader r to m.
> 
> Another property equivalent to the first three laws is that there is a 
> monad morphism phi :: forall a. ReaderT r m a -> m a such that phi ask = 
> ask and phi . lift = id.
> 
> ---
> 
> ## MonadError
> 
> See also Exceptionally Monadic Error Handling, by Jan Malakhovski: 
> https://arxiv.org/abs/1810.13430
> 
> catchError and throwError form a monad, with (>>=) interpreted as 
> catchError and pure as throwError.
> 
>  > catchError (throwError e) h   = h e
>  > catchError m throwError       = m
>  > catchError (catchError m k) h = catchError m (\e -> catchError (k e) h)
> 
> pure and throwError are left zeros for catchError and (>>=) respectively.
> 
>  > catchError (pure a) h         = pure a
>  > throwError e >>= k            = throwError e
> 
> catchError commutes with fmap (it is a natual transformation).
> 
>  > fmap f (catchError u h)       = catchError (fmap f u) (fmap f . h)
> 
> 
> ---
> 
> Li-yao Xia
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