Re: [Haskell-cafe] Do type classes have a partial order?
It seems like you would, going by semantics of System F, where types with type variables name a certain subset of types, = constraints further restrict the types of the same shape (are they an independent kind of restriction?), so typeclass declarations with/without = specify a partial order over types because the subset relation is. On Mon, Nov 14, 2011 at 3:47 AM, Patrick Browne patrick.bro...@dit.ie wrote: Is there a partial order on Haskell type classes? If so, does it induce any quasi-order relation on types named in the instances? In the example below types C and D have the same operation f Thanks, Pat data C = C deriving Show data D = D deriving Show class A t where f::t-t f t = t instance A C where instance A D where class A t = B t where instance B C where instance B D where ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Wondering if this could be done.
Haskell does not play as well with overloading as one would do it in C++; every name used must be fully qualified. Indeed, if we try something like Indeed, if we try something like data A = A Int deriving (Show, Eq) test = A 3 unA (A i) = i class Group a where (+) :: a - a - a instance Group A where (+) x y = A $ unA x + unA y we will get Ambiguous occurrence `+' It could refer to either `Main.+', defined at .hs:7:1 or `Prelude.+', imported from Prelude Failed, modules loaded: none. Haskell has its own brand of 'overloading': type classes. Every (+) sign used assumes that the operands are of the Num typeclass in particular. In order to define (+) on something else you will need to instance the Num typeclass over your A type. I am not sure what you mean by the stuff defined in class Num is meanless to A. Strictly speaking nothing needs to be defined in a typeclass declaration other than the required type signatures. To instance the Num typeclass with A, though, assuming that A constructors take something that works with Num, you would do something similar to what Miguel posted: data A = A Int deriving (Show, Eq) test = A 3 unA (A i) = i instance Num A where (+) x y = A $ (unA x) + (unA y) (-) x y = A $ (unA x) - (unA y) (*) x y = A $ (unA x) * (unA y) abs x = A $ (unA $ abs x) signum y = A $ (unA $ signum y) fromInteger i = A (fromInteger i) Look at fromInteger, which must take Integer as as argument. That may be inconvenient for you. The Awesome Prelude, referenced in Chris's post, is a way of defining less specific version of basic types like Bool so that you have more choices in defining things like fromInteger in the Num typeclass (which must take an Integer; it is 'sad' if that Integer refers to a grounded, specific type). Still, if not every one of the Num operations make sense for your A type, you can leave them blank and get a warning. On Sun, Nov 21, 2010 at 10:48 PM, Magicloud Magiclouds magicloud.magiclo...@gmail.com wrote: Hi, For example, I have a data A defined. Then I want to add (+) and (-) operators to it, as a sugar (compared to addA/minusA). But * or other stuff defined in class Num is meanless to A. So I just do: (+) :: A - A - A (+) a b = A (elem1 a + elem1 b) (elem2 a + elem2 b) -- I got errors here, for the (+) is ambiguous. So, just wondering, does this way work in Haskell? -- 竹密岂妨流水过 山高哪阻野云飞 ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] Re: Making monadic code more concise
Thank you for highlighting these problems; I should really test my own code
more thoroughly. After reading these most recent examples, the translation
to
existing monads is definitely too neat, and a lot of semantics of the monad
are
'defaulted' on. In particular for the probability monad examples I see I had
the mistaken impression that it would preserve the random-world semantics of
the do-notation whereas autolifting actually imposes a random-evaluation
semantics, which would not be how I envision an autolifted probabilistic
DSL.
Overall, I think I pretty much got caught up in how cool it was going to be
to
use $, *, join/enter/exit as primitives to make any monad-based DSL work
'concisely' in an environment of existing typeclasses. That is kind of the
thing I want to do at a high level; implement DSLs like probabilistic
programming languages as primitives that that play transparently with
existing
expressions.
But now it seems clear to me that this autolifting approach will not be
useful
with any monad where it is important to control sharing and effects, which
is
critical in the probability monad (and all others I can think of); in fact
it
seems necessary to incur the do-notation 'overhead' (which I now see is not
overhead at all!) to work with them meaningfully at all, no matter how
'pure'
they look in other settings. Because of this we see that the Prob monad as
it
is defined here is mostly unusable with autolifting. Again, thanks for the
examples; I think I now have a much better intuition for do/bind and why
they
are required.
At this point, however, I would still want to see if it is possible to do
the
autolifting in a more useful manner, so that the user still has some control
over effects. Things like the share combinator in the paper you linked will
probably be very useful. I will definitely go over it in detail.
From my previous experience however, this might also be accomplished by
inserting something between the autolifting and the target monad.
I think it would be more helpful now to talk more about where I'm coming
from.
Indeed, the probability monad examples feature heavily here because I'm
coming
off of implementing a probabilistic programming language in Python that
worked
through autolifting, so expressions in it looked like host language
expressions. It preserved the random-world semantics because it was using a
quote-like applicative functor to turn a function composition in the
language
into an expression-tree rep of the same. I am not sure yet how to express it
in
Haskell (as I need to get more comfortable with GADTs), but pure would take
a
term to an abstract version of it, and fmap would take a function and
abstract
term to an abstract term representing the answer. One would then have the
call
graph available after doing this on lifted functions. I think of this as an
automatic way of performing the 'polymorphic embedding' referenced in
[Hofer et al 2008] Polymorphic Embedding of DSLs.
By keeping IDs on different abstract terms, expressions like X + X (where X
=
coin 0.5) would take the proper distributions under random-world semantics.
In general for monads where the control of sharing is important, it can be
seen
as limiting the re-running of effects to one per name. Each occurence of a
name
using the same unwrapped value.
I had the initial impression, now corrected, that I could just come up with
an
autolifting scheme in Haskell, use it with the usual probability monad and
somehow get this random-world semantics for free. No; control of sharing and
effects is in fact critical, but could be done through using the autolifting
as
a way to turn expressions into a form where control of them is possible.
For now, though, it looks like I have a lot of things to read through.
Again, thanks Oleg and everyone else for all the constructive feedback. This
definitely sets a personal record for misconceptions corrected / ideas
clarified per day.
On Wed, Nov 17, 2010 at 12:08 AM, o...@okmij.org wrote:
Let me point out another concern with autolifting: it makes it easy to
overlook sharing. In the pure world, sharing is unobservable; not so
when effects are involved.
Let me take the example using the code posted in your first message:
t1 = let x = 1 + 2 in x + x
The term t1 is polymorphic and can be evaluated as an Int or as a
distribution over Int:
t1r = t1 ::Int-- 6
t1p = t1 ::Prob Int -- Prob {getDist = [(6,1.0)]}
That looks wonderful. In fact, too wonderful. Suppose later on we
modify the code to add a non-trivial choice:
t2 = let x = coin 0.5 + 1 in x + x
-- Prob {getDist = [(4,0.25),(3,0.25),(3,0.25),(2,0.25)]}
The result isn't probably what one expected. Here, x is a shared
computation rather than a shared value. Therefore, in (x + x)
the two occurrences of 'x' correspond to two _independent_ coin flips.
Errors like that are insidious and very difficult to find. There are
no overt problems, no exceptions are thrown, and the
[Haskell-cafe] Re: Making monadic code more concise
Thanks Oleg for the feedback. If I understand right, there is a hard (as in computability) limit for automatic instancing due to the general requirement of deriving functions off of the behavior of another. At this point, I wonder: How good are we at doing this? There's languages that reside in the more expressive corners of the lambda cube, such as Epigram. Some of the concepts have been translated to Haskell, such as Djinn. Are only 'trivial' results possible, or that the incomputability problems are just moved into type space? In any case, it's a good reason to limit the scope of autolifting. On Tue, Nov 16, 2010 at 2:49 AM, o...@okmij.org wrote: Ling Yang wrote I think a good question as a starting point is whether it's possible to do this 'monadic instance transformation' for any typeclass, and whether or not we were lucky to have been able to instance Num so easily (as Num, Fractional can just be seen as algebras over some base type plus a coercion function, making them unusually easy to lift if most typeclasses actually don't fit this description). In general, what this seems to be is a transformation on functions that also depends explicitly on type signatures. For example in Num: Things start to break down when we get to the higher order. In the first order, it is indeed easy to see that the monadification of the term Int - Int - Int should/could be m Int - m Int - m Int Indeed, liftM2 realizes this transformation. But what about (Int - Int) - Int ? Should it be (m Int - m Int) - m Int ? A moment of thought shows that there is no total function of the type Monad m = ((Int - Int) - Int) - ((m Int - m Int) - m Int) because there is no way, in general, to get from (m Int - m Int) to the pure function Int-Int. That is, we can't write Monad m = (m Int - m Int) - m (Int-Int) One may try tabulation (for finite domains) tf f = do vt - f (return True) vf - f (return False) return $ \x - if x then vt else vf but that doesn't quite work: what if the resulting function is never invoked on the True argument. We have needlessly computed that value, vt. Worse, we have incurred the effect of computing vt; that effect could be failure. We have needlessly failed. One can say: we need runnable class (Monad m) = Runnable m where exit : m a - a are there many monads that are members of that typeclass? For example, Maybe cannot be Runnable; otherwise, what is (exit Nothing)? Any Error or MonadPlus monad can't be Runnable. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] Making monadic code more concise
Hi,
I'm fairly new to Haskell and recently came across some programming
tricks for reducing monadic overhead, and am wondering what
higher-level concepts they map to. It would be great to get some
pointers to related work.
Background:
I'm a graduate student whose research interests include methods for
implementing domain specific languages. Recently, I have been trying
to get more familiar with Haskell and implementing DSLs in it. I'm
coming from having a fair bit of experience in Python so I know the
basics of functional programming.
However, I'm much less familiar with Haskell. In particular I have
little to no internal map from existing DSL implementation techniques
to the Haskell extensions that would no doubt make DSL implementations
easier (and when they are *not* needed).
I also don't have a complete picture of the functional programming
research that would inform these techniques. I would greatly
appreciate it if I could get pointers to the appropriate references so
I can really get going on this.
Specifically: There are some DSLs that can be largely expressed as monads,
that inherently play nicely with expressions on non-monadic values.
We'd like to use the functions that already work on the non-monadic
values for monadic values without calls to liftM all over the place.
The probability monad is a good example.
import Control.Monad
import Data.List
newtype Prob a = Prob { getDist :: [(a, Float)] } deriving (Eq, Show)
multiply :: Prob (Prob a) - Prob a
multiply (Prob xs) = Prob $ concat $ map multAll xs
where multAll (Prob innerxs, p) = map (\(x, r) - (x, p * r)) innerxs
instance Functor Prob where
fmap f (Prob xs) = Prob $ map (\(x, p) - (f x, p)) xs
instance Monad Prob where
return x = Prob [(x, 1.0)]
x = f = multiply (fmap f x)
In this monad, = hides the multiplying-out of conditional
probabilities that happen during the composition of a random variable
with a conditional distribution.
coin x = Prob [(1, x), (0, 1 - x)]
test = do
x - (coin 0.5)
y - (coin 0.5)
return $ x + y
*Main test
Prob {getDist = [(2,0.25),(1,0.25),(1,0.25),(0,0.25)]}
We can use a 'sum out' function to get more useful results:
sumOut :: (Ord a) = Prob a - Prob a
sumOut (Prob xs) = Prob $ map (\kvs - foldr1 sumTwoPoints kvs) eqValues
where
eqValues = groupBy (\x y - (fst x == fst y)) $ sortBy compare
xs
sumTwoPoints (v1, p1) (v2, p2) = (v1, p1 + p2)
*Main sumOut test
Prob {getDist = [(0,0.25),(1,0.5),(2,0.25)]}
I'm interested in shortening the description of 'test', as it is
really just a 'formal addition' of random variables. One can use liftM
for that:
test = liftM2 (+) (coin 0.5) (coin 0.5)
It seems what I'm leading into here is making functions on ordinary
values polymorphic over their monadic versions; I think this is the
desire for 'autolifting' or 'monadification' that has been mentioned
in works such as HaRE
http://www.haskell.org/pipermail/haskell/2005-March/015557.html
One alternate way of doing this, however, is instancing the
typeclasses of the ordinary values with their monadic versions:
instance (Num a) = Num (Prob a) where
(+) = liftM2 (+)
(*) = liftM2 (*)
abs = liftM abs
signum = liftM signum
fromInteger = return . fromInteger
instance (Fractional a) = Fractional (Prob a) where
fromRational = return . fromRational
(/) = liftM2 (/)
Note that already, even though each function in the typeclass had to
be manually lifted, this eliminates more overhead compared to lifting
every function used, because any function with a general enough type
bound can work with both monadic and non-monadic values, not just the
ones in the typeclass:
*Main sumOut $ (coin 0.5) + (coin 0.5) + (coin 0.5)
Prob {getDist = [(0,0.125),(1,0.375),(2,0.375),(3,0.125)]}
*Main let foo x y z = (x + y) * z
*Main sumOut $ foo (coin 0.5) (coin 0.5) (coin 0.5)
Prob {getDist = [(0,0.625),(1,0.25),(2,0.125)]}
Because of the implementation of fromInteger as return . fromInteger,
we also 'luck out' and have the ability to mix ordinary and
non-monadic values in the same expression:
*Main 1 + coin 0.5 / 2
Prob {getDist = [(1.5,0.5),(1.0,0.5)]}
My question is, what are the higher-level concepts at play here? The
motivation is that it should be possible to automatically do this
typeclass instancing, letting us get the benefits of concise monadic
expressions without manually instancing the typeclasses.
Indeed, I didn't have this idea in Haskell; I'm coming from Python
where one can realize the automatic instances: if we take the view
that classes in Python are combined datatypes and instanced
typeclasses, we can use the meta-object protocol to look inside one
class's representation and output another class with liftM-ed (or
return . -ed) methods and a custom constructor. I realize Template
Haskell gives you the ability to reify instance/class declarations,
but perhaps there's a
Re: [Haskell-cafe] Making monadic code more concise
See my reply to Alex's post for my perspective on how this relates to applicative functors, reproduced here: This, to me, is a big hint that applicative functors could be useful. Indeed, the ideas here also apply to applicative functors; it is just the lifting primitives that will be different; instead of having liftMN, we can use $ and * to lift the functions. We could have done this for Num and Maybe (suppose Maybe is an instance of Applicative): instance (Num a) = Num (Maybe a) where (+) = \x y - (+) $ x * y (-) = \x y - (-) $ x * y (*) = \x y - (+) $ x * y abs = abs $ signum = signum $ fromInteger = pure . fromInteger The larger goal remains the same: autolifting in a principled manner. However, you actually bring up a very good point; what if it is really only the applicative functors that this method works on in general, that there is no 'use case' for considering this autolifting for monads in particular? I think the answer lies in the fact that monads can be 'flattened;' that is, realizations of the type m (m a) - m a are mechanical (in the form of 'join') given that = is defined. This is important when we have a typeclass that also has monadic signatures. To be more concrete, consider how this function could be used in a 'monadic DSL': enter x = case x of 0 - Nothing _ - Just hi The type of 'enter' is one case of the general from 'a - M b'. If we were instancing a typeclass that had an 'a - M b' function, we'd need a function of type 'M a - M b'. This would be accomplished by enter' = join . liftM enter So the set of lifting primitives must include at least some way to get M a - M b from 'a - M b'---which requires that M is a monad, not just an applicative functor. Thanks for the mention of applicative functors; I should have included them in the original post. Lingfeng Yang lyang at cs dot stanford dot edu I should have included a mention of Applicative in my original post. Part of the reason Num was so easy is that all the functions produce values whose type is the class parameter. Your Num instance could almost be completely generic for any ((Applicative f, Num a) = f a), except that Num demands instances of Eq and Show, neither of which can be blindly lifted the way the numeric operations can. I imagine it should be fairly obvious why you can't write a non-trivial generic instance (Show a) = Show (M a) that would work for any possible monad M--you'd need a function (show :: M a - String) which is impossible for abstract types like IO, as well as function types like the State monad. The same applies to (==), of course. Trivial instances are always possible, e.g. show _ = [not showable], but then you don't get sensible behavior when a non-trivial instance does exist, such as for Maybe or []. Good point. This is where we can start defining restrictions for when this automatic lifting can or cannot take place. I reference the concept of 'runnable monads' here, from [Erwig and Ren 2004] Monadification of Functional Programs A 'runnable monad' is a monad with an exit function: class (Monad m) = Runnable m where exit : m a - a And yes, for monads like IO, no one would really have a need for 'exit' outside of the cases where they need unsafePerformIO. However, for Maybe and Prob, 'exit' is extremely useful. In fact, in the probability monad, if you could not exit the monad, you could not get anything done, as the real use is around sampling and computing probabilities, which are of non-monadic types. Provided M is a runnable monad, class (Show a) = Show (M a) where show = show . exit I'm aware of the limitations of this approach; I just want to come up with a set of primitives that characterize the cases where autolifting/monadic instancing is useful. On Mon, Nov 15, 2010 at 11:19 AM, C. McCann c...@uptoisomorphism.net wrote: On Mon, Nov 15, 2010 at 12:43 PM, Ling Yang ly...@cs.stanford.edu wrote: Specifically: There are some DSLs that can be largely expressed as monads, that inherently play nicely with expressions on non-monadic values. We'd like to use the functions that already work on the non-monadic values for monadic values without calls to liftM all over the place. It's worth noting that using liftM is possibly the worst possible way to do this, aesthetically speaking. To start with, liftM is just fmap with a gratuitous Monad constraint added on top. Any instance of Monad can (and should) also be an instance of Functor, and if the instances aren't buggy, then liftM f = (= return . f) = fmap f. Additionally, in many cases readability is improved by using ($), an operator synonym for fmap, found in Control.Applicative, I believe. The probability monad is a good example. [snip] I'm interested in shortening the description of 'test', as it is really just a 'formal addition' of random variables. One can use liftM for that: test = liftM2 (+) (coin 0.5
