[Haskell-cafe] Semantic Domain, Function,
and denotational model.....
Conal Elliott
conal at conal.net
Tue Sep 16 09:47:31 EDT 2008
Hi Ryan,
Thanks very much for these explanations. Clear and right on!
Best regards, - Conal
P.S. I'll be at ICFP and am looking forward to seeing folks there.
2008/9/16 Ryan Ingram <ryani.spam at gmail.com>
> The key insight is that Behavior a is not necessarily a time function;
> it's abstract. But you can treat it as if it was one by observing it
> with "at".
>
> In Conal's paper, the internal type of behavior is:
>
> > -- composition of types; like (.) at the type level
> > newtype O h g a = O (h (g a))
>
> > -- function type that can directly observe some constant functions
> > data Fun t a = K a | Fun (t -> a)
>
> > -- Behavior a ~~ Reactive (Fun Time a)
> > type Behavior = Reactive `O` Fun Time
>
> > -- Reactive has a current value and an event stream of values to switch
> to at particular times
> > -- Then an event is just a reactive that might not have a current value
> until some time in the future.
> > data Reactive a = Stepper a (Event a)
> > newtype Event a = Ev (Future (Reactive a))
>
> Now, at the internal level, you can write the primitive "time" as
>
> > time :: Behavior Time
> > time = O (pure (Fun id))
>
> with "pure" from the Applicative instance for Reactive:
>
> > pure x = Stepper x never
>
> "never" is a future that never occurs, so the reactive value never changes.
>
> From a users' point of view, all this is invisible--you only get a few
> observation functions (including "at"). Internally, however, constant
> behaviors, or behaviors that contain "steps" that are constant, can be
> evaluated extremely quickly; once the behavior returns K x, you know
> that the result can't change until the next event in the reactive
> stream. You only need to continuously evaluate the behavior if you
> get a "Fun" result. See sinkB on page 9 of the paper to understand
> how this is used to improve performance.
>
> The semantic function "at" drives the model; it allows you to describe
> the laws for the library to fulfill very succinctly:
>
> at (fmap f x) = fmap f (at x)
> at (pure x) = pure x
> at (f <*> x) = at f <*> at x
> at (return x) = return x
> at (m >>= f) = at m >>= at . f
> etc.
>
> Similarily, for Futures, we have "force :: Future a -> (Time, a)"
>
> force (fmap f z) = (t, f x) where (t,x) = force z
> force (pure x) = (minBound, x)
> force (ff <*> fx) = (max tf tx, f x) where (tf, f) = force ff ; (tx,
> x) = force fx
> force (return x) = (minBound, x)
> force (m >>= f) = (max tm tx, x) where (tm, v) = force m; (tx, x) = force
> (f v)
> etc.
>
> This gives the library user solid ground to stand on when reasoning
> about their code; it should do what they expect. And it gives the
> library author a very strong goal to shoot for: just fulfill these
> laws, and the code is correct! This allows radical redesigns of the
> internals of the system while maintaining a consistent and intuitive
> interface that reuses several classes that the user is hopefully
> already familiar with: monoids, functors, applicative functors, and
> monads.
>
> -- ryan
>
> 2008/9/16 Daryoush Mehrtash <dmehrtash at gmail.com>:
> > ّ I don't follow the "at" and "type B a". "Behavior a" itself is a
> > time function. At least in the version of the code that was
> > developed in Pual Hudak's Haskell School of Expression it was defined
> > as:
> >
> >> newtype Behavior a
> >> = Behavior (([Maybe UserAction],[Time]) -> [a])
> >
> > In a function like time you can see that the "at" function makes things
> simpler.
> >
> > In the original version time was defined as:
> >
> >> time :: Behavior Time
> >> time = Behavior (\(_,ts) -> ts)
> >
> > In Conal's paper
> >
> > time :: Behavior Time
> > at time = id
> >
> > Comparing the two implementation of the time, it seems to me that "at"
> > and "type B a" has put the design on a more solid ground. But I don't
> > quite understand the thought process, or the principal behind what is
> > happening.
> >
> > daryoush
> >
> >
> > On Mon, Sep 15, 2008 at 10:46 AM, Ryan Ingram <ryani.spam at gmail.com>
> wrote:
> >> Here's a quick overview that might help you.
> >>
> >> For a reactive behavior, we have two types to think about:
> >>
> >> type B a = Time -> a
> >> (the semantic domain)
> >>
> >> data Behavior a = ?
> >> (the library's implementation).
> >> at :: Behavior a -> B a
> >> (observation function)
> >>
> >> This is really just classic "information hiding" as you would do with
> >> any abstract data type. Consider a simple "stack" data structure that
> >> supports push and pop.
> >>
> >>> data S a = S
> >>> { popS :: Maybe (a, S a)
> >>> , pushS :: a -> S a
> >>> }
> >>
> >>> data Stack a = ?
> >>> observeStack :: Stack a -> S a
> >>
> >> As a library user, you don't really care about the implementation of
> >> Stack, just as a user of Conal's library doesn't really care about the
> >> implementation of Behavior. What you *do* care about is that you can
> >> think about it in the simpler terms of "Time -> a" which is the model
> >> he has chosen.
> >>
> >> The rest of the library design comes from taking that model and
> >> thinking about what typeclasses and operations "Time -> a" should
> >> support, and creating typeclass morphisms between Behavior a and B a
> >> where necessary. For example:
> >>
> >>> -- This makes (r -> a) into a functor over a; it is a generalization of
> Time -> a
> >>> instance Functor ((->) r) where
> >>> -- fmap :: (a -> b) -> (r -> a) -> (r -> b)
> >>> fmap f x = \r -> f (x r)
> >>> -- or, "fmap = (.)", if you're golfing :)
> >>
> >> In order for the morphism between B and Behavior to make sense, you
> >> want this law to hold:
> >> fmap f (at behavior) = at (fmap f behavior)
> >> for all behavior :: Behavior a.
> >>
> >> The fmap on the left applies to B which is (Time ->); the fmap on the
> >> right applies to Behavior.
> >>
> >> Conal writes this law more elegantly like this:
> >>> instance(semantic) Functor Behavior where
> >>> fmap f . at = at . fmap f
> >>
> >> As long as you as the user can think about behaviors generally as
> >> functions of Time, you can ignore the implementation details, and
> >> things that you expect to work should work. This drives the design of
> >> the entire library, with similar morphisms over many typeclasses
> >> between Event and E, Reactive and B, etc.
> >>
> >> -- ryan
> >>
> >> On Mon, Sep 15, 2008 at 10:13 AM, Daryoush Mehrtash <
> dmehrtash at gmail.com> wrote:
> >>> Interestingly, I was trying to read his paper when I realized that I
> >>> needed to figure out the meaning of denotational model, semantic
> >>> domain, semantic functions. Other Haskell books didn't talk about
> >>> design in those terms, but obviously for him this is how he is driving
> >>> his design. I am looking for a simpler tutorial, text book like
> >>> reference on the topic.
> >>>
> >>> Daryoush
> >>>
> >>> On Mon, Sep 15, 2008 at 1:33 AM, Ryan Ingram <ryani.spam at gmail.com>
> wrote:
> >>>> I recommend reading Conal Elliott's "Efficient Functional Reactivity"
> >>>> paper for an in-depth real-world example.
> >>>>
> >>>> http://www.conal.net/papers/simply-reactive
> >>>>
> >>>> -- ryan
> >>>>
> >>>> On Sun, Sep 14, 2008 at 11:31 AM, Daryoush Mehrtash <
> dmehrtash at gmail.com> wrote:
> >>>>> I have been told that for a Haskell/Functional programmer the process
> >>>>> of design starts with defining Semantic Domain, Function, and
> >>>>> denotational model of the problem. I have done some googling on the
> >>>>> topic but haven't found a good reference on it. I would appreciate
> >>>>> any good references on the topic.
> >>>>>
> >>>>> thanks,
> >>>>>
> >>>>> daryoush
> >>>>>
> >>>>> ps. I have found referneces like
> >>>>> http://en.wikibooks.org/wiki/Haskell/Denotational_semantics which
> >>>>> talks about semantic domain for "the Haskell programs 10, 9+1, 2*5"
> >>>>> which doesn't do any good for me. I need something with a more
> real
> >>>>> examples.
> >>>>> _______________________________________________
> >>>>> Haskell-Cafe mailing list
> >>>>> Haskell-Cafe at haskell.org
> >>>>> http://www.haskell.org/mailman/listinfo/haskell-cafe
> >>>>>
> >>>>
> >>>
> >>
> >
>
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