I have a type of regular expressions:
data RegExp = Empty
| Letter Char
| Plus RegExp RegExp
| Cat RegExp RegExp
| Star RegExp
and a type of (deterministic) finite state machines that is
polymorphic in the state type:
data FSM a = FSM {
states :: [a],
start :: a,
finals :: [a],
delta :: [(a,Char,a)]
}
I've fixed an alphabet sigma :: [Char], and I want to write the
function that converts a regular expression to its associated FSM. The
machines associated with Empty and Letter are given by
data EmptyFSM = Etrap
emptyFSM :: FSM EmptyFSM
emptyFSM = FSM {
states = [Etrap],
start = Etrap,
finals = [],
delta = [(Etrap, c, Etrap) | c <- sigma]
}
data LetterFSM = Lstart | Lfinal | Ltrap
letterFSM :: Char -> FSM LetterFSM
letterFSM c = FSM {
states = [Lstart, Lfinal, Ltrap],
start = Lstart,
finals = [Lfinal],
delta = [(Lstart, c', if c' == c then Lfinal else Ltrap) | c' <- sigma] ++
[(q, c', Ltrap) | q <- [Lfinal, Ltrap], c' <- sigma]
}
Suppose I can code the constructions of the union machine,
concatenation machine, and star machine so that they have the types
unionFSM :: FSM a -> FSM b -> FSM (a,b)
catFSM :: FSM a -> FSM b -> FSM (a,[b])
starFSM :: FSM a -> FSM [a]
Now what I want to do is to put all of this together into a function
that takes a regular expression and returns the associated FSM. In
effect, my function should have a dependent type like
reg2fsm :: {r : RegExp} -> FSM (f r)
where f is the function
f :: RegExp -> *
f Empty = EmptyFSM
f (Letter a) = LetterFSM
f (Plus r1 r2) = (f r1, f r2)
f (Cat r1 r2) = (f r1, [f r2])
f (Star r) = [f r]
and be given by
reg2fsm Empty = emptyFSM
reg2fsm (Letter c) = letterFSM c
reg2fsm (Plus r1 r2) = unionFSM (reg2fsm r1) (reg2fsm r2)
reg2fsm (Cat r1 r2) = catFSM (reg2fsm r1) (reg2fsm r2)
reg2fsm (Star r) = starFSM (reg2fsm r)
What is the suggested approach to achieving this in Haskell?
--
Todd Wilson A smile is not an individual
Computer Science Department product; it is a co-product.
California State University, Fresno -- Thich Nhat Hanh
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Sorry, I will not answer your question, but discuss some points.
On Mon, Sep 26, 2016 at 7:48 AM, Todd Wilson <twi...@csufresno.edu> wrote:
> Suppose I can code the constructions of the union machine,
> concatenation machine, and star machine so that they have the types
>
> unionFSM :: FSM a -> FSM b -> FSM (a,b)
> catFSM :: FSM a -> FSM b -> FSM (a,[b])
> starFSM :: FSM a -> FSM [a]
What is the type of the union of an union ? For example :
union fsmA (union fsmB fsmC)
where fsmX are of type FSM x.
Are you waiting for a triple (i.e. FSM (a, b, c)) or does a recursive
tuple structures is ok ? (Such as FSM (a, (b, c))) such as your
example depict ?
> Now what I want to do is to put all of this together into a function
> that takes a regular expression and returns the associated FSM. In
> effect, my function should have a dependent type like
>
> reg2fsm :: {r : RegExp} -> FSM (f r)
As far as I understand depend type in Haskell, you cannot do that.
You need to "pack" your "typed" FSM inside a existentially qualified
generic fsm, such as :
data AnyFSM = forall t. AnyFsm (FSM t)
And then, later, "unpack" it using a case. However I think you also
need a constraint on `t` else you will not be able to recover anything
from it.
Sorry, my knowledge stops here.
> where f is the function
>
> f :: RegExp -> *
> f Empty = EmptyFSM
> f (Letter a) = LetterFSM
> f (Plus r1 r2) = (f r1, f r2)
> f (Cat r1 r2) = (f r1, [f r2])
> f (Star r) = [f r]
As far as I know, and I'll be happy to be proven wrong here, you
cannot write a function from a value to a type. However you can
"promote" your value level RegExp to a type level 'RegExp (using the
DataKind extension), and in this case, you will need to use tick in
from of you promoted kind (such as 'Empty, or 'Plus), but as far as I
know it is still impossible to write function this way, you need to
write type families, something such as :
type family f (t :: RegexExp) = * where
f 'Empty = EmptyFSM
f ('Letter a) = LetterFSM
I apologies for this partial answer, but I'm interested by your
question so I wanted to contribute at my level.
--
Guillaume.
This seems like a neat place to use GADTs!
Instead of trying to define f :: RegExp -> * by fully lifting RegExp to
the type level, you can use GADTs to reflect the structure of a RegExp
in its type:
data RegExp a where
-- a is the state type of the FSM associated with the regexp
Empty :: RegExp EmptyFSM
Letter :: RegExp LetterFSM
Plus :: RegExp a -> RegExp b -> RegExp (a, b)
Cat :: RegExp a -> RegExp b -> RegExp (a, [b])
Star :: RegExp a -> RegExp [a]
reg2fsm :: Eq a => RegExp a -> FSM a
-- ^ I assume you'll need an Eq constraint here to normalize
-- the "non-deterministic states".
It may be much less flexible than what you originally intended, because
the RegExp type is now tied to your FSM implementation. RegExps might
also become painful to pass around because of their too explicit type.
If that happens to be the case, you can clean up an API which uses this
regexp with an existential wrapper.
data SomeRegExp where
SomeRegExp :: Eq a => RegExp a -> SomeRegExp
data SomeFSM where
SomeFSM :: Eq a => FSM a -> SomeFSM
Another simpler solution is to only use such a wrapper around FSM (i.e.,
your output), keeping the RegExp type as you first wrote it. You'd have:
reg2fsm :: RegExp -> SomeFSM
Here you totally lose the relationship between the RegExp and the FSM,
so it seems to move away from what you were looking for. But if
ultimately you don't care about the actual type of the state of a FSM,
this gets the job done.
Regards,
Li-yao
--Todd
Thanks for this suggestion. I wasn't aware of the singletons library;
I'll have to take a look.
To you and the other readers of this thread: are there other
approaches that have not yet been suggested? Since I'm using this for
a series of assignments for students that have seen Haskell before but
are still relative beginners, I'm looking for something that will be
relatively easy to explain, won't get in the way of them writing the
constructions in the straightforward way, and yet will provide some
additional type-checking to help catch their errors.
--Todd
> -=-=-=-=-=-=-=-=-=-=-
> Richard A. Eisenberg
> Asst. Prof. of Computer Science
> Bryn Mawr College
> Bryn Mawr, PA, USA
> cs.brynmawr.edu/~rae
I’m afraid dependent types in today’s Haskell are not for the faint-of-heart, or any but the most advanced students. Indeed, as I’ve been thinking about my new course on typed functional programming for the spring, I’ve debated on whether I should skip Haskell and go straight to Idris/Agda because of this. I won’t do that, I’ve decided, but this means I have to give up on some of the dependent-type material I’d like to cover.
To be fair, *some* slices of dependently typed programming are accessible. Stephanie Weirich’s red-black tree GADT has much of the flavor of dependently typed programming without as much of the Haskell-induced awkwardness. There is a decent description of this example in https://themonadreader.files.wordpress.com/2013/08/issue221.pdf
I hope this helps!
Richard
-=-=-=-=-=-=-=-=-=-=-
Richard A. Eisenberg
Asst. Prof. of Computer Science
Bryn Mawr College
Bryn Mawr, PA, USA
cs.brynmawr.edu/~rae