Hi folks,
Inspired by Ralf's post, I thought I'd just GADTize a dependently typed
program I wrote in 2001.
Wolfgang Jeltsch wrote:
Now lets consider using an algebraic datatype for regexps:
data RegExp
= Empty | Single Char | RegExp :+: RegExp | RegExp :|: RegExpt
| Iter RegExp
Manipulating regular expressions now becomes easy and safe – you are just not
able to create "syntactically incorrect regular expressions" since during
runtime you don't deal with syntax at all.
A fancier variation on the same theme...
> data RegExp :: * -> * -> * where
> Zero :: RegExp tok Empty
> One :: RegExp tok ()
> Check :: (tok -> Bool) -> RegExp tok tok
> Plus :: RegExp tok a -> RegExp tok b -> RegExp tok (Either a b)
> Mult :: RegExp tok a -> RegExp tok b -> RegExp tok (a, b)
> Star :: RegExp tok a -> RegExp tok [a]
> data Empty
The intuition is that a RegExp tok output is a regular expression
explaining how to parse a list of tok as an output. Here, Zero is the
regexp which does not accept anything, One accepts just the empty
string, Plus is choice and Mult is sequential composition; Check lets
you decide whether you like a single token.
Regular expressions may be seen as an extended language of polynomials
with tokens for variables; this parser works by repeated application of
the remainder theorem.
> parse :: RegExp tok x -> [tok] -> Maybe x
> parse r [] = empty r
> parse r (t : ts) = case divide t r of
> Div q f -> return f `ap` parse q ts
Example
*RegExp> parse (Star (Mult (Star (Check (== 'a'))) (Star (Check (==
'b'))))) "abaabaaabbbb"
Just [("a","b"),("aa","b"),("aaa","bbbb")]
The 'remainder' explains if a regular expression accepts the empty
string, and if so, how. The Star case is a convenient
underapproximation, ruling out repeated empty values.
> empty :: RegExp tok a -> Maybe a
> empty Zero = mzero
> empty One = return ()
> empty (Check _) = mzero
> empty (Plus r1 r2) = (return Left `ap` empty r1) `mplus`
> (return Right `ap` empty r2)
> empty (Mult r1 r2) = return (,) `ap` empty r1 `ap` empty r2
> empty (Star _) = return []
The 'quotient' explains how to parse the tail of the list, and how to
recover the meaning of the whole list from the meaning of the tail.
> data Division tok x = forall y. Div (RegExp tok y) (y -> x)
Here's how it's done. I didn't expect to need scoped type variables, but
I did...
> divide :: tok -> RegExp tok x -> Division tok x
> divide t Zero = Div Zero naughtE
> divide t One = Div Zero naughtE
> divide t (Check p) | p t = Div One (const t)
> | otherwise = Div Zero naughtE
> divide t (Plus (r1 :: RegExp tok a) (r2 :: RegExp tok b)) =
> case (divide t r1, divide t r2) of
> (Div (q1 :: RegExp tok a') (f1 :: a' -> a),
> Div (q2 :: RegExp tok b') (f2 :: b' -> b)) ->
> Div (Plus q1 q2) (f1 +++ f2)
> divide t (Mult r1 r2) = case (empty r1, divide t r1, divide t r2) of
> (Nothing, Div q1 f1, _) -> Div (Mult q1 r2) (f1 *** id)
> (Just x1, Div q1 f1, Div q2 f2) ->
> Div (Plus (Mult q1 r2) q2) (either (f1 *** id) (((,) x1) . f2))
> divide t (Star r) = case (divide t r) of
> Div q f -> Div (Mult q (Star r)) (\ (y, xs) -> (f y : xs))
Bureaucracy.
> (***) :: (a -> b) -> (c -> d) -> (a, c) -> (b, d)
> (f *** g) (a, c) = (f a, g c)
> (+++) :: (a -> b) -> (c -> d) -> Either a c -> Either b d
> (f +++ g) (Left a) = Left (f a)
> (f +++ g) (Right c) = Right (g c)
> naughtE :: Empty -> x
> naughtE = undefined
It's not the most efficient parser in the world (doing some algebraic
simplification on the fly wouldn't hurt), but it shows the sort of stuff
you can do.
Have fun
Conor
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