title	date	synopsys
You could have invented Parsec	August 17, 2016 01:29 AM	2

As most of us should know, Parsec is a relatively fast, lightweight monadic parser combinator library.

In this post I aim to show that monadic parsing is not only useful, but a simple concept to grok.

We shall implement a simple parsing library with instances of common typeclasses of the domain, such as Monad, Functor and Applicative, and some example combinators to show how powerful this abstraction really is.

---

Getting the buzzwords out of the way, being monadic just means that Parsers instances of Monad{.haskell}. Recall the Monad typeclass, as defined in Control.Monad{.haskell},

class Applicative m => Monad m where
  return :: a -> m a
  (>>=)  :: m a -> (a -> m b) -> m b
  {- Some fields omitted -}

How can we fit a parser in the above constraints? To answer that, we must first define what a parser is.

A naïve implementation of the Parser{.haskell} type would be a simple type synonym.

type Parser a = String -> (a, String)

This just defines that a parser is a function from a string to a result pair with the parsed value and the resulting stream. This would mean that parsers are just state transformers, and if we define it as a synonym for the existing mtl State{.haskell} monad, we get the Monad, Functor and Applicative instances for free! But alas, this will not do.

Apart from modeling the state transformation that a parser expresses, we need a way to represent failure. You already know that Maybe a{.haskell} expresses failure, so we could try something like this:

type Parser a = String -> Maybe (a, String)

But, as you might have guessed, this is not the optimal representation either: Maybe{.haskell} does model failure, but in a way that is lacking. It can only express that a computation was successful or that it failed, not why it failed. We need a way to fail with an error message. That is, the Either{.haskell} monad.

type Parser e a = String -> Either e (a, String)

Notice how we have the Maybe{.haskell} and Either{.haskell} outside the tuple, so that when an error happens we stop parsing immediately. We could instead have them inside the tuple for better error reporting, but that's out of scope for a simple blag post.

This is pretty close to the optimal representation, but there are still some warts things to address: String{.haskell} is a bad representation for textual data, so ideally you'd have your own Stream{.haskell} class that has instances for things such as Text{.haskell}, ByteString{.haskell} and String{.haskell}.

One issue, however, is more glaring: You can't define typeclass instances for type synonyms! The fix, however, is simple: make Parser{.haskell} a newtype.

newtype Parser a
  = Parser { parse :: String -> Either String (a, String) }

---

Now that that's out of the way, we can actually get around to instancing some typeclasses.

Since the AMP landed in GHC 7.10 (base 4.8), the hierarchy of the Monad typeclass is as follows:

class Functor (m :: * -> *) where
class Functor m     => Applicative m where
class Applicative m => Monad m where

That is, we need to implement Functor and Applicative before we can actually implement Monad.

We shall also add an Alternative{.haskell} instance for expressing choice.

First we need some utility functions, such as runParser{.haskell}, that runs a parser from a given stream.

runParser :: Parser a -> String -> Either String a
runParser (Parser p) s = fst <$> p s

We could also use function for modifying error messages. For convenience, we make this an infix operator, <?>{.haskell}.

(<?>) :: Parser a -> String -> Parser a
(Parser p) <?> err = Parser go where
  go s = case p s of
    Left _ -> Left err
    Right x -> return x
infixl 2 <?>

Functor

Remember that Functor models something that can be mapped over (technically, fmap-ed over).

We need to define semantics for fmap on Parsers. A sane implementation would only map over the result, and keeping errors the same. This is a homomorphism, and follows the Functor laws.

However, since we can't modify a function in place, we need to return a new parser that applies the given function after the parsing is done.

instance Functor Parser where
  fn `fmap` (Parser p) = Parser go where
    go st = case p st of
      Left e            -> Left e
      Right (res, str') -> Right (fn res, str')

Applicative

While Functor is something that can be mapped over, Applicative defines semantics for applying a function inside a context to something inside a context.

The Applicative class is defined as

class Functor m => Applicative m where
  pure  :: a -> m a
  (<*>) :: f (a -> b) -> f a -> f b

Notice how the pure{.haskell} and the return{.haskell} methods are equivalent, so we only have to implement one of them.

Let's go over this by parts.

instance Applicative Parser where
  pure x = Parser $ \str -> Right (x, str)

The pure{.haskell} function leaves the stream untouched, and sets the result to the given value.

The (<*>){.haskell} function needs to to evaluate and parse the left-hand side to get the in-context function to apply it.

  (Parser p) <*> (Parser p') = Parser go where
    go st = case p st of
      Left e -> Left e
      Right (fn, st') -> case p' st' of
        Left e' -> Left e'
        Right (v, st'') -> Right (fn v, st'')

Alternative

Since the only superclass of Alternative is Applicative, we can instance it without a Monad instance defined. We do, however, need an import of Control.Applicative{.haskell}.

instance Alternative Parser where
  empty = Parser $ \_ -> Left "empty parser"
  (Parser p) <|> (Parser p') = Parser go where
    go st = case p st of
      Left _  -> p' st
      Right x -> Right x

Monad

After almost a thousand words, one would be excused for forgetting we're implementing a monadic parser combinator library. That means, we need an instance of the Monad{.haskell} typeclass.

Since we have an instance of Applicative, we don't need an implementation of return: it is equivalent to pure, save for the class constraint.

instance Monad Parser where
  return = pure

The (>>=){.haskell} implementation, however, needs a bit more thought. Its type signature is

(>>=) :: m a -> (a -> m b) -> m b

That means we need to extract a value from the Parser monad and apply it to the given function, producing a new Parser.

  (Parser p) >>= f = Parser go where
    go s = case p s of
      Left e -> Left e
      Right (x, s') -> parse (f x) s'

While some people think that the fail{.haskell} is not supposed to be in the Monad typeclass, we do need an implementation for when pattern matching fails. It is also convenient to use fail{.haskell} for the parsing action that returns an error with a given message.

  fail m = Parser $ \_ -> Left m

---

We now have a Parser{.haskell} monad, that expresses a parsing action. But, a parser library is no good when actual parsing is made harder than easier. To make parsing easier, we define combinators, functions that modify a parser in one way or another.

But first, we should get some parsing functions.

any, satisfying

any is the parsing action that pops a character off the stream and returns that. It does no further parsing at all.

any :: Parser Char
any = Parser go where
  go []     = Left "any: end of file"
  go (x:xs) = Right (x,xs)

satisfying tests the parsed value against a function of type Char -> Bool{.haskell} before deciding if it's successful or a failure.

satisfy :: (Char -> Bool) -> Parser Char
satisfy f = d
  x <- any
  if f x
    then return x
    else fail "satisfy: does not satisfy"

We use the fail{.haskell} function defined above to represent failure.

oneOf, char

These functions are defined in terms of satisfying, and parse individual characters.

char :: Char -> Parser Char
char c = satisfy (c ==) <?> "char: expected literal " ++ [c]

oneOf :: String -> Parser Char
oneOf s = satisfy (`elem` s) <?> "oneOf: expected one of '" ++ s ++ "'"

string

This parser parses a sequence of characters, in order.

string :: String -> Parser String
string [] = return []
string (x:xs) = do
  char   x
  string xs
  return $ x:xs

---

And that's it! In a few hundred lines, we have built a working parser combinator library with Functor, Applicative, Alternative, and Monad instances. While it's not as complex or featureful as Parsec in any way, it is powerful enough to define grammars for simple languages.

A transcription (with syntax highlighting) of this file is available as runnable Haskell. The transcription also features some extra combinators for use.