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

As most of us should know, [Parsec](https://hackage.haskell.org/package/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},

```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.

```haskell
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:

```haskell
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.

```haskell
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.

```haskell
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:

```haskell
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.

```haskell
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}.

```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.

```haskell
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

```haskell
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.

```haskell
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.

```haskell
  (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}.

```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.

```haskell
instance Monad Parser where
  return = pure
```


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

```haskell
(>>=) :: 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.

```haskell
  (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.

```haskell
  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.

```haskell
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.

```haskell
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.

```haskell
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.

```haskell
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](/static/Parser.hs) ([with syntax
highlighting](/static/Parser.hs.html)) of this file is available as runnable
Haskell. The transcription also features some extra combinators for use.