Can you explain the concept of "Monadic Parsing" in Haskell? Give an example.?

Monadic parsing is a technique in Haskell (and other functional programming languages) for building complex parsers by composing simpler, more focused parsers together using monads. A parser is a function that takes some input (usually a string or list of tokens) and produces a structured output (such as abstract syntax trees) if the parsing is successful, or an error if the parsing fails.

Before diving into monadic parsing, let’s first understand what a monad is in Haskell. A monad is an abstraction that allows us to chain computations together while implicitly passing around some hidden context. The basic definition of the ‘Monad‘ type class in Haskell is as follows:

class Monad m where
  return :: a -> m a
  (>>=) :: m a -> (a -> m b) -> m b

Now, let’s see how monads can be used for parsing. Consider a simple parser ‘Parser‘ defined as follows:

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

Here, the ‘Parser‘ is a wrapper around a function that takes a ‘String‘ as input and returns a list of tuples, where each tuple contains a parsed value of type ‘a‘ and the remaining input ‘String‘. An empty list ‘[]‘ represents a parsing failure.

To make ‘Parser‘ an instance of the ‘Monad‘ type class, we need to provide implementations for ‘return‘ and ‘(>>=)‘:

instance Monad Parser where
  return a = Parser $ input -> [(a, input)]
  p >>= f = Parser $ input ->
    case parse p input of
      [] -> []
      [(a, remaining)] -> parse (f a) remaining

Now that we have a ‘Monad‘ instance for our ‘Parser‘, let’s create some basic parsers and see how to use monadic composition to create more complex parsers.

1. Item parser: This parser consumes one character from the input:

item :: Parser Char
item = Parser $ input -> case input of
  [] -> []
  (x:xs) -> [(x, xs)]

2. Filtering parser: This higher-order parser takes a predicate function ‘p :: a -> Bool‘ and a parser ‘m‘ and applies the predicate to the parsed value using the ‘guard‘ function from ‘Control.Monad‘.

import Control.Monad (guard)

sat :: (a -> Bool) -> Parser a -> Parser a
sat p m = do
  x <- m
  guard (p x)
  return x

Now let’s create a parser for digits using the aforementioned building blocks:

isDigit :: Char -> Bool
isDigit c = '0' <= c && c <= '9'

digit :: Parser Char
digit = sat isDigit item

To create a parser for signed integers, we can use the monadic composition:

integer :: Parser Int
integer = do
  sign <- optional (sat (== '-') item)
  digits <- some digit
  return $ (if sign == Just '-' then negate else id) (read digits)

Here, ‘optional‘ and ‘some‘ are helper functions from ‘Control.Applicative‘. In this example, we use the monadic ‘do‘ notation to glue together parsers for optional sign characters, followed by one or more digit characters. Finally, we convert the parsed digits to an integer, taking the optional sign into account.

To parse a simple arithmetic expression (e.g., ‘1 + 2 * 3‘) with support for parentheses, we can use the following:

data Expr
  = Val Int
  | Add Expr Expr
  | Mul Expr Expr
  deriving (Show, Eq)

expr :: Parser Expr
expr = chainl1 term (symbol "+" >> return Add)

term :: Parser Expr
term = chainl1 factor (symbol "*" >> return Mul)

factor :: Parser Expr
factor = integer
      <|> (symbol "(" *> expr <* symbol ")")

In this example, we use ‘chainl1‘ (a helper function from ‘Control.Monad‘) to compose parsers for expressions and terms, and monadic combinators like ‘(<|>)‘, ‘(*>)‘ and ‘(<*)‘ from ‘Control.Applicative‘.

To summarize, monadic parsing is a powerful way to build parsers in Haskell by combining smaller parsers using monadic composition. The ability to chain and glue together parsers while abstracting away error handling, backtracking, and other common parsing concerns makes it an elegant and expressive technique for writing parsers in a functional programming style.