How to implement monadic parsing?

Question

I'm stuck on understanding how Monad works. I'm writing a Parser by using Grammar S->aSb. Input is "a^n b^n for n>=0", e.g "aabb"; Return value for this Parser is a Boolean Value.

Previously by using temporary variables for this Grammar. Now I want to implement this parser by using Monad instead of temporary Variables. But i tried many times und still stuck on this problem.

object Parser {
  case class Parser[+A](parse: List[Char] => Option[(A, List[Char])]) {
    def accepts(l: List[Char]): Boolean
    // If List[Char] is empty after parsing it was successful
    = parse(l).map(_._2.isEmpty).getOrElse(false)

    // Not sure that this exactly does
    def flatMap[B](f: A => Parser[B]): Parser[B] = Parser {
      input => parse(input).flatMap { case (x, midput) => f(x).parse(midput) }
    }

    // a.map( i => i + 1) parses using Parser a and applies function i+1
    def map[B](f: A => B): Parser[B] = Parser {
      input => parse(input) match {
        case Some((t, remains)) => Some((f(t), remains))
        case _ => None
      }
    }

    // a.orElse(b) tries to use Parser a first
    // If a returns None it tries Parser b
    // If b also returns None all rules are exhausted
    def orElse[B](that: Parser[B]): Parser[Either[A, B]] = Parser {
      input => parse(input) match {
        case Some((p, remains)) => Some((Left(p), remains))
        case None => that.parse(input) match {
          case Some((p, remains)) => Some((Right(p), remains))
          case None => None
        }
      }
    }
  }

  // Not sure if this is correct
  def unit[T](x: T): Parser[T] = Parser {
    _ => Some((x, List()))
  }

  // Consumes c if possible
  def char(c: Char): Parser[Unit] = Parser {
    case x :: rest if c == x => Some(((), rest))
    case _ => None
  }

  def main(args: Array[String]): Unit = {
    val S: Parser[Int]
    = char('a').flatMap { _ =>
      S.flatMap { i =>
        char('b').map { _ =>
          i + 1
        }
      }
    }.orElse(unit(0)).map(_.merge)

    S.accepts("".toList) // true
    S.accepts("aaabbb".toList) // true
    S.accepts("aaa".toList) // false
    S.accepts("bbbaaa".toList) // false
  }
}

Answer 1

Usually, when we say "monadic parsing", we mean making Parser a monad. We write

class Parser[+A] { ... }

A Parser[A] takes input and returns the parsed A, or maybe it fails, or maybe there will be some input left over. Let's keep it really simple: a Parser[A] takes a List[Char], and Optionally returns an A and the remaining List[Char].

case class Parser[+A](parse: List[Char] => Option[(A, List[Char])]) {
  def accepts(l: List[Char]): Boolean
    = parse(l).map(_._2.isEmpty).getOrElse(false)
  // do not bother with the List('#') stuff
}

You build up a Parser by using combinators. a.flatMap(b) is a parser that matches a followed by b

// case class Parser[+A](...) {
  def flatMap[B](f: A => Parser[B]): Parser[B] = Parser { input =>
    parse(input).flatMap { case (x, midput) => f(x).parse(midput) }
  }
// }

and Parser.unit(x) returns x without consuming any input, which is why Monad is important. You should also have map, which alters the returned value without changing what's matched. You also need a combinator for alternation. I'll leave those for you to implement.

object Parser {
    def unit[T](x: T): Parser[T] = ???
}
// case class Parser[+A](...) {
    def map[B](f: A => B): Parser[B] = ???

    // left-biased greedy: if this parser succeeds (produces Some) then
    // that parser is never tried (i.e. no backtracking)
    // replacing Option with Seq is the easiest way to get backtracking
    // but we don't need it to define S
    def orElse[B](that: Parser[B]): Parser[Either[A, B]] = ???
// }

You also want some basic Parsers to build more complicated ones from. Parser.char(x) matches the single char x and returns nothing useful.

// object Parser {
  def char(c: Char): Parser[Unit] = Parser {
    case x :: rest if c == x => Some(((), rest))
    case _ => None
  }
// }

Then you can define S in a pretty natural manner. You can even make the parser return an Int for how many as/how many bs were matched:

lazy val S: Parser[Int]
  = (for { _ <- Parser.char('a')
           i <- S
           _ <- Parser.char('b')
         } yield (i + 1)).orElse(Parser.unit(0)).map(_.merge)
// i.e
lazy val S: Parser[Int]
  = Parser.char('a').flatMap { _ =>
      S.flatMap { i =>
        Parser.char('b').map { _ =>
          i + 1
        }
      }
    }.orElse(Parser.unit(0)).map(_.merge)

S.accepts("".toList) // true
S.accepts("aaabbb".toList) // true
S.accepts("aaa".toList) // false
S.accepts("bbbaaa".toList) // false

You do not have to move the List[Char] around in the definition of S, because the combinators we've written do that for you, leaving behind only the logic of the grammar itself.