Parsing

In this case study, we show how to implement a parser, using the lexer from the Lexer case study.


Again, we require some imports – in particular, we reuse the lexer implementation.

module examples/casestudies/parser

import examples/casestudies/lexer
import immutable/option
import text/string

Parsers can be expressed by using the lexer effect and process the token stream. To model different alternatives in the grammar, we use the following effect for non-determinism:

effect Nondet {
  def alt(): Boolean
  def fail[A](msg: String): A
}

effect Parser = { Nondet, Lexer }

Parser Combinators

Given these two effects, we can readily define a host of (imperative) parser combinators. We start by the simplest one, which applies a predicate to the next element in the input stream and fails, if it does not match.

def accept { p: Token => Boolean } : Token / Parser = {
  val got = next();
  if (p(got)) got
  else fail("Unexpected token " ++ show(got))
}

Using accept, we can define parsers for the different token types.

def any() = accept { t => true }
def accept(exp: TokenKind) = accept { t => t.kind == exp }
def ident() = accept(Ident()).text
def number() = accept(Number()).text
def punct(p: String) = {
  val tok = accept(Punct())
  if (tok.text == p) ()
  else fail("Expected " ++ p ++ " but got " ++ tok.text)
}
def kw(exp: String): Unit / Parser = {
  val got = ident();
  if (got == exp) ()
  else fail("Expected keyword " ++ exp ++ " but got " ++ got)
}

Using the effect for non-deterministic choice alt, we can model alternatives, optional matches and various repetitions:

def or[R] { p: R } { q: R } =
  if (alt()) { p() } else { q() }

def opt[R] { p: R }: Option[R] / Parser =
  or { Some(p()) } { None() }

def many { p: Unit }: Unit / Parser =
  or { some { p() } } { () }

def some { p: Unit }: Unit / Parser =
  { p(); many { p() } }

Example: A Simple Expression Language

We illustrate the usage of the above parser combinators by parsing a simple expressions language.

type Tree {
  Lit(value: Int)
  Var(name: String)
  Let(name: String, binding: Tree, body: Tree)
  App(name: String, arg: Tree)
}

Let us start by defining the parser for numeric literals.

def parseNum(): Tree / Parser = {
  val numText = number()
  val num = toInt(numText).getOrElse {
    fail("Expected number, but cannot convert input to integer: " ++ numText)
  }
  Lit(num)
}

We simply call the parser for number() and try to convert the resulting string to an intenger.

def parseVar(): Tree / Parser =
  Var(ident())

def parseAtom() = or { parseVar() } { parseNum() }

Parsing variables is simply a matter of reusing the ident parser and changing the result. Users of monadic parser combinator libraries might be delighted to see, that we do not need to use map or something similar to transform the result. The parser is simply written in direct style, still offering great flexibility in modifying the semantics of effects.

Similarly, we can write parsers for let bindings, by sequentially composing our existing parsers:

def parseLet(): Tree / Parser = {
  kw("let");
  val name = ident();
  punct("=");
  val binding = parseExpr();
  kw("in");
  val body = parseExpr();
  Let(name, binding, body)
}

Again, note how naturally the result can be composed from the individual results, much like manually writing a recursive descent parser. Compared to handcrafted parsers, the imperative parser combinators presented here offer a similar flexibility. At the same time, the semantics of alt and fail is still left open, offering flexibility in the implementation of the actual underlying parsing algorithm.

We proceed to implement the remaining parsers for our expression language:

def parseGroup() = or { parseAtom() } {
  punct("(");
  val res = parseExpr();
  punct(")");
  res
}

def parseApp(): Tree / Parser = {
  val funName = ident();
  punct("(");
  val arg = parseExpr();
  punct(")");
  App(funName, arg)
}

def parseExpr(): Tree / Parser =
  or { parseLet() } { or { parseApp() } { parseGroup() } }

Example: Combining Parsers and Local Mutable State

It is possible to combine the imperative parser combinators with local mutable state. The implementation of local variables in Effekt is designed to interact well with continuation capture and multiple resumptions. This is important for use cases like the parser example where the continuation is potentially called multiple times.

The following example implements an example

// <EXPR> ::= <NUMBER> | <IDENT> `(` <EXPR> (`,` <EXPR>)*  `)`

It uses local (mutable) variables to count the number of leafs as semantic action.

def parseCalls(): Int / Parser =
  or { number(); 1 } {
    var count = 1;
    ident();
    punct("(");
    count = count + parseCalls();
    many {
        punct(",");
        count = count + parseCalls()
    };
    punct(")");
    count
  }

Notice how the user defined combinator many feels like a built-in control operator while.

Backtracking Parsers

As mentioned above, so far we used the non-determinism effects, but did not specify what they mean. We could implement depth-first parsing corresponding to recursive descent. Alternatively, we also could implement breadth-first parsing. Further, in case of ambiguities, we have the choice whether we want to recognize only the first result or compute all possible alternative ways to recognize the input.

For this case study, we implement a depth-first parser, computing the first successful result. Results are represented by the following datatype:

type ParseResult[R] {
  Success(t: R);
  Failure(msg: String)
}

The parsing algorithm is simply implemented as a handler for Parser.

def parse[R](input: String) { p: R / Parser }: ParseResult[R] = try {
  lexer(input) { skipWhitespace { Success(p()) } }
} with Nondet {
  def alt() = resume(true) match {
    case Failure(msg) => resume(false)
    case Success(res) => Success(res)
  }
  def fail(msg) = Failure(msg)
} with LexerError { (msg, pos) =>
  Failure(msg)
}

The handler reuses the lexer implementation of the Lexer case study. The lexer raises a LexerError in case of an unexpected enf of the input stream or if it cannot recognize a token. Those lexer errors are simply converted into failures, which the parser can backtrack. To establish the backtracking behavior, it is important that the lexer is executed under the parser handler and not the other way around. Only this way the lexer positions will be restored when calling the continuation a second time with resume(false).

Running the Examples

Having implemented a handler for the Parser effect, we can run our example “grammars” on some inputs.

def main() = {
  println(parse("42") { parseCalls() })
  println(parse("foo(1)") { parseCalls() })
  println(parse("foo(1, 2)") { parseCalls() })
  println(parse("foo(1, 2, 3, 4)") { parseCalls() })
  println(parse("foo(1, 2, bar(4, 5))") { parseCalls() })
  println(parse("foo(1, 2,\nbar(4, 5))") { parseCalls() })

  println(parse("}42") { parseExpr() })
  println(parse("42") { parseExpr() })
  println(parse("let x = 4 in 42") { parseExpr() })
  println(parse("let x = let y = 2 in 1 in 42") { parseExpr() })
  println(parse("let x = (let y = 2 in 1) in 42") { parseExpr() })
  println(parse("let x = (let y = f(42) in 1) in 42") { parseExpr() })
  println(parse("let x = (let y = f(let z = 1 in z) in 1) in 42") { parseExpr() })
}