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Adventure in parserland – parsing lambda expressions in F# – Part IV

Let’ now look at the parser. First let’s review the grammar:

        <expression> ::= <name> | <function> | <application>
        <name> ::= non­blank character sequence
        <function> ::= \ <name> . <body>
        <body> ::= <expression>
        <application> ::= ( <function expression> <argument expression> )
        <function expression> ::= <expression>
        <argument expression> ::= <expression>

And the data type to represent it:

type Name = string
and Body = Expression
and Function = Name * Expression
and FunctionExpression = Expression
and ArgumentExpression = Expression
and Expression =
| EName of string
| Function of Expression * Body
| Application of FunctionExpression * ArgumentExpression

In essence, the data type need to store all the information needed for subsequent stages of computation (i.e. beta reductions and such). The closer it is to the grammar, the better. In this case it looks pretty close.

Remember what is the main goal of our parser:

let parseTextReader: TextReader -> seq<Expression> =
                    textReaderToLazyList >> tokenStream >> parseExpressions

We have already looked at TextReaderToLazyList and tokenStream. Now it is the time to look at parseExpressions. It’s goal is to  parse the LazyList and return a sequence of expressions. The choice of returning a sequence at this point is to make the parseTextReader, which is the main function in the program, return a more ‘standard’ type.

and parseExpressions tokens = seq {
   let tokens = parseOptionalWs tokens
   let expr, tokens = parseExpr tokens
   let tokens = parseOptionalWs tokens
   match expr with
    | EOT   -> yield EOT
    | exp   -> yield exp; yield! parseExpressions tokens }

parseOtionalWs simply skips ahead whatever whitespaces it finds.

and parseOptionalWs tokens = match tokens with
                                | LazyList.Nil -> LazyList.empty
                                | LazyList.Cons(h, t) ->
                                    match h with
                                       | Ws _ -> parseOptionalWs t
                                       | _ -> tokens

parseExpr is more interesting. It is the main switch that creates expression kinds.

let rec parseExpr tokens = match tokens with
                            | LazyList.Nil -> EOT, LazyList.empty
                            | LazyList.Cons(h, t) ->
                                match h with
                                    | EOF -> parseEOF tokens
                                    | Name _ -> parseName  tokens
                                    | Lambda -> parseFunction tokens
                                    | OpenParens -> parseApplication tokens
                                    | token -> errorAtStart "Expression" token

parseEOF is not.

and parseEOF tokens = EOT, LazyList.empty

parseName just returns a EName, unwrapping it from Name.

and parseName tokens = EName (head tokens |> unwrapName), tail tokens

Unwrap just unwraps it.

let unwrapName = function
    | Name(s) -> s
    | tok -> errorExpecting "a Name" <| writeToken tok

parseFunction just conumes a Lambda, a name, a Dot token, a body (i.e. \x.x)and assembles them in a Function:

and parseFunction tokens =
    let tokens = consumeToken Lambda tokens
    let name, tokens = parseName tokens
    let tokens = consumeToken Dot tokens
    let body, tokens = parseExpr tokens
    Function(name, body), tokens

consumeToken tries to consume a token generating an error if it doesn’t find it:

let consumeToken token =
    genericConsumeToken (fun token' _ -> errorExpecting (writeToken token') (writeToken token)) token

genericConsumeToken is just a generalization of the function above:

let genericConsumeToken noMatch token = function
    | LazyList.Nil -> LazyList.empty
    | LazyList.Cons(h, t) as originalTokens ->
        match h with
        | tok when tok = token -> t
        | tok -> noMatch token originalTokens

The last thing left to consume is an application which is in this form (func args):

and parseApplication tokens =
    let tokens = consumeToken OpenParens tokens
    let funExpr, tokens = parseExpr tokens
    let tokens = parseOptionalWs tokens
    let argExpr, tokens = parseExpr tokens
    let tokens = consumeToken CloseParens tokens
    Application(funExpr, argExpr), tokens

Various error and utility functions are defined below:

let errorEOF expecting = failwith  ("Expected " + expecting + ", got EOF")
let errorExpecting expecting gotToken = failwith ("Expected " + expecting + ", got" + gotToken)
let errorAtStart expecting gotToken = failwith ("Expected " + expecting + " which cannot start with" + writeToken gotToken)
let tail = LazyList.tail
let head = LazyList.head

And that is the parser. All 100+ lines of it. As you can tell it is rather formulaic to go from a grammar to a lexer and a parser, which is why you shouldn’t do it, but instead let a tool generate the code for you given the grammar or use FParsec.

We have written 200+ code and I don’t think we can be too proud of our achievement. It is:

So let’s look next at a better way to do it.