How to write a recursive descent parser from scratch?

Question

As a purely academic exercise, I'm writing a recursive descent parser from scratch -- without using ANTLR or lex/yacc.

I'm writing a simple function which converts math expressions into their equivalent AST. I have the following:

// grammar
type expr =
    | Lit of float
    | Add of expr * expr
    | Mul of expr * expr
    | Div of expr * expr
    | Sub of expr * expr

// tokens
type tokens =
    | Num of float
    | LParen | RParen
    | XPlus | XStar | XMinus | XSlash

let tokenize (input : string) =
    Regex.Matches(input.Replace(" ", ""), "\d+|[+/*\-()]")
    |> Seq.cast<Match>
    |> Seq.map (fun x -> x.Value)
    |> Seq.map (function
        | "+" -> XPlus
        | "-" -> XMinus
        | "/" -> XSlash
        | "*" -> XStar
        | "(" -> LParen
        | ")" -> RParen
        | num -> Num(float num))
    |> Seq.to_list

So, tokenize "10 * (4 + 5) - 1" returns the following token stream:

[Num 10.0; XStar; LParen; Num 4.0; XPlus; Num 5.0; RParen; XMinus; Num 1.0]

At this point, I'd like to map the token stream to its AST with respect to operator precedence:

Sub(
    Mul(
        Lit 10.0
        ,Add(Lit 4.0, Lit 5.0)
       )
    ,Lit 1.0
   )

However, I'm drawing a blank. I've never written a parser from scratch, and I don't know even in principle how to begin.

How do I convert a token stream its representative AST?

Answer 1

Do you know about language grammars?

Assuming yes, you have a grammar with rules along the lines

...
addTerm := mulTerm addOp addTerm
         | mulTerm

addOp   := XPlus | XMinus

mulTerm := litOrParen mulOp mulTerm
         | litOrParen
...

which ends up turning into code like (writing code in browser, never compiled)

let rec AddTerm() =
    let mulTerm = MulTerm() // will parse next mul term (error if fails to parse)
    match TryAddOp with  // peek ahead in token stream to try parse
    | None -> mulTerm    // next token was not prefix for addOp rule, stop here
    | Some(ao) ->        // did parse an addOp
         let rhsMulTerm = MulTerm()
         match ao with
         | XPlus -> Add(mulTerm, rhsMulTerm)
         | XMinus -> Sub(mulTerm, rhsMulTerm)
and TryAddOp() =
    let next = tokens.Peek() 
    match next with
    | XPlus | XMinus ->
        tokens.ConsumeNext()
        Some(next)
    | _ -> None
...

Hopefully you see the basic idea. This assumes a global mutable token stream that allows both 'peek at next token' and 'consume next token'.