Rinse and Recurse

[Part 4 of the FScheme series ]  

Recursive ‘let’

Normal ‘let’ can’t be used to bind names to recursive expressions (ones referring back to the names) because the expressions end up being evaluated in the calling environment before it’s been extended with the names themselves. It’s something of a chicken and egg problem. There are a few ways to solve it but the technique I’ve seen often (in SICP, LispKit, Lisp in Small Pieces, and this Bill Hails book we’re following) is to:

  1. Create a dummy environment with names but unusable values
  2. Eval expressions in this dummy environment
  3. Swap out the dummy values with these actual results

This trick works as long as the expressions don’t actually crack open the dummy values. That is exactly the case with ‘lambda’. Remember it creates a new function; capturing the environment but not actually using it until the function is called. So we’ll make a new ‘letrec’ for binding recursive lambda expressions.

and LetRec env = function
    | [List(bindings); body] – >
        let dummy = Function(fun _ -> failwith "Cannot evaluate dummy values.")
        let bind = function List([Symbol(s); _]) -> s, ref dummy | _ -> failwith "Malformed 'letrec' binding."
        let env' = List.map bind bindings |> extend env
// now update dummy env - assumes dummy env will be captured but not actually accessed (e.g. lambda)
        let update = function List([Symbol(s); e]) -> (env'.Head.Item s) := (eval env' e)
| _ -> failwith
"Malformed 'letrec' binding."
        List.iter update bindings
        eval env' body | _ -> failwith "Malformed LetRec."

Notice that in ‘update’ we have to do a destructive update (with :=) to swap out the dummy values! Maybe we’ll come back later and redo without resorting to mutation…

Mutable Environment

As much as I hate mutation, I think we need it to accomplish the swapping out in step 3 above. There are ways to do it without mutation but it’s by far the most straight forward and will come in handy when we later add ‘set!’ to the language itself. First we’ll go through and change the environment bindings to refs, patch up a couple places to expect that and go ahead and add our new ‘letrec’ to the global environment:

type Expression =
    | Number of BigInteger
    | String of string
    | Symbol of string
    | List of Expression list
    | Function of (Expression list -> Expression)
    | Special of (Map<string, Expression ref> list -> Expression list -> Expression)

and Let env = function
    | [List(bindings); body] –>
        let bind = function List([Symbol(s); e]) -> s, ref( eval env e ) | _ -> failwith "Malformed 'let' binding."
        let env' = List.map bind bindings |> extend env
        eval env' body
    | _ -> failwith "Malformed Let."

and Lambda env = function
    | [List(parameters); body] –>
        let closure env' args =
            // bind parameters to actual arguments (evaluated in the caller's environment)
            let bindings = List.zip parameters args
            let bind = function Symbol(p), a -> p, ref( eval env' a ) | _ -> failwith "Malformed 'lambda' parameter." 
            let env'' = List.map bind bindings |> extend env // extend the captured definition-time environment
        eval env'' body
    | _ -> failwith "Malformed Lambda."

and environment =
    extend [] [
        "*", ref( Function(Multiply) )
        "-", ref( Function(Subtract) )
        "if", ref( Special(If) )
        "let", ref( Special(Let) )
        "letrec", ref (Special(LetRec))
        "lambda", ref ( Special(Lambda) ) ]

and eval env expression =
    match expression with
    | Expression.Number(_) as lit –> lit
    | Expression.String(_) as lit –> lit
    | Expression.Symbol(s) -> ( lookup env s ).Value
    | Expression.List(h :: t) –>
        match eval env h with
        | Function(f) -> apply env f t
        | Special(f) -> f env t
        | _ -> failwith "Malformed expression."
    | _ -> failwith "Malformed expression."


All the old tests still pass. Just one new one:

    case "(letrec ((factorial (lambda (n) (if n (* n (factorial (- n 1))) 1)))) (factorial 4))" "24" // letrec and recursion