1. Implementing Recursion Efficiently¶

We next consider how to implement recursive functions in SLang2. In the lambda calculus and SLang1, all functions are anonymous and cannot call themselves. We could use a fixed-point combinator, like the \(Y\) combinator.

let
     Y = fn (h) => (fn (x) => (h (x x)) fn (x) => (h (x x)))
     f = fn (g) => fn (n) => if (n === 0) then 1 else (n * (g (n - 1)))
in
     ((Y f) 5)
end

What is the problem with this approach?

In SLang2, we can take advantage of references and the assignment statement to implement recursion in an efficient way with a technique called "tying the knot."

To get a sense of how this technique works, ask yourself what is the value of the following program?

let
     dummyClosure = fn (n) => (n + 1)
in
     let
           f = fn (n) => if (n===0) then 1 else (n * (dummyClosure (n - 1)))
     in
           (f 5)
     end
end

How can we modify this program to turn the function f above into the (recursive) function that we know under the name factorial so that the value of the program above is 120?

Hint: add a single assignment statement, but which one? and where? Answer these questions and you will see why this technique is called "tying the knot".

2. Practice TTK¶

This problem will help you get comfortable with the TTK technique. To earn credit for it, you must complete this randomized problem correctly three times in a row.

When you provide your answer, remember to include the full denoted values, for example [ "Num", 0 ] and not just 0.