Rewrite rules

I like rewrite rules. They are simple to understand and easy to implement. If you have a rewrite rule (or a set of them) that makes incremental progress in making an expression "simpler" in some sense, you can keep applying it (or them) repeatedly until you have finished the calculation.

If a function directly returns the result of calling another function, we say that the call is in "tail position". We can view this as a simple rewrite rule. Consider the fact-mul function which computes the factorial of a number n and multiplies it by another number m:

fact-mul (n, m) = n! * m

We can define two rewrite rules for this function. If n is 0, then the factorial is 1, so we just rewrite this as m. If n is not 0, we can rewrite this as fact-mul (n-1, n*m). This translates directly into Lisp:

(defun fact-mul (n m)
  (if (zerop n)
      m
     (fact-mul (- n 1) (* n m))))

The rewrite rule doesn't have to rewrite into the same function. For instance, we can rewrite odd? (x) to even? (x-1) and even? (x) to odd? (x-1):

(defun odd? (x)
  (if (zerop x)
      nil
      (even? (1- x))))

(defun even? (x)
  (if (zerop x)
      t  
      (odd? (1- x))))

Yes, this is bog-simple tail recursion, but somehow it seems even simpler to me if I think of it as a rewrite rule.

Even if you don't have tail recursion, or if you have disabled it, it still works, but the number of rewrites is limited by the stack depth.