11.3.8 No Deferment Solution

The solution to the problem of deferred operations is to write in a manner that does not defer operations¹⁵. This requires writing to a different pattern, often one that involves writing two function definitions, an initialization function and a helper function.

The initialization function sets up the job; the helper function does the work.

Here are the two function definitions for adding up numbers. They are so simple, I find them hard to understand.

(defun triangle-initialization (number)
  "Return the sum of the numbers 1 through NUMBER inclusive.
This is the initialization component of a two function
duo that uses recursion."
  (triangle-recursive-helper 0 0 number))

(defun triangle-recursive-helper (sum counter number)
  "Return SUM, using COUNTER, through NUMBER inclusive.
This is the helper component of a two function duo
that uses recursion."
  (if (> counter number)
      sum
    (triangle-recursive-helper (+ sum counter)  ; sum
                               (1+ counter)     ; counter
                               number)))        ; number

Install both function definitions by evaluating them, then call triangle-initialization with 2 rows:

(triangle-initialization 2)
    ⇒ 3

The initialization function calls the first instance of the helper function with three arguments: zero, zero, and a number which is the number of rows in the triangle.

The first two arguments passed to the helper function are initialization values. These values are changed when triangle-recursive-helper invokes new instances.¹⁶

Let’s see what happens when we have a triangle that has one row. (This triangle will have one pebble in it!)

triangle-initialization will call its helper with the arguments 0 0 1. That function will run the conditional test whether (> counter number):

(> 0 1)

and find that the result is false, so it will invoke the else-part of the if clause:

    (triangle-recursive-helper
     (+ sum counter)  ; sum plus counter ⇒ sum
     (1+ counter)     ; increment counter ⇒ counter
     number)          ; number stays the same

which will first compute:

(triangle-recursive-helper (+ 0 0)  ; sum
                           (1+ 0)   ; counter
                           1)       ; number

which is:

(triangle-recursive-helper 0 1 1)

Again, (> counter number) will be false, so again, the Lisp interpreter will evaluate triangle-recursive-helper, creating a new instance with new arguments.

This new instance will be;

    (triangle-recursive-helper
     (+ sum counter)  ; sum plus counter ⇒ sum
     (1+ counter)     ; increment counter ⇒ counter
     number)          ; number stays the same

which is:

(triangle-recursive-helper 1 2 1)

In this case, the (> counter number) test will be true! So the instance will return the value of the sum, which will be 1, as expected.

Now, let’s pass triangle-initialization an argument of 2, to find out how many pebbles there are in a triangle with two rows.

That function calls (triangle-recursive-helper 0 0 2).

In stages, the instances called will be:

                          sum counter number
(triangle-recursive-helper 0    1       2)

(triangle-recursive-helper 1    2       2)

(triangle-recursive-helper 3    3       2)

When the last instance is called, the (> counter number) test will be true, so the instance will return the value of sum, which will be 3.

This kind of pattern helps when you are writing functions that can use many resources in a computer.