Lisp Number Representations

This exercise partly comes from a couple different chapters in the book The Little Schemer. The book is an introduction to the Scheme programming language, a dialect of Lisp. The purpose to to teach basic programming concepts in a way that anyone can follow along just as well as someone with a degree in, say, computer science. It is still very useful for us programmer types because there are some good practice you get from reading and playing along.

First of all, Lisp is famous (infamous?) for lacking syntax. Any Lisp program is simply an S-expression, put simply, a list of lists. There is no operator precedence because operators are treated just like functions. This leads to prefix notation for mathematical expressions,

(+ 4 5)
=> 9

where the => indicates the result of evaluating the expression. We can apply as many operands as we want,

(+ 2 3 4 5 10)
=> 24

We can put another list right in there as an operand,

(+ 3 (* 2 5) 4)
=> 17

You get the idea. In a function, the value of the last expression is the return value. For example, here is the square function in Scheme, which squares its input,

(define (square x)
  (* x x))

Then we can use it,

(+ (square 2) (square 5))
=> 29

There are three important list operators to understand as well: car, cdr, and cons. car returns the first element in a list. In the example below, the ', a single quote, tells the interpreter or compiler that the list is to be treated as data and not to be executed. This is shorthand, or syntactic sugar, for the quote operator: (quote (stallman moglen)) is the same as '(stallman moglen).

(car '(stallman moglen lessig))
=> stallman

cdr returns the "rest" of a list (everything but the car of the list). When passing a list with only one element cdr returns the empty list: ().

(cdr '(stallman moglen lessig))
=> (moglen lessig)
(cdr '(stallman))
=> ()

We can ask if a list is empty or not with null?. #t and #f are true and false.

(null? '(stallman moglen lessig))
=> #f
(null? '())
=> #t

And finally, for lists, we have cons. This function allows us to build a list. It glues the first argument to the front of the list in the second argument,

(cons 'stallman '(moglen lessig))
=> (stallman moglen lessig)
(cons 'stallman '())
=> (stallman)

And one last function you need to know: eq?. It determines the two atoms are the same atom,

(eq? 'stallman 'moglen)
=> #f
(eq? 'stallman 'stallman)
=> #t

Now, for this exercise we will pretend that the basic arithmetic functions have not been defined for us. Instead all we have is add1 and sub1, each of which adds or subtracts 1 from its argument respectively.

(add1 5)
=> 6
(sub1 5)
=> 4

Oh, I almost forgot. We also have the zero? function defined for us, which tells us if its argument is 0 or not. Notice that functions that return true or false, called predicates, have a ? on the end.

(zero? 2)
=> #f
(zero? 0)
=> #t

To make things simple, these definitions will only consider positive numbers. We can define the + function (for only two arguments) in terms of the three basic functions shown above. It might be interesting to try to write this yourself before you look any further. (Hint: define it recursively!)

;; Adds together n and m
(define (+ n m)
  (if (zero? m) n
      (add1 (+ n (sub1 m)))))

If the second argument is 0 we are done and simply return the first argument. If not, we add 1 to n + (m - 1). The - function is defined similarly.

;; Subtracts m from n
(define (- n m)
  (if (zero? m) n
      (sub1 (- n (sub1 m)))))

Multiplication is the act of performing addition many times. We can go on defining it in terms of addition,

(define (* n m)
  (if (zero? m) 0
      (+ n (* n (sub1 m)))))

(We'll leave division as an exercise for the reader as it gets a little more complicated than I need to go in order to get my overall point across.)

We will leave math behind for a moment take a look at The Roots of Lisp. In that link is an excellent paper written by Paul Graham about John McCarthy, the inventor (or perhaps discoverer?) of Lisp, and how Lisp came to be. It turns out that in order to have a fully functional Lisp engine we only need seven primitive operators: operators defined outside of the language itself as building blocks for the language. For Lisp these seven operators are (Scheme-ized for our purposes): eq?, atom?, car, cdr, cons, quote, and if.

Notice how none of these are math operators. You may wonder how we can possibly perform mathematical operations when we lack these facilities. The answer: we have to define our own representation for numbers! Let's try this, define a number as a list of empty lists. So, the number 3 is,

'(() () ())

And here is 0, 2, and 4,

'(() ())
'(() () () ())

See how that works? Before, when we wanted to define addition and subtraction, we needed three other functions: zero?, add1, and sub1. With our number representation, how could we define add1 with our seven primitive operators? Our numbers are defined as lists, so we can use our list operators. To add 1 to a number, we append another empty list. Hey, that sounds a lot like cons!

(define (add1 n)
  (cons '() n))

Subtraction is removing an element from the list, which sounds a lot like cdr,

(define (sub1 n)
  (cdr n))

And to define zero? we need to check for an empty list. Notice this will also be the definition for null?.

(define (zero? n)
  (eq? '() n))

And now we are back where we started. In fact, you can use the exact definitions above to define +, -, and *. Our entire method number representation depends on how we define add1, sub1, and zero?. Let's try it out,

;; 3 + 4
(+ '(() () ()) '(() () () ()))
=> (() () () () () () ())

;; 5 - 2
(- '(() () () () ()) '(() ()))
=> (() () ())

;; 2 * 2
(* '(() ()) '(() ()))
=> (() () () ())

;; 3 + 4 * 2   bolded for clarity
(+ (* '(() () () ()) '(() ())) '(() () ()))
=> (() () () () () () () () () () ())

Pretty cool, huh? We just added arithmetic (albeit extremely simple) to our basic Lisp engine. With some modifications we should be able to define and operate on negative integers and even define any rational number (limited by how much memory your computer's hardware can provide).

Now, thank goodness this isn't how real Lisp implementations actually handle numbers. It would be incredibly slow and impractical, not to mention annoying to read. Normally, numbers and math operators are primitive so that they are fast.

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null program

Chris Wellons