What Is Meant by Data?

We began the rational-number implementation in earlier by implementing the rational-number operations add-rat, sub-rat and so on in terms of three unspecified functions: make-rat, numer, and denom. At that point, we could think of the operations as being defined in terms of data objects - numerators, denominators, and rational numbers - whose behavior was specified by the latter three functions.

But exactly what is meant by data? It is not enough to say “whatever is implemented by the given selectors and constructors.” Clearly, not every arbitrary set of three functions can serve as an appropriate basis for the rational-number implementation. We need to guarantee that, if we construct a rational number x from a pair of integers n and d , then extracting the numer and the denom of x and dividing them should yield the same result as dividing n by d. In other words, make-rat, numer, and denom must satisfy the condition that, for any integer n and any non-zero integer d, if x is (make-rat n d), then:

> (= (/ (numer x)
        (demom y))
     (/ n d))
true

In fact, this is the only condition make-rat, numer, and denom must fulfill in order to form a suitable basis for a rational-number representation. In general, we can think of data as defined by some collection of selectors and constructors, together with specified conditions that these functions must fulfill in order to be a valid representation ¹

This point of view can serve to define not only “high-level” data objects, such as rational numbers, but lower-level objects as well.

Everything From (Almost) Nothing

Consider the notion of a pair, which we used in order to define our rational numbers. We never actually said what a pair was, only that the language supplied functions cons , car , and cdr for operating on pairs. But the only thing we need to know about these three operations is that if we glue two objects together using cons we can retrieve the objects using car and cdr . That is, the operations satisfy the condition that, for any objects x and y, if z is (cons x y) then (car z)is x and (cdr z) is y .

Indeed, we imagined that these three functions are included as primitives in our language. However, any triple of functions that satisfies the above condition can be used as the basis for implementing pairs. This point is illustrated strikingly by the fact that we could implement cons , car , and cdr without using any data structures at all but only using functions.

Here are the definitions:

(defn cons [x y]
  (fn [m]
    (cond (= m 0) x
          (= m 1) y)))

(defn car [z]
  (z 0))

(defn cdr [z]
  (z 1))

This use of functions corresponds to nothing like our intuitive notion of what data should be. Nevertheless, all we need to do to show that this is a valid way to represent pairs is to verify that these functions satisfy the condition given above.

The subtle point to notice is that the value returned by (cons x y) is a function, which takes one argument and returns either x or y depending on whether the argument is 0 or 1. Correspondingly, (car z) is defined to apply z to 0. Hence, if z is the function formed by (cons x y), then z applied to 0 will yield x . Thus, we have shown that (car (cons x y)) yields x, as desired. Similarly, (cdr (cons x y)) applies the function returned by (cons x y) to 1, which returns y. Therefore, this functional implementation of pairs is a valid implementation, and if we access pairs using only cons, car, and cdr we cannot distinguish this implementation from one that uses “real” data structures.

The point of exhibiting the functional representation of pairs is not that our language works this way (Scheme, Clojure, and Lisp systems in general, implement pairs directly, for efficiency reasons) but that it could work this way. The functional representation, although obscure, is a perfectly adequate way to represent pairs, since it fulfills the only conditions that pairs need to fulfill. This example also demonstrates that the ability to manipulate functions as objects automatically provides the ability to represent compound data. This may seem a curiosity now, but procedu- ral representations of data will play a central role in our programming repertoire.

Numbers from functions

In case representing pairs as functions wasn’t mind-boggling enough, consider that, in a language that can manipulate functions, we can get by without numbers (at least insofar as nonnegative integers are concerned) by implementing 0 and the operation of adding 1 as

(def zero (fn [f] (fn [x] x)))

(defn inc [n]
  (fn [f] (fn [x] (f ((n f) x)))))

Can you figure out what one and two would be? (use substitution to evaluate (inc zero))

This representation is known as Church Numerals, after its inventor, Alonzo Church, the logician who invented the λ-calculus, and you will have lots of fun using them in the project

So in Chapter One we looked at functions of numbers, we then introduced data structures and saw that we could create them from functions, lastly we created numbers from functions. I'd say that was pretty magic.