Abstraction barriers shape the way in which we think about data. A valid representation of a rational number is not restricted to any particular implementation (such as a two-element list); it is a value returned by rational that can be passed to numer, and denom. In addition, the appropriate relationship must hold among the constructor and selectors. That is, if we construct a rational number x from integers n and d, then it should be the case that numer(x)/denom(x) is equal to n/d.

In general, we can express abstract data using a collection of selectors and constructors, together with some behavior conditions. As long as the behavior conditions are met (such as the division property above), the selectors and constructors constitute a valid representation of a kind of data. The implementation details below an abstraction barrier may change, but if the behavior does not, then the data abstraction remains valid, and any program written using this data abstraction will remain correct.

This point of view can be applied broadly, including to the pair values that we used to implement rational numbers. We never actually said much about what a pair was, only that the language supplied the means to create and manipulate lists with two elements. The behavior we require to implement a pair is that it glues two values together. Stated as a behavior condition,

• If a pair p was constructed from values x and y, then select(p, 0) returns x, and select(p, 1) returns y.

We don't actually need the list type to create pairs. Instead, we can implement two functions pair and select that fulfill this description just as well as a two-element list.

>>> def pair(x, y):
        """Return a function that represents a pair."""
        def get(index):
            if index == 0:
                return x
            elif index == 1:
                return y
        return get

>>> def select(p, i):
        """Return the element at index i of pair p."""
        return p(i)

With this implementation, we can create and manipulate pairs.

>>> p = pair(20, 14)
>>> select(p, 0)
20
>>> select(p, 1)
14

This use of higher-order functions corresponds to nothing like our intuitive notion of what data should be. Nevertheless, these functions suffice to represent pairs in our programs. Functions are sufficient to represent compound data.

The point of exhibiting the functional representation of a pair is not that Python actually works this way (lists are implemented more 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. The practice of data abstraction allows us to switch among representations easily.