Before continuing with more examples of compound data and data
abstraction, let us consider some of the issues raised by the
rational-number example. We defined the rational-number operations in
terms of a constructor
`make_rat`
and selectors `numer` and
`denom`. In general, the underlying idea of data
abstraction is to identify for each type of data object a basic set of
operations in terms of which all manipulations of data objects of that type
will be expressed, and then to use only those operations in manipulating the
data.

We can envision the structure of the rational-number system as
shown in
Figure 2.1.
The horizontal lines represent *abstraction barriers* that isolate
different levels

of the system. At each level, the barrier
separates the programs (above) that use the data abstraction from the
programs (below) that implement the data abstraction. Programs that
use rational numbers manipulate them solely in terms of the
functions
supplied for public use

by the rational-number package:
`add_rat`,
`sub_rat`,
`mul_rat`,
`div_rat`,
and
`equal_rat`.
These, in turn, are implemented solely in terms of the
constructor and selectors
`make_rat`,
`numer`, and `denom`,
which themselves are implemented in terms of pairs. The details of how
pairs are implemented are irrelevant to the rest of the rational-number
package so long as pairs can be manipulated by the use of
`pair`,
`head`,
and
`tail`.
In effect,
functions
at each level are the interfaces that define the abstraction barriers and
connect the different levels.

This simple idea has many advantages. One advantage is that it makes programs much easier to maintain and to modify. Any complex data structure can be represented in a variety of ways with the primitive data structures provided by a programming language. Of course, the choice of representation influences the programs that operate on it; thus, if the representation were to be changed at some later time, all such programs might have to be modified accordingly. This task could be time-consuming and expensive in the case of large programs unless the dependence on the representation were to be confined by design to a very few program modules.

For example, an alternate way to address the problem of reducing rational numbers to lowest terms is to perform the reduction whenever we access the parts of a rational number, rather than when we construct it. This leads to different constructor and selector functions:

function make_rat(n, d) { return pair(n, d); } function numer(x) { const g = gcd(head(x), tail(x)); return head(x) / g; } function denom(x) { const g = gcd(head(x), tail(x)); return tail(x) / g; }

The difference between this implementation and the previous one lies in when
we compute the `gcd`. If in our typical use of
rational numbers we access the numerators and denominators of the same
rational numbers many times, it would be preferable to compute the
`gcd` when the rational numbers are constructed.
If not, we may be better off waiting until access time to compute the
`gcd`. In any case, when we change from one
representation to the other, the
functions
`add_rat`,
`sub_rat`,
and so on do not have to be modified at all.

Constraining the dependence on the representation to a few interface
functions
helps us design programs as well as modify them, because it allows us to
maintain the flexibility to consider alternate implementations. To continue
with our simple example, suppose we are designing a rational-number package
and we can't decide initially whether to perform the
`gcd` at construction time or at selection time.
The data-abstraction methodology gives us a way to defer that decision
without losing the ability to make progress on the rest of the system.

function print_point(p) { display("("); display(x_point(p)); display(","); display(y_point(p)); display(")"); }

function x_point(x) { return head(x); } function y_point(x) { return tail(x); } function make_point(x, y) { return pair(x, y); } function make_segment(start_point, end_point) { return pair(start_point,end_point); } function start_segment(x) { return head(x); } function end_segment(x) { return tail(x); } function average(a, b) { return (a + b) / 2; } function mid_point_segment(x) { const a = start_segment(x); const b = end_segment(x); return make_point(average(x_point(a), x_point(b)), average(y_point(a), y_point(b))); }

function make_point(x,y){ return pair(x,y); } function x_point(x){ return head(x); } function y_point(x){ return tail(x); } function make_rect(bottom_left, top_right){ return pair(bottom_left, top_right); } function top_right(rect){ return tail(rect); } function bottom_right(rect){ return make_point(x_point(tail(rect)), y_point(head(rect))); } function top_left(rect){ return make_point(x_point(head(rect)), y_point(tail(rect))); } function bottom_left(rect){ return head(rect); } function abs(x){ return x < 0 ? - x : x; } function width_rect(rect){ return abs(x_point(bottom_left(rect)) - x_point(bottom_right(rect))); } function height_rect(rect){ return abs (y_point(bottom_left(rect)) - y_point(top_left(rect))); } function area_rect(rect){ return width_rect(rect) * height_rect(rect); } function perimeter_rect(rect){ return 2 * (width_rect(rect) + height_rect(rect)); }

function make_point(x,y){ return pair(x,y); } function make_rect(bottom_left, width, height){ return pair(bottom_left, pair(width, height)); } function height_rect(rect){ return tail(tail(rect)); } function width_rect(rect){ return head(tail(rect)); } function area_rect(rect){ return width_rect(rect) * height_rect(rect); } function perimeter_rect(rect){ return 2 * (width_rect(rect) + height_rect(rect)); }

2.1.2 Abstraction Barriers