II. Notes on Data Structuring * - Cornell University

NOTES ON DATA STRUCTURING 103
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4. THE CARTESIAN PRODUCT
Defined enumerations and subranges, like primitive data types, are in principle
unstructured. Of course, any particular representation of these types will be
structured, for example, as a collection of consecutive binary digits; but
from the abstract point of view, this structuring is essentially irrelevant. No
operators are provided for accessing the individual bits, or for building up a
value from them. In fact, it is essential to the successful use of an abstraction
that such a possibility should be ignored; since it is only thus that detailed
decisions can be postponed, and data representations can be decided in the
light of the characteristics of the computer, as well as the manner in which
the data is to be manipulated.
We now turn to deal with data types for which the structure is meaningful
to the programmer, at least at some stage in the development of his program.
The basis of our approach is that, as in the case of enumerations, the pro-
grammer should be able by declaration to introduce new data types; but for
structured data, the definition of a new type will refer to other primitive or
previously defined types, namely the types of the components of the structure.
Thus the declaration of a new type will be somewhat similar to the declara-
tion of a new function in a language such as ALGOL and FORTRAN. A
function declaration defines the new function in terms of existing or pre-
viously declared functions and operations. Just as a declared function can be
invoked on many occasions from within statements of the program or other
function declarations, so the new type can be "invoked" many times from
within other declarations of the program; these may be either declarations
of variables specified to range over the newly declared type, or they may be
declarations of yet another new type.
We will deal first with elementary data structures, Cartesian products and
unions. These elementary structures are almost as simple and familiar to
mathematicians and logicians as the natural numbers. Furthermore, from