II. Notes on Data Structuring * - Cornell University

166 c.A.R. HOARE
12. AXIOMATISATION
The preceding sections have introduced a number of methods of constructing
data spaces (types), and have explained some useful operations defined over
these spaces. But the description has been essentially intuitive and informal,
and the question arises whether all the relevant information about the data
spaces has been communicated, or whether there remains some possibility of
misunderstanding of the details.
In order to remove such misunderstanding, or check that it has not
occurred, it is desirable to give a rigorous mathematical specification of
each data space, and the operators defined over it; and we follow what is
now a customary mathematical practice of defining rigorously the subject
matter of our reasoning, not by traditional definitions, but by sets of axioms.
In view of the role which axioms play in the theory of data structuring,
it may be helpful to summarise their intended properties.
(1) Axioms are a formal statement of those properties which are shared
by the real world and by its representation inside a computer, in virtue of
which manipulation of the representation by a computer program will yield
results which can be successfully applied back to the real world.
(2) They establish a conceptual framework covering those aspects of the
real world which are believed to be relevant to the programmer's task, and
thereby assist in his constructive and inventive thinking.
(3) They state rigorously those assumptions about the real world on which
the computer program will be based.
(4) They state the necessary properties which must be possessed by any
computer representation of the data, in a manner free from detail which is in
initial stages irrelevant.
(5) They offer a carefully circumscribed freedom to the programmer or
high-level language implementor to choose a representation most suitable
for his application and hardware available.
(6) They form the basis of any proof of correctness of a program.
The axioms given here are not intended to be used directly in the proof
of non-trivial programs, since such proofs would be excessively long-winded.
Rather they may be used to establish the familiar properties of the data
spaces they describe, and these properties can then be used informally in
proofs. Eventually it may be possible to get computers to check such proofs;
but this will require the development of much more powerful formal languages
for expressing proofs than are at present provided by logicians, and the
use of powerful decision procedures for large subclasses of theorem, to assist
in verification of the individual steps of a proof.