II. Notes on Data Structuring * - Cornell University

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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.
NOTES ON DATA STRUCTURING 167 The axioms applicable to a given type depend on how that type has been defined. Thus it is not possible to give in each case a fixed set of axioms like those for integers; instead we give a pattern or schema which shows how a particular axiom set may be derived from the general form of the corres- ponding type definition. 12.1. ENUMERATIONS AND SUBRANGES The following axioms are common to both enumerations and subranges. They are modelled on the familiar axioms for natural numbers. The type name is assumed to be T, and all variables are assumed to be of this type. (1) T.min is a T (2) If x is a T, and x 4: T. max then succ (x) is a T (3) The only elements of T are as specified in (1) and (2) (4) succ (x) = succ (y) ~ x = y (5) succ (x) -¢ T. min (6) pred (succ (x)) = x The following three axioms apply only to ordered types (7) T.min
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II. Notes on Data Structuring * - Cornell University

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