y := t

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{ x = Y A y = X} .

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A sequence of statements, parenthesized in this way by assertions, is called a tableau. We can read such a tableau as a collection of three claims, one for each of the individual assignments as shown below.

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9.3 Hoare Triples as Program Specifications

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(1) {x = X A y = Y} Ifx = t := x t := x is executed {t = x = X A y = Yj then afterwards t = x = X and y - Y.

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{t = x = X A ; x = Y} If t = x = X and y = Y when x := y x := y is executed {t-X A x = y = Y} then afterwards t = X and x = y = Y. {t = X A x = y = Y} If t = X and x = y = Y when y := t y := t is executed {x = Y A y = X} then afterwards x = Y and _y = X.

Additionally, we can read the tableau as making claims about subsequences of the three assignments. For instance, omitting the last-but-one assertion, the tableau asserts the following about the last two assignments. (4) {t = x = X A y = Y} If t = x = X and y = Y when x '= y ; y := t x := y followed by y := t is executed {x = Y A y = X} then afterwards x = Y and y = X. We shall see in 10 that, provided the claims about each of the individual assignments (claims (1), (2) and (3) above) are valid, any claim about a composition of assignments, obtained by omitting the intermediate assertions as we did in (4) above, is also valid.

Hoare Triples as Program Specifications

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The specification of a program, in its simplest form, is a relation between input values and output values. The specification of a sorting program, for example, would require that the input is an array of values and the output is the same array but sorted according to some given function on the elements of the array. The precise and clear specification of computer programs is a demanding task that needs to be done with great care and attention to detail. Here, we only consider very simple specifications. It is important to note that specifications are, commonly, non-deterministic. That is, the output values are typically not completely determined by the input values, thus giving some latitude in what is acceptable output for given input. For example, when sorting the entries in a table by, say, date of last access, the program is free to list entries last accessed on the same day in an arbitrary order. In mathematical terms, specifications are truly relations and not functions. The emphasis in this text is on methods for constructing programs to meet their specifications (so-called correct-by-construction design methods). A program S is specified by stating a precondition P and a postcondition Q (possibly involving

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9: The Assignment Statement ghost variables in order to relate input and output values) and requiring that S be constructed to satisfy {P}S{Q}. If so, we say that S establishes (postcondition) Q under the assumption of precondition P. The notation { P } S { Q } is so important that it is worth repeating its meaning and making it stand out. { P } S { Q } means that, for all possible values of the variables in P, 5 and Q, if, initially, the state of the program variables satisfies the predicate P and the statement S is executed, 5 is guaranteed to terminate and, on termination of S, the final state will satisfy the predicate Q. Although we use Hoare triples exclusively in this text to specify programs, they do have a number of inadequacies which we need to overcome. The reason for choosing Hoare triples as specification mechanism is that they fit in well with the simple process of adding comments to the program text. Four problems with the use of Hoare triples are (a) we are forced to name the variables to be used in the program (whereas the names are irrelevant to the specification), (b) there is no way of saving which variables may be altered in the course of execution of the program and which should remain constant (that is, there is no distinction between input and output variables), (c) there is no way of limiting the mechanisms for updating the values of the output variables, (d) an artificial mechanism the 'ghost' variables discussed above often needs to be employed to relate the input values of variables to their desired output values. We ignore the first problem. It is a nuisance rather than a serious issue, and one that is impossible to avoid entirely. The second problem is more serious. It is illustrated by a very simple example. If we require that program S satisfies {true}5{ i = j ] , then this can be achieved by the assignments

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