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9.3 Hoare Triples as Program Specifications
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there being no way to distinguish between the two variables. In reality, however, one of i and j would store the input value and the other the output value, and the requirement would be to assign a value to the output variable so as to meet the specification, leaving the value of the input variable unchanged. A program to determine whether a student is registered for a module would not be acceptable if it altered the register in the process! It is possible to avoid this problem formally using a so-called specification statement. (See the bibliographic notes for references.) The problem is resolved here informally we state which are the input and which are the output variables in the text accompanying the formal specification. Problem (c) is also resolved informally. The issue here is that implementations are limited by the implementation language. The limitations are sometimes imposed to improve efficiency. For example, when sorting a very large collection of data, it is undesirable to make copies of large portions. Sometimes, a requirement is that the implementation be limited to functions supplied by a given library. For example, a hardware implementation of a given specification might require that the implementation be composed of circuits from a small base collection. Problem (d) is exemplified by the specification of sorting programs. Sorting a list of values involves permuting the values in the list. But, if we simply require that the output list is sorted, the requirement can be met by outputting any list that is sorted without regard to the contents of the input list. The remedy for this problem, as already discussed in Section 9.2, is to introduce ghost variables. (Some authors call these rigid variables because their values do not change throughout the execution of the program. Some authors call them auxiliary variables.) It is worthwhile giving another illustration of the use of ghost variables. Suppose we want to specify that the sum of two variables i and j should remain constant. For example, i and j may be initially zero. The requirement is then that, on termination, i+j should also be zero. If, however, i and j are initially 1 and 2, we require that i+j is 3 on termination of the program. We specify this by introducing a ghost variable C. This variable should not be used anywhere else in the program. Then the program S is specified by
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This says that if the sum of i and j has the value C before execution of statement S, execution of statement S is guaranteed to terminate in a state in which the sum of i and j still has the value C. In summary, we specify a program S by supplying a precondition P and a postcondition Q and requiring that S be constructed to satisfy {P}S{Q} . In addition, we (informally) specify the input variables and the output variables of the program and their types. Ghost variables may be used to link the precondition to the postcondition. Ghost variables are treated just like ordinary program
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9: The Assignment Statement
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variables but the program code may not refer to them in any way. To distinguish ghost variables from normal program variables we use a sans serif type. Finally, we sometimes impose limitations on the primitives that may be used in the implementation of the specification. These limitations are stated informally.
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9.4 Assignment Statements
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The very simplest specifications can be met by assignment statements. An assignment statement has the form
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where x is a program variable and e is an expression. It is executed by evaluating the expression e and then updating the value of the variable x to the value obtained. For example,
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