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evaluates twice the value of variable y and 'assigns' this value to x. It is convenient to allow simultaneous assignments. In a simultaneous assignment, the left side is a list of variables and the right side is a list of expressions of the same length as the list of variables. A simultaneous assignment to three variables x, y and z is, for example, x,y,z := 2xy,x+y,3xz . A simultaneous assignment xo,xi,...,xn := eo,ei,...,en is executed by evaluating all of the expressions eo, e\,..., en and then, for each i, updating the value of the variable Xi to the value obtained for expression ei. In the above example, 2xy, x+y and 3xz are all evaluated before assigning their values to x, y and z, respectively. Simultaneous assignments are not allowed in many programming languages (C and Java, for example) but they are allowed in some (Perl, for example). It is unfortunate when they are not allowed because it is very simple to program a compiler to handle them, and their use contributes to the avoidance of error. An example is the assignment x,y := y,x , which has the effect of swapping the values stored in variables x and y. Barred from using simultaneous assignments, the programmer is forced to introduce a local variable t, say, and write
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t := x ; x := y ; y := t .
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9. 5 The Assignment Axiom
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The number of times that programmers have had to write such a sequence of statements and made a mistake, for example in the order, must be countless. There are, of course, restrictions on the use of simultaneous assignments. Basically, the variables on the left side should be pairwise distinct. For example, the assignment x,x := 0,1 does not make sense (because it tries to assign the distinct values 0 and 1 simultaneously to the variable x) and is disallowed. Very occasionally it is useful to relax this requirement. The statement
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swaps the array elements indexed by i and j. The statement makes sense whatever the values of i and j, including when they are equal. (When i and j are equal the statement means 'do nothing'.) Because it is convenient to do so, we also allow such statements. The rule is that, if a variable occurs more than once on the left side of an assignment, the corresponding expressions on the right side must be guaranteed to denote equal values.
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9.5 The Assignment Axiom
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We have already used the notation of simultaneous assignment in hints accompanying proof steps. There we used the notation to explain how a law is instantiated to a particular case. The coincidence of notations is deliberate. Rather than understanding the way an assignment is executed, we can also understand an assignment in terms of syntactic substitution. The key insight involves working backwards from postconditions to preconditions. Suppose the assignment x := e is required to establish the postcondition Q. The postcondition is any boolean-valued expression in the program variables, one of which is x. We can regard it as a local law governing the program variables immediately after execution of the assignment statement. After the assignment, x will have the value of expression e before the assignment. So, if Q is to apply to x after the assignment, it should apply to e before the assignment. That is, the property Q[x := e] the property Q but with all occurrences of 'x' replaced by 'e' must hold in advance of executing the assignment. This is stated in the assignment axiom: [Assignment Axiom] { Q[x := e] } x := e { Q } . (9.2)
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The assignment axiom is a very straightforward rule, at least after having seen a number of examples! The dynamics of its use, something that is difficult to convey in the static pages of a textbook, is to work backwards from the postcondition.
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