2. 3. , AR B and par while (B) {C} B in Software Assign Code39 in Software 2. 3. , AR B and par while (B) {C} B

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1. 2. 3. , AR B and par while (B) {C} B using none toinclude none for web,windows application ISBN - 13 are all valid, w none for none here the latter is shown by means of Partial-while. Then, Implied infers that (4.10) is a valid partial-correctness triple.

The crucial thing, then, is the discovery of a suitable invariant . It is a necessary step in order to use the proof rule Partial-while and in general it requires intelligence and ingenuity. This contrasts markedly with the case of the proof rules for if-statements and assignments, which are purely mechanical in nature: their usage is just a matter of symbol-pushing and does not require any deeper insight.

Discovery of a suitable invariant requires careful thought about what the while-statement is really doing. Indeed the eminent computer scientist, the late E. Dijkstra, said that to understand a while-statement is tantamount to knowing what its invariant is with respect to given preconditions and postconditions for that while-statement.

This is because a suitable invariant can be interpreted as saying that the intended computation performed by the while-statement is correct up to the current step of the execution. It then follows that, when the execution. 4 Program verification terminates, the none none entire computation is correct. Let us formalize invariants and then study how to discover them. De nition 4.

15 An invariant of the while-statement while (B) {C} is a formula such that par B C holds; i.e. for all states l, if and B are true in l and C is executed from state l and terminates, then is again true in the resulting state.

Note that does not have to be true continuously during the execution of C; in general, it will not be. All we require is that, if it is true before C is executed, then it is true (if and) when C terminates. For any given while-statement there are several invariants.

For example, is an invariant for any while-statement; so is , since the premise of the implication if B is true, then . . .

is false, so that implication is true. The formula B is also an invariant of while (B) do {C}; but most of these invariants are useless to us, because we are looking for an invariant for which the sequents AR and AR B , are valid, where and are the preconditions and postconditions of the while-statement. Usually, this will single out just one of all the possible invariants up to logical equivalence.

A useful invariant expresses a relationship between the variables manipulated by the body of the while-statement which is preserved by the execution of the body, even though the values of the variables themselves may change. The invariant can often be found by constructing a trace of the while-statement in action. Example 4.

16 Consider the program Fac1 from page 262, annotated with location labels for our discussion: y = 1; z = 0; l1: while (z != x) { z = z + 1; y = y * z; l2: } Suppose program execution begins in a store in which x equals 6. When the program ow rst encounters the while-statement at location l1, z equals 0 and y equals 1, so the condition z = x is true and the body is executed. Thereafter at location l2, z equals 1 and y equals 1 and the boolean guard is still true, so the body is executed again.

Continuing in this way, we obtain. 4.3 Proof calculus for partial correctness the following tr none none ace: after iteration z at l1 y at l1 B at l1 0 0 1 true 1 1 1 true 2 2 2 true 3 3 6 true 4 24 true 4 5 120 true 5 6 6 720 false The program execution stops when the boolean guard becomes false. The invariant of this example is easy to see: it is y = z! . Every time we complete an execution of the body of the while-statement, this fact is true, even though the values of y and z have been changed.

Moreover, this invariant has the needed properties. It is. r weak enough to be implied by the precondition of the while-statement, which we will shortly discover to be y = 1 z = 0 based on the initial assignments and def their precondition 0! = 1, r but also strong enough that, together with the negation of the boolean guard, it implies the postcondition y = x! ..
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