This chapter illustrates some uses of the kind of semantic definition given in Chapter 19 via a series of examples. We continue to use the notations in Notation 19.2.1.

20.1 Some properties of declarations and commands

Example 20.1.1 (Eliminating an unused variable) Consider the skeleton class declaration in Table 20.1. If none of Ci,…, C*, contains either the expression X or a command of the form X := E, then deleting the declaration of X should not change the behaviour of objects of the class, and hence should not affect the behaviour of any program in which the class appears. To prove this, we use the straightforward equivalence

provided. (For notational convenience, in what follows we assume that T is Int.)

Now suppose that X is the jth variable of A, and write Vj for V〈gj, pj, 0〉 and abbreviate gj to g and pj to p. Then there are processes G and P such that

Let A’ be the class obtained from A by deleting the declaration of X. Then, since none of the method bodies contains the expression X or an assignment to X,

Hence

using (20.2), the equivalence in (20.1), and the observation in (20.3). This completes Example 20.1.1.

Example 20.1.2 (Replacing a variable by a constant) Consider again the skeleton class declaration in Table 20.1. If none of C1,… ,Ck contains a command of the form X : = E, then replacing each occurrence of the expression X in C1,…, Ck by 0 (the initial value of X) and deleting the declaration of X should not change the behaviour of objects of the class, and hence should not affect the behaviour of any program in which the class appears.

Let us reuse the notations Vj, MA, etc. from Example 20.1.1, and write A” for the class obtained from A by applying the transformation.