Modelling Distributed Systems

Data Types In this chapter we give an introduction to the algebraic specification of abstract data types by means of a set of equations. See [10, 80] for lucid overviews of this field. 2.1 Algebraic specification We start with an example. Example 2.1 As a standard example we specify the natural numbers with addition and multiplication. The signature consists of the function 0 of arity zero, the unary successor function S, and the binary functions addition plus and multiplication mul. The equality relation on terms is specified by four axioms: 1. plus(x, 0) = x 2. plus(x, S(y)) = S(plus(x, y)) 3. mul(x, 0) = 0 4. mul(x, S(y)) = plus(mul(x, y), x) The initial model of this axiomatisation consists of the distinct classes [[0]], [[S(0)]], [[S(0)]], [[S(0)]], . . . . The first three classes, with some typical representatives of each of these classes, are depicted in Figure 2.1. For example, the equation plus(S(S(S(0))), S(0)) = mul(S(S(0)), S(S(0))) (i.e., 3 + 1 = 2 · 2) can be derived from the axiomatisation of the natural numbers as follows. plus(S(S(S(0))), S(0)) = S(plus(S(S(S(0))), 0)) = S(S(S(S(0))))