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## 5.2  Semantics Of An Object-Oriented  Model

The meaning of an object-oriented schema is given by specifying the characteristics of an instance of the schema. The definition of instance makes use of the notions of object identifier and value.

Let us first characterize the set of values that can be constructed from a set of symbols, called object identifiers. Given a finite set O of symbols denoting real world objects, the set Vc) of values over O is inductively defined as follows:
- O C Vo.

- If v 1 , . . , v k ∈ V O then {| v 1 , . . , v k |} ∈ V O .

- If v1, . . . , vk E VO then [[A1: v1, . . . , Ak : vk ]] E VO.

- Nothing else is in Vo.
A database instance J of a schema S = (Cs, As,Ds) is constituted by
- a finite set OJ of object identifiers;

- a mapping π T assigning to each class in C S a subset of O T ;
- a mapping ρ T assigning a value in V OJ to each object in OJ
Although the set Voj of values that can be constructed from a set Oj of object identifiers is infinite, for a database instance one needs only to consider a finite subset of Vog Definition 5.3  Given an object-oriented schema S and an instance J of S, the set VJ of active values with respect to J is constituted by:
- the set OJ of object identifiers.

- the set of values assigned by ρ T to the elements of O T , including those values that are not explicitly associated with object identifiers, but are used to form other values.
The interpretation of type expressions in J is defined through an interpretation function I that assigns to each type expression a subset of VOI such that the following conditions are satisfied:

$$\begin{array}{l c l}{{C^{\mathcal{I}}}}&{{=}}&{{\pi^{\mathcal{I}}(C)}}\end{array}$$
