A Graph-Based Generic Type System for Object-Oriented Programs

Type Operations 25Proof. By Definition 12 and the definition of substitution composition, we can directly establishthe instantiation function.□In addition, an instantiation of a conjunction of a set of classes can be distributed over theconjunction.Lemma 6 Given a type graph Γ, nodes t and a type substitution s, if for nodes j, j ′ , r, r ′ ,then Γ ⊢ r ′ ∼ j ′ .Γ ⊢ j ≃ t〈s〉, r ′ ∼ ⋓ j, r ∼ ⋓ t, j ′ ≃ r〈s〉,Proof. For a navigation path a in the form of a or σ.m.x, and a path j ′ a −→ u ∈ Γ, by instantiation,there is either a path r −→ a u ∈ Γ, or a path r −→ a α ∈ Γ if α ↦→ u ∈ s. Thus, by conjunction, there⊲exists a node t 0 ∈ t such that t 0 −→∗−→ a ⊲u ∈ Γ, or t 0 −→∗−→ a α ∈ Γ. Let j 0 ∈ j be the instantiation⊲node of t 0 . There exists a path j 0 −→∗−→ a u 0 ∈ Γ such that Γ ⊢ u 0 ≃ u, either directly mapped,or by the substitution α ↦→ u. Thus there exists a path r ′ a −→ u′0 ∈ Γ such that Γ ⊢ u ′ 0 ∼ u. Wehave proven that for such a navigation path a, j ′ a −→ u ∈ Γ =⇒ ∃r′ a −→ u′0 • Γ ⊢ u ′ 0 ∼ u. Theopposite is similar.□4.5 Recursive Generic ClassesFor a non-recursive generic class D〈β〉, we draw the graph of D〈β〉 directly from its textualdefinition. Based on this graph, we can construct any instantiation of D〈β〉 in the way stated inthe previous subsection. However, for a recursive generic class C〈α〉, a class instantiation C〈s〉,where s is a type substitution, may be used as the type of an attribute of C〈α〉. In this case,there is an edge from the node C to the node representing C〈s〉, thus the graph of C〈α〉 dependson the graph of C〈s〉. For the same reason the graph of C〈s〉 depends on the graph of C〈s 2 〉,and so on, where s k ̂= s k−1 ◦ s. Notice that Lemma 5 ensures that the notation of instantiationC〈s k 〉 is well defined for a positive number k. Because of the recursive instantiation, a genericclass C〈α〉 with an attribute of type C〈s〉 is defined only if the type substitution s is convergent.A type substitution s is called convergent if the set {s i | i > 0} is finite, otherwise it is calleddivergent. For example, 〈α ↦→ α〉 and 〈α ↦→ β, β ↦→ α〉 are convergent substitutions, while〈α ↦→ C〈α ↦→ α〉〉 and 〈α ↦→ γ, γ ↦→ D〈β ↦→ α〉〉 are divergent ones. A type substitution s isdivergent if and only if there is a list of mappings α 1 ↦→ Te 1 , . . . , α k ↦→ Te k (k ≥ 1) in s suchthat α i+1 occurs as an actual type in Te i for 1 ≤ i < k and α 1 occurs as an actual type in aclass instantiation in Te k .When s is convergent, the set {C〈s i 〉 | i > 0} of instantiations that the generic class C〈α〉depends on is finite, i.e., the graph of C〈α〉 is finite. Since the types C〈s i 〉 mutually refer to eachReport No. 448, June 2011UNU-IIST, P.O. Box 3058, Macao