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

Type Operations 194.2 Conjunction GraphsA class contains all the attributes and methods of its direct superclass as well as the new membersintroduced in the class itself. Inductively along the inheritance hierarchy, the effective memberstructure of a class is a conjunction of all the members of its superclasses. The effective memberstructure is also called the interface of the class.The notion of conjunctions can be extended to a set of arbitrary classes. A conjunction of aset of classes consists of all the effective members of these classes. For the conjunction of a setof classes to be defined, these classes must be consistent that members with the same name indifferent classes must have equivalent types. We often use a conjunction to combine a numberof interfaces to define a structural requirement that is to be implemented by a concrete class,hence the term “conjunction”. The requirement is satisfied only when all of the interfaces areimplemented.Definition 9 (Conjunction Graph) Given a list of rooted graphs G, a rooted graph G ′ is aconjunction graph of G, denoted as G ′ ∼ ⋓ G, if1. G ′ . ∈ S ,2. all attributes in G are included in G ′ ,G ∈ G ∧ a ∈ A ∧ ⊲ −→ ∗ a −→ n ∈ G =⇒ ∃ a −→ n ′ ∈ G ′ • G n ∼ G ′ n ′ ,3. G ′ only contains attributes in G,a ≠ σ ∧ a −→ · ∈ G ′ =⇒ a ∈ A ∧ ∃G ∈ G • ⊲ −→ ∗ a −→ · ∈ G,4. all method signatures in G are included in G ′ ,G ∈ G ∧ m ∈ A ∧ ⊲ −→ ∗ σ.m −−→ n ∈ G =⇒ ∃ σ.m −−→ n ′ ∈ G ′ • G n ∼ G ′ n ′ ,5. G ′ only contains method signatures in G, −−→ σ.m · ∈ G ′ =⇒ m ∈ A ∧ ∃G ∈ G • −→ ⊲ ∗ −−→ σ.m · ∈ G.Fig. 3 (left) shows a conjunction graph rooted at r, of two classes C 1 and C 2 . A conjunctiongraph can be considered as a shallow structure of multiple classes.In a type graph Γ, it is often the case that two nodes n and n ′ lead to equivalent nodes n 1 andn ′ 1 through the same attribute navigation path a. We can thus merge n 1 and n ′ 1 to obtain asimpler graph. Formally, let n be a list of nodes of Γ and a a navigation path, provided that∀t 1 , t 2 ∈ {t | n ⊲ −→ ∗ a −→ t ∈ Γ, n ∈ n} • Γ ⊢ t1 ∼ t 2 ,we define co-target(Γ, n, a) ̂= ts, where{⊲{tts = 0 } if ∃n 0 ∈ n • n 0 −→∗−→ a t 0 ∈ Γ,∅ otherwise.Report No. 448, June 2011UNU-IIST, P.O. Box 3058, Macao