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

Type Operations 211. all effective attributes in G are included in G ′ ,a ≠ σ ∧ a −→ n ∈ clo(G) =⇒ ∃ a −→ n ′ ∈ clo(G ′ ) • G n ∼ G ′ n ′ ,2. all effective method signatures in G are included in G ′ , −−→ σ.m n ∈ clo(G) =⇒ ∃ −−→ σ.m n ′ ∈ clo(G ′ ) • G n ∼ G ′ n ′ .If a rooted graph G ′ contains an ⊲-chain from the root to a node n, G ′ is a subclass of thesubgraph rooted at n of G ′ . We call G ′ a superclass cover of a rooted graph that encodes asuperclass constraint ⊲ −→ n.Definition 11 (Superclass Cover) A rooted graph G ′ is a superclass cover of a rooted graphG, denoted as G ′ ̂⊲ G, if ⊲ −→ n ∈ G =⇒ G ′ G n.4.4 Instantiation GraphsGiven a generic class C〈α〉 with a list α of type variables, C〈α ↦→ Te〉 is a class instantiationof C〈α〉, where Te is a list of type expressions. We call 〈α ↦→ Te〉 the type substitution of theinstantiation. The root of the graph of the generic class C〈α〉 is named with C. However, theroot of the graph of an instantiation C〈α ↦→ Te〉 is a structural node. And in this instantiationgraph, all the subgraphs rooted at the type variables α in the graph of C〈α〉 are replaced by thecorresponding graphs of the actual types Te. Let u be the roots of the graphs of Te. We alsocall 〈α ↦→ u〉 the type substitution of the instantiation of the graph of C〈α〉 with the graph ofC〈α ↦→ Te〉.Generally, any rooted graph containing type variables is generic, and can be used as a templatefor instantiation. Type variables only appear in the graph of a generic class, or an instantiationgraph resulted by using type variables as actual types. We unify the two cases by using theshallow structure of a generic class as the template for the instantiation of the generic class. Infact, the shallow structure is type equivalent to an instantiation graph of the generic class overan identity type substitution 〈α ↦→ α〉. Moreover, since a generic class itself cannot be used as atype, there is no edge pointing to the root of its graph. Thus, in a template graph, nodes on apath to a type variable can only be structural nodes or type variables.To use a homomorphism to define the instantiation of a template G over a type substitution〈α ↦→ u〉, we define the subgraph of G terminating at the α-nodes. Given a list n of nodes ina rooted graph G, a subgraph of G terminating at n is obtained from G by removing all theoutgoing edges from n,G n ̂= G out-es(G, n).Notice that G n has the same root as G.Report No. 448, June 2011UNU-IIST, P.O. Box 3058, Macao