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

Type Operations 17Table 2: Type Equivalencet-eq 0 (X, Y, Q) ̂=(s, t) ∈ Q x ∈ s y ∈ t x −→ a x ′ ∈ X x ′ /∈ Q.sy −→ a y ′ ∈ Y y ′ ∈ t ′ (s ′ , t ′ ) ∈ Q y −→ a y ′ ∈ Y y ′ /∈ Q.tt-eq 0 (X, Y, Q {(s ′ , t ′ )} ∪ {(s ′ ∪ {x ′ }, t ′ )} t-eq 0 (X, Y, Q ∪ {({x ′ }, {y ′ })})false(s, t) ∈ Q x ∈ s y ∈ t y a −→ y ′ ∈ Y y ′ /∈ Q.tt-eq 0 (Y, X, {(v, u) | (u, v) ∈ Q})(s, t), (s ′ 1, t ′ 1), (s ′ 2, t ′ 2) ∈ Q(s ′ 1, t ′ 1) ≠ (s ′ 2, t ′ 2) x 1 , x 2 ∈ s x 1a−→ x′1 , x 2a−→ x′2 ∈ X x ′ 1 ∈ s ′ 1 x ′ 2 ∈ s ′ 2t-eq 0 (X, Y, Q {(s ′ 1, t ′ 1)} {(s ′ 2, t ′ 2)} ∪ {(s ′ 1 ∪ s ′ 2, t ′ 1 ∪ t ′ 2)})(s, t), (s ′ 1, t ′ 1), (s ′ 2, t ′ 2) ∈ Q(s ′ 1, t ′ 1) ≠ (s ′ 2, t ′ 2) y 1 , y 2 ∈ t y 1a−→ y′1 , y 2a−→ y′2 ∈ Y y ′ 1 ∈ t ′ 1 y ′ 2 ∈ t ′ 2t-eq 0 (X, Y, Q {(s ′ 1, t ′ 1)} {(s ′ 2, t ′ 2)} ∪ {(s ′ 1 ∪ s ′ 2, t ′ 1 ∪ t ′ 2)})(s, t) ∈ Qn 1 ≠ n 2n 1 , n 2 ∈ s ∪ t{n 1 , n 2 } Sfalse(s, t) ∈ Q x ∈ s y ∈ t{a | x a −→ · ∈ X} ≠ {b | y b −→ · ∈ Y }false (Q, true) ,where Q is a set of pairs and each pair consists of two sets of equivalent nodes, Q.s = ⋃ (s,t)∈Q s andQ.t = ⋃ (s,t)∈Q t.4 Type OperationsIn an rcos/g program, types are specified by type expressions. Besides atomic type expressions,such as primitive types, classes and type variables, a type expression also consists of typeoperations.We have two type operations, class conjunction and class instantiation. Since types are representedby rooted graphs, type operations are defined in terms of graph operations. In thissection, we define conjunction graphs and instantiation graphs for class conjunctions and classinstantiations, respectively.The graph operations share a common idea — the result graph is constructed by introducing newstructural nodes and new edges pointing into the operand graphs without changing any existingrooted graphs.Report No. 448, June 2011UNU-IIST, P.O. Box 3058, Macao