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

Execution Environment 42Table 9: Construction of Environment Graphsf e (P, Γ, R) ̂=(attributes) t σ −→ · ∈ Γ isCc(Γ, t) t /∈ R es = {t a −→ ini(n) | a −→ n ∈ clo(Γ t), a ≠ σ}f e (P, Γ, 〈R.ns ∪ {t} ∪ {l | t a −→ l ∈ es}, R.es ∪ es, R.sm〉)(static tables) t ∈ R t σ −→ s ∈ Γ t σ −→ s /∈ R ts = {t ′ | Γ ⊢ t t ′ }f e (P, Γ, 〈R.ns ∪ {s}, R.es ∪ {t σ −→ s}, R.sm ∪ {s ↦→ ts}〉)t −→ σ s ∈ R −−→ σ.m· ∈ clo(Γ t) t −−→ σ.m· /∈ R (Γ ′ , k) = insc(Γ, mk(P, Γ, t, m), ssj (Γ, t))(methods)f e (P, Γ ′ , 〈R.ns ∪ {k}, R.es ∪ {s −→ m k}, R.sm〉) R ,where isCc(Γ, t) ̂= t-var(Γ, t) = ∅ ∧ t ∈ U ∨ t-var(Γ, t) ≠ ∅ ∧ t ∈ S is a condition for t to be a concreteclass node.In (attributes), the function f e builds the class frame that represents the structure of the initialobject for each concrete class node t in Γ. It adds to the environment graph the type nodesrepresenting t, the default values of all attributes, and the edges labeled by attributes outgoingfrom t to the default values. We use ini(t ′ ) to denote the default value of a type node t ′ , i.e.,ini(Z) = 0, ini(B) = false, ini(S) = ε and ini(C) = null for any class type C. In (static tables),f e adds the static tables from Γ to the environment graph, and associates each static table withthe set of supertype nodes. Finally, f e constructs outgoing edges from static tables labeled bymethod names, targeting at the instantiated body commands of the methods respectively, asshown in (methods).In order to get the instantiated method body commands, we introduce two functions. First, fora class instantiation j ∈ Γ, when we instantiate a command for j, the type substitution can berecovered from the edges labeled by type variables in the generic static table of j. The functionssj returns the type substitution. Formally,ssj (Γ, j) ̂= 〈α ↦→ t | α ∈ X , j σ.α −−→ t ∈ Γ〉.Second, for method body commands, we define a function mk to lookup in program P for bodycommand c of method m of type t. If t is a class instantiation, we track for its template genericclass in Γ. Formally, mk(P, Γ, t, m) ̂= c, where⎧⎪⎨ k ′ if ∃t :: m{k ′ } ∈ P,c = mk(P, Γ, t⎪⎩′ , m) if ∃t ′ = gcls(Γ, t),mk(P, Γ, t ′ , m) otherwise, if ∃t −→ ⊲ t ′ ∈ Γ.When t is directly defined in P , mk returns the body command by the method definition, otherwiseit looks up the command in the template generic class or the superclass. Thus, for a typeReport No. 448, June 2011UNU-IIST, P.O. Box 3058, Macao