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

Operational Semantics 44RanulllnkCalnknullxfalsetruebblnkb a iσ σσ...σσAbbfalsez$y0x $i0Figure 8: A State Graph and a Corresponding Environment GraphG6. a static table in G must also be a static table in dom(R.sm).A state graph G is shown as solid lines in Fig. 8.Besides extending the stack operations on type contexts to state graphs, we define the swingoperation to change the targets of a list of edges in a state graph. Formally, given G = 〈N, E, r〉,d = n ′ a −→ · ∈ E, and n ∈ N ∪ L , we define swing(G, d, n) ̂= 〈N ′ , E ′ 〉 r, whereN ′ = N ∪ {n}, E ′ = E n ′ a −→ n.The M e map operation sequentially adds mappings e to M, and overrides the existing mappings.Formally,M (a ↦→ b, s ↦→ t) ̂= ({x ↦→ y ∈ M | x ≠ a} ∪ {a ↦→ b}) s ↦→ t.7.2 Execution Environment LookupThe static table nodes connect objects to their runtime class information in the environmentgraph. Via static tables, we can lookup superclasses, method names and method body commands.A static table is introduced to a state graph by object creation, which can be simplified as shallowcopying all the outgoing edges of a type node in the environment graph to the state graph withthe substitution of a new object node for the type node.Besides the static table of an object, we can also access the environment graph through theenvironment dependent ⌈t⌋.new command. Given an environment graph R, a type node t ∈ R,a state graph G and an edge d ∈ G, we define the function new to add a new instance of t tothe state graph and swing d to the instance. Formally, given G = 〈N, E, r〉 and d = n a −→ · ∈ E,we define new(R, G, t, d) ̂= 〈N ′ , E ′ 〉 r, whereN ′ = N ∪ {o} ∪ {l | t b −→ l ∈ R},E ′ = E ∪ {o b −→ l | t b −→ l ∈ R} n a −→ o,o ∈ O N.Report No. 448, June 2011UNU-IIST, P.O. Box 3058, Macao