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

Operational Semantics 47The transition rules for assignment, object creation, variable declaration and undeclaration aredefined by the simple graph operations of edge swing, new instance adding, stack push and pop.By contrast, the rule for method invocation is more dedicate and deserves some explanation.In a method invocation, a result argument re is an l-expression. We have to remember the l-valueof re initially before it is possibly changed during the invocation. If re = e.a is a navigationpath, we call the object referred to by e the parent object of the attribute a. Our approach isto assign this parent object po(G, re) to an auxiliary variable x ∗ when entering a method. Andwhen exiting the method, we recover the initial l-value of re by spo(G, re, q) that returns theoutgoing edge labeled by a of the parent object q retrieved from x ∗ . For a variable w, the notionof parent object is not significant since its l-value cannot be changed. For unification, however,we define the parent object of w as the null object.7.5 Type-Safety of ProgramsThe type-safety of a program ensures that a well-typed expression can be evaluated and a welltypedcommand can be executed. We reason about the type-safety by showing that given awell-typed state graph, there exists a semantic rule which applies, and that the resulted stategraph of the semantic rule is also well-typed.There are exceptional cases when an expression cannot be evaluated, or a command cannot beexecuted.1. Null reference: the evaluation of an expression e.a or the execution of a command e.m(. . .)fails, if e is evaluated to null.2. Illegal downcast: the evaluation of an expression (⌈t⌋)e fails, if e is evaluated to a node nand the type of n is not a subtype of t.These two cases can not be checked statically.Lemma 12 Let Γ be a type graph, ∆ a type context, e an expression, and G a well-typed stategraph w.r.t. (Γ, ∆). If Γ, ∆ ⊢ e : t, we have1. n = eval(G, e) exists and Γ ⊢ Γ(G, n) t, and2. if e is an l-expression, l-eval(G, e) exists,unless one of the exceptional cases happens.Proof. The proof is given by induction on the structure of the expression e. Suppose none of theexceptional cases happens.Report No. 448, June 2011UNU-IIST, P.O. Box 3058, Macao