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

Type System 31∆ 1 ∆ 2 = pop(∆ 1 ) push (∆ 2 , (y, z), (A, B))x zz$$lnk Blnk BlnkB ybxb ybx$b ybx$b y $CaA ZC A ZiaC A Zia iFigure 6: Stack Push and Popbe ill-typed (undefined) with respect to Γ. To address this issue, we introduce a graph notationof type context that describes the the scopes of local variables. A type context graph consistsof nodes representing scopes, and edges labeled by the variables from their scope nodes to thecorresponding type nodes in the underlying type graph.Let O be a new infinite set of nodes disjoint with N .Definition 15 (Type Context) A type context is a rooted, directed and labeled graph ∆ =〈N, E, r〉, where– N ⊆ N ∪ O is the set of nodes, containing type nodes N ∩ N and scope nodes N ∩ O,– E : N × (A ∪ {$, self }) → N is the set of edges,– r ∈ N ∩ O is the root of the graph.A type context ∆ = 〈N, E, r〉 is well-formed if the following conditions hold,1. each outgoing edge from a scope node is labeled by a variable (or self ), and targets at a typenode,2. all the scope nodes are on a path starting from the root r, with edges labeled by $, and3. except the root r, which has no incoming edge, each scope node has exactly one incomingedge.In Fig. 6, the dashed subgraphs are examples of type contexts.A type context represents a snapshot of the type environment at a time of type-checking, recordingthe types of variables, including self , declared in different scopes at that time. A type contexthas a stack structure. When the type-checking enters a new scope, a node with outgoing edgesrecording the variables in the scope is pushed onto the top of the stack. When the type-checkingexits a scope, the top node of the stack, together with its outgoing edges, is popped out, so thatthe type context recovers to the one exactly before entering the scope.Report No. 448, June 2011UNU-IIST, P.O. Box 3058, Macao