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

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An Object Oriented Language 10Table 1: rCos/g Syntaxprogram ::= cdecl • Mainprogramcdecl ::= class Cclass declaration[〈α cnstr〉]type parameters[extends Cc]direct superclass{adef ; mdef }memberscnstr ::=constraints[extends Cu]superclass constraint[implements Cn] member constraintadef ::= visib Te a attribute definitionvisib ::= private | protect | public visibilitymdef ::= m(Te x) {c} method definitionCu ::= Cc | αsuperclass upper boundTe ::=type expressionZ | B | Sprimitive types| CeCe ::= Cc | Cn | αclass expressionsCn ::= ⋓ Ccclass conjunctionCc ::= C | Cjconcrete classesCj ::= C〈α ↦→ Te〉class instantiationc ::=commandskip| var Te x variable declaration| end x variable undeclaration| Cc.new(le) object creation| le := e assignment| e.m(ve • x • re) method invocation| c; c | c ⊳ e ⊲ c | e ∗ c structural commandsve ::= evalue argumentre ::= leresult argumente ::=expressionle | self | null | l| (Ce)e type cast| f(e) built-in function appl.le ::=l-expressionx| e.a object attributeextv ::= Te xexternal variable decl.Main ::= {extv; c}main method definitionAn attribute definition adef consists of a visibility specification visib and an attribute typing.A method definition mdef is a method signature consisting of a method name and a formalparameter typing list, followed by a command as the method body.A type is specified by a type expression Te, which is either a primitive type or a class expression.Class expressions Ce are formed from classes, type variables, class conjunctions and class instantiations.Among class expressions, classes and class instantiations are called concrete classes,Report No. 448, June 2011UNU-IIST, P.O. Box 3058, Macao
Type Graphs 11denoted by Cc. Only concrete classes can have class instances, thus we can only invoke the newcommand on concrete classes. A class conjunction Cn is a collection of member signatures of aset of concrete classes. Class conjunctions are used as class interfaces to describe memberwiserequirements and assumptions. A class instantiation Cj represents a new instance of a genericclass, in the form of the generic class name followed by a substitution of type variables with typeexpressions.A command c in rcos/g is similar to those in common imperative languages. We provide structuralconstructors of commands including sequential compositions c; c, conditionals c ⊳ e ⊲ cand while-loops e ∗ c. We require that commands, including assignments, object creations andmethod invocations, should not occur in expressions, so that the evaluation of expressions has noside-effects. We also explicitly formulate variable declarations and undeclarations as commandsfor local scope introductions and cleanups, respectively.A method invocation specifies the method name and the object upon which the method isinvoked, as well as how the value arguments ve are passed to and how the result arguments reare retrieved from the formal parameters x. Upon entering the method, the value of a valueargument is copied to the corresponding formal parameter. And upon leaving the method, thevalue of a formal parameter is copied to the corresponding result argument. For example, themethod invocationo.m(true, 4 • x, y • p.a, q.b)invokes method m upon object o, passing true to x, 4 to y when entering, and retrieving p.afrom x, q.b from y when leaving.Expressions are formed from variables, attributes and literals through applications of membernavigations e.a, typecasts (Ce)e and primitive-type functions f(e). We denote the null objectby null, and use the special variable self to point to the currently active object. The set ofl-expressions, ranged over by le, includes variables and navigation paths, which are allowed tooccur on the left-hand side of assignments.To simplify our discussion, all variables, parameters and attributes are initialized with the defaultvalue of their types, specifically, 0 for integers, false for booleans, ε (empty) for text strings andnull for all class types.3 Type GraphsThe class declarations cdecl of an oo program define the structure of the objects of the programin terms of their types and relations. This structure can be formally represented by a directedand labeled graph [9], called a type graph. In this paper, we develop a graph-based type systemfor rcos/g. For this, we extend the model in [9] to capture the extended features of the language.Report No. 448, June 2011UNU-IIST, P.O. Box 3058, Macao
Page 1 and 2: UNU-IISTInternational Institute for
Page 3 and 4: UNU-IISTInternational Institute for
Page 5: Contents 5Contents1 Introduction 72
Page 9: An Object Oriented Language 9system
Page 13 and 14: Type Graphs 13aa generic classa sta
Page 15 and 16: Type Graphs 15u ∼ v ∼ w ∼ ru,
Page 17 and 18: Type Operations 17Table 2: Type Equ
Page 19 and 20: Type Operations 194.2 Conjunction G
Page 21 and 22: Type Operations 211. all effective
Page 23 and 24: Type Operations 23Lemma 3 For a typ
Page 25 and 26: Type Operations 25Proof. By Definit
Page 27: Type System 27αaγPairaList⊲Pair
Page 30 and 31: Type System 30c ::= . . . other com
Page 32 and 33: Type System 32For a list x of varia
Page 34 and 35: Type System 34Table 5: Visibility o
Page 36 and 37: Type System 36Table 6: Type-Checkin
Page 38 and 39: Type System 38Table 8: Type-Checkin
Page 40 and 41: Execution Environment 40We extract
Page 42 and 43: Execution Environment 42Table 9: Co
Page 44 and 45: Operational Semantics 44RanulllnkCa
Page 46 and 47: Operational Semantics 46Table 10: O
Page 48 and 49: Operational Semantics 48- Case e is
Page 50 and 51: Operational Semantics 50push(G, (se
Page 52 and 53: References 52References[1] J. Gosli

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

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