F-Bounded Polymorphism for Object-Oriented Programming Peter Canning, William Cook, Walter Hill, Walter Oltho® Hewlett-Packard Laboratories P.O. Box 10490, Palo Alto, CA 94303-0971 John C. Mitchell Department of Computer Science Stanford University, Stanford, CA 94309 Abstract ing was identi¯ed as an important form of polymor- phism in object-oriented programs. This lead to the Bounded quanti¯cation was introduced by Cardelli and development of \bounded quanti¯cation" in [7]. If we Wegner as a means of typing functions that operate uni- view objects as elements of non-recursive record types, formly over all subtypes of a given type. They de¯ned then bounded quanti¯cation provides a useful form of a simple \object" model and used bounded quanti¯ca- polymorphism over objects. However, a more sophis- tion to type-check functions that make sense on all ob- ticated presentation of object-oriented constructs (as jects having a speci¯ed set of \attributes." A more re- in [5, 11, 10]) would allow objects that are elements alistic presentation of object-oriented languages would of recursively-de¯ned types. With recursive types, the allow objects that are elements of recursively-de¯ned Cardelli-Wegner form of bounded quanti¯cation is not types. In this context, bounded quanti¯cation no longer su±ciently expressive to meet its original goal. serves its intended purpose. It is easy to ¯nd func- F-bounded quanti¯cation is a natural extension of tions that makes sense on all objects having a speci- bounded quanti¯cation that seems particularly useful ¯ed set of methods, but which cannot be typed in the in connection with recursive types. The essential idea Cardelli-Wegner system. To provide a basis for typed may be illustrated by comparison with Cardelli-Wegner polymorphic functions in object-oriented languages, we bounded quanti¯cation. Using \⊆" for the subtype re- introduce F-bounded quanti¯cation. Some applications lation, a simple example of a bounded-quanti¯ed type of F-bounded quanti¯cation are presented and seman- is the type 8t ⊆ ¿: t ! t. This is the type of func- tic issues are discussed. Although our original moti- tions which map t to t, for every subtype t of ¿. In vation was to type polymorphic functions over objects, a setting where all structurally similar objects belong F-bounded quanti¯cation is a general form of polymor- to subtypes of a given type, many useful polymorphic phism that seems useful whenever recursive type de¯n- functions will have bounded quanti¯ed types. For ex- itions and subtyping are used. ample, if we type an object by listing its methods and their types, an object with a print method may have type f:::; print: void ! string;:::g, indicating that the 1 Introduction method print produces a print representation of the object. In the view of subtyping presented in [8], Although object-oriented programming has attracted every type of this form will be a subtype of the type increasing interest in recent years, the development fprint: void ! stringg of objects having only a print of polymorphic type systems for object-oriented lan- method. For example, guages has progressed slowly. One reason is that object- oriented languages are often described using terminol- fA: int ! void; print: void ! stringg ogy that sets them apart from functional languages. In ⊆ fprint: void ! stringg: addition, there has been a lack of formal models for object-oriented languages. As a result, it has been A function taking a list of printable objects, a string and di±cult to see how practical polymorphic type sys- returning an object whose print representation matches tems should be adapted for typing object-oriented con- the given string will have the bounded-quanti¯ed type structs. In Cardelli's seminal paper [8], record subtyp- 8t ⊆ fprint: stringg: list[t] ! string ! t. 1 The flexibility of bounded quanti¯cation is dramati- may be imposed on the quanti¯ed type variable. Thus cally reduced when objects belong to recursively-de¯ned Curtis' system subsumes our more modest proposal. types. For example, consider the type However, we believe that our form of quanti¯cation is both su±cient to type-check practically useful object- PartialOrder = f lesseq: PartialOrder ! Bool g. oriented programs, and more tractable. Some evidence for the naturality of F-bounded quanti¯cation is the Each object of this type has a method lesseq which may intriguing connection with F-coalgebras, the dual of a be applied to another object of the same type. We in- standard construction in the category-theoretic charac- tend the result of x:lesseq(y) to be true if x does not terization of type recursion [22]. exceed y in some ordering, and false otherwise. Using The next section surveys relevant background on lesseq, it should be possible to sort lists of PartialOrder strongly-typed object-oriented programming. Section 3 elements, for example. We may easily write a sort- illustrates in detail problems with previous techniques ing function with type 8t ⊆ PartialOrder: list[t] ! list[t]. for dealing with recursive object types. Section 4 intro- However, as we shall see later in the paper, object- duces F-bounded quanti¯cation and demonstrates that oriented versions of the usual partially-ordered types it solves the typing problems presented in the previous such as integers are not subtypes of PartialOrder. There- section. some speculative observations on the semantic fore, our generic sorting function cannot be used in the aspects of F-bounded quanti¯cation. Section 6 summa- way we might expect. To solve this problem and related rizes our contribution and indicates directions for future di±culties with other forms of recursive types we intro- research. duce a generalization of bounded quanti¯cation which we call F-bounded quanti¯cation, for want of better ter- minology. For the example at hand, consider the func- 2 Background tion F from types to types given by 2.1 Objects, Records and Recursive F[t] = f lesseq: t ! bool g. Types We may write a polymorphic sorting function of type A fundamental premise underlying most models of 8t ⊆ F[t]:list[t] ! list[t], where this function is de¯ned on object-oriented programming is that objects may be re- any type t such that t ⊆ F[t]. Since integer ⊆ F[integer], garded as record whose components are functions rep- as explained in Section 3.3, we may apply such a sorting resenting methods [8, 7, 25, 11, 5]. In this model, \mes- function to lists of integers or lists of any other type of sage sending" is implemented by simple component se- objects having lesseq methods. lection. One practical application of F-bounded quanti¯cation Record types are used to describe the protocols [14] is for understanding some problems in the type system or interfaces of objects. A record type is a map- of the Ei®el programming language [18]. Ei®el intro- ping of labels to types. A record consisting of labels duces a special type expression, like Current, to represent l1; : : : ; lj with associated values in types σ1; : : : ; σj has recursion at the type level. Like Current always repre- type fl1: σ1; : : : ; lj: σjg. The ¯elds of the record type sents the `current class': in a class P it simply stands describe messages together with the types of their ar- for P , but when this class is inherited by a subclass C, guments and return values. This view is adopted in the like Current in an inherited attribute is reinterpreted to programming languages Amber [6], Modula-3 [10], and stand for the subclass C. The subclasses of such a class TS [17]. are usefully understood as having interfaces that sat- Even the very simplest \textbook" examples isfy an F-bound. However, as mentioned above, there from object-oriented programming produce recursively- is no necessary subtype relation among types satisfying de¯ned record types [21, 1, 6, 16, 10]. For example, an F-bound. This analysis explains some insecurities in the type of a planar point with four methods is de¯ned the Ei®el type-system, which always allows subclasses recursively as follows. to act as if they are subtypes of their parents. Further Point = f details are given in [12], along with some suggestions for x : void ! Real, correcting Ei®el's problems using the results in this pa- y : void ! Real, per. The error in the Ei®el type system illustrates the move : Real £ Real ! Point, subtlety involved in designing flexible and sound type equal : Point ! Boolean systems for object-oriented languages. g Another type system with a generalized form of bounded quanti¯cation is presented by Curtis [13]. In The body of the recursive type is a record type indi- this system, arbitrary collections of subtype constraints cated by braces f:::g. The use of Point in the return 2 type of move indicates that the move method returns This means, informally, that if assuming s is a sub- an object that has the same type as the original point. type of t allows one to prove that σ is a subtype of ¿, Its use as the argument type of equal indicates that the then the recursive type Rec s: σ is a subtype of Rec t: ¿. equal method takes an argument with the same type. For example, the following type is a supertype of Point: In other words, equal is a `binary' operation on points, viz. for two points p and q, the expression p.equal(q) is Movable = Rec mv. f move : Real £ Real ! mv g meaningful. because the body of Point is a subtype of the body of A useful notation for recursively de¯ned types is the Movable, given the assumption pnt ⊆ mv: form Rec t:A .