Pizza is an extension to Java based on F-bounded polymorphism [Odersky & Wadler 97]. Like Cecil, Pizza supports classes with mutually recursive bounds, crucial for supporting interrelated families of classes such as the
view example from section 4.4. Also like Cecil, Pizza automatically infers instantiating type parameters of polymorphic methods and constructors, although the instantiating parameters must match the actual argument types exactly, which is more restrictive than Cecil which can infer appropriate supertypes of the argument types. Pizza lacks signature constraints and the resulting implicit structural subtyping. Pizza does not support any subtyping between different instances of a parameterized type, such as the desirable and legal subtyping between different read-only interfaces to collection types as in our
i_vector example. Pizza also inherits several restrictions from its Java base, including that it does not allow contravariant method overriding. Pizza extends Java with first-class, lexically nested functions and with algebraic data types and pattern-matching. The authors justify introducing algebraic data types by claiming that classes allow new representations to be added easily but not new operations, while algebraic data types support the reverse. Cecil's multi-methods enable both new representations and new operations to be added easily, avoiding the need for new language constructs.
Bruce, Odersky, and Wadler [Bruce et al. 98] recently proposed to extend Pizza with special support for declaring families of mutually recursive classes. They argue that pure F-bounded polymorphism is too cumbersome for programmers to use in practice. We have not found pure F-bounded polymorphism to be untenable, however; the
view example from section 4.4 illustrates our approach. Our experience may be better than theirs because our multi-method framework encourages us to treat each argument and parameter symmetrically and uniformly, while their model is complicated by the asymmetry between the implicit receiver and the explicit arguments. Nevertheless, we are working on syntactic sugars that would make the more sophisticated uses of F-bounded polymorphism simpler.
Agesen, Freund, and Mitchell propose a similar extension to Java [Agesen et al. 97]. It differs from Pizza and Cecil in being able to parameterize a class over its superclass. However, this feature cannot be typechecked when the abstraction is declared, but instead must be rechecked at each instantiation.
Haskell's type classes can be viewed as a kind of F-bounded polymorphism [Wadler & Blott 89]. Haskell automatically infers the most-general parameterization and constraints on functions that take polymorphic arguments, as well as automatically inferring instantiations on calls to such functions; Cecil requires polymorphic methods to explicitly declare type variables and constraints over these variables. (In some cases, Haskell cannot unambiguously infer instantiations.) However, Haskell is not truly object-oriented, in that after instantiation, no subtype polymorphism remains; values of different classes but a common supertype cannot be mixed together at run-time, preventing for instance lists of mixed integers and floats.
ML£ is a powerful polymorphic object-oriented language supporting multi-methods [Bourdoncle & Merz 97]. ML£ supports subtyping directly, but treats inheritance as a separate syntactic sugar (which must follow the subtyping relation). Similarly to Cecil, ML£ constrains type variables using sets of potentially recursive subtype constraints, supports inference of type parameters to methods, and supports least-upper-bound type expressions (although not greatest-lower-bound type expressions). ML£ also supports parameterization over type constructors, while in Cecil type constructors must be instantiated before use. ML£ supports explicit declarations of co- and contravariant type parameters of type constructors, while Cecil uses polymorphic subtype declarations to achieve more general effects. ML£ only allows subtyping between types in the same type constructor "class," however, which for instance restricts subtyping to be between types with the same number of type parameters with the same variance properties, and ML£ does not support other forms of constrained subtyping, conformance, or inheritance. Cecil supports multiple polymorphic signature declarations for the same message, while ML£ allows only a single signature declaration per message. ML£ is purely functional and side-effect-free.