By Martín Abadi, Luca Cardelli
Procedural languages are usually good understood and their formal foundations forged within the varieties of quite a few lambda-calculi. For object-oriented languages but the state of affairs isn't as simple. during this publication the authors suggest and increase a unique process by means of constructing item calculi within which items are taken care of as primitives. utilizing item calculi, the authors may be able to clarify either the semantics of items and their typing principles and exhibit how you can advance the entire most crucial thoughts of object-oriented programming languages: self, dynamic dispatch, sessions, inheritance, secure and personal tools, prototyping, subtyping, covariance and contravariance, and strategy specialization. Many researchers and graduate scholars will locate this a huge improvement of the underpinnings of object-oriented programming.
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To explain these techniques, we first review the basic subtyping properties of product and function types. Let us consider product types. The type AxB is the type of pairs with left component of type A and right component of type B. The operations fst(c) and snd(c) extract the left and right components, respectively, of an element c of type AxB. We say that x is a covariant operator (in both arguments), because AxB varies in the same sense as A or B: AxB <: A'xB' provided that A <: A' and B <: B' We can justify this property as follows.
5 Object Protocols Even when subclasses do not induce subtypes, we can find a relation between the type induced by a class and the type induced by one of its subclasses. It just so happens that, unlike subtyping, this relation does not enjoy the subsumption property. We now examine this new relation between object types. 4, we cannot usefully quantify over the subtypes of Max because of the failure of subtyping. A parametric definition such as: ObjectOperator P[M <: Max] is ... end; is not very useful; we can instantiate P to P[Max], but P[MinMax] is not well-formed.
Subclassing still implies subtyping, so all the previous uses of subsumption are still allowed. But, since subsumption is based on sub typing and not subclassing, we now have even more freedom in subsumption. In conclusion, the notion of subclassing-is-subtyping can be weakened to subclassing-implies-subtyping without loss of expressiveness, and with a gain in separation between interfaces and implementations. 3 Type Parameters Type parameterization is a general technique for reusing the same piece of code at different types.