An algebraic semantics approach to the effective resolution of type equations

Type Systems for Programming Languages

One purpose of type checking in programming languages is to guarantee a degree of "representation independence:" programs should not depend on the way stacks are represented, only on the behavior of stacks with respect to push and pop operations. In languages with abstract data type declarations, representation independence should hold for user-defined types as well as built-in types. We study the representation independence properties of a typed functional language (second-order lambda calculus) with polymorphic functions and abstract data type declarations in which data type implementations (packages) may be passed as function parameters and returned as results. The type checking rules of the language guarantee that two data type implementations P and Q are equivalence whenever there is a correspondence between the behavior of the operations of P and the behavior of the operations of Q.

Representation independence and data abstraction

Dedicated to the memory of Edsger W.Dijkstra. 
Representation independence or relational parametricity formally characterizes the encapsulation provided by language constructs for data abstraction and justifies reasoning by simulation. Representation independence has been shown for a variety of languages and constructs but not for shared references to mutable state; indeed it fails in general for such languages. This paper formulates representation independence for classes, in an imperative, object-oriented language with pointers, subclassing and dynamic dispatch, class oriented visibility control, recursive types and methods, and a simple form of module. An instance of a class is considered to implement an abstraction using private fields and so-called representation objects. Encapsulation of representation objects is expressed by a restriction, called confinement, on aliasing. Representation independence is proved for programs satisfying the confinement condition. A static analysis is given for confinement that accepts common designs such as the observer and factory patterns. The formalization takes into account not only the usual interface between a client and a class that provides an abstraction but also the interface (often called ``protected'') between the class and its subclasses.

Ownership Confinement Ensures Representation Independence for Object-Oriented Programs

A theory of data abstraction in modular programming is presented that explains why this technique leads to correct programs. Data abstraction allows users and implementers to take different views of a specification: While users may depend on a specification as it is, implementers need not provide pràgram entities that satisfy the specification, but merely a "representation" of such entities. This means that the users may be supplied with program entities that do not satisfy the specification, and so an explanation is needed that their code functions correctly nevertheless. it is shown that data abstraction leads to correct programs if the modules of the programs are "stable", and it is suggested that programming languages for data abstraction should guarantee stability of their modules. The stability criterion corresponds closely to the intuitive idea of "limited access" to encapsulated data types, and to "representation independence" properties of the typed A-calculus. The theory is developed in the general framework of an "institution" and uses an abstract notion of "representation". Specifically, the institution of partial many-sorted algebras is considered and the representation relations "behavioural inclusion", "behavioural equivalence" and "standard representation" (the popular concept based on abstraction functions) are studied. These relations are characterized by certain kinds of relations between algebras, called "correspondences", which provide useful practical proof methods for the correctness of data representations. Behavioural equivalence is found to be superior to standard representation in that "representation bias" of a specification no longer restricts its range of implementations, and in that it allows more constructs to be included in a data abstraction programming language.

Data abstraction and the correctness of modular programming

This paper considers the general problem of specifying the meaning of programming languages which include "data type definition facilities". The fundamental question posed in attempting to define such languages is: "what meaning should be given to a data type definition," or more simply, "what does data type mean?". In this paper we describe a new approach to defining the meaning of data types and give its application to the definition of a typed lambda calculus extension. We also prove a theorem stating that our language is "strongly typed".

On the Semantics of Data Type

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On the Semantics of Data Type