Data Abstraction and General Recursion

The theory of algebraic specifications-one of the most important mathematical approaches to the specification of abstract data types and software systems-is reviewed from a mathematical and a computer science point of view. The important role of category theory in this area is discussed and it is shown how the following selected problems are treated using category theory: First, a unified framework for specification logics, second compositional semantics, third partial algebras and their specification, and fourth specifications and models for concurrent systems. For the solution of two of the problems classifying categories are used. They allow to present categories of algebras as functor categories and to derive a number of important properties from well known results for functor categories.

[1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science, 1991

The confluence of the union of two sets of rewrite rules which have no function symbols in common follows from the confluence of each set [24], a property called modularity of confluence. The confluence of the combination of the typed lambda calculus with first-order rewrite rules is also modular, although both systems share implicitely the application operator [3]. In this case, the termination property is modular as well [4,19]. These results extend to the combination of polymorphic lambda-calculus with (firstorder) rewrite rules [4,19]. In this paper, we consider the combination of (polymorphically) typed lambdacalculi with first-order as well as higher-order rewrite rules. We show the need of such a combination for exploiting the benefits of algebraically defined data types within functional programming. We prove a general modularity result, which allows as particular cases primitive recursive functionals of higher types, transfinite recursion of higher types as well as inheritance for all types. The proof depends on the only assumption that the original typed functional language has a termination proof based on the use of a Girard and Tait's computability predicate. 'This work was partly supported by the "Greco de programmation du CNRS", NSERC (Canada) and FCAR (QuCbec). It was achieved thanks to several visits of the authors to the other's institution under the France-Quebec Cooperation Program

Journal of Computer and System Sciences, 1987

Implementations of abstract data types are defined via enrichments of a target type. We propose to use an extended typed I-calculus for enrichments in order to meet the conceptual requirement that an implementation has to bring us closer to a (functional) program. Composability of implementations is investigated, the main result being that composition of correct implementations is correct if terminating programs are implemented by terminating programs. Moreover, we provide syntactical criteria to guarantee correctness of composition. The proof is based on strong normalization and Church-Rosser results of the extended I-calculus which seem to be of interest in their own right. q" IMPLEMENTATION OF ABSTRACT DATA TYPES 341 algebraic specifications. There are two conceptual requirements we believe to be important:

Mathematical Structures in Computer Science, 1996

In this paper we try to shed some light on the similarities and differences in the different approaches denning the notions of implementation and implementation correctness. For obvious reasons, we do not discuss all existing approaches individually. Instead, a formal framework is introduced in order to discuss the most important ones. Additionally, we discuss some issues, which in our opinion are often misunderstood, concerning transitivity of implementation correctness and its role in the software development process. In particular, on the one hand, we show that for reasonable notions of implementation, it is almost impossible to prove transitivity of implementation correctness at the specification level. On the other hand, we show that this is not really important if the programming language satisfies the properties of horizontal and vertical composition.

Lecture Notes in Computer Science, 1983

A kernel specification language called ASL is presented. ASL comprises five fundamental but powerful specificationbuilding operations and has a simple semantics. Behavioural abstraction with respect to a set of observable sorts san be expressed, and (recurstve) parameterisad specifications can be defined using a more powerful and more expressive parameterisation mechanism than usual. A simple notion of implementation permitting vertical and horizontal composition (i.e. it is transitive and monotonic) is adopted and compared with previous more elaborate notions. A collection of identities is given which san provide a foundation for the development of programs by transformation.

Theoretical Computer Science, 1981

h&act. Thz question of implementability and expressive power of equational axiom definitions of data abstr:&ons is faced in the paper from the point of view of computability theory. A definition of implementable algebra is given, which looks reasonable and very general. With respect to the given definition it is proved tha:, if the least congruence semantics is accepted, an equationally defined data algebra is implemertable if and only if the least congruence on terms induced by the equational definition is decida5le. Moreover, the paper shows that there are: (a) equationally defined data algebras that cannot be implemented; (b) implementable algebras th,at cannot be expressed in any way by equational axioms.. Inlroduction

This paper presents in an informal way the main ideas underlying our work on algebraic speci cation. The central idea, due to Goguen and Burstall, is that much work on algebraic speci cation can be done independently of the particular logical system (or institution) on which the speci cation formalism is based. We also examine the nature of speci cations and speci cation languages, the problem of proving that a statement follows from a speci cation, the important notion of behavioural equivalence, and the evolution of programs from speci cations by stepwise re nement. Although many of the issues discussed are motivated by technically complicated problems, in this paper the technicalities have been suppressed in an attempt to make the ideas more accessible. The same ideas are presented with full technical details in ST 85c]. We assume that the reader is convinced as we are that formal speci cations are not only theoretically interesting but are also practically important. Throughout the paper we also assume some familiarity with the basic concepts of algebraic speci cation, although we do not rely on any speci c technical knowledge. Many of the ideas expressed here were evolved under the in uence of Rod Burstall and Martin Wirsing, but this remains a personal statement.

Acta Informatica, 2006

The use of abstraction in the context of abstract data types, is investigated. Properties to be checked are formulas in a first order logic under Kleene's 3-valued interpretation. Abstractions are defined as pairs consisting of a congruence and a predicate interpretation. Three types of abstractions are considered, ∀∀, ∀∃, and ∃ 0,1 ∀, and for each of them corresponding property preservation results are established. An abstraction refinement property is also obtained. It shows how one can pass from an existing abstraction to a (less) finer one. Finally, equationally specified abstractions in the context of equationally specified abstract data types are discussed and exemplified.

Lecture Notes in Computer Science, 1980

Lecture Notes in Computer Science, 1980