Bernd Mahr

ALGEBRAIC IMPLEMENTATION OF ABSTRACT DATA TYPES

Abstract Starting with a review of the theory of algebraic specifications in the sense of the ADJ-group a new theory for algebraic implementations of abstract data types is presented. While main concepts of this new theory were given already at several conferences this paper provides the full theory of algebraic implementations developed in Berlin except of complexity considerations which are given in a separate paper. The new concept of algebraic implementations includes implementations for algorithms in specific programming languages and on the other hand it meets also the requirements for stepwise refinement of structured programs and software systems as introduced by Dijkstra and Wirth. On the syntactical level an algebraic implementation corresponds to a system of recursive programs while the semantical level is defined by algebaic constructions, called synthesis, restriction and identification . Moreover the concept allows composition of implementations and a rigorous study of correctness. The main results of the paper are different kinds of correctness criteria which are applied to a number of illustrating examples including the implementation of sets by hash-tables. Algebraic implementations of larger systems like a histogram or a parts system are given in separate case studies which, however, are not included in this paper.

Characterizing specification languages which admit initial semantics

Abstract The paper proposes an axiomatic approach to specification languages, and introduces notions of reducibility and equivalence as tools for their study and comparison. Algebraic specification languages are characterized up to equivalence. They are shown to be limited in expressive power by implicational languages.

Information science and the logic of models

ion is a fundamental human capacity that permits us to deal with complexity . . . We build models of complex systems because we cannot comprehend such systems in their entirety. There are limits to human capacity to understand complexity . . .Models help us organize, visualize, understand 34 The view of entity as ,something of interest’ is not consistent with established ontology, according to which entities are something which does not necessarily depend on the presence of a conceiving subject. However, in the model of conception on which the model of modelbeing presented here is based, this dependence of entities on a conceiving subject is absolutely necessary (cf. also footnote 33).

Characterizing Specification Languages Which Admit Initial Semantics

The paper proposes a unified approach to study and compare specification techniques and languages. It is devided into two parts: (1) A general notion of specification language is given which provides an axiomatic framework for the various known specification techniques. (2) Algebraic specification languages which admit initial semantics are characterized up to equivalence. They are shown to be exactly the implicational languages.

Introduction to algebraic specification. Part I: formal methods for software development

The intention of this part 1 of an overview paper on algebraic specifications is an informal introduction to formal methods for software development in general and to applications of algebraic specifications in particular. Horizontal structuring and vertical refinement techniques for algebraic specifications are shown to support the general software development process. Moreover, a short overview of case studies and tools in the ESPRIT projects LOTOSPHERE and PROSPECTRA is given

Iteration and Summability in Semirings

An algebraic treatment of various questions in linear algebra, automata and formal language theory, combinatorial optimization, data flow analysis, graph theory, semantics and others has exhibited the importance of transitive closure over semirings and the broad applicability of Gauss-Jordan elimination method. In view of these applications we study the solvability of equations of the form x=l+ax, called iteration, and introduct a concept of summability in positive semirings.

Applications of Type Theory

The paper discusses the setting of type theory and proposes a uniform framework for disciplines of declarations and types. The framework combines fundamental concepts found in the setting of type theory and supports both its denotational and its constructive view. In the denotational view e-structures are introduced as semantic models and in the constructive view the general forms of judgements and rules are given. Finally applications arc discussed which build on the ideas of this framework, namely models of untyped λ-calculus which truly solve the equation of reflexive domains, and eT-logic, a first-order theory of propositions with selfrcfcrcncc and imprcdicative quantification allowing for intcnsional models of truth.

Complexity of algebraic implementations for abstract data types

Abstract A notion of complexity for algebraic implementations of abstract data types is introduced and studied. The main results concern the expressive power of algebraic specifications and implementations as well as upper and lower bounds on the complexity of implementations in terms of time on Turing Machines.

Compound Algebraic Implementations: An Approach to Stepwise Refinement of Software Systems

A precise mathematical approach to stepwise refinement of software systems is given within the framework of algebraic specifications. Since our new concept for the implementation of abstract data types — recently introduced in another paper — corresponds to a single refinement step, the composition problem for algebraic implementations is studied in this paper. It is shown that in general algebraic implementations are not closed under composition unless we have the special case of “persistent implementations”. For other types of implementations sufficient consistency conditions are given to achieve closure under composition. These results can be extended to compound algebraic implementations which are syntactically defined to be sequences of (weak) implementations according to the idea of stepwise refinement of software systems.

Die Informatik und die Logik der Modelle

Modelle in der Informatik Es gibt verschiedene Weisen eine Wissenschaft zu verstehen: uber die wissenschaftlichen Gemeinschaften2, die sie pflegen, uber die Fragen, an deren Antworten sie ihren wissenschaftlichen Anspruch stellt, uber die Gegenstande, auf die sie sich bezieht, uber die Methoden undWerkzeuge, die sie verwendet, uber die Techniken der Begrundungen, die sie gibt, durch ihre Einordnung in eine akademische Klassifikation, durch ihre Aufteilung in abgegrenzte Disziplinen, durch die Analogie zu einer anderen schon bekannten Wissenschaft, oder auch durch die masgebende Zielsetzung, an der sie sich orientiert. Die Unterschiede dieser Verstehensweisen liegen in der Verschiedenheit der Perspektiven, durch die jeweils andere Aspekte ihrer Erscheinung in den Vordergrund treten. Das zeigt sich auch,