Walter Dosch

On hierarchies of abstract data types

SummaryHierarchical abstract data types are algebraic specifications of computation structures where certain sorts, function symbols, and axioms are designated as being primitive. On hierarchical abstract data types additional structure is imposed. An algebraic specification is thus decomposed into several well-separated levels, such that both the understanding and the independent implementation of the levels is supported. This paper provides both model-theoretic and deduction-oriented conditions guaranteeing the soundness of a hierarchical specification. Furthermore necessary and sufficient conditions for the existence of initial and terminal models are investigated, and their close connection to the soundness of a hierarchy is demonstrated. In order to provide freedom and flexibility for specifications a wide class of axioms — namely universal-existential formulas — are admitted.

Existential Quantifiers in Abstract Data Types

Hierarchies of abstract data types are specified by axioms which are positive formulas consisting of universally and existentially quantified disjunctions and conjunctions of equations. Necessary and sufficient conditions for the existence of terminal algebras are investigated. Furthermore, some advantages of disjunctions and existential quantifiers within the laws are discussed and the usefulness of terminal algebras is demonstrated by a few examples.

On the Algebraic Specification of Domains

We explore the possibilities for the algebraic specification of semantic domains by continuous abstract types, that is, by abstract types with monotonicity, continuity and completeness constraints. In this framework, a domain results from the ideal completion of a term-generated algebra constituting its finite part. Then by construction, all finite elements of a domain can be denoted by finite terms. These finite elements are specified algebraically by conditional order relations describing the characteristic properties of the operations manipulating them. Properties of infinite elements can be inferred by continuity from the properties that hold for their finite approximations. The approach is illustrated by giving algebraic specifications of various domains and domain constructions.

Polynomials - The Specification, Analysis and Development of an Abstract Data Type

Abstract data types provide a powerful tool for the representation independent specification of data structures. This paper applies theoretical results to a simple example and demonstrates how to work with this algebraic specification technique: Polynomials are defined by a hierarchical type. For this specification the class of all models is analyzed and properties of initial and terminal models are discussed.

An Algebraic Semantics for Bachus' Functional Programming Language with Infinite Objects

For a variant of Backus’ functional programming language FP using non-strict operations a structured algebraic specification is given. First the basic data structure of finite nested sequences, i.e. trees with arbitrary branching, is generalized to infinite trees by allowing non-strict constructor functions. Then for the language constructs an extensional semantics is defined, and a matching operational semantics is outlined. At each level of the hierarchical specification the class of semantic models is analysed. This axiomatic definition clarifies the foundations of Backus’ algebra of functional programs. Moreover, the essential concepts of the language are clearly separated from notational extensions.

Calculating a Functional Module for Binary Search Trees

We formally derive a functional module for binary search trees comprising search, insert, delete, minimum and maximum operations. The derivation starts from an extensional specification that refers only to the multiset of elements stored in the tree. The search tree property is systematically derived as an implementation requirement.

Zur Didaktik der Datenstrukturen

Die vorliegende Arbeit beschreibt grundlegende fachdidaktische Ansatze zur funktionalen Behandlung von Datenstrukturen in der Sekundarstufe II. Der Vorgehensweise „vom Problem zur Maschine“ folgend, wird eine Einfuhrung maschinenunabhangiger Grundbegriffe skizziert und der Bezug zur Programmiermethodik hergestellt.

GOTOs — A Study in the Algebraic Specification of Programming Languages (Extended Abstract)

During recent years the concept of algebraic specification (cf. e.g. /Guttag 75/, /Burstall, Goguen 77/, /Goguen et al. 78/, /Wirsing et al. 80/) has proved to be a powerful and flexible tool for the formal definition of data structures. Algebraic concepts have also been employed for the specification of programming language semantics, e.g. first-order identities (/Wand 77/) or continuous algebras (/Courcelle, Nivat 78/, /Goguen et al. 77/). In contrast to these “explicit” constructions of semantics, /Broy, Wirsing 80a/ have introduced a technique for characterizing the semantic models of a language by the axioms of an algebraic type without resorting to (the isomorphism class of) a fixed model. This approach is characterized by the following peculiarities (cf. /Wirsing et al. 80/, /Broy, Wirsing 80c/): (1) As semantic models, finitely generated (cf. /Bauer, Wossner 81/) heterogeneous algebras with partial operations are considered. (2) Their properties are specified in algebraic types using positive conditional equations and a definedness predicate D on the terms of the type. (3) Within the equations, the metasymbol = is interpreted as strong equality, and D is total, so that the underlying logic remains two-valued. (4) In general, the types are hierarchical, i.e. a type may be based on a subtype which is considered as the specification of primitive objects and operations. Models of hierarchical types are required to preserve the properties of the primitive type.