Fixed Points on Abstract Structures without the Equality Test

In this paper we present a study of definability properties of fixed points of effective operators on abstract structures without the equality test. In particular we prove that Gandy theorem holds for abstract structures. This provides a useful tool for dealing with recursive definitions using Sigma-formulas. One of the applications of Gandy theorem in the case of the reals without the equality test is that it allows us to define universal Sigma-predicates. It leads to a topological characterisation of Sigma-relations on |R

Recent Advances in Σ-definability over Continuous Data Types

The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Sigma-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the reals by the structure of hereditarily finite sets. In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals. We prove Engeler's Lemma for Sigma-definability over the reals without the equality test which relates Sigma-definability with definability in the constructive infinitary language L_{omega_1 omega}. Thus, a relation over the real numbers is Sigma-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas. This result reveals computational aspects of Sigma-definability and also gives topological characterisation of Sigma-definable relations over the reals without the equality test. We also illustrate how computability over the real numbers can be expressed in the language of Sigma-formulas

Executable formal specifications of complex distributed systems with CoreASM

Formal specifications play a crucial role in the design of reliable complex software systems. Executable formal specifications allow the designer to attain early validation and verification of design using static analysis techniques and accurate simulation of the runtime behavior of the system-to-be. With increasing complexity of software-intensive computer-based systems and the challenges of validation and verification of abstract software models prior to coding, the need for interactive software tools supporting executable formal specifications is even more evident. In this paper, we discuss how CoreASM, an environment for writing and running executable specifications according to the ASM method, provides flexibility and manages the complexity by using an innovative extensible language architecture

Logische Grundlagen von Datenbanktransformationen für Datenbanken mit komplexen Typen

Database transformations consist of queries and updates which are two fundamental types of computations in any databases - the first provides the capability to retrieve data and the second is used to maintain databases in light of ever-changing application domains. With the rising popularity of web-based applications and service-oriented architectures, the development of database transformations must address new challenges, which frequently call for establishing a theoretical framework that unifies both queries and updates over complex-value databases. This dissertation aims to lay down the foundations for establishing a theoretical framework of database transformations in the context of complex-value databases. We shall use an approach that has successfully been used for the characterisation of sequential algorithms. The sequential Abstract State Machine (ASM) thesis captures semantics and behaviour of sequential algorithms. The thesis uses the similarity of general computations and database transformations for characterisation of the later by five postulates: sequential time postulate, abstract state postulate, bounded exploration postulate, background postulate, and the bounded non-determinism postulate. The last two postulates reflect the specific form of transformations for databases. The five postulates exactly capture database transformations. Furthermore, we provide a logical proof system for database transformations that is sound and complete.Datenbanktransformationen sind Anfragen an ein Datenbanksystem oder Modifikationen der Daten des Datenbanksystemes. Diese beiden grundlegenden Arten von Berechnungen auf Datenbanksystemen erlauben zum einem den Zugriff auf Daten und zum anderen die Pflege der Datenbank. Eine theoretische Fundierung von Datenbanktransformationen muss so flexibel sein, dass auch neue web-basierten Anwendungen und den neuen serviceorientierte Architekturen reflektiert sind, sowie auch die komplexeren Datenstrukturen. Diese Dissertation legt die Grundlagen für eine Theoriefundierung durch Datenbanktransformationen, die auch komplexe Datenstrukturen unterstützen. Wir greifen dabei auf einen Zugang zurück, der eine Theorie der sequentiellen Algorithmen bietet. Die sequentielle ASM-These (abstrakte Zustandsmaschinen) beschreibt die Semantik und das Verhalten sequentieller Algorithmen. Die Dissertation nutzt dabei die Gleichartigkeit von allgemeinen Berechnungen und Datenbanktransformationen zur Charakterisierung durch fünf Postulate bzw. Axiome: das Axiom der sequentiellen Ausführung, das Axiom einer abstrakten Charakterisierbarkeit von Zuständen, das Axiom der Begrenzbarkeit von Zustandsänderungen und Zustandssicht, das Axiom der Strukturierung von Datenbanken und das Axiom der Begrenzbarkeit des Nichtdeterminismus. Die letzten beiden Axiome reflektieren die spezifische Seite der Datenbankberechnungen. Die fünf Axiome beschreiben vollständig das Verhalten von Datenbanktransformationen. Weiterhin wird eine Beweiskalkül für Datenbanktransformationen entwickelt, der vollständig und korrekt ist

Background, Reserve, and Gandy Machines

Abstract. The reserve of a state of an abstract state machine was de-fined to be a “naked set”. In applications, it may be convenient to have tuples, sets, lists, arrays, etc. defined ahead of time on all elements, in-cluding the reserve elements. We generalize the notion of reserve appro-priately. As an application, we solve a foundational problem in Gandy’s formalization of mechanical devices