Proof Transformation with Built-in Equality Predicate

One of the main reasons why computer generated proofs are not widely accepted is often their complexity and incomprehensibility. Especially proofs of mathematical theorems with equations are normally presented in an inadequate and not intuitive way. This is even more of a problem for the presentation of inferences drawn by automated reasoning components in other AI systems. For first order logic, proof transformation procedures have been designed in order to structure proofs and state them in a formalism that is more familiar to human mathematicians. In this report we generalize these approaches, so that proofs involving equational reasoning can also be handled. To this end extended refutation graphs are introduced to represent combined resolution and paramodulation proofs. In the process of transforming these proofs into natural deduction proofs with equality, the inherent structure can also be extracted by exploiting topological properties of refutation graphs

Computer supported mathematics with Ωmega

AbstractClassical automated theorem proving of today is based on ingenious search techniques to find a proof for a given theorem in very large search spaces—often in the range of several billion clauses. But in spite of many successful attempts to prove even open mathematical problems automatically, their use in everyday mathematical practice is still limited.The shift from search based methods to more abstract planning techniques however opened up a paradigm for mathematical reasoning on a computer and several systems of that kind now employ a mix of interactive, search based as well as proof planning techniques.The Ωmega system is at the core of several related and well-integrated research projects of the Ωmega research group, whose aim is to develop system support for a working mathematician as well as a software engineer when employing formal methods for quality assurance. In particular, Ωmega supports proof development at a human-oriented abstract level of proof granularity. It is a modular system with a central proof data structure and several supplementary subsystems including automated deduction and computer algebra systems. Ωmega has many characteristics in common with systems like NuPrL, CoQ, Hol, Pvs, and Isabelle. However, it differs from these systems with respect to its focus on proof planning and in that respect it is more similar to the proof planning systems Clam and λClam at Edinburgh

Formal methods for real-time requirements engineering

Timed model checking turned out to be a very successful technique for the verification of real-time systems. In general, however, large-scale systems require more than a mere real-time perspective: They utilise, for example, Abstract Data Types and Fairness Aspects. VSE-II (Verification Support Environment) is a general tool which supports the design and the verification process of such large-scale systems. The basic machinery within VSE-II is theorem proving rather than model checking and one of its underlying formalisms is close to TLA (Temporal Logic of Actions), i.e. it is based on linear discrete time. In this thesis we develop a technique to perform an exact discretisation of dense real-time aspects, i.e. a discretisation that is not just an approximation but rather mirrors dense behaviour exactly. This discretisation is achieved without an explicit or implicit introduction of rational numbers. With the help of the exact discretisation we define an embedding of Hybrid Automata into VSE-II such that model checking strategies for Hybrid Automata can be used in VSE-II. Vice versa, the embedding allows the model checking strategies to benefit from the proof work done in VSE-II. This thesis introduces a general methodology for formal requirements analysis, namely observer models, that deals with particular perspectives on a system rather than with particular aspects of it. This way, different specialised approaches can be integrated and used to describe the overall system requirements. One such view, for example, is a real-time which uses a new discretisation technique.In der Verifikation von Realzeit-Systemen haben sich Model-Checking Verfahren bewährt. Im Allgemeinen kann man jedoch sagen, dass große industrielle Anwendungen nicht nur die Realzeit Dimension aufweisen. Sie bestehen vielmehr aus einer Vielzahl weiterer Dimensionen (Sichten) wie eine Informationsflusssicht oder eine Security-Sicht. Zur Spezifikation dieser Sichten werden beispielsweise Abstrakte Datentypen oder auch Fairness Aspekte verwendet. VSE-II (Verification Support Environment) ist ein Werkzeug, welches den formalen Entwicklungsprozess vom Design bis hin zur Verifikation solcher Anwendungen unterstützt. Der Kern des VSE-IIWerkzeugs ist ein interaktives Beweissystem, das auf einem Sequenzenkalkül basiert, der neben der Logik erster Stufe und Dynamischer Logik auch die Temporale Logik der Aktionen (TLA) beinhaltet. TLA beruht auf einem Zeitmodell, welches linear und diskret ist. In dieser Arbeit beschreiben wir eine Technik, die eine exakte Diskretisierung von dichten Realzeitaspekten erlaubt, so dass das VSE-II System diese Aspekte mit den vorhandenen Verfahren und Regeln behandeln kann. Die Diskretisierung ist so definiert, dass sie nicht nur eine Approximation ist, sondern sie spiegelt vielmehr das dichte Verhalten exakt wider. Dies wird ohne die explizite oder implizite Einführung von rationalen Zahlen erreicht. Mit Hilfe der exakten Diskretisierung wird eine Einbettung von Hybriden Automaten in VSE-II definiert, die es ermöglicht Teilbeweise, die von Modelcheckingverfahren für Hybride Automaten gefunden wurden, ohne weiteren Beweis in VSE-II zu verwenden und umgekehrt. Weiterhin wird eine Methodologie zur formalen Anforderungsanalyse eingeführt, die verschiedene Sichten auf ein System und nicht nur verschiedene Aspekte eines Systems behandelt. Diese Methodologie, genannt Observer Models, ermöglicht die Integration unterschiedlicher spezieller Werkzeuge bzw. Verfahren zur Beschreibung der einzelnen Sichten und somit zur Beschreibung der gesamten Systemanforderungen. Eine solche Sicht stellt beispielsweise eine Realzeit-Sicht dar, welche auf der oben erwähnten Einbettung beruht

Methodology of Algorithm Engineering

Research on algorithms has drastically increased in recent years. Various sub-disciplines of computer science investigate algorithms according to different objectives and standards. This plurality of the field has led to various methodological advances that have not yet been transferred to neighboring sub-disciplines. The central roadblock for a better knowledge exchange is the lack of a common methodological framework integrating the perspectives of these sub-disciplines. It is the objective of this paper to develop a research framework for algorithm engineering. Our framework builds on three areas discussed in the philosophy of science: ontology, epistemology and methodology. In essence, ontology describes algorithm engineering as being concerned with algorithmic problems, algorithmic tasks, algorithm designs and algorithm implementations. Epistemology describes the body of knowledge of algorithm engineering as a collection of prescriptive and descriptive knowledge, residing in World 3 of Popper's Three Worlds model. Methodology refers to the steps how we can systematically enhance our knowledge of specific algorithms. The framework helps us to identify and discuss various validity concerns relevant to any algorithm engineering contribution. In this way, our framework has important implications for researching algorithms in various areas of computer science

A Machine-Checked, Type-Safe Model of Java Concurrency : Language, Virtual Machine, Memory Model, and Verified Compiler

The Java programming language provides safety and security guarantees such as type safety and its security architecture. They distinguish it from other mainstream programming languages like C and C++. In this work, we develop a machine-checked model of concurrent Java and the Java memory model and investigate the impact of concurrency on these guarantees. From the formal model, we automatically obtain an executable verified compiler to bytecode and a validated virtual machine

Approaches to the reuse of plan schemata in planning formalisms

Planning in complex domains is normally a resource and time consuming process when it is purely based on first principles. Once a plan is generated it represents problem solving knowledge. It implicitly describes knowledge used by the planning system to achieve a given goal state from a particular initial state. In classical planning systems, this knowledge is often lost after the plan has been successfully executed. If such a planner has to solve the same problem again, it will spend the same planning effort to solve it and is not capable of "learning\u27; from its "experience\u27;. Therefore it seems to be useful to save generated plans for a later reuse and thus, extending the problem solving knowledge possessed by the planner. The planning knowledge can now be applied to find out whether a problem can be solved by adapting an already existing plan. The aim of this paper is to analyze the problem of plan reuse and to describe the state of the art based on a variety of approaches which might contribute to a solution of the problem. It describes the main problems and results that could be of some relevance for the integration of plan reuse into a deductive planning formalism. As a result, this description of the state of the art leads to a deeper insight into the complex problem of plan reuse, but also shows that the problem itself is still far from being solved