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Termination of Rewriting
(1994)

More and more, term rewriting systems are applied in computer science aswell as in mathematics. They are based on directed equations which may be used as non-deterministic functional programs. Termination is a key property for computing with termrewriting systems.In this thesis, we deal with different classes of so-called simplification orderings which areable to prove the termination of term rewriting systems. Above all, we focus on the problemof applying these termination methods to examples occurring in practice. We introduce aformalism that allows clear representations of orderings. The power of classical simplifica-tion orderings - namely recursive path orderings, path and decomposition orderings, Knuth-Bendix orderings and polynomial orderings - is improved. Further, we restrict these orderingssuch that they are compatible with underlying AC-theories by extending well-known methodsas well as by developing new techniques. For automatically generating all these orderings,heuristic-based algorithms are given. A comparison of these orderings with respect to theirpowers and their time complexities concludes the theoretical part of this thesis. Finally, notonly a detailed statistical evaluation of examples but also a brief introduction into the designof a software tool representing the integration of the specified approaches is given.

In dieser Dissertation wird das Konzept der Gröbnerbasen für endlich erzeugte Monoid-und Gruppenringe verallgemeinert. Dabei werden Reduktionsmethoden sowohl zurDarstellung der Monoid- beziehungsweise Gruppenelemente, als auch zur Beschreibungder Rechtsidealkongruenz in den entsprechenden Monoid- beziehungsweise Gruppenrin-gen benutzt. Da im allgemeinen Monoide und insbesondere Gruppen keine zulässigenOrdnungen mehr erlauben, treten bei der Definition einer geeigneten Reduktionsrela-tion wesentliche Probleme auf: Zum einen ist es schwierig, die Terminierung einer Re-duktionsrelation zu garantieren, zum anderen sind Reduktionsschritte nicht mehr mitMultiplikationen verträglich und daher beschreiben Reduktionen nicht mehr unbedingteine Rechtsidealkongruenz. In dieser Arbeit werden verschiedene Möglichkeiten Reduk-tionsrelationen zu definieren aufgezeigt und im Hinblick auf die beschriebenen Problemeuntersucht. Dabei wird das Konzept der Saturierung, d.h. eine Polynommenge so zu er-weitern, daß man die von ihr erzeugte Rechtsidealkongruenz durch Reduktion erfassenkann, benutzt, um Charakterisierungen von Gröbnerbasen bezüglich der verschiedenenReduktionen durch s-Polynome zu geben. Mithilfe dieser Konzepte ist es gelungenfür spezielle Klassen von Monoiden, wie z.B. endliche, kommutative oder freie, undverschiedene Klassen von Gruppen, wie z.B. endliche, freie, plain, kontext-freie odernilpotente, unter Ausnutzung struktureller Eigenschaften spezielle Reduktionsrelatio-nen zu definieren und terminierende Algorithmen zur Berechnung von Gröbnerbasenbezüglich dieser Reduktionsrelationen zu entwickeln.

Automated theorem proving is a search problem and, by its undecidability, a very difficult one. The challenge in the development of a practically successful prover is the mapping of the extensively developed theory into a program that runs efficiently on a computer. Starting from a level-based system model for automated theorem provers, in this work we present different techniques that are important for the development of powerful equational theorem provers. The contributions can be divided into three areas: Architecture. We present a novel prover architecture that is based on a set-based compression scheme. With moderate additional computational costs we achieve a substantial reduction of the memory requirements. Further wins are architectural clarity, the easy provision of proof objects, and a new way to parallelize a prover which shows respectable speed-ups in practice. The compact representation paves the way to new applications of automated equational provers in the area of verification systems. Algorithms. To improve the speed of a prover we need efficient solutions for the most time-consuming sub-tasks. We demonstrate improvements of several orders of magnitude for two of the most widely used term orderings, LPO and KBO. Other important contributions are a novel generic unsatisfiability test for ordering constraints and, based on that, a sufficient ground reducibility criterion with an excellent cost-benefit ratio. Redundancy avoidance. The notion of redundancy is of central importance to justify simplifying inferences which are used to prune the search space. In our experience with unfailing completion, the usual notion of redundancy is not strong enough. In the presence of associativity and commutativity, the provers often get stuck enumerating equations that are permutations of each other. By extending and refining the proof ordering, many more equations can be shown redundant. Furthermore, our refinement of the unfailing completion approach allows us to use redundant equations for simplification without the need to consider them for generating inferences. We describe the efficient implementation of several redundancy criteria and experimentally investigate their influence on the proof search. The combination of these techniques results in a considerable improvement of the practical performance of a prover, which we demonstrate with extensive experiments for the automated theorem prover Waldmeister. The progress achieved allows the prover to solve problems that were previously out of reach. This considerably enhances the potential of the prover and opens up the way for new applications.

The validity of formulas w.r.t. a specification over first-order logic with a semantics based on all models is semi-decidable. Therefore, we may implement a proof procedure which finds a proof for every valid formula fully automatically. But this semantics often lacks intuition: Some pathological models such as the trivial model may produce unexpected results w.r.t. validity. Instead, we may consider just a class of special models, for instance, the class of all data models. Proofs are then performed using induction. But, inductive validity is not semi-decidable -- even for first-order logic. This theoretical drawback manifests itself in practical limitations: There are theorems that cannot be proved by induction directly but only generalizations can be proved. For their definition, we may have to extend the specification. Therefore, we cannot expect to prove interesting theorems fully automatically. Instead, we have to support user-interaction in a suitable way. In this thesis, we aim at developing techniques that enhance automatic proof control of (inductive) theorem provers and that enable user-interaction in a suitable way. We integrate our new proof techniques into the inductive theorem prover QuodLibet and validate them with various case studies. Essentially, we introduce the following three proof techniques: -We integrate a decision procedure for linear arithmetic into QuodLibet in a close way by defining new inference rules that perform the elementary steps of the decision procedure. This allows us to implement well-known heuristics for automatic proof control. Furthermore, we are able to provide special purpose tactics that support the manual speculation of lemmas if a proof attempt gets stuck. The integration improves the ability of the theorem prover to prove theorems automatically as well as its efficiency. Our approach is competitive with other approaches regarding efficiency; it provides advantages regarding the speculation of lemmas. -The automatic proof control searches for a proof by applying inference rules. The search space is not only infinite, but grows dramatically with the depth of the search. In contrast to this, checking and analyzing performed proofs is very efficient. As the search space also has a high redundancy, it is reasonable to reuse subproofs found during proof search. We define new notions for the contribution of proof steps to a proof. These notions enable the derivation of pruned proofs and the identification of superfluous subformulas in theorems. A proof may be reused in two ways: upward propagation prunes a proof by eliminating superfluous proof steps; sideward reuse closes an open proof obligation by replaying an already found proof. -For interactive theorem provers, it is essential not only to prove automatically as many lemmas as possible but also to restrict proof search in such a way that the proof process stops within a reasonable amount of time. We introduce different markings in the goals to be proved and the lemmas to be applied to restrict proof search in a flexible way: With a forbidden marking, we can simulate well-known approaches for applying conditional lemmas. A mandatory marking provides a new heuristics which is inspired by local contribution of proof steps. With obligatory and generous markings, we can fine-tune the degree of efficiency and extent of proof search manually. With an elaborate case study, we show the benefits of the different techniques, in particular the synergetic effect of their combination.

Embedded systems have become ubiquitous in everyday life, and especially in the automotive industry. New applications challenge their design by introducing a new class of problems that are based on a detailed analysis of the environmental situation. Situation analysis systems rely on models and algorithms of the domain of computational geometry. The basic model is usually an Euclidean plane, which contains polygons to represent the objects of the environment. Usual implementations of computational geometry algorithms cannot be directly used for safety-critical systems. First, a strict analysis of their correctness is indispensable and second, nonfunctional requirements with respect to the limited resources must be considered. This thesis proposes a layered approach to a polygon-processing system. On top of rational numbers, a geometry kernel is formalised at first. Subsequently, geometric primitives form a second layer of abstraction that is used for plane sweep and polygon algorithms. These layers do not only divide the whole system into manageable parts but make it possible to model problems and reason about them at the appropriate level of abstraction. This structure is used for the verification as well as the implementation of the developed polygon-processing library.

Modelling languages are important in the process of software development. The suitability of a modelling language for a project depends on its applicability to the target domain. Here, domain-specific languages have an advantage over more general modelling languages. On the other hand, modelling languages like the Unified Modeling Language can be used in a wide range of domains, which supports the reuse of development knowledge between projects. This thesis treats the syntactical and semantical harmonisation of modelling languages and their combined use, and the handling of complexity of modelling languages by providing language subsets - called language profiles - with tailor-made formal semantics definitions, generated by a profile tool. We focus on the widely-used modelling languages SDL and UML, and formal semantics definitions specified using Abstract State Machines.

Nowadays, accounting, charging and billing users' network resource consumption are commonly used for the purpose of facilitating reasonable network usage, controlling congestion, allocating cost, gaining revenue, etc. In traditional IP traffic accounting systems, IP addresses are used to identify the corresponding consumers of the network resources. However, there are some situations in which IP addresses cannot be used to identify users uniquely, for example, in multi-user systems. In these cases, network resource consumption can only be ascribed to the owners of these hosts instead of corresponding real users who have consumed the network resources. Therefore, accurate accountability in these systems is practically impossible. This is a flaw of the traditional IP address based IP traffic accounting technique. This dissertation proposes a user based IP traffic accounting model which can facilitate collecting network resource usage information on the basis of users. With user based IP traffic accounting, IP traffic can be distinguished not only by IP addresses but also by users. In this dissertation, three different schemes, which can achieve the user based IP traffic accounting mechanism, are discussed in detail. The inband scheme utilizes the IP header to convey the user information of the corresponding IP packet. The Accounting Agent residing in the measured host intercepts IP packets passing through it. Then it identifies the users of these IP packets and inserts user information into the IP packets. With this mechanism, a meter located in a key position of the network can intercept the IP packets tagged with user information, extract not only statistic information, but also IP addresses and user information from the IP packets to generate accounting records with user information. The out-of-band scheme is a contrast scheme to the in-band scheme. It also uses an Accounting Agent to intercept IP packets and identify the users of IP traffic. However, the user information is transferred through a separated channel, which is different from the corresponding IP packets' transmission. The Multi-IP scheme provides a different solution for identifying users of IP traffic. It assigns each user in a measured host a unique IP address. Through that, an IP address can be used to identify a user uniquely without ambiguity. This way, traditional IP address based accounting techniques can be applied to achieve the goal of user based IP traffic accounting. In this dissertation, a user based IP traffic accounting prototype system developed according to the out-of-band scheme is also introduced. The application of user based IP traffic accounting model in the distributed computing environment is also discussed.

In urban planning, sophisticated simulation models are key tools to estimate future population growth for measuring the impact of planning decisions on urban developments and the environment. Simulated population projections usually result in large, macro-scale, multivariate geospatial data sets. Millions of records have to be processed, stored, and visualized to help planners explore and analyze complex population patterns. We introduce a database driven framework for visualizing geospatial multidimensional simulation data based on the output from UrbanSim, a software for the analysis and planning of urban developments. The designed framework is extendable and aims at integrating empirical-stochastic methods and urban simulation models with techniques developed for information visualization and cartography. First, we develop an empirical model for the estimation of residential building types based on demographic household characteristics. The predicted dwelling type information is important for the analysis of future material use, carbon footprint calculations, and for visualizing simultaneously the results of land usage, density, and other significant parameters in 3D space. Our model uses multinomial logistic regression to derive building types at different scales. The estimated regression coefficients are applied to UrbanSim output in order to predict residential building types. The simulation results and the estimated building types are managed in an object-relational geodatabase. From the database, density, building types, and significant demographic variables are visually encoded as scalable, georeferenced 3D geometries and displayed on top of aerial photographs in a Google Earth visual synthesis. The geodatabase can be accessed and the visualization parameters can be chosen through a web-based user interface. The geometries are encoded in KML, Google's markup language, as ready-to-visualize data sets. The goal is to enhance human cognition by displaying abstract representations of multidimensional data sets in a realistic context and thus to support decision making in planning processes.

The visualization of numerical fluid flow datasets is essential to the engineering processes that motivate their computational simulation. To address the need for visual representations that convey meaningful relations and enable a deep understanding of flow structures, the discipline of Flow Visualization has produced many methods and schemes that are tailored to a variety of visualization tasks. The ever increasing complexity of modern flow simulations, however, puts an enormous demand on these methods. The study of vortex breakdown, for example, which is a highly transient and inherently three-dimensional flow pattern with substantial impact wherever it appears, has driven current techniques to their limits. In this thesis, we propose several novel visualization methods that significantly advance the state of the art in the visualization of complex flow structures. First, we propose a novel scheme for the construction of stream surfaces from the trajectories of particles embedded in a flow. These surfaces are extremely useful since they naturally exploit coherence between neighboring trajectories and are highly illustrative in nature. We overcome the limitations of existing stream surface algorithms that yield poor results in complex flows, and show how the resulting surfaces can be used a building blocks for advanced flow visualization techniques. Moreover, we present a visualization method that is based on moving section planes that travel through a dataset and sample the flow. By considering the changes to the flow topology on the plane as it moves, we obtain a method of visualizing topological structures in three-dimensional flows that are not accessible by conventional topological methods. On the same algorithmic basis, we construct an algorithm for the tracking of critical points in such flows, thereby enabling the treatment of time-dependent datasets. Last, we address some problems with the recently introduced Lagrangian techniques. While conceptually elegant and generally applicable, they suffer from an enormous computational cost that we significantly use by developing an adaptive approximation algorithm. This allows the application of such methods on very large and complex numerical simulations. Throughout this thesis, we will be concerned with flow visualization aspect of general practical significance but we will particularly emphasize the remarkably challenging visualization of the vortex breakdown phenomenon.