This book constitutes the thoroughly refereed post-proceedings of the Second International Conference on Rough Sets and Current Trends in Computing, RSCTC 2000, held in Banff, Canada in October 2000.The 80 revised papers presented together with an introduction and three keynote presentations have gone through two rounds of reviewing and revision. The papers are organized in topical sections on granual computing, rough sets and systems, fuzzy sets and systems, rough sets and data mining, nonclassical logics and reasoning, pattern recognition and image processing, neural networks and genetic algorithms, and current trends in computing.

Since their introduction nearly 40 years ago, research on Petri nets has diverged in many different directions. Various classes of Petri net, motivated either by theory or applications, with its own specific features and methods of analysis, have been proposed and studies in depth.These successful developments have led to a very heterogeneous landscape of diverse models, and this, in turn, has stimulated research on concepts and approaches that contribute to unifying and structuring the diverse landscape. This state-of-the-art survey presents the most relevant approaches to unifying Petri nets in a systematic and coherent way. The 14 chapters written by leading researchers are organized in topical sections on application-oriented approaches, unifying frameworks, and theoretical approaches.

This volume contains the revised versions of papers presented at the fourth international Workshop on Implementing Automata (WIA), held 17-19 July, 1999, at Potsdam University, Germany. As for its predecessors, the theme of WIA99 was the implementation of au- mata and grammars of all types and their application in other ?elds. The papers contributed to this volume address, among others, algorithmic issues regarding automata, image and dictionarystorage byautomata, and natural language p- cessing. In addition to the papers presented in these proceedings, the workshop - cluded a paper on quantum computing byC. Calude, E. Calude, and K. Svozil (published elsewhere), an invited lecture byW. Thomas on Algorithmic P- blems in the Theory of ?-Automata, a tutorial byM. Silberztein on the INTEX linguistic development environment, and several demonstrations of systems. The local arrangements for WIA99 were conducted byHelmut Jurgensen, Suna Aydin, Oliver Boldt, Carsten Haustein, Beatrice Mix, and Lynda R- bins. The meeting was held in the Communs building, now the main university building, of the New Palace in the park of Sanssouci, Potsdam. The program committee for WIA99 was: A. Bruggemann-Klein .. Technische Universitat Munc .. hen J.-M. Champarnaud Universit'e de Rouen F. Gun .. thner Universitat Munc .. hen H. Jurgensen .. Universitat Potsdam and Universityof Western Ontario D. Maurel Universit'e de Tours D. Raymond Gateway Group Inc. K. Salomaa Universityof Western Ontario W. Thomas Rheinisch-Westfalische Technische Hochschule Aachen B. Watson Ribbit Software Systems Inc.

This book provides a self-contained exposition of the theory of plane Cremona maps, reviewing the classical theory. The book updates, correctly proves and generalises a number of classical results by allowing any configuration of singularities for the base points of the plane Cremona maps. It also presents some material which has only appeared in research papers and includes new, previously unpublished results. This book will be useful as a reference text for any researcher who is interested in the topic of plane birational maps.

This volume gives an overview of linear logic in five parts: category theory; complexity and expressivity; proof theory; proof nets; and the geometry of interaction. The book includes a general introduction to linear logic that will ensure this book's use by the novice as well as the expert. Mathematicians and computer scientists will learn much from this book.

This is the most comprehensive treatment available in book form of the classical decision problem of mathematical logic and of the role of the classical decision problem in modern computer science. A revealing analysis of the natural order of decidable and undecidable cases is given. The complete classification of the solvable and unsolvable standard cases of the classical decision problem will be of particular interest to the reader. The classification comes complete with the complexity analysis of the solvable cases, with the comprehensive treatment of the reduction method, and with the model-theoretical analysis of solvable cases. Many cases are treated here for the first time, and a great number of simple proofs and exercises have been included. The results and methods of the book are extensively used in logic, computer science and artificial intelligence.

This book constitutes the refereed proceedings of the International Seminar on Proof Theory in Computer Science, PTCS 2001, held in Dagstuhl Castle, Germany, in October 2001.The 13 thoroughly revised full papers were carefully reviewed and selected for inclusion in the book. Among the topics addressed are higher type recursion, lambda calculus, complexity theory, transfinite induction, categories, induction-recursion, post-Turing analysis, natural deduction, implicit characterization, iterate logic, and Java programming.

This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency (1979). Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention.

As society comes to rely increasingly on software for its welfare and prosperity there is an urgent need to create systems in which it can trust. Experience has shown that confidence can only come from a more profound understanding of the issues, which in turn can come only if it is based on logically sound foundations.
This volume contains contributions from leading researchers in the critical disciplines of computing and information science, mathematics, logic, and complexity. All contributions are self-contained, aiming at comprehensibility as well as comprehensiveness. The volume also contains introductory hints to technical issues, concise surveys, introductions, and various fresh results and new perspectives.

This volume contains the papers presented at the Third Discrete Mathematics and Theoretical Computer Science Conference (DMTCS1), which was held at 'Ovidius'University Constantza, Romania in July 2001.The conference was open to all areas of discrete mathematics and theoretical computer science, and the papers contained within this volume cover topics such as: abstract data types and specifications; algorithms and data structures; automata and formal languages; computability, complexity and constructive mathematics; discrete mathematics, combinatorial computing and category theory; logic, nonmonotonic logic and hybrid systems; molecular computing.

This volume contains the proceedings of the 22nd International Conference on Application and Theory of Petri Nets. The aim of the Petri net conferences is to create a forum for discussing progress in the application and theory of Petri nets. Typically, the conferences have 100{150 participants { one third of these coming from industry while the rest are from universities and research institutions. The conferences always take place in the last week of June. This year the conference was organized jointly with the 2nd International Conference on Application of Concurrency to System Design (ICACSD 2001). The two conferences shared the invited lectures and the social program. The conference and a number of other activities are co-ordinated by a steering committee with the following members: G. Balbo (Italy), J. Billington (Aust- lia), G. De Michelis (Italy), C. Girault (France), K. Jensen (Denmark), S. - magai (Japan), T. Murata (USA), C.A. Petri (Germany; honorary member), W. Reisig (Germany), G. Rozenberg (The Netherlands; chairman), and M. Silva (Spain). Other activities before and during the 2001 conference included tool dem- strations, a meeting on \XML Based Interchange Formats for Petri Nets," - tensive introductory tutorials, two advanced tutorials on \Probabilistic Methods in Concurrency" and \Model Checking," and two workshops on \Synthesis of Concurrent Systems" and \Concurrency in Dependable Computing." The tu- rial notes and workshop proceedings are not published in these proceedings, but copies are available from the organizers.

Diagrams are widely used in reasoning about problems in physics, mathematics and logic, but have traditionally been considered to be only heuristic tools and not valid elements of mathematical proofs. This book challenges this prejudice against visualisation in the history of logic and mathematics and provides a formal foundation for work on natural reasoning in a visual mode. The author presents Venn diagrams as a formal system of representation equipped with its own syntax and semantics and specifies rules of transformation that make this system sound and complete. The system is then extended to the equivalent of a first-order monadic language. The soundness of these diagrammatic systems refutes the contention that graphical representation is misleading in reasoning. The validity of the transformation rules ensures that the correct application of the rules will not lead to fallacies. The book concludes with a discussion of some fundamental differences between graphical systems and linguistic systems. This groundbreaking work will have important influence on research in logic, philosophy and knowledge representation.

This is a first course in propositional modal logic, suitable for mathematicians, computer scientists and philosophers. Emphasis is placed on semantic aspects, in the form of labelled transition structures, rather than on proof theory. The book covers all the basic material - propositional languages, semantics and correspondence results, proof systems and completeness results - as well as some topics not usually covered in a modal logic course. It is written from a mathematical standpoint. To help the reader, the material is covered in short chapters, each concentrating on one topic. These are arranged into five parts, each with a common theme. An important feature of the book is the many exercises, and an extensive set of solutions is provided.

This book constitutes the refereed proceedings of the 5th International Conference on Typed Lambda Calculi and Applications, TLCA 2001, held in Krakow, Poland in May 2001. The 28 revised full papers presented were carefully reviewed and selected from 55 submissions. The volume reports research results on all current aspects of typed lambda calculi. Among the topics addressed are type systems, subtypes, coalgebraic methods, pi-calculus, recursive games, various types of lambda calculi, reductions, substitutions, normalization, linear logic, cut-elimination, prelogical relations, and mu calculus.

This book constitutes the thoroughly refereed post-proceedings of the 6th International Workshop on DNA-Based Computers, DNA 2000, held in Leiden, The Netherlands, in June 2000.The 16 revised full papers presented together with two invited papers were carefully reviewed and selected from 33 submissions. All current aspects of DNA computing, ranging from theoretical and foundational issues to algorithms, are addressed, from the computer science point of view as well as from the molecular biology point of view.

Stat Labs: Mathematical Statistics Through Applications blends mathematical statistics with modern statistical practice. It turns the traditional teaching of mathematical statistics on its head by making a case study the centerpiece of each chapter. Chapters begin with the introduction of a real problem followed by a description of the data collected to address the problem, rich background material to put the problem in context, and suggestions for investigating the problem. This novel approach to bringing data analysis into the theoretical course is ideal for motivating and illustrating standard statistical techniques, for helping students understand mathematical statistics, and for showing how statistics can be useful in a wide variety of contexts. Stat Labs is designed for use in a calculus-based introductory statistics course. It would be equally effective as a stand-alone text or as a supplement to a traditional mathematical statistics text. Deborah Nolan received her Ph.D in Statistics from Yale University, and she is now Professor of Statistics at the University of California, Berkeley. Her research interests include high-dimensional modeling, the teaching of statistics, and the use of technology in education. She has also been involved in encouraging women into research careers in the mathematical sciences, and in 1997 edited the volume, Women in Mathematics: Scaling the Heights for the Mathematical Association of America. Terry Speed's professional activities center around teaching and research involving th application of statistics to genetics and molecular biology. He spends 50% of his time in the Department of Statistics of the University of California at Berkeley, and the other 50% with the Genetics and Bioinformatics Group of the Walter & Eliza Hall Institute of Medical Research in Melbourne, Australia.

This book constitutes the thoroughly refereed post-workshop proceedings of the 15th International Workshop on Algebraic Development Techniques, WADT 2001, held jointly with the General Workshop of the ESPRIT Working Group CoFI in Genova, Italy, in April 2001.The book presents 16 papers selected from 32 workshop presentations. Among the topics addressed are formal specification, specification languages, term rewriting, and proof systems.

This volumecontains the papers presentedatthe SeventhInternationalC- ference on Logicfor Programmingand Automated Reasoning (LPAR 2000)held onReunionIsland, France,6 10November2000, followedbythe ReunionWo- shop on Implementation of Logic. Sixty-?ve papers were submitted to LPAR 2000 of which twenty-six papers were accepted. Submissions by the program committee members were not - lowed. There was a special category of experimental papers intended to describe implementations of systems, to report experiments with implemented systems, orto compareimplementedsystems.Eachof thesubmissionswasreviewedbyat least three program committee members and an electronic program committee meeting was held via the Internet. In addition to the refereed papers, this volume contains full papers by two of the four invited speakers, Georg Gottlob and Micha] el Rusinowitch, along with an extended abstract of Bruno Courcelle s invited lecture and an abstract of Erich Gr] adel s invited lecture. WewouldliketothankthemanypeoplewhohavemadeLPAR2000possible. We are grateful to the following groups and individuals: the program and or- nizing committees; the additional referees; the local arrangements chair Teodor Knapik; PascalManoury, who was in chargeof accommodation; Konstantin - rovin, whomaintainedthe programcommittee Webpage;andBillMcCune, who implemented the program committee management software."

This book contains all full papers presented at ACRI 2000, the Fourth International Conference on Cellular Automata for Research and Industry, held at the University of Karlsruhe (Germany), 4 - 6 October, 2000. The continuation of and growing interest in research on Cellular Automata models for real world phenomena indicates the feasibility of this approach. A quick glance at the table contents of this book shows that results came from such different areas as biology, economics, physics, traffic flow and urban development. This work is complemented by contributions on the implementation and evaluation of software for Cellular Automata simulation, which is a necessary (but of course in no way sufficient) ingredient for the successful application of Cellular Automata. Applying Cellular Automata without trying to understand their behavior, in depth would be an unfortunate development. But as properties and power in earlier years it was again one of the strong points of ACRI to bring together researchers not only from different application areas but also from theory. Of course, this is reflected by the list of accepted contributions which also comprise theoretical papers and even papers which certainly belong to the intersection of several fields. Examples are the generation and recognition of geometrical patters and the influence of possible failures on the power of CA which obviously are of relevance also to applications.

This textbook explains the basic principles of categorical type theory and the techniques used to derive categorical semantics for specific type theories. It introduces the reader to ordered set theory, lattices and domains, and this material provides plenty of examples for an introduction to category theory, which covers categories, functors, natural transformations, the Yoneda lemma, cartesian closed categories, limits, adjunctions and indexed categories. Four kinds of formal system are considered in detail, namely algebraic, functional, polymorphic functional, and higher order polymorphic functional type theory. For each of these the categorical semantics are derived and results about the type systems are proved categorically. Issues of soundness and completeness are also considered. Aimed at advanced undergraduates and beginning graduates, this book will be of interest to theoretical computer scientists, logicians and mathematicians specializing in category theory.

This textbook explains the basic principles of categorical type theory and the techniques used to derive categorical semantics for specific type theories. It introduces the reader to ordered set theory, lattices and domains, and this material provides plenty of examples for an introduction to category theory, which covers categories, functors, natural transformations, the Yoneda lemma, cartesian closed categories, limits, adjunctions and indexed categories. Four kinds of formal system are considered in detail, namely algebraic, functional, polymorphic functional, and higher order polymorphic functional type theory. For each of these the categorical semantics are derived and results about the type systems are proved categorically. Issues of soundness and completeness are also considered. Aimed at advanced undergraduates and beginning graduates, this book will be of interest to theoretical computer scientists, logicians and mathematicians specializing in category theory.

This volume contains the proceedings of FroCoS2000, the 3rd International WorkshoponFrontiersofCombiningSystems, heldMarch22-24,2000, inNancy, France. Like its predecessors organized in Munich (1996) and in Amsterdam (1998), FroCoS2000 is intended to o?er a common forum for research activities related to the combination and the integration of systems in the areas of logic, automateddeduction, constraintsolving, declarativeprogramming, andarti?cial intelligence. There were 31 submissions of overall high quality, authored by researchers from countries including Australia, Brasil, Belgium, Chili, France, Germany, - pan, Ireland, Italy, Portugal, Spain, Switzerland, The Netherlands, the United Kingdom, and the United States of America. All submissions were thoroughly evaluatedonthebasisofatleastthreerefereereports, andanelectronicprogram committeemeetingwasheldthroughtheInternet.Theprogramcommitteesel- ted 14 research contributions. The topics covered by the selected papers include: combinationoflogics;combinationofconstraintsolvingtechniques, combination of decision procedures; modular properties for theorem proving; combination of deduction systems and computer algebra; integration of decision procedures and other solving processes into constraint programming and deduction systems. We welcomed ?ve invited lectures by Alexander Bockmayr on "Combining Logic and Optimization in Cutting Plane Theory," Gilles Dowek on "Axioms vs. Rewrite Rules: From Completeness to Cut Elimination," Klaus Schulz on "Why Combined Decision Problems Are Often Intractable," Tomas Uribe on "Combinations of Theorem Proving and Model Checking," and Richard Zippel on "Program Composition Techniques for Numerical PDE Codes." Full papers of these lectures, except the last one, are also included in this volume.

This is an up-to-date and integrated introduction to model theory, designed to be used for graduate courses (for students who are familiar with first-order logic), and as a reference for more experienced logicians and mathematicians. Model theory is concerned with the notions of definition, interpretation and structure in a very general setting, and is applied to a wide variety of other areas such as set theory, geometry, algebra (in particular group theory), and computer science (e.g. logic programming and specification). Professor Hodges emphasises definability and methods of construction, and introduces the reader to advanced topics such as stability. He also provides the reader with much historical information and a full bibliography, enhancing the book's use as a reference.

By considering the size of the logical network needed to perform a given computational task, the intrinsic difficulty of that task can be examined. Boolean function complexity, the combinatorial study of such networks, is a subject that started back in the 1950s and has today become one of the most challenging and vigorous areas of theoretical computer science. The papers in this book stem from the London Mathematical Society Symposium on Boolean Function Complexity held at Durham University in July 1990. The range of topics covered will be of interest to the newcomer to the field as well as the expert, and overall the papers are representative of the research presented at the Symposium. Anyone with an interest in Boolean Function complexity will find that this book is a necessary purchase.

The papers contained in this volume were presented at the third international Workshop on Implementing Automata, held September 17{19,1998, at the U- versity of Rouen, France. Automata theory is the cornerstone of computer science theory. While there is much practical experience with using automata, this work covers diverse - eas, includingparsing, computationallinguistics, speechrecognition, textsear- ing, device controllers, distributed systems, andprotocolanalysis.Consequently, techniques that have been discovered in one area may not be known in another. In addition, there is a growing number of symbolic manipulation environments designed to assist researchers in experimenting with and teaching on automata and their implementation; examples include FLAP, FADELA, AMORE, Fire- Lite, Automate, AGL, Turing's World, FinITE, INR, and Grail. Developers of such systems have not had a forum in which to expose and compare their work. The purpose of this workshop was to bring together members of the academic, research, andindustrialcommunitieswithaninterestinimplementingautomata, to demonstrate their work and to explain the problems they have been solving. These workshops started in 1996 and 1997 at the University of Western Ontario, London, Ontario, Canada, prompted by Derick Wood and Sheng Yu. The major motivation for starting these workshops was that there had been no single forum in which automata-implementation issues had been discussed. The interest shown in the r st and second workshops demonstrated that there was a need for such a forum. The participation at the third workshop was very interesting: we counted sixty-three registrations, four continents, ten countries, twenty-three universities, and three companie