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 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 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.

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.

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.

Combinatorics and Number Theory of Counting Sequences is an introduction to the theory of finite set partitions and to the enumeration of cycle decompositions of permutations. The presentation prioritizes elementary enumerative proofs. Therefore, parts of the book are designed so that even those high school students and teachers who are interested in combinatorics can have the benefit of them. Still, the book collects vast, up-to-date information for many counting sequences (especially, related to set partitions and permutations), so it is a must-have piece for those mathematicians who do research on enumerative combinatorics. In addition, the book contains number theoretical results on counting sequences of set partitions and permutations, so number theorists who would like to see nice applications of their area of interest in combinatorics will enjoy the book, too. Features The Outlook sections at the end of each chapter guide the reader towards topics not covered in the book, and many of the Outlook items point towards new research problems. An extensive bibliography and tables at the end make the book usable as a standard reference. Citations to results which were scattered in the literature now become easy, because huge parts of the book (especially in parts II and III) appear in book form for the first time.

Are mathematical equations the best way to model nature? For many years it had been assumed that they were. But in the early 1980s, Stephen Wolfram made the radical proposal that one should instead build models that are based directly on simple computer programs. Wolfram made a detailed study of a class of such models known as cellular automata, and discovered a remarkable fact: that even when the underlying rules are very simple, the behavior they produce can be highly complex, and can mimic many features of what we see in nature. And based on this result, Wolfram began a program of research to develop what he called ?A Science of Complexity.?The results of Wolfram's work found many applications, from the so-called Wolfram Classification central to fields such as artificial life, to new ideas about cryptography and fluid dynamics. This book is a collection of Wolfram's original papers on cellular automata and complexity. Some of these papers are widely known in the scientific community; others have never been published before. Together, the papers provide a highly readable account of what has become a major new field of science, with important implications for physics, biology, economics, computer science and many other areas.

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

Computability theory is a branch of mathematical logic and computer science that has become increasingly relevant in recent years. The field has developed growing connections in diverse areas of mathematics, with applications in topology, group theory, and other subfields. In A Hierarchy of Turing Degrees, Rod Downey and Noam Greenberg introduce a new hierarchy that allows them to classify the combinatorics of constructions from many areas of computability theory, including algorithmic randomness, Turing degrees, effectively closed sets, and effective structure theory. This unifying hierarchy gives rise to new natural definability results for Turing degree classes, demonstrating how dynamic constructions become reflected in definability. Downey and Greenberg present numerous construction techniques involving high-level nonuniform arguments, and their self-contained work is appropriate for graduate students and researchers. Blending traditional and modern research results in computability theory, A Hierarchy of Turing Degrees establishes novel directions in the field.

Inverse problems are concerned with determining causes for observed or desired effects. Problems of this type appear in many application fields both in science and in engineering. The mathematical modelling of inverse problems usually leads to ill-posed problems, i.e., problems where solutions need not exist, need not be unique or may depend discontinuously on the data. For this reason, numerical methods for solving inverse problems are especially difficult, special methods have to be developed which are known under the term "regularization methods." This volume contains twelve survey papers about solution methods for inverse and ill-posed problems and about their application to specific types of inverse problems, e.g., in scattering theory, in tomography and medical applications, in geophysics and in image processing. The papers have been written by leading experts in the field and provide an up-to-date account of solution methods for inverse problems.

This volume contains the proceedings of the 20th International Conference on Application and Theory of Petri Nets. The aim of the Petri net conferences is to create a forum for the dissemination of the latest results in the application and theory of Petri nets. Typically there are some 150-200 participants and usually one third of these come from industry, while the rest are from universities and research institutions. The conferences and a number of other activitiesare co- dinatedbyasteering committeeformedby: G.Balbo (Italy), J. Billington(A- tralia), C. Girault (France), K. Jensen (Denmark), S. Kumagai (Japan), G. De Michelis (Italy), T. Murata (U.S.A.), C.A. Petri (Germany; honorary member) W. Reisig (Germany), G. Roucairol (France), G. Rozenberg (The Netherlands; chair), M. Silva (Spain). The 1999 Petri net conference took place in Williamsburg, Virginia, and was organized by the Department of Computer Science of The College of William and Mary, Williamsburg. This was the second time the conference had been organized in the United States. We received 45 submissions from 15 countries on 5 continents of which 21 accepted for presentation. The submitted papers were evaluated by a program committee with the following members: W. van der Aalst (The Netherlands), P. Azema (France), W. Brauer (Germany), S. Christensen (Denmark), A. Desrochers (U.S.A.), S. Donatelli (Italy; co-chair), C. Girault (France), L. Gomes (Portugal), J. Hillston (United Kingdom), E

The 1998Annual Conference of the EuropeanAssociation for Computer Science Logic, CSL'98, was held in Brno, Czech Republic, during August 24-28, 1998. CSL'98wasthe12thinaseriesofworkshopsandthe7thtobeheldasthe Annual Conference of the EACSL. The conference was organized at Masaryk University in Brno by the Faculty of Informatics in cooperation with universities in Aaachen, Caen, Haagen, Linz, Metz, Pisa, Szeged, Vienna, and other institutions. CSL'98 formed one part of a federated conferences event, the other part being MFCS'98, the 23rd Int- national Symposium on the Mathematical Foundations of Computer Science. This federated conferences event consisted of common plenary sessions, invited talks, several parallel technical programme tracks, a dozen satellite workshops organized in parallel, and tutorials. The Federated CSL/MFCS'98 Conferences event included 19 invited talks, four of them joint CSL/MFCS'98 talks (D. Harel, W. Maass, Y. Matiyasevic, and M. Yannakakis), four for CSL (P. Hajek, J. Mitchell, Th. Schwentick, and J. Tiuryn), and eleven for MFCS. Last but not least, two tutorials were organized by CSL on the day preceding the symposium on "Inference Rules in Fragments of Arithmetic" by Lev Beklemishev and on "Proofs, Types, and Safe Mobile Code" by Greg Morrisett. A total of 345 persons attended the Federated CSL/MFCS'98 Conference which was a great success. The program committee of CSL'98 selected 27 of 74 papers submitted for the conference.From the 27 papers selected for presentation,25 havebeen accepted, following the standard refereeeing procedure, for publication in the present p- ceedings. Three invited speakers submitted papers, that were likewise refereeed and accepted.

This book constitutes the refereed proceedings of the 4th International Conference on Typed Lambda Calculi and Applications, TLCA'99, held in L'Aquila, Italy in April 1999. The 25 revised full papers presented were carefully reviewed and selected from a total of 50 submissions. Also included are two invited demonstrations. The volume reports research results on various aspects of typed lambda calculi. Among the topics addressed are noncommutative logics, type theory, algebraic data types, logical calculi, abstract data types, and subtyping.

Stewart Shapiro presents a distinctive original view of the foundations of mathematics, arguing that second-order logic has a central role to play in laying these foundations. He gives an accessible account of second-order and higher-order logic, paying special attention to philosophical and historical issues. Foundations without Foundationalism is a key contribution both to philosophy of mathematics and to mathematical logic.

'In this excellent treatise Shapiro defends the use of second-order languages and logic as frameworks for mathematics. His coverage of the wide range of logical and philosophical . . . is thorough, clear, and persuasive.' Michael D. Resnik, History and Philosophy of Logic

Lattice-gas cellular automata (LGCA) and lattice Boltzmann models (LBM) are relatively new and promising methods for the numerical solution of nonlinear partial differential equations. The book provides an introduction for graduate students and researchers. Working knowledge of calculus is required and experience in PDEs and fluid dynamics is recommended. Some peculiarities of cellular automata are outlined in Chapter 2. The properties of various LGCA and special coding techniques are discussed in Chapter 3. Concepts from statistical mechanics (Chapter 4) provide the necessary theoretical background for LGCA and LBM. The properties of lattice Boltzmann models and a method for their construction are presented in Chapter 5.

Cellular Automata (CA), about to enter their fifties, are coming of age, seen by the breadth and quality of CA-related research carried out worldwide, as well as by the appearance of interesting applications to real world problems. The papers collected in this book, presented at ACRI 98 (Third Conference on Cellular Automata for Research and Industry -7-9 October 1998), further demonstrate the vitality of this line ofresearch. Until some years ago, a researcher interested in dynamical modelling of spatially of the partial extended systems had only one language at his disposal, namely that differential equations (PDE). These are wonderful tools to use when an analytical solution can be found or a perturbative approach can provide a good approximation of the observed phenomena. The use of digital computers has enormously expanded the explanatory and predictive power of partial differential equations by allowing one to treat cases which had been outside the scope of a "pen and pencil" approach. However, it has also opened up a way to new formalisms which are able to describe interesting phenomena and are, at the same time, well-suited for digital simulation.

Simplicity theory is an extension of stability theory to a wider class of structures, containing, among others, the random graph, pseudo-finite fields, and fields with a generic automorphism. Following Kim's proof of forking symmetry' which implies a good behaviour of model-theoretic independence, this area of model theory has been a field of intense study. It has necessitated the development of some important new tools, most notably the model-theoretic treatment of hyperimaginaries (classes modulo type-definable equivalence relations). It thus provides a general notion of independence (and of rank in the supersimple case) applicable to a wide class of algebraic structures. The basic theory of forking independence is developed, and its properties in a simple structure are analyzed. No prior knowledge of stability theory is assumed; in fact many stability-theoretic results follow either from more general propositions, or are developed in side remarks. Audience: This book is intended both as an introduction to simplicity theory accessible to graduate students with some knowledge of model theory, and as a reference work for research in the field.

Introduction Model theorists have often joked in recent years that the part of mathemat- ical logic known as "pure model theory" (or stability theory), as opposed to the older and more traditional "model theory applied to algebra" , turns out to have more and more to do with other subjects ofmathematics and to yield gen- uine applications to combinatorial geometry, differential algebra and algebraic geometry. We illustrate this by presenting the very striking application to diophantine geometry due to Ehud Hrushovski: using model theory, he has given the first proof valid in all characteristics of the "Mordell-Lang conjecture for function fields" (The Mordell-Lang conjecture for function fields, Journal AMS 9 (1996), 667-690). More recently he has also given a new (model theoretic) proof of the Manin-Mumford conjecture for semi-abelian varieties over a number field. His proofyields the first effective bound for the cardinality ofthe finite sets involved (The Manin-Mumford conjecture, preprint). There have been previous instances of applications of model theory to alge- bra or number theory, but these appl~cations had in common the feature that their proofs used a lot of algebra (or number theory) but only very basic tools and results from the model theory side: compactness, first-order definability, elementary equivalence...

Quantum Logic deals with the foundations of quantum mechanics and, related to it, the behaviour of finite, discrete deterministic systems. The quantum logical approach is particulalry suitable for the investigation and exclusion of certain hidden parameter models of quantum mechanics. Conversely, it can be used to embed quantum universes into classical ones. It is also highly relevant for the characterization of finite automation. This book has been written with a broad readership in mind. Great care has been given to the motivation of the concepts and to the explicit and detailed discussions of examples.

This book comprises revised full versions of lectures given during the 9th European Summer School in Logic, Languages, and Information, ESSLLI'97, held in Aix-en-Provence, France, in August 1997. The six lectures presented introduce the reader to the state of the art in the area of generalized quantifiers and computation. Besides an introductory survey by the volume editor various aspects of generalized quantifiers are studied in depth.

This book is concerned with techniques for formal theorem-proving, with particular reference to Cambridge LCF (Logic for Computable Functions). Cambridge LCF is a computer program for reasoning about computation. It combines the methods of mathematical logic with domain theory, the basis of the denotational approach to specifying the meaning of program statements. Cambridge LCF is based on an earlier theorem-proving system, Edinburgh LCF, which introduced a design that gives the user flexibility to use and extend the system. A goal of this book is to explain the design, which has been adopted in several other systems. The book consists of two parts. Part I outlines the mathematical preliminaries, elementary logic and domain theory, and explains them at an intuitive level, giving reference to more advanced reading; Part II provides sufficient detail to serve as a reference manual for Cambridge LCF. It will also be a useful guide for implementors of other programs based on the LCF approach.

The study of permutation groups has always been closely associated with that of highly symmetric structures. The objects considered here are countably infinite, but have only finitely many different substructures of any given finite size. They are precisely those structures which are determined by first-order logical axioms together with the assumption of countability. This book concerns such structures, their substructures and their automorphism groups. A wide range of techniques are used: group theory, combinatorics, Baire category and measure among them. The book arose from lectures given at a research symposium and retains their informal style, whilst including as well many recent results from a variety of sources. It concludes with exercises and unsolved research problems.

This book constitutes the refereed proceedings of the 18th International Conference on the Application and Theory of Petri Nets, ICATPN'97, held in Toulouse, France, in June 1997.
The 22 revised full papers presented in the volume were selected from a total of 61 submissions; also included are three invited contributions. All relevant topics in the area are addressed. Besides a variety of Petri net classes, workflow management, telecommunication networking, constraint satisfaction, program semantics, concurrency, and temporal logic are among the topics addressed.

This book covers blockchain from the underlying principles to how it enables applications to survive and surf on its shoulder. Having covered the fundamentals of blockchain, the book turns to cryptocurrency. It thoroughly examines Bitcoin before presenting six other major currencies in a rounded discussion. The book then bridges between technology and finance, concentrating on how blockchain-based applications, including cryptocurrencies, have pushed hard against mainstream industries in a bid to cement their positions permanent. It discusses blockchain as underlying banking technology, crypto mining and offering, cryptocurrency as investment instruments, crypto regulations, and markets.