Finite Fields are fundamental structures of Discrete Mathematics. They serve as basic data structures in pure disciplines like Finite Geometries and Combinatorics, and also have aroused much interest in applied disciplines like Coding Theory and Cryptography. A look at the topics of the proceed ings volume of the Third International Conference on Finite Fields and Their Applications (Glasgow, 1995) (see [18]), or at the list of references in I. E. Shparlinski's book [47] (a recent extensive survey on the Theory of Finite Fields with particular emphasis on computational aspects), shows that the area of Finite Fields goes through a tremendous development. The central topic of the present text is the famous Normal Basis Theo rem, a classical result from field theory, stating that in every finite dimen sional Galois extension E over F there exists an element w whose conjugates under the Galois group of E over F form an F-basis of E (i. e. , a normal basis of E over F; w is called free in E over F). For finite fields, the Nor mal Basis Theorem has first been proved by K. Hensel [19] in 1888. Since normal bases in finite fields in the last two decades have been proved to be very useful for doing arithmetic computations, at present, the algorithmic and explicit construction of (particular) such bases has become one of the major research topics in Finite Field Theory.

Scheduling in Parallel Computing Systems: Fuzzy and Annealing Techniques advocates the viability of using fuzzy and annealing methods in solving scheduling problems for parallel computing systems. The book proposes new techniques for both static and dynamic scheduling, using emerging paradigms that are inspired by natural phenomena such as fuzzy logic, mean-field annealing, and simulated annealing. Systems that are designed using such techniques are often referred to in the literature as intelligent' because of their capability to adapt to sudden changes in their environments. Moreover, most of these changes cannot be anticipated in advance or included in the original design of the system. Scheduling in Parallel Computing Systems: Fuzzy and Annealing Techniques provides results that prove such approaches can become viable alternatives to orthodox solutions to the scheduling problem, which are mostly based on heuristics. Although heuristics are robust and reliable when solving certain instances of the scheduling problem, they do not perform well when one needs to obtain solutions to general forms of the scheduling problem. On the other hand, techniques inspired by natural phenomena have been successfully applied for solving a wide range of combinatorial optimization problems (e.g. traveling salesman, graph partitioning). The success of these methods motivated their use in this book to solve scheduling problems that are known to be formidable combinatorial problems. Scheduling in Parallel Computing Systems: Fuzzy and Annealing Techniques is an excellent reference and may be used for advanced courses on the topic.

The main theme in classical ring theory is the structure theory of rings of a particular kind. For example, no one text book in ring theory could miss the Wedderburn-Artin theorem, which says that a ring R is semisimple Artinian iffR is isomorphic to a finite direct sum of full matrix rings over skew fields. This is an example of a finiteness condition which, at least historically, has dominated in ring theory. Ifwe would like to consider a requirement of a lattice-theoretical type, other than being Artinian or Noetherian, the most natural is uni-seriality. Here a module M is called uni-serial if its lattice of submodules is a chain, and a ring R is uni-serial if both RR and RR are uni-serial modules. The class of uni-serial rings includes commutative valuation rings and closed under homomorphic images. But it is not closed under direct sums nor with respect to Morita equivalence: a matrix ring over a uni-serial ring is not uni-serial. There is a class of rings which is very close to uni-serial but closed under the constructions just mentioned: serial rings. A ring R is called serial if RR and RR is a direct sum (necessarily finite) of uni-serial modules. Amongst others this class includes triangular matrix rings over a skew field. Also if F is a finite field of characteristic p and G is a finite group with a cyclic normal p-Sylow subgroup, then the group ring FG is serial.

Logic and Complexity looks at basic logic as it is used in Computer Science, and provides students with a logical approach to Complexity theory. With plenty of exercises, this book presents classical notions of mathematical logic, such as decidability, completeness and incompleteness, as well as new ideas brought by complexity theory such as NP-completeness, randomness and approximations, providing a better understanding for efficient algorithmic solutions to problems. Divided into three parts, it covers: - Model Theory and Recursive Functions - introducing the basic model theory of propositional, 1st order, inductive definitions and 2nd order logic. Recursive functions, Turing computability and decidability are also examined. - Descriptive Complexity - looking at the relationship between definitions of problems, queries, properties of programs and their computational complexity. - Approximation - explaining how some optimization problems and counting problems can be approximated according to their logical form. Logic is important in Computer Science, particularly for verification problems and database query languages such as SQL. Students and researchers in this field will find this book of great interest.

Aggregation plays a central role in many of the technological tasks we are faced with. The importance of this process will become even greater as we move more and more toward becoming an information-cent.ered society, us is happening with the rapid growth of the Internet and the World Wirle Weh. Here we shall be faced with many issues related to the fusion of information. One very pressing issue here is the development of mechanisms to help search for information, a problem that clearly has a strong aggregation-related component. More generally, in order to model the sophisticated ways in which human beings process information, as well as going beyond the human capa bilities, we need provide a basket of aggregation tools. The centrality of aggregation in human thought can be be very clearly seen by looking at neural networks, a technology motivated by modeling the human brain. One can see that the basic operations involved in these networks are learning and aggregation. The Ordered Weighted Averaging (OWA) operators provide a parameter ized family of aggregation operators which include many of the well-known operators such as the maximum, minimum and the simple average."

Fuzzy hardware developments have been a major force driving the applications of fuzzy set theory and fuzzy logic in both science and engineering. This volume provides the reader with a comprehensive up-to-date look at recent works describing new innovative developments of fuzzy hardware. An important research trend is the design of improved fuzzy hardware. There is an increasing interest in both analog and digital implementations of fuzzy controllers in particular and fuzzy systems in general. Specialized analog and digital VLSI implementations of fuzzy systems, in the form of dedicated architectures, aim at the highest implementation efficiency. This particular efficiency is asserted in terms of processing speed and silicon utilization. Processing speed in particular has caught the attention of developers of fuzzy hardware and researchers in the field.The volume includes detailed material on a variety of fuzzy hardware related topics such as: * Historical review of fuzzy hardware research * Fuzzy hardware based on encoded trapezoids * Pulse stream techniques for fuzzy hardware * Hardware realization of fuzzy neural networks * Design of analog neuro-fuzzy systems in CMOS digital technologies * Fuzzy controller synthesis method * Automatic design of digital and analog neuro-fuzzy controllers * Electronic implementation of complex controllers * Silicon compilation of fuzzy hardware systems * Digital fuzzy hardware processing * Parallel processor architecture for real-time fuzzy applications * Fuzzy cellular systems Fuzzy Hardware: Architectures and Applications is a technical reference book for researchers, engineers and scientists interested in fuzzy systems in general and in building fuzzy systems in particular.

In writing this book, our goal was to produce a text suitable for a first course in mathematical logic more attuned than the traditional textbooks to the re cent dramatic growth in the applications oflogic to computer science. Thus, our choice oftopics has been heavily influenced by such applications. Of course, we cover the basic traditional topics: syntax, semantics, soundnes5, completeness and compactness as well as a few more advanced results such as the theorems of Skolem-Lowenheim and Herbrand. Much ofour book, however, deals with other less traditional topics. Resolution theorem proving plays a major role in our treatment of logic especially in its application to Logic Programming and PRO LOG. We deal extensively with the mathematical foundations ofall three ofthese subjects. In addition, we include two chapters on nonclassical logics - modal and intuitionistic - that are becoming increasingly important in computer sci ence. We develop the basic material on the syntax and semantics (via Kripke frames) for each of these logics. In both cases, our approach to formal proofs, soundness and completeness uses modifications of the same tableau method in troduced for classical logic. We indicate how it can easily be adapted to various other special types of modal logics. A number of more advanced topics (includ ing nonmonotonic logic) are also briefly introduced both in the nonclassical logic chapters and in the material on Logic Programming and PROLOG.

Basic Real Analysis demonstrates the richness of real analysis, giving students an introduction both to mathematical rigor and to the deep theorems and counter examples that arise from such rigor. In this modern and systematic text, all the touchstone results and fundamentals are carefully presented in a style that requires little prior familiarity with proofs or mathematical language. With its many examples, exercises and broad view of analysis, this work is ideal for senior undergraduates and beginning graduate students, either in the classroom or for self-study.

A basic issue in computer science is the complexity of problems. Computational complexity measures how much time or memory is needed as a function of the input problem size. Descriptive complexity is concerned with problems which may be described in first-order logic. By virtue of the close relationship between logic and relational databses, it turns out that this subject has important applications to databases such as analysing the queries computable in polynomial time, analysing the parallel time needed to compute a query, and the analysis of nondeterministic classes. This book is written as a graduate text and so aims to provide a reasonably self-contained introduction to this subject. The author has provided numerous examples and exercises to further illustrate the ideas presented.

An Introduction to Fuzzy Logic Applications in Intelligent Systems consists of a collection of chapters written by leading experts in the field of fuzzy sets. Each chapter addresses an area where fuzzy sets have been applied to situations broadly related to intelligent systems. The volume provides an introduction to and an overview of recent applications of fuzzy sets to various areas of intelligent systems. Its purpose is to provide information and easy access for people new to the field. The book also serves as an excellent reference for researchers in the field and those working in the specifics of systems development. People in computer science, especially those in artificial intelligence, knowledge-based systems, and intelligent systems will find this to be a valuable sourcebook. Engineers, particularly control engineers, will also have a strong interest in this book. Finally, the book will be of interest to researchers working in decision support systems, operations research, decision theory, management science and applied mathematics. An Introduction to Fuzzy Logic Applications in Intelligent Systems may also be used as an introductory text and, as such, it is tutorial in nature.

Intelligent Hybrid Systems: Fuzzy Logic, Neural Networks, and Genetic Algorithms is an organized edited collection of contributed chapters covering basic principles, methodologies, and applications of fuzzy systems, neural networks and genetic algorithms. All chapters are original contributions by leading researchers written exclusively for this volume. This book reviews important concepts and models, and focuses on specific methodologies common to fuzzy systems, neural networks and evolutionary computation. The emphasis is on development of cooperative models of hybrid systems. Included are applications related to intelligent data analysis, process analysis, intelligent adaptive information systems, systems identification, nonlinear systems, power and water system design, and many others. Intelligent Hybrid Systems: Fuzzy Logic, Neural Networks, and Genetic Algorithms provides researchers and engineers with up-to-date coverage of new results, methodologies and applications for building intelligent systems capable of solving large-scale problems.

ClDo _ IIIIIIIoaIIIics bu _ die 'EI JDDi, *** sij'_ ...-..._. je _ ...lIbupalaJllllllllll __ D'y_poa~: wbae it beIoap...die . . "...,. . _ DOD to dlecluly __ * __ . ~ 1110 _ is dioapaI; -. . e _ may be EricT. BeD IbIetodo--'_iL O. 1feaoriIide Mathematics is a tool for dloogIrt. A bighly necessary tool in a world where both feedback and noolineari- ties abound. Similarly, all kinds of parts of IIIIIIhcmatiI:s serve as tools for odIcr parts and for ocher sci- eoccs. Applying a simple rewriting rule to the quote on the right above one finds suc:h stalements as: 'One ser- vice topology has rcncIerM mathematical physics ...'; 'One service logic has rendered computer science . * . '; 'One service category theory has rmdcn:d mathematics ...'. All arguably true. And all statements obrainable this way form part of the raison d'etm of this series. This series, Mathmlatics tDIII Its Applications, saaned in 1977. Now that over one hundred volumcs have appeared it seems opportune to reexamine its scope. AI. the time I wrote "Growing spccialization and divenification have brought a host of monographs and textbooks on incJeasingly specialized topics. However, the 'tree' of knowledge of JJJatbcmatics and reIatcd ficIds docs not grow only by putting forth new bnDdIcs. It also happens, quite often in fact, that brancbes which were thought to be comp1etcly disparate am suddenly seen to be rdatcd.

Science involves descriptions of the world we live in. It also depends on nature exhibiting what we can best describe as a high aLgorithmic content. The theme running through this collection of papers is that of the interaction between descriptions, in the form of formal theories, and the algorithmic content of what is described, namely of the modeLs of those theories. This appears most explicitly here in a number of valuable, and substantial, contributions to what has until recently been known as 'recursive model theory' - an area in which researchers from the former Soviet Union (in particular Novosibirsk) have been pre-eminent. There are also articles concerned with the computability of aspects of familiar mathematical structures, and - a return to the sort of basic underlying questions considered by Alan Turing in the early days of the subject - an article giving a new perspective on computability in the real world. And, of course, there are also articles concerned with the classical theory of computability, including the first widely available survey of work on quasi-reducibility. The contributors, all internationally recognised experts in their fields, have been associated with the three-year INTAS-RFBR Research Project "Com putability and Models" (Project No. 972-139), and most have participated in one or more of the various international workshops (in Novosibirsk, Heidelberg and Almaty) and otherresearch activities of the network.

Reasoning and Unification over Conceptual Graphs is an exploration of automated reasoning and resolution in the expanding field of Conceptual Structures. Designed not only for computing scientists researching Conceptual Graphs, but also for anyone interested in exploring the design of knowledge bases, the book explores what are proving to be the fundamental methods for representing semantic relations in knowledge bases. While it provides the first comprehensive treatment of Conceptual Graph unification and reasoning, the book also addresses fundamental issues of graph matching, automated reasoning, knowledge bases, constraints, ontology and design. With a large number of examples, illustrations, and both formal and informal definitions and discussions, this book is excellent as a tutorial for the reader new to Conceptual Graphs, or as a reference book for a senior researcher in Artificial Intelligence, Knowledge Representation or Automated Reasoning.

Domain theory is a rich interdisciplinary area at the intersection of logic, computer science, and mathematics. This volume contains selected papers presented at the International Symposium on Domain Theory which took place in Shanghai in October 1999. Topics of papers range from the encounters between topology and domain theory, sober spaces, Lawson topology, real number computability and continuous functionals to fuzzy modelling, logic programming, and pi-calculi. This book is a valuable reference for researchers and students interested in this rapidly developing area of theoretical computer science.

One criterion for classifying books is whether they are written for a single pur pose or for multiple purposes. This book belongs to the category of multipurpose books, but one of its roles is predominant-it is primarily a textbook. As such, it can be used for a variety ofcourses at the first-year graduate or upper-division undergraduate level. A common characteristic of these courses is that they cover fundamental systems concepts, major categories of systems problems, and some selected methods for dealing with these problems at a rather general level. A unique feature of the book is that the concepts, problems, and methods are introduced in the context of an architectural formulation of an expert system referred to as the general systems problem solver or aSPS-whose aim is to provide users ofall kinds with computer-based systems knowledge and methodo logy. Theasps architecture, which is developed throughout the book, facilitates a framework that is conducive to acoherent, comprehensive, and pragmaticcoverage ofsystems fundamentals-concepts, problems, and methods. A course that covers systems fundamentals is now offered not only in sys tems science, information science, or systems engineering programs, but in many programs in other disciplines as well. Although the level ofcoverage for systems science or engineering students is surely different from that used for students in other disciplines, this book is designed to serve both of these needs."

Computational intelligence paradigms have attracted the growing interest of researchers, scientists, engineers and application engineers in a number of everyday applications. These applications are not limited to any particular field and include engineering, business, banking and consumer electronics. Computational intelligence paradigms include artificial intelligence, artificial neural networks, fuzzy systems and evolutionary computing. Artificial neural networks can mimic the biological information processing mechanism in a very limited sense. Evolutionary computing algorithms are used for optimisation applications, and fuzzy logic provides a basis for representing uncertain and imprecise knowledge. Practical Applications of Computational Intelligence Techniques contains twelve chapters providing actual application of these techniques in the real world. Such examples include, but are not limited to, intelligent household appliances, aerial spray models, industrial applications and medical diagnostics and practice. This book will be useful to researchers, practicing engineers/scientists and students, who are interested in developing practical applications in a computational intelligence environment.

For more than 30 years, the author has studied the model-theoretic aspects of the theory of valued fields and multi-valued fields. Many of the key results included in this book were obtained by the author whilst preparing the manuscript. Thus the unique overview of the theory, as developed in the book, has been previously unavailable. The book deals with the theory of valued fields and mutli-valued fields. The theory of Prufer rings is discussed from the `geometric' point of view. The author shows that by introducing the Zariski topology on families of valuation rings, it is possible to distinguish two important subfamilies of Prufer rings that correspond to Boolean and near Boolean families of valuation rings. Also, algebraic and model-theoretic properties of multi-valued fields with near Boolean families of valuation rings satisfying the local-global principle are studied. It is important that this principle is elementary, i.e., it can be expressed in the language of predicate calculus. The most important results obtained in the book include a criterion for the elementarity of an embedding of a multi-valued field and a criterion for the elementary equivalence for multi-valued fields from the class defined by the additional natural elementary conditions (absolute unramification, maximality and almost continuity of local elementary properties). The book concludes with a brief chapter discussing the bibliographic references available on the material presented, and a short history of the major developments within the field.

When solving real-life engineering problems, linguistic information is often encountered that is frequently hard to quantify using "classical" mathematical techniques. This linguistic information represents subjective knowledge. Through the assumptions made by the analyst when forming the mathematical model, the linguistic information is often ignored. On the other hand, a wide range of traffic and transportation engineering parameters are characterized by uncertainty, subjectivity, imprecision, and ambiguity. Human operators, dispatchers, drivers, and passengers use this subjective knowledge or linguistic information on a daily basis when making decisions. Decisions about route choice, mode of transportation, most suitable departure time, or dispatching trucks are made by drivers, passengers, or dispatchers. In each case the decision maker is a human. The environment in which a human expert (human controller) makes decisions is most often complex, making it difficult to formulate a suitable mathematical model. Thus, the development of fuzzy logic systems seems justified in such situations. In certain situations we accept linguistic information much more easily than numerical information. In the same vein, we are perfectly capable of accepting approximate numerical values and making decisions based on them. In a great number of cases we use approximate numerical values exclusively. It should be emphasized that the subjective estimates of different traffic parameters differs from dispatcher to dispatcher, driver to driver, and passenger to passenger.

In The Foundations of Quantum Mechanics - Historical Analysis and Open Questions, leading Italian researchers involved in different aspects of the foundations and history of quantum mechanics are brought together in an interdisciplinary debate. The book therefore presents an invaluable overview of the state of Italian work in the field at this moment, and of the open problems that still exist in the foundations of the theory. Audience: Physicists, logicians, mathematicians and epistemologists whose research concerns the historical analysis of quantum mechanics.

This volume consists of papers delivered at the conference 'The Lvov-Warsaw School and Contemporary Philosophy', organised in celebration of the hun- dredth anniversary ofKazimierz Twardowski's first lecture as Professor of Phi- losophy at Lvov University. This lecture can be regarded as the starting point of the development of analytic philosophy in Poland, which culminated in the Warsaw School of Logic. The conference was held in Lvov (15-17 November) and Warsaw (19-21 November 1995). It was organised jointly by the Ukrainian Academy of Sci- ences and the Polish Academy of Sciences. The general organisation was un- dertaken by Professor Andrzej Grzegorczyk (polish Academy of Sciences) and Professor Marat Vernikov (Ukrainian Academy of Sciences). Professors Jaroslaw Isaievich (Ukrainian Academy of Sciences) and Jan Wolenski (Jagiel- Ionian University) were responsible for the scientific programme. Over 100 philosophers participated in the conference. Papers published in this volume are organised according to the sections of the conference and rep- of the papers delivered. resent a selection The editors would like to express their gratitude to Professor Andrzej Grze- gorczyk, spiritus movens of the conference, who, by including the present vol- ume in a programme of publications connected with the hundredth anniversary of the Lvov-Warsaw School, provided financial support for its preparation. Fi- nally, we express our gratitude to Dr Timothy Childers of the Academy of Sciences of the Czech Republic for correcting the English of the papers.

Mathematics is often considered as a body of knowledge that is essen tially independent of linguistic formulations, in the sense that, once the content of this knowledge has been grasped, there remains only the problem of professional ability, that of clearly formulating and correctly proving it. However, the question is not so simple, and P. Weingartner's paper (Language and Coding-Dependency of Results in Logic and Mathe matics) deals with some results in logic and mathematics which reveal that certain notions are in general not invariant with respect to different choices of language and of coding processes. Five example are given: 1) The validity of axioms and rules of classical propositional logic depend on the interpretation of sentential variables; 2) The language dependency of verisimilitude; 3) The proof of the weak and strong anti inductivist theorems in Popper's theory of inductive support is not invariant with respect to limitative criteria put on classical logic; 4) The language-dependency of the concept of provability; 5) The language dependency of the existence of ungrounded and paradoxical sentences (in the sense of Kripke). The requirements of logical rigour and consistency are not the only criteria for the acceptance and appreciation of mathematical proposi tions and theories.

Fuzzy knowledge and fuzzy systems affect our lives today as systems enter the world of commerce. Fuzzy systems are incorporated in domestic appliances (washing machine, air conditioning, microwave, telephone) and in transport systems (a pilotless helicopter has recently completed a test flight). Future applications are expected to have dramatic implications for the demand for labor, among other things. It was with such thoughts in mind that this first international survey of future applications of fuzzy logic has been undertaken. The results are likely to be predictive for a decade beyond the millenium. The predictive element is combined with a bibliography which serves as an historical anchor as well as being both extensive and extremely useful. Analysis and Evaluation of Fuzzy Systems is thus a milestone in the development of fuzzy logic and applications of three representative subsystems: Fuzzy Control, Fuzzy Pattern Recognition and Fuzzy Communications.

Adaptive Resonance Theory Microchips describes circuit strategies resulting in efficient and functional adaptive resonance theory (ART) hardware systems. While ART algorithms have been developed in software by their creators, this is the first book that addresses efficient VLSI design of ART systems. All systems described in the book have been designed and fabricated (or are nearing completion) as VLSI microchips in anticipation of the impending proliferation of ART applications to autonomous intelligent systems. To accommodate these systems, the book not only provides circuit design techniques, but also validates them through experimental measurements. The book also includes a chapter tutorially describing four ART architectures (ART1, ARTMAP, Fuzzy-ART and Fuzzy-ARTMAP) while providing easily understandable MATLAB code examples to implement these four algorithms in software. In addition, an entire chapter is devoted to other potential applications for real-time data clustering and category learning.

3. Textbook for a course in expert systems, if an emphasis is placed on Chapters 1 to 3 and on a selection of material from Chapters 4 to 7. There is also the option of using an additional commercially available sheU for a programming project. In assigning a programming project, the instructor may use any part of a great variety of books covering many subjects, such as car repair. Instructions for mostofthe "weekend mechanic" books are close stylisticaUy to expert system rules. Contents Chapter 1 gives an introduction to the subject matter; it briefly presents basic concepts, history, and some perspectives ofexpert systems. Then itpresents the architecture of an expert system and explains the stages of building an expert system. The concept of uncertainty in expert systems and the necessity of deal ing with the phenomenon are then presented. The chapter ends with the descrip tion of taxonomy ofexpert systems. Chapter 2 focuses on knowledge representation. Four basic ways to repre sent knowledge in expert systems are presented: first-order logic, production sys tems, semantic nets, and frames. Chapter 3 contains material about knowledge acquisition. Among machine learning techniques, a methodofrule learning from examples is explained in de tail. Then problems ofrule-base verification are discussed. In particular, both consistency and completeness oftherule base are presented."