This is a compact mtroduction to some of the pnncipal tOpICS of mathematical logic . In the belief that beginners should be exposed to the most natural and easiest proofs, I have used free-swinging set-theoretic methods. The significance of a demand for constructive proofs can be evaluated only after a certain amount of experience with mathematical logic has been obtained. If we are to be expelled from "Cantor's paradise" (as nonconstructive set theory was called by Hilbert), at least we should know what we are missing. The major changes in this new edition are the following. (1) In Chapter 5, Effective Computability, Turing-computabIlity IS now the central notion, and diagrams (flow-charts) are used to construct Turing machines. There are also treatments of Markov algorithms, Herbrand-Godel-computability, register machines, and random access machines. Recursion theory is gone into a little more deeply, including the s-m-n theorem, the recursion theorem, and Rice's Theorem. (2) The proofs of the Incompleteness Theorems are now based upon the Diagonalization Lemma. Lob's Theorem and its connection with Godel's Second Theorem are also studied. (3) In Chapter 2, Quantification Theory, Henkin's proof of the completeness theorem has been postponed until the reader has gained more experience in proof techniques. The exposition of the proof itself has been improved by breaking it down into smaller pieces and using the notion of a scapegoat theory. There is also an entirely new section on semantic trees.

Fuzzy Algorithms for Control gives an overview of the research results of a number of European research groups that are active and play a leading role in the field of fuzzy modeling and control. It contains 12 chapters divided into three parts. Chapters in the first part address the position of fuzzy systems in control engineering and in the AI community. State-of-the-art surveys on fuzzy modeling and control are presented along with a critical assessment of the role of these methodologists in control engineering. The second part is concerned with several analysis and design issues in fuzzy control systems. The analytical issues addressed include the algebraic representation of fuzzy models of different types, their approximation properties, and stability analysis of fuzzy control systems. Several design aspects are addressed, including performance specification for control systems in a fuzzy decision-making framework and complexity reduction in multivariable fuzzy systems. In the third part of the book, a number of applications of fuzzy control are presented. It is shown that fuzzy control in combination with other techniques such as fuzzy data analysis is an effective approach to the control of modern processes which present many challenges for the design of control systems. One has to cope with problems such as process nonlinearity, time-varying characteristics for incomplete process knowledge. Examples of real-world industrial applications presented in this book are a blast furnace, a lime kiln and a solar plant. Other examples of challenging problems in which fuzzy logic plays an important role and which are included in this book are mobile robotics and aircraft control. The aim of this book is to address both theoretical and practical subjects in a balanced way. It will therefore be useful for readers from the academic world and also from industry who want to apply fuzzy control in practice.

Since the advent of the Semantic Web, interest in the dynamics of ontologies (ontology evolution) has grown significantly. Belief revision presents a good theoretical framework for dealing with this problem; however, classical belief revision is not well suited for logics such as Description Logics.

"Belief Revision in Non-Classical Logics" presents a framework which can be applied to a wide class of logics that include - besides most Description Logics such as the ones behind OWL - Horn Logic and Intuitionistic logic, amongst others. The author also presents algorithms for the most important constructions in belief bases. Researchers and practitioners in theoretical computing will find this an invaluable resource.

Between the two world wars, Stanislaw Lesniewski (1886-1939), created the famous and important system of foundations of mathematics that comprises three deductive theories: Protothetic, Ontology, and Mereology. His research started in 1914 with studies on the general theory of sets (later named `Mereology'). Ontology followed between 1919 and 1921, and was the next step towards an integrated system. In order to combine these two systematically he constructed Protothetic - the system of `first principles'. Together they amount to what Z. Jordan called `... most thorough, original, and philosophically significant attempt to provide a logically secure foundation for the whole of mathematics'. The volume collects many of the most significant commentaries on, and contributions to, Protothetic. A Protothetic Bibliography is included.

Although several books cover the coding theory of wireless communications and the hardware technologies and coding techniques of optical CDMA, no book has been specifically dedicated to optical coding theory-until now. Written by renowned authorities in the field, Optical Coding Theory with Prime gathers together in one volume the fundamentals and developments of optical coding theory, with a focus on families of prime codes, supplemented with several families of non-prime codes. The book also explores potential applications to coding-based optical systems and networks. Learn How to Construct and Analyze Optical Codes The authors use a theorem-proof approach, breaking down theories into digestible form so that readers can understand the main message without searching through tedious proofs. The book begins with the mathematical tools needed to understand and apply optical coding theory, from Galois fields and matrices to Gaussian and combinatorial analytical tools. Using a wealth of examples, the authors show how optical codes are constructed and analyzed, and detail their performance in a variety of applications. The book examines families of 1-D and 2-D asynchronous and synchronous, multilength, and 3-D prime codes, and some non-prime codes. Get a Working Knowledge of Optical Coding Theory to Help You Design Optical Systems and Networks Prerequisites include a basic knowledge of linear algebra and coding theory, as well as a foundation in probability and communications theory. This book draws on the authors' extensive research to offer an authoritative reference on the emerging field of optical coding theory. In addition, it supplies a working knowledge of the theory and optical codes to help readers in the design of coding-based optical systems and networks. For more on the technological aspects of optical CDMA, see Optical Code Division Multiple Access: Fundamentals and Applications (CRC Press 2005).

Mathematical Logic and Model Theory: A Brief Introduction offers a streamlined yet easy-to-read introduction to mathematical logic and basic model theory. It presents, in a self-contained manner, the essential aspects of model theory needed to understand model theoretic algebra. As a profound application of model theory in algebra, the last part of this book develops a complete proof of Ax and Kochen's work on Artin's conjecture about Diophantine properties of p-adic number fields. The character of model theoretic constructions and results differ quite significantly from that commonly found in algebra, by the treatment of formulae as mathematical objects. It is therefore indispensable to first become familiar with the problems and methods of mathematical logic. Therefore, the text is divided into three parts: an introduction into mathematical logic (Chapter 1), model theory (Chapters 2 and 3), and the model theoretic treatment of several algebraic theories (Chapter 4). This book will be of interest to both advanced undergraduate and graduate students studying model theory and its applications to algebra. It may also be used for self-study.

Weighted finite automata are classical nondeterministic finite automata in which the transitions carry weights. These weights may model, for example, the cost involved when executing a transition, the resources or time needed for this, or the probability or reliability of its successful execution. Weights can also be added to classical automata with infinite state sets like pushdown automata, and this extension constitutes the general concept of weighted automata. Since their introduction in the 1960s they have stimulated research in related areas of theoretical computer science, including formal language theory, algebra, logic, and discrete structures. Moreover, weighted automata and weighted context-free grammars have found application in natural-language processing, speech recognition, and digital image compression.

This book covers all the main aspects of weighted automata and formal power series methods, ranging from theory to applications. The contributors are the leading experts in their respective areas, and each chapter presents a detailed survey of the state of the art and pointers to future research. The chapters in Part I cover the foundations of the theory of weighted automata, specifically addressing semirings, power series, and fixed point theory. Part II investigates different concepts of weighted recognizability. Part III examines alternative types of weighted automata and various discrete structures other than words. Finally, Part IV deals with applications of weighted automata, including digital image compression, fuzzy languages, model checking, and natural-language processing.

Computer scientists and mathematicians will find this book an excellent survey and reference volume, and it will also be a valuable resource for students exploring this exciting research area.

Point-counting results for sets in real Euclidean space have found remarkable applications to diophantine geometry, enabling significant progress on the Andre-Oort and Zilber-Pink conjectures. The results combine ideas close to transcendence theory with the strong tameness properties of sets that are definable in an o-minimal structure, and thus the material treated connects ideas in model theory, transcendence theory, and arithmetic. This book describes the counting results and their applications along with their model-theoretic and transcendence connections. Core results are presented in detail to demonstrate the flexibility of the method, while wider developments are described in order to illustrate the breadth of the diophantine conjectures and to highlight key arithmetical ingredients. The underlying ideas are elementary and most of the book can be read with only a basic familiarity with number theory and complex algebraic geometry. It serves as an introduction for postgraduate students and researchers to the main ideas, results, problems, and themes of current research in this area.

Spatial Reasoning Puzzles That Make Kids Think! engages even the most reluctant math learner. In this fun and challenging book, students must conquer four types of logical and spatial reasoning puzzles (Slitherlink, Hashiwokakero, Masyu, and Yajilin). The rules for each type of puzzle are very different, but easy to understand. The challenge is for students to apply their critical thinking skills to new situations and develop new strategies for solving each puzzle. Teacher support is provided for solving the puzzles and also for helping students to create puzzles of their own. Students will be begging for more of these unique spatial reasoning puzzles! Grades 6-8

This book provides a broad-ranging, but detailed overview of the basics of Fuzzy Logic. The fundamentals of Fuzzy Logic are discussed in detail, and illustrated with various solved examples. The book also deals with applications of Fuzzy Logic, to help readers more fully understand the concepts involved. Solutions to the problems are programmed using MATLAB 6.0, with simulated results. The MATLAB Fuzzy Logic toolbox is provided for easy reference.

Als mehrbandiges Nachschlagewerk ist das Springer-Handbuch der Mathematik in erster Linie fur wissenschaftliche Bibliotheken, akademische Institutionen und Firmen sowie interessierte Individualkunden in Forschung und Lehregedacht. Es erganzt das einbandige themenumfassende Springer-Taschenbuch der Mathematik (ehemaliger Titel Teubner-Taschenbuch der Mathematik), das sich in seiner begrenzten Stoffauswahl besonders an Studierende richtet.Teil IV des Springer-Handbuchs enthalt die folgenden Zusatzkapitel zum Springer-Taschenbuch: Hohere Analysis, Lineare sowie Nichtlineare Funktionalanalysis und ihre Anwendungen, Dynamische Systeme, Nichtlineare partielle Differentialgleichungen, Mannigfaltigkeiten, Riemannsche Geometrie und allgemeine Relativitatstheorie, Liegruppen, Liealgebren und Elementarteilchen, Topologie, Krummung und Analysis.

Now in it's fourth edition, this classic work on logic presents the student with a clear, concise introduction to the subject of logic and its apllications. The first part of the book introduces the concepts and principles which make up the elements of logic, demonstrating that the concepts of logic are found in all branches of mathematics, and that logical laws are constantly applied in mathematical reasoning. The book goes on to show the applications of logic in mathematical theory building using concrete examples, drawing upon the concepts and principles presented in the first section. An introduction to the theory of real numbers is also presented. Exercises are included, designed to assist in the assimilation of the concepts and principles. Throughout the conceptual side or logic is stressed.

Thoroughly revised by the author's son, the book remains a fundametal guide to modern mathematica logic and is a very important addition to this highly successful series.

This volume is based on the talks given at the Workshop on Infinity and Truth held at the Institute for Mathematical Sciences, National University of Singapore, from 25 to 29 July 2011. The chapters cover topics in mathematical and philosophical logic that examine various aspects of the foundations of mathematics. The theme of the volume focuses on two basic foundational questions: (i) What is the nature of mathematical truth and how does one resolve questions that are formally unsolvable within the Zermelo-Fraenkel Set Theory with the Axiom of Choice, and (ii) Do the discoveries in mathematics provide evidence favoring one philosophical view over others? These issues are discussed from the vantage point of recent progress in foundational studies.The final chapter features questions proposed by the participants of the Workshop that will drive foundational research. The wide range of topics covered here will be of interest to students, researchers and mathematicians concerned with issues in the foundations of mathematics.

This third edition, now available in paperback, is a follow up to the author's classic Boolean-Valued Models and Independence Proofs in Set Theory,. It provides an exposition of some of the most important results in set theory obtained in the 20th century: the independence of the continuum hypothesis and the axiom of choice. Aimed at graduate students and researchers in mathematics, mathematical logic, philosophy, and computer science, the third edition has been extensively updated with expanded introductory material, new chapters, and a new appendix on category theory. It covers recent developments in the field and contains numerous exercises, along with updated and increased coverage of the background material. This new paperback edition includes additional corrections and, for the first time, will make this landmark text accessible to students in logic and set theory.

This book started with "Lattice Theory, First Concepts," in 1971. Then came "General Lattice Theory," First Edition, in 1978, and the Second Edition twenty years later. Since the publication of the first edition in 1978, "General Lattice Theory" has become the authoritative introduction to lattice theory for graduate students and the standard reference for researchers. The First Edition set out to introduce and survey lattice theory. Some 12,000 papers have been published in the field since then; so "Lattice Theory: Foundation" focuses on introducing the field, laying the foundation for special topics and applications. "Lattice Theory: Foundation," based on the previous three books, covers the fundamental concepts and results. The main topics are distributivity, congruences, constructions, modularity and semimodularity, varieties, and free products. The chapter on constructions is new, all the other chapters are revised and expanded versions from the earlier volumes. Almost 40 diamond sections, many written by leading specialists in these fields, provide a brief glimpse into special topics beyond the basics. Lattice theory has come a long way... For those who appreciate lattice theory, or who are curious about its techniques and intriguing internal problems, Professor Gratzer's lucid new book provides a most valuable guide to many recent developments. Even a cursory reading should provide those few who may still believe that lattice theory is superficial or naive, with convincing evidence of its technical depth and sophistication. "Bulletin of the American Mathematical Society" Gratzer s book General Lattice Theory has become the lattice theorist s bible. "Mathematical Reviews"

Als mehrbandiges Nachschlagewerk ist das Springer-Handbuch der Mathematik in erster Linie fur wissenschaftliche Bibliotheken, akademische Institutionen und Firmen sowie interessierte Individualkunden in Forschung und Lehregedacht. Es erganzt das einbandige themenumfassende Springer-Taschenbuch der Mathematik (ehemaliger Titel Teubner-Taschenbuch der Mathematik), das sich in seiner begrenzten Stoffauswahl besonders an Studierende richtet. Teil I des Springer-Handbuchs enthalt neben dem einfuhrenden Kapitel und dem Kapitel 1 des Springer-Taschenbuchs zusatzliches Material zur hoheren komplexen Funktionentheorie und zur allgemeinen Theorie der partiellen Differentialgleichungen.

Circular analyses of philosophical, linguistic, or computational phenomena have been attacked on the assumption that they conflict with mathematical rigour. Barwise and Moss have undertaken to prove this assumption false. This volume is concerned with extending the modelling capabilities of set theory to provide a uniform treatment of circular phenomena. As a means of guiding the reader through the concrete examples of the theory, the authors have included many exercises and solutions: these exercises range in difficulty and ultimately stimulate the reader to come up with new results. Vicious Circles is intended for use by researchers who want to use hypersets; although some experience in mathematics is necessary, the book is accessible to people with widely differing backgrounds and interests.

This book is a collection of articles, some introductory, some extended surveys, and some containing previously unpublished research, on a range of topics linking infinite permutation group theory and model theory. Topics covered include: oligomorphic permutation groups and omega-categorical structures; totally categorical structures and covers; automorphism groups of recursively saturated structures; Jordan groups; Hrushovski's constructions of pseudoplanes; permutation groups of finite Morley rank; applications of permutation group theory to models of set theory without the axiom of choice. There are introductory chapters by the editors on general model theory and permutation theory, recursively saturated structures, and on groups of finite Morley rank. The book is almost self-contained, and should be useful to both a beginning postgraduate student meeting the subject for the first time, and to an active researcher from either of the two main fields looking for an overview of the subject.

"Philosophie der Mathematik" wird in diesem Buch verstanden als ein Bemuhen um die Klarung solcher Fragen, die die Mathematik selber aufwirft, aber mit ihren eigenen Methoden nicht beantworten kann. Dazu gehoeren beispielsweise die Fragen nach dem ontologischen Status der mathematischen Objekte (z.B.: was ist die Natur der mathematischen Objekte?) und dem epistemologischen Status der mathematischen Theoreme (z.B.: aus welchen Quellen schoepfen wir, wenn wir mathematische Theoreme beweisen?). Die Antworten, die Platon, Aristoteles, Euklid, Descartes, Locke, Leibniz, Kant, Frege, Dedekind, Hilbert und andere gegeben haben, sollen im Detail studiert werden. Dies fuhrt zu tiefen Einsichten, nicht nur in die Geschichte der Mathematik, sondern auch in die Konzeption der Mathematik, so wie sie in der Gegenwart allgemein vertreten wird.

The annual conference of the European Association for Computer Science Logic (EACSL), CSL 2009, was held in Coimbra (Portugal), September 7-11, 2009. The conference series started as a programme of International Workshops on Computer Science Logic, and then at its sixth meeting became the Annual C- ference of the EACSL. This conference was the 23rd meeting and 18th EACSL conference; it was organized at the Department of Mathematics, Faculty of S- ence and Technology, University of Coimbra. In response to the call for papers, a total of 122 abstracts were submitted to CSL 2009of which 89 werefollowedby a full paper. The ProgrammeCommittee selected 34 papers for presentation at the conference and publication in these proceedings. The Ackermann Award is the EACSL Outstanding Dissertation Award for Logic in Computer Science. The awardrecipient for 2009 was Jakob Nordstr om. Citation of the award, abstract of the thesis, and a biographical sketch of the recipient may be found at the end of the proceedings. The award was sponsored for the years 2007-2009 by Logitech S.A.

The language of -categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an -category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of -categories from first principles in a model-independent fashion using the axiomatic framework of an -cosmos, the universe in which -categories live as objects. An -cosmos is a fertile setting for the formal category theory of -categories, and in this way the foundational proofs in -category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.

The ability to reason correctly is critical to most aspects of computer science and to software development in particular. This book teaches readers how to better reason about software development, to communicate reasoning, to distinguish between good and bad reasoning, and to read professional literature that presumes knowledge of elementary logic. The reader 's knowledge and understanding can be assessed through numerous examples and exercises. This book provides a reader-friendly foundation to logic and offers valuable insight into the topic, thereby serving as a helpful reference for practitioners, as well as students studying software development.

How you can become better at solving real-world problems by learning creative puzzle-solving skills We solve countless problems-big and small-every day. With so much practice, why do we often have trouble making simple decisions-much less arriving at optimal solutions to important questions? Are we doomed to this muddle-or is there a practical way to learn to think more effectively and creatively? In this enlightening, entertaining, and inspiring book, Edward Burger shows how we can become far better at solving real-world problems by learning creative puzzle-solving skills using simple, effective thinking techniques. Making Up Your Own Mind teaches these techniques-including how to ask good questions, fail and try again, and change your mind-and then helps you practice them with fun verbal and visual puzzles. The goal is not to quickly solve each challenge but to come up with as many different ways of thinking about it as possible. As you see the puzzles in ever-greater depth, your mind will change, helping you become a more imaginative and creative thinker in daily life. And learning how to be a better thinker pays off in incalculable ways for anyone-including students, businesspeople, professionals, athletes, artists, leaders, and lifelong learners. A book about changing your mind and creating an even better version of yourself through mental play, Making Up Your Own Mind will delight and reward anyone who wants to learn how to find better solutions to life's innumerable puzzles. And the puzzles extend to the thought-provoking format of the book itself because one of the later short chapters is printed upside down while another is printed in mirror image, further challenging the reader to see the world through different perspectives and make new meaning.

This volume contains the papers presented at the 11th International Conference on Theory and Applications of Satis?ability Testing (SAT 2008). The series of International Conferences on Theory and Applications of S- is?ability Testing (SAT) has evolved from a ?rst workshop on SAT in 1996 to an annual international conference which is a platform for researchers studying various aspects of the propositional satis?ability problem and its applications. In the past, the SAT conference venue alternated between Europe and North America. For the ?rst time, the conference venue was in Asia, more precisely at the Zhudao Guest House, near Sun Yat-Sen University in Guangzhou, P. R. China. Many hard combinatorial problems can be encoded into SAT. Therefore - provementsonheuristics onthe practicalside, as wellastheoreticalinsightsinto SAT apply to a large range of real-world problems. More speci?cally, many - portant practical veri?cation problems can be rephrased as SAT problems. This applies to veri?cation problems in hardware and software. Thus SAT is bec- ing one of the most important core technologies to verify secure and dependable systems. The topics of the conference span practical and theoretical research on SAT and its applications and include but are not limited to proof systems, proof complexity, search algorithms, heuristics, analysis of algorithms, hard instances, randomized formulae, problem encodings, industrial applications, solvers, s- pli?ers, tools, case studies, and empirical results. SAT is interpreted in a rather broad sense: besides propositional satis?ability, it includes, for example, the - main of quanti?ed Boolean formulae (QBF) and satis?ability modulo theories (SMT)

This book constitutes the refereed proceedings of the Fourth International Conference on Concept Lattices and their Applications, CLA 2006, held in Tunis, Tunisia, October 30-November 1, 2006.

The 18 revised full papers together with 3 invited contributions presented were carefully reviewed and selected from 41 submissions. The topics include formal concept analysis, foundations of FCA, mathematical structures related to FCA, relationship of FCA to other methods of data analysis, visualization of data in FCA, and applications of FCA.