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Nonstandard Analysis
The book is an introduction with emphasis on those more advanced applications in analysis which are hardly accessible by other methods. Examples of such topics are a deeper analysis of certain functionals like Hahn-Banach limits or of finitely additive measures: From the viewpoint of classical analysis these are strange objects whose mere existence is even hard to prove. From the viewpoint of nonstandard analysis, these are rather 'explicit' objects.
New Computational Paradigms : Changing Conceptions of What is Computable
This book examines new developments in the theory and practice of computation from a mathematical perspective, with topics ranging from classical computability to complexity, from biocomputing to quantum computing. The book opens with an introduction by Andrew Hodges, the Turing biographer, who analyzes the pioneering work that anticipated recent developments concerning computation’s allegedly new paradigms. The remaining material covers traditional topics in computability theory such as relative computability, theory of numberings, and domain theory, in addition to topics on the relationships between proof theory, computability, and complexity theory.
Model and Mathematics : From the 19th to the 21st Century
This book collects the historical and medial perspectives of a systematic and epistemological analysis of the complicated, multifaceted relationship between model and mathematics, ranging from, for example, the physical mathematical models of the 19th century to the simulation and digital modelling of the 21st century. The aim of this anthology is to showcase the status of the mathematical model between abstraction and realization, presentation and representation, what is modeled and what models.
Institution-independent Model Theory
A model theory that is independent of any concrete logical system allows a general handling of a large variety of logics. This generality can be achieved by applying the theory of institutions that provides a precise mathematical formulation for the intuitive concept of a logical system. Especially in computer science, where the development of a huge number of specification logics is observable, institution-independent model theory simplifies and sometimes even enables a concise model-theoretic analysis of the system. Besides incorporating important methods and concepts from conventional model theory, the proposed top-down methodology allows for a structurally clean understanding of model-theoretic phenomena. As a consequence, results from conventional concrete model theory can be understood more easily, and sometimes even new results are obtained.
Foundations of information and knowledge systems ; 5th International Symposium, FoIKS 2008, Pisa, Italy, February 11-15, 2008. Proceedings
This book constitutes the refereed proceedings of the 5th International Symposium on Foundations of Information and Knowledge Systems, FoIKS 2008 held in Pisa, Italy, in February 2008. The 13 revised full papers presented together with 9 revised short papers and 3 invited lectures were carefully selected during two rounds of reviewing and improvement from from 79 submissions. The papers deal with any foundational aspect of information and knowledge systems, including submissions from researchers working in fields such as discrete mathematics, logic and algebra, model theory, information theory, complexity theory, algorithmics and computation, geometry, analysis, statistics and optimisation who are interested in applying their ideas, theories and methods to research on information and knowledge systems.
Foundations of information and knowledge systems ; 4th International Symposium, FoIKS 2006, Budapest, Hungary, February 14-17, 2006, Proceedings
This book constitutes the refereed proceedings of the 4th International Symposium on Foundations of Information and Knowledge Systems, held in February 2006. The 14 revised full papers presented together with three revised short papers and one invited paper were carefully reviewed and selected from 54 submissions. Among the topics covered are the theoretical foundations of information and knowledge systems, as well as mathematical fields such as discrete mathematics, combinatorics, logics and finite model theory, and applications thereof for research on database and knowledge base theory.
Foundations and applications of MIS : A model theory approach
Foundations and Applications of MIS presents a unique systems theory approach to management information system (MIS) development. The development is driven by the need to eliminate ambiguity in specification, design and construction of the application software. Further, the authors show that the considerable effort being expanded nowadays on validation, verification and testing, as required in current software engineering practices, will be reduced. The approach also reinforces the belief that MIS development is independent of software development. The work presents an approach that provides a theoretical foundation for MIS development from the systems theoretic viewpoint along with practical applications ranging from a transaction processing system to a solver system. Both formal systems theory and automatic system generation based on the authors' newly extended Prolog offer a significant increase in the efficiency of specification, design and production of the application software, as well as an increase in the functional reliability of the software produced. The book assumes a working knowledge of elementary set theory, logic, and familiarity with some systems concepts, such as the automaton model.
Finite model theory and its applications
This book gives a broad overview of core topics of finite model theory: expressive power, descriptive complexity, and zero-one laws, together with selected applications to database theory and artificial intelligence, especially, constraint databases and constraint satisfaction problems. The final chapter provides a concise modern introduction to modal logic, which emphasizes the continuity in spirit and technique with finite model theory. This underlying spirit involves the use of various fragments of, and hierarchies within, first order, second order, fixed point, and infinitary logics to gain insight into phenomena in complexity theory and combinatorics.
Finite Model Theory ; 2nd ed.
The book presents the main results of descriptive complexity theory, that is, the connections between axiomatizability of classes of finite structures and their complexity with respect to time and space bounds. The logics that are important in this context include fixed-point logics, transitive closure logics, and also certain infinitary languages; their model theory is studied in full detail. Other topics include DATALOG languages, quantifiers and oracles, 0-1 laws, and optimization and approximation problems. The book is written in such a way that the respective parts on model theory and descriptive complexity theory may be read independently. This second edition is a thoroughly revised and enlarged version of the original text.
Computer science logic ; Vol. 4207 ; 20th International Workshop, CSL 2006, 15th Annual Conference of the EACSL, Szeged, Hungary, September 25-29, 2006, Proceedings
Coverage includes automated deduction and interactive theorem proving, constructive mathematics and type theory, equational logic and term rewriting, automata and formal logics, modal and temporal logic, model checking, finite model theory, and more.
Computer Science Logic ; 21 International Workshop, CSL 2007, 16th Annual Conference of the EACSL, Lausanne, Switzerland, September 11-15, 2007, Proceedings
This book covers logic and games, expressiveness, games and trees, logic and deduction, lambda calculus, finite model theory, linear logic, proof theory, and game semantics.
Logica Universalis : Towards a General Theory of Logic
Modern logic has been intimately connected with algebra since its origins in figures such as Boole, De Morgan, and Peirce. But while universal algebra is a long recognized field, universal logic has only recently been named as such. This is perhaps because classical logic was until relatively recently taken by many as the "one true logic". But with the proliferation of special purpose non-classical logics in recent years, universal logic is clearly a field whose time has come. This book contains many excellent papers demonstrating the value of this approach.
Logica Universalis : Towards a General Theory of Logic
Signifies the arrival of a new renaissance in logic, a new revival not only of logic, but of the vision of logic as a unifying tool for science as a whole, including mathematics, physics, cosmology, computer science and AI. The book and the vision behind it give logic, conceived as a scientific study of rationality, new unifying power, new perspectives, and new horizons.Universal Logic is not a new logic, but a general theory of logics, considered as mathematical structures. The name was introduced about ten years ago, but the subject is as old as the beginning of modern logic: Alfred Tarski and other Polish logicians such as Adolf Lindenbaum developed a general theory of logics at the end of the 1920s based on consequence operations and logical matrices. The subject was revived after the flowering of thousands of new logics during the last thirty years: there was a need for a systematic theory of logics to put some order in this chaotic multiplicity.
An introduction to description logics
Designed so that domain knowledge can be described and so that computers can reason about this knowledge. DLs have recently gained increased importance since they form the logical basis of widely used ontology languages, in particular the web ontology language OWL. Written by four renowned experts, this is the first textbook on description logics. It is suitable for self-study by graduates and as the basis for a university course. Starting from a basic DL, the book introduces the reader to their syntax, semantics, reasoning problems and model theory and discusses the computational complexity of these reasoning problems and algorithms to solve them.
Advances in proof-theoretic semantics
This volume is the first ever collection devoted to the field of proof-theoretic semantics. Contributions address topics including the systematics of introduction and elimination rules and proofs of normalization, the categorial characterization of deductions, the relation between Heyting's and Gentzen's approaches to meaning, knowability paradoxes, proof-theoretic foundations of set theory, Dummett's justification of logical laws, Kreisel's theory of constructions, paradoxical reasoning, and the defence of model theory.
A Course on Mathematical Logic
This is a short, distinctive, modern, and motivated introduction to mathematical logic for senior undergraduate and beginning graduate students in mathematics and computer science. Any mathematician who is interested in knowing what logic is concerned with and who would like to learn Gödel’s incompleteness theorems should find this book particularly convenient. The treatment is thoroughly mathematical, and the entire subject has been approached like a branch of mathematics. Serious efforts have been made to make the book suitable for the classroom as well as for self-reading. The book does not strive to be a comprehensive encyclopedia of logic. Still, it gives essentially all the basic concepts and results in mathematical logic. The book prepares students to branch out in several areas of mathematics related to foundations and computability such as logic, axiomatic set theory, model theory, recursion theory, and computability.
A Concise Introduction to Mathematical Logic
This book is unique in that it is more concise than most others; the material is treated in a streamlined fashion. This allows the lecturer to select the material for a one-semester course on a topic more easily. These initial chapters cover just the material for an introductory course on mathematical logic combined with the necessary material from set theory. Chapter 3 is partly of a descriptive nature, providing a view towards decision problems, automated theorem proving, non-standard models and related subjects. The other chapters contain material on logic programming for computer scientists, model theory, recursion theory, Gödel's Incompleteness Theorems, and applications of mathematical logic. Philosophical and foundational problems of mathematics are discussed where appropriate.
