Foundations of the Formal Sciences V

Infinite Games

Author: Stefan Bold,B Loewe,T Rasch

Publisher: N.A

ISBN: 9781904987758

Category: Computers

Page: 351

View: 9793

Infinity can feature in games in various forms: we can play games of infinite length, with infinitely many players, or allow for infinitely many moves or strategies. Games of infinite length have been thoroughly investigated by mathematicians and have played a central role in mathematical logic. However, their applications go far beyond mathematics: they feature prominently in theoretical computer science, philosophical Gedankenexperiments, as limit cases in economical applications, and in many other applications. The conference Foundations of the Formal Sciences V focused on games of infinite length, but was very opn to include other notions of infinity in games as well.

Combinatorial Set Theory

With a Gentle Introduction to Forcing

Author: Lorenz J. Halbeisen

Publisher: Springer Science & Business Media

ISBN: 9781447121732

Category: Mathematics

Page: 456

View: 355

This book provides a self-contained introduction to modern set theory and also opens up some more advanced areas of current research in this field. The first part offers an overview of classical set theory wherein the focus lies on the axiom of choice and Ramsey theory. In the second part, the sophisticated technique of forcing, originally developed by Paul Cohen, is explained in great detail. With this technique, one can show that certain statements, like the continuum hypothesis, are neither provable nor disprovable from the axioms of set theory. In the last part, some topics of classical set theory are revisited and further developed in the light of forcing. The notes at the end of each chapter put the results in a historical context, and the numerous related results and the extensive list of references lead the reader to the frontier of research. This book will appeal to all mathematicians interested in the foundations of mathematics, but will be of particular use to graduates in this field.

Language, Culture, Computation: Computing - Theory and Technology

Essays Dedicated to Yaacov Choueka on the Occasion of His 75 Birthday

Author: Nachum Dershowitz,Ephraim Nissan

Publisher: Springer

ISBN: 364245321X

Category: Computers

Page: 604

View: 4389

This Festschrift volume is published in Honor of Yaacov Choueka on the occasion of this 75th birthday. The present three-volumes liber amicorum, several years in gestation, honours this outstanding Israeli computer scientist and is dedicated to him and to his scientific endeavours. Yaacov's research has had a major impact not only within the walls of academia, but also in the daily life of lay users of such technology that originated from his research. An especially amazing aspect of the temporal span of his scholarly work is that half a century after his influential research from the early 1960s, a project in which he is currently involved is proving to be a sensation, as will become apparent from what follows. Yaacov Choueka began his research career in the theory of computer science, dealing with basic questions regarding the relation between mathematical logic and automata theory. From formal languages, Yaacov moved to natural languages. He was a founder of natural-language processing in Israel, developing numerous tools for Hebrew. He is best known for his primary role, together with Aviezri Fraenkel, in the development of the Responsa Project, one of the earliest fulltext retrieval systems in the world. More recently, he has headed the Friedberg Genizah Project, which is bringing the treasures of the Cairo Genizah into the Digital Age. This first part of the three-volume set covers a range of topics in computer science. The papers are grouped in topical sections on: the jubilaris: Yaacov and his oeuvre; theory of computation; science computing and tools for engineering; information retrieval.

Logical Foundations of Computer Science

International Symposium, LFCS 2013, San Diego, CA, USA, January 6-8, 2013. Proceedings

Author: Sergei Artemov,Anil Nerode

Publisher: Springer

ISBN: 3642357229

Category: Mathematics

Page: 415

View: 6288

This book constitutes the refereed proceedings of the International Symposium on Logical Foundations of Computer Science, LFCS 2013, held in San Diego, CA, USA in January 2013. The volume presents 29 revised refereed papers carefully selected by the program committee. The scope of the Symposium is broad and includes constructive mathematics and type theory; logic, automata and automatic structures; computability and randomness; logical foundations of programming; logical aspects of computational complexity; logic programming and constraints; automated deduction and interactive theorem proving; logical methods in protocol and program verification; logical methods in program specification and extraction; domain theory logic; logical foundations of database theory; equational logic and term rewriting; lambda and combinatory calculi; categorical logic and topological semantics; linear logic; epistemic and temporal logics; intelligent and multiple agent system logics; logics of proof and justification; nonmonotonic reasoning; logic in game theory and social software; logic of hybrid systems; distributed system logics; mathematical fuzzy logic; system design logics; and other logics in computer science.

Roads to Infinity

The Mathematics of Truth and Proof

Author: John C. Stillwell

Publisher: CRC Press

ISBN: 1439865507

Category: Mathematics

Page: 250

View: 5716

Winner of a CHOICE Outstanding Academic Title Award for 2011! This book offers an introduction to modern ideas about infinity and their implications for mathematics. It unifies ideas from set theory and mathematical logic, and traces their effects on mainstream mathematical topics of today, such as number theory and combinatorics. The treatment is historical and partly informal, but with due attention to the subtleties of the subject. Ideas are shown to evolve from natural mathematical questions about the nature of infinity and the nature of proof, set against a background of broader questions and developments in mathematics. A particular aim of the book is to acknowledge some important but neglected figures in the history of infinity, such as Post and Gentzen, alongside the recognized giants Cantor and Gödel.

The Principles of Mathematics Revisited

Author: Jaakko Hintikka

Publisher: Cambridge University Press

ISBN: 9780521624985

Category: Mathematics

Page: 288

View: 8563

This book, written by one of philosophy's pre-eminent logicians, argues that many of the basic assumptions common to logic, philosophy of mathematics and metaphysics are in need of change. It is therefore a book of critical importance to logical theory. Jaakko Hintikka proposes a new basic first-order logic and uses it to explore the foundations of mathematics. This new logic enables logicians to express on the first-order level such concepts as equicardinality, infinity, and truth in the same language. The famous impossibility results by Gödel and Tarski that have dominated the field for the last sixty years turn out to be much less significant than has been thought. All of ordinary mathematics can in principle be done on this first-order level, thus dispensing with the existence of sets and other higher-order entities.

Discovering Modern Set Theory: The basics

Author: Winfried Just,Martin Weese

Publisher: American Mathematical Soc.

ISBN: 0821802666

Category: Mathematics

Page: 210

View: 644

This book is an introduction to set theory for beginning graduate students who want to get a sound grounding in those aspects of set theory used extensively throughout other areas of mathematics. Topics covered include formal languages and models, the power and limitation of the Axiomatic Method, the Axiom of Choice, including the fascinating Banach-Tarski Paradox, applications of Zorn's Lemma, ordinal arithmetic, including transfinite induction, and cardinal arithmetic. The style of writing, more a dialogue with the reader than that of the Master indoctrinating the pupil, makes this also very suitable for self-study.

Mathematics for Computer Science

Author: Eric Lehman,F. Thomson Leighton,Albert R. Meyer

Publisher: N.A

ISBN: 9789888407064


Page: 979

View: 4803

This book covers elementary discrete mathematics for computer science and engineering. It emphasizes mathematical definitions and proofs as well as applicable methods. Topics include formal logic notation, proof methods; induction, well-ordering; sets, relations; elementary graph theory; integer congruences; asymptotic notation and growth of functions; permutations and combinations, counting principles; discrete probability. Further selected topics may also be covered, such as recursive definition and structural induction; state machines and invariants; recurrences; generating functions.

FSTTCS 2007: Foundations of Software Technology and Theoretical Computer Science

27th International Conference, New Delhi, India, December 12-14, 2007, Proceedings

Author: V. Arvind,Sanjiva Prasad

Publisher: Springer Science & Business Media

ISBN: 3540770496

Category: Computers

Page: 558

View: 3679

This book constitutes the refereed proceedings of the 27th International Conference on the Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2007, held in New Delhi, India, in December 2007. The 40 revised full papers presented together with five invited papers were carefully reviewed. They provide original research results in fundamental aspects of computer science and reports from the frontline of software technology and theoretical computer science.

Second-order Quantifier Elimination

Foundations, Computational Aspects and Applications

Author: Dov M. Gabbay,Renate A. Schmidt,Andrzej Szałas

Publisher: N.A

ISBN: 9781904987567

Category: Computers

Page: 308

View: 1820

In recent years there has been an increasing use of logical methods and significant new developments have been spawned in several areas of computer science, ranging from artificial intelligence and software engineering to agent-based systems and the semantic web. In the investigation and application of logical methods there is a tension between: * the need for a representational language strong enough to express domain knowledge of a particular application, and the need for a logical formalism general enough to unify several reasoning facilities relevant to the application, on the one hand, and * the need to enable computationally feasible reasoning facilities, on the other hand. Second-order logics are very expressive and allow us to represent domain knowledge with ease, but there is a high price to pay for the expressiveness. Most second-order logics are incomplete and highly undecidable. It is the quantifiers which bind relation symbols that make second-order logics computationally unfriendly. It is therefore desirable to eliminate these second-order quantifiers, when this is mathematically possible; and often it is. If second-order quantifiers are eliminable we want to know under which conditions, we want to understand the principles and we want to develop methods for second-order quantifier elimination. This book provides the first comprehensive, systematic and uniform account of the state-of-the-art of second-order quantifier elimination in classical and non-classical logics. It covers the foundations, it discusses in detail existing second-order quantifier elimination methods, and it presents numerous examples of applications and non-standard uses in different areas. These include: * classical and non-classical logics, * correspondence and duality theory, * knowledge representation and description logics, * commonsense reasoning and approximate reasoning, * relational and deductive databases, and * complexity theory. The book is intended for anyone interested in the theory and application of logics in computer science and artificial intelligence.

The Continuum

A Critical Examination of the Foundation of Analysis

Author: Hermann Weyl

Publisher: Courier Corporation

ISBN: 0486679829

Category: Mathematics

Page: 130

View: 6742

Concise classic by great mathematician and physicist deals with logic and mathematics of set and function, concept of number and the continuum. Bibliography. Originally published 1918.

The Foundations of Mathematics

Author: Kenneth Kunen

Publisher: N.A

ISBN: 9781904987147

Category: Mathematics

Page: 251

View: 8459

Mathematical logic grew out of philosophical questions regarding the foundations of mathematics, but logic has now outgrown its philosophical roots, and has become an integral part of mathematics in general. This book is designed for students who plan to specialize in logic, as well as for those who are interested in the applications of logic to other areas of mathematics. Used as a text, it could form the basis of a beginning graduate-level course. There are three main chapters: Set Theory, Model Theory, and Recursion Theory. The Set Theory chapter describes the set-theoretic foundations of all of mathematics, based on the ZFC axioms. It also covers technical results about the Axiom of Choice, well-orderings, and the theory of uncountable cardinals. The Model Theory chapter discusses predicate logic and formal proofs, and covers the Completeness, Compactness, and Lowenheim-Skolem Theorems, elementary submodels, model completeness, and applications to algebra. This chapter also continues the foundational issues begun in the set theory chapter. Mathematics can now be viewed as formal proofs from ZFC. Also, model theory leads to models of set theory. This includes a discussion of absoluteness, and an analysis of models such as H( ) and R( ). The Recursion Theory chapter develops some basic facts about computable functions, and uses them to prove a number of results of foundational importance; in particular, Church's theorem on the undecidability of logical consequence, the incompleteness theorems of Godel, and Tarski's theorem on the non-definability of truth.

Set Theory and the Continuum Hypothesis

Author: Paul J. Cohen,Martin Davis

Publisher: Courier Corporation

ISBN: 0486469212

Category: Mathematics

Page: 154

View: 7648

This exploration of a notorious mathematical problem is the work of the man who discovered the solution. Written by an award-winning professor at Stanford University, it employs intuitive explanations as well as detailed mathematical proofs in a self-contained treatment. This unique text and reference is suitable for students and professionals. 1966 edition. Copyright renewed 1994.

Formal Models of Domestic Politics

Author: Scott Gehlbach

Publisher: Cambridge University Press

ISBN: 0521767156

Category: Mathematics

Page: 228

View: 3908

A unified and accessible treatment of important formal models of domestic politics appropriate for students in political science and economics.

Relation Algebras by Games

Author: Robin Hirsch,Ian Hodkinson

Publisher: Elsevier

ISBN: 9780080540450

Category: Mathematics

Page: 710

View: 8837

Relation algebras are algebras arising from the study of binary relations. They form a part of the field of algebraic logic, and have applications in proof theory, modal logic, and computer science. This research text uses combinatorial games to study the fundamental notion of representations of relation algebras. Games allow an intuitive and appealing approach to the subject, and permit substantial advances to be made. The book contains many new results and proofs not published elsewhere. It should be invaluable to graduate students and researchers interested in relation algebras and games. After an introduction describing the authors' perspective on the material, the text proper has six parts. The lengthy first part is devoted to background material, including the formal definitions of relation algebras, cylindric algebras, their basic properties, and some connections between them. Examples are given. Part 1 ends with a short survey of other work beyond the scope of the book. In part 2, games are introduced, and used to axiomatise various classes of algebras. Part 3 discusses approximations to representability, using bases, relation algebra reducts, and relativised representations. Part 4 presents some constructions of relation algebras, including Monk algebras and the 'rainbow construction', and uses them to show that various classes of representable algebras are non-finitely axiomatisable or even non-elementary. Part 5 shows that the representability problem for finite relation algebras is undecidable, and then in contrast proves some finite base property results. Part 6 contains a condensed summary of the book, and a list of problems. There are more than 400 exercises. The book is generally self-contained on relation algebras and on games, and introductory text is scattered throughout. Some familiarity with elementary aspects of first-order logic and set theory is assumed, though many of the definitions are given. Chapter 2 introduces the necessary universal algebra and model theory, and more specific model-theoretic ideas are explained as they arise.


The Laws of Truth

Author: Nicholas Jeremy Josef Smith

Publisher: Princeton University Press

ISBN: 0691151636

Category: Philosophy

Page: 528

View: 3479

Provides an essential introduction to classical logic.

Approaches and Methodologies in the Social Sciences

A Pluralist Perspective

Author: Donatella Della Porta,Michael Keating

Publisher: Cambridge University Press

ISBN: 1139474596

Category: Political Science

Page: N.A

View: 581

A revolutionary textbook introducing masters and doctoral students to the major research approaches and methodologies in the social sciences. Written by an outstanding set of scholars, and derived from successful course teaching, this volume will empower students to choose their own approach to research, to justify this approach, and to situate it within the discipline. It addresses questions of ontology, epistemology and philosophy of social science, and proceeds to issues of methodology and research design essential for producing a good research proposal. It also introduces researchers to the main issues of debate and contention in the methodology of social sciences, identifying commonalities, historic continuities and genuine differences.

Error and the Growth of Experimental Knowledge

Author: Deborah G. Mayo

Publisher: University of Chicago Press

ISBN: 9780226511979

Category: Mathematics

Page: 493

View: 7722

This text provides a critique of the subjective Bayesian view of statistical inference, and proposes the author's own error-statistical approach as an alternative framework for the epistemology of experiment. It seeks to address the needs of researchers who work with statistical analysis.

Foundations of Modern Probability

Author: Olav Kallenberg

Publisher: Springer Science & Business Media

ISBN: 0387227040

Category: Mathematics

Page: 523

View: 5144

Unique for its broad and yet comprehensive coverage of modern probability theory, ranging from first principles and standard textbook material to more advanced topics. In spite of the economical exposition, careful proofs are provided for all main results. After a detailed discussion of classical limit theorems, martingales, Markov chains, random walks, and stationary processes, the author moves on to a modern treatment of Brownian motion, L=82vy processes, weak convergence, It=93 calculus, Feller processes, and SDEs. The more advanced parts include material on local time, excursions, and additive functionals, diffusion processes, PDEs and potential theory, predictable processes, and general semimartingales. Though primarily intended as a general reference for researchers and graduate students in probability theory and related areas of analysis, the book is also suitable as a text for graduate and seminar courses on all levels, from elementary to advanced. Numerous easy to more challenging exercises are provided, especially for the early chapters. From the author of "Random Measures".