For propositional logic it can be decided whether a formula has a deduction from a finite set of other formulas. This volume begins with a method to decide this for the quantified formulas of those fragments of arithmetic which express the properties of order-plus-successor and of order-plus-addition (Pressburger arithmetic). It makes use of an algorithm eliminating quantifiers which, in turn, is also applied to obtain consistency proofs for these fragments.

This book gives an introduction to theories of computability from a mathematically sophisticated point of view. It treats not only 'the' theory of computability (created by Alan Turing and others in the 1930s), but also a variety of other theories (of Boolean functions, automata and formal languages). These are addressed from the classical perspective of their generation by grammars and from the modern perspective as rational cones. The treatment of the classical theory of computable functions and relations takes the form of a tour through basic recursive function theory, starting with an axiomatic foundation and developing the essential methods in order to survey the most memorable results of the field. This authoritative account by one of the leading lights of the subject will prove exceptionally useful reading for graduate students, and researchers in theoretical computer science and mathematics.

Fuzzy social choice theory is useful for modeling the uncertainty and imprecision prevalent in social life yet it has been scarcely applied and studied in the social sciences. Filling this gap, Application of Fuzzy Logic to Social Choice Theory provides a comprehensive study of fuzzy social choice theory. The book explains the concept of a fuzzy maximal subset of a set of alternatives, fuzzy choice functions, the factorization of a fuzzy preference relation into the "union" (conorm) of a strict fuzzy relation and an indifference operator, fuzzy non-Arrowian results, fuzzy versions of Arrow's theorem, and Black's median voter theorem for fuzzy preferences. It examines how unambiguous and exact choices are generated by fuzzy preferences and whether exact choices induced by fuzzy preferences satisfy certain plausible rationality relations. The authors also extend known Arrowian results involving fuzzy set theory to results involving intuitionistic fuzzy sets as well as the Gibbard-Satterthwaite theorem to the case of fuzzy weak preference relations. The final chapter discusses Georgescu's degree of similarity of two fuzzy choice functions.

Intellectual property owners must continually exploit new ways of reproducing, distributing, and marketing their products. However, the threat of piracy looms as a major problem with digital distribution and storage technologies. Multimedia Encryption and Authentication Techniques and Applications covers current and future trends in the design of modern systems that use encryption and authentication to protect multimedia content. Containing the works of contributing authors who are worldwide experts in their fields, this volume is intended for researchers and practitioners, as well as for those who want a broad understanding of multimedia security. In the wake of the explosive growth of digital entertainment and Internet applications, this book is a definitive resource for scientists, researchers, programmers, engineers, business managers, entrepreneurs, and investors. Features Describes and evaluates the state of the art in multimedia encryption and authentication techniques and related technologies, architectures, standards, and applications Includes advanced topics, such as chaotic encryption techniques for digital images and video, as well as streaming media encryption Focuses on digital rights management issues for video and for consumer devices Covers key management and protection for IP multimedia, digital media fingerprinting, and signature-based media authentication

‘Another terrific book by Rob Eastaway’ SIMON SINGH ‘A delightfully accessible guide to how to play with numbers’ HANNAH FRY How many cats are there in the world? What's the chance of winning the lottery twice? And just how long does it take to count to a million? Learn how to tackle tricky maths problems with nothing but the back of an envelope, a pencil and some good old-fashioned brain power. Join Rob Eastaway as he takes an entertaining look at how to figure without a calculator. Packed with amusing anecdotes, quizzes, and handy calculation tips for every situation, Maths on the Back of an Envelope is an invaluable introduction to the art of estimation, and a welcome reminder that sometimes our own brain is the best tool we have to deal with numbers.

Originally published in 1973. This book presents a valid mode of reasoning that is different to mathematical probability. This inductive logic is investigated in terms of scientific investigation. The author presents his criteria of adequacy for analysing inductive support for hypotheses and discusses each of these criteria in depth. The chapters cover philosophical problems and paradoxes about experimental support, probability and justifiability, ending with a system of logical syntax of induction. Each section begins with a summary of its contents and there is a glossary of technical terms to aid the reader.

This book addresses the argument in the history of the philosophy of science between the positivists and the anti-positivists. The author starts from a point of firm conviction that all science and philosophy must start with the given... But that the range of the given is not definite. He begins with an examination of science from the outside and then the inside, explaining his position on metaphysics and attempts to formulate the character of operational acts before a general theory of symbolism is explored. The last five chapters constitute a treatise to show that the development from one stage of symbolismto the next is inevitable, consequently that explanatory science represents the culmination of knowledge.

Originally published in 1931. This inquiry investigates and develops John Cook Wilson's view of the province of logic. It bases the study on the posthumous collected papers Statement and Inference. The author seeks to answer questions on the nature of logic using Cook Wilson's thought. The chapters introduce and consider topics from metaphysics to grammar and from psychology to knowledge. An early conception of logic in the sciences and presenting the work of an important twentieth century philosopher, this is an engaging work.

Originally published in 1965. This is a textbook of modern deductive logic, designed for beginners but leading further into the heart of the subject than most other books of the kind. The fields covered are the Propositional Calculus, the more elementary parts of the Predicate Calculus, and Syllogistic Logic treated from a modern point of view. In each of the systems discussed the main emphases are on Decision Procedures and Axiomatisation, and the material is presented with as much formal rigour as is compatible with clarity of exposition. The techniques used are not only described but given a theoretical justification. Proofs of Consistency, Completeness and Independence are set out in detail. The fundamental characteristics of the various systems studies, and their relations to each other are established by meta-logical proofs, which are used freely in all sections of the book. Exercises are appended to most of the chapters, and answers are provided.

Originally published in 1981. This is a book for the final year undergraduate or first year graduate who intends to proceed with serious research in philosophical logic. It will be welcomed by both lecturers and students for its careful consideration of main themes ranging from Gricean accounts of meaning to two dimensional modal logic. The first part of the book is concerned with the nature of the semantic theorist's project, and particularly with the crucial concepts of meaning, truth, and semantic structure. The second and third parts deal with various constructions that are found in natural languages: names, quantifiers, definite descriptions, and modal operators. Throughout, while assuming some familiarity with philosophical logic and elementary formal logic, the text provides a clear exposition. It brings together related ideas, and in some places refines and improves upon existing accounts.

Originally published in 1966. This is a self-instructional course intended for first-year university students who have not had previous acquaintance with Logic. The book deals with "propositional" logic by the truth-table method, briefly introducing axiomatic procedures, and proceeds to the theory of the syllogism, the logic of one-place predicates, and elementary parts of the logic of many-place predicates. Revision material is provided covering the main parts of the course. The course represents from eight to twenty hours work. depending on the student's speed of work and on whether optional chapters are taken.

For computer scientists, especially those in the security field, the use of chaos has been limited to the computation of a small collection of famous but unsuitable maps that offer no explanation of why chaos is relevant in the considered contexts. Discrete Dynamical Systems and Chaotic Machines: Theory and Applications shows how to make finite machines, such as computers, neural networks, and wireless sensor networks, work chaotically as defined in a rigorous mathematical framework. Taking into account that these machines must interact in the real world, the authors share their research results on the behaviors of discrete dynamical systems and their use in computer science. Covering both theoretical and practical aspects, the book presents: Key mathematical and physical ideas in chaos theory Computer science fundamentals, clearly establishing that chaos properties can be satisfied by finite state machines Concrete applications of chaotic machines in computer security, including pseudorandom number generators, hash functions, digital watermarking, and steganography Concrete applications of chaotic machines in wireless sensor networks, including secure data aggregation and video surveillance Until the authors' recent research, the practical implementation of the mathematical theory of chaos on finite machines raised several issues. This self-contained book illustrates how chaos theory enables the study of computer security problems, such as steganalysis, that otherwise could not be tackled. It also explains how the theory reinforces existing cryptographically secure tools and schemes.

Researchers and practitioners of cryptography and information security are constantly challenged to respond to new attacks and threats to information systems. Authentication Codes and Combinatorial Designs presents new findings and original work on perfect authentication codes characterized in terms of combinatorial designs, namely strong partially balanced designs (SPBD). Beginning with examples illustrating the concepts of authentication schemes and combinatorial designs, the book considers the probability of successful deceptions followed by schemes involving three and four participants, respectively. From this point, the author constructs the perfect authentication schemes and explores encoding rules for such schemes in some special cases. Using rational normal curves in projective spaces over finite fields, the author constructs a new family of SPBD. He then presents some established combinatorial designs that can be used to construct perfect schemes, such as t-designs, orthogonal arrays of index unity, and designs constructed by finite geometry. The book concludes by studying definitions of perfect secrecy, properties of perfectly secure schemes, and constructions of perfect secrecy schemes with and without authentication. Supplying an appendix of construction schemes for authentication and secrecy schemes, Authentication Codes and Combinatorial Designs points to new applications of combinatorial designs in cryptography.

The huge number and broad range of the existing and potential applications of fuzzy logic have precipitated a veritable avalanche of books published on the subject. Most, however, focus on particular areas of application. Many do no more than scratch the surface of the theory that holds the power and promise of fuzzy logic. Fuzzy Automata and Languages: Theory and Applications offers the first in-depth treatment of the theory and mathematics of fuzzy automata and fuzzy languages. After introducing background material, the authors study max-min machines and max-product machines, developing their respective algebras and exploring properties such as equivalences, homomorphisms, irreducibility, and minimality. The focus then turns to fuzzy context-free grammars and languages, with special attention to trees, fuzzy dendrolanguage generating systems, and normal forms. A treatment of algebraic fuzzy automata theory follows, along with additional results on fuzzy languages, minimization of fuzzy automata, and recognition of fuzzy languages. Although the book is theoretical in nature, the authors also discuss applications in a variety of fields, including databases, medicine, learning systems, and pattern recognition. Much of the information on fuzzy languages is new and never before presented in book form. Fuzzy Automata and Languages incorporates virtually all of the important material published thus far. It stands alone as a complete reference on the subject and belongs on the shelves of anyone interested in fuzzy mathematics or its applications.

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.

An engaging, accessible introduction into how numbers work and why we shouldn't be afraid of them, from maths expert Rachel Riley. Do you know your fractions from your percentages? Your adjacent to your hypotenuse? And who really knows how to do long division, anyway? Puzzled already? Don't blame you... But fret not! You won't be At Sixes and Sevens for long. In this brilliant, well-rounded guide, Countdown's Rachel Riley will take you back to the very basics, allow you to revisit what you learnt at school (and may have promptly forgotten, *ahem*), build your understanding of maths from the get-go and provide you with the essential toolkit to gain confidence in your numerical abilities. Discover how to divide and conquer, make your decimal debut, become a pythagoras professional and so much more with these easy-to-learn tips and tricks. Packed full of working examples, fool-proof methods, quirky trivia and brainteasers to try from puzzle-pro Dr Gareth Moore, this book is an absolute must-read for anyone and everyone who ever thought maths was 'above' them. Because the truth is: you can do it. What's more, it can be pretty fun too!

Information security has a major gap when cryptography is implemented. Cryptographic algorithms are well defined, key management schemes are well known, but the actual deployment is typically overlooked, ignored, or unknown. Cryptography is everywhere. Application and network architectures are typically well-documented but the cryptographic architecture is missing. This book provides a guide to discovering, documenting, and validating cryptographic architectures. Each chapter builds on the next to present information in a sequential process. This approach not only presents the material in a structured manner, it also serves as an ongoing reference guide for future use.

The language of ends and (co)ends provides a natural and general way of expressing many phenomena in category theory, in the abstract and in applications. Yet although category-theoretic methods are now widely used by mathematicians, since (co)ends lie just beyond a first course in category theory, they are typically only used by category theorists, for whom they are something of a secret weapon. This book is the first systematic treatment of the theory of (co)ends. Aimed at a wide audience, it presents the (co)end calculus as a powerful tool to clarify and simplify definitions and results in category theory and export them for use in diverse areas of mathematics and computer science. It is organised as an easy-to-cite reference manual, and will be of interest to category theorists and users of category theory alike.

Transition to Real Analysis with Proof provides undergraduate students with an introduction to analysis including an introduction to proof. The text combines the topics covered in a transition course to lead into a first course on analysis. This combined approach allows instructors to teach a single course where two were offered. The text opens with an introduction to basic logic and set theory, setting students up to succeed in the study of analysis. Each section is followed by graduated exercises that both guide and challenge students. The author includes examples and illustrations that appeal to the visual side of analysis. The accessible structure of the book makes it an ideal refence for later years of study or professional work. Combines the author's previous works Elements of Advanced Mathematics with Foundations of Analysis Combines logic, set theory and other elements with a one-semester introduction to analysis. Author is a well-known mathematics educator and researcher Targets a trend to combine two courses into one

Model theory investigates mathematical structures by means of formal languages. These so-called first-order languages have proved particularly useful. The text introduces the reader to the model theory of first-order logic, avoiding syntactical issues that are not too relevant to model-theory. In this spirit, the compactness theorem is proved via the algebraically useful ultraproduct technique, rather than via the completeness theorem of first-order logic. This leads fairly quickly to algebraic applications, like Malcev's local theorems (of group theory) and, after a little more preparation, also to Hilbert's Nullstellensatz (of field theory). Steinitz' dimension theory for field extensions is obtained as a special case of a much more general model-theoretic treatment of strongly minimal sets. The final chapter is on the models of the first-order theory of the integers as an abelian group. This material appears here for the first time in a textbook of introductory level, and is used to give hints to further reading and to recent developments in the field, such as stability (or classification) theory. The latter itself is not touched upon. The undergraduate or graduate, is assumed t

With rapid progress in Internet and digital imaging technology, there are more and more ways to easily create, publish, and distribute images. Considered the first book to focus on the relationship between digital imaging and privacy protection, Visual Cryptography and Secret Image Sharing is a complete introduction to novel security methods and sharing-control mechanisms used to protect against unauthorized data access and secure dissemination of sensitive information. Image data protection and image-based authentication techniques offer efficient solutions for controlling how private data and images are made available only to select people. Essential to the design of systems used to manage images that contain sensitive data-such as medical records, financial transactions, and electronic voting systems-the methods presented in this book are useful to counter traditional encryption techniques, which do not scale well and are less efficient when applied directly to image files. An exploration of the most prominent topics in digital imaging security, this book discusses: Potential for sharing multiple secrets Visual cryptography schemes-based either on the probabilistic reconstruction of the secret image, or on different logical operations for combining shared images Inclusion of pictures in the distributed shares Contrast enhancement techniques Color-image visual cryptography Cheating prevention Alignment problems for image shares Steganography and authentication In the continually evolving world of secure image sharing, a growing number of people are becoming involved as new applications and business models are being developed all the time. This contributed volume gives academicians, researchers, and professionals the insight of well-known experts on key concepts, issues, trends, and technologies in this emerging field.

Convergence of Blockchain, AI and IoT: A Digital Platform discusses the convergence of three powerful technologies that play into the digital revolution and blur the lines between biological, digital, and physical objects. This book covers novel algorithms, solutions for addressing issues in applications, security, authentication, and privacy. Discusses innovative technological upgradation and significant challenges in the current era Gives an overview of clinical scientific research that enables smart diagnosis through artificial intelligence Provides an insight into how disruptive technology enabled with the self-running devices and protection mechanism is involved in an augmented reality with blockchain mechanism Talks about neural science being capable of enhancing deep brain waves to predict an overall improvement in human thoughts and behaviours Covers the digital currency mechanism in detail Enhances the knowledge of the readers about smart contract and ledger mechanism with artificial intelligence and blockchain mechanism Targeted audiences range from those interested in the technical revolution of blockchain, big data and the Internet of Things, to research scholars and the professional market.

Logic Works is a critical and extensive introduction to logic. It asks questions about why systems of logic are as they are, how they relate to ordinary language and ordinary reasoning, and what alternatives there might be to classical logical doctrines. The book covers classical first-order logic and alternatives, including intuitionistic, free, and many-valued logic. It also considers how logical analysis can be applied to carefully represent the reasoning employed in academic and scientific work, better understand that reasoning, and identify its hidden premises. Aiming to be as much a reference work and handbook for further, independent study as a course text, it covers more material than is typically covered in an introductory course. It also covers this material at greater length and in more depth with the purpose of making it accessible to those with no prior training in logic or formal systems. Online support material includes a detailed student solutions manual with a running commentary on all starred exercises, and a set of editable slide presentations for course lectures. Key Features Introduces an unusually broad range of topics, allowing instructors to craft courses to meet a range of various objectives Adopts a critical attitude to certain classical doctrines, exposing students to alternative ways to answer philosophical questions about logic Carefully considers the ways natural language both resists and lends itself to formalization Makes objectual semantics for quantified logic easy, with an incremental, rule-governed approach assisted by numerous simple exercises Makes important metatheoretical results accessible to introductory students through a discursive presentation of those results and by using simple case studies

A lively and engaging look at logic puzzles and their role in mathematics, philosophy, and recreation Logic puzzles were first introduced to the public by Lewis Carroll in the late nineteenth century and have been popular ever since. Games like Sudoku and Mastermind are fun and engrossing recreational activities, but they also share deep foundations in mathematical logic and are worthy of serious intellectual inquiry. Games for Your Mind explores the history and future of logic puzzles while enabling you to test your skill against a variety of puzzles yourself. In this informative and entertaining book, Jason Rosenhouse begins by introducing readers to logic and logic puzzles and goes on to reveal the rich history of these puzzles. He shows how Carroll's puzzles presented Aristotelian logic as a game for children, yet also informed his scholarly work on logic. He reveals how another pioneer of logic puzzles, Raymond Smullyan, drew on classic puzzles about liars and truthtellers to illustrate Kurt Goedel's theorems and illuminate profound questions in mathematical logic. Rosenhouse then presents a new vision for the future of logic puzzles based on nonclassical logic, which is used today in computer science and automated reasoning to manipulate large and sometimes contradictory sets of data. Featuring a wealth of sample puzzles ranging from simple to extremely challenging, this lively and engaging book brings together many of the most ingenious puzzles ever devised, including the "Hardest Logic Puzzle Ever," metapuzzles, paradoxes, and the logic puzzles in detective stories.