Mathematical Labyrinths. Pathfinding provides an overview of various non-standard problems and the approaches to their solutions. The essential idea is a framework laid upon the reader on how to solve nonconventional problems - particularly in the realm of mathematics and logic. It goes over the key steps in approaching a difficult problem, contemplating a plan for its solution, and discusses set of mental models to solve math problems.The book is not a routine set of problems. It is rather an entertaining and educational journey into the fascinating world of mathematical reasoning and logic. It is about finding the best path to a solution depending on the information given, asking and answering the right questions, analyzing and comparing alternative approaches to problem solving, searching for generalizations and inventing new problems. It also considers as an important pedagogical tool playing mathematical and logical games, deciphering mathematical sophisms, and interpreting mathematical paradoxes.It is suitable for mathematically talented and curious students in the age range 10-20. There are many 'Eureka'- type, out of the ordinary, fun problems that require bright idea and insight. These intriguing and thought-provoking brainteasers and logic puzzles should be enjoyable by the audience of almost any age group, from 6-year-old children to 80-year-old and older adults.

The present book aims to provide systematic and reliable techniques, called the global solution, for Sudoku puzzles. Any proper Sudoku puzzle, which has one and only one solution of Sudoku, can be solved by anyone following the techniques provided in this book. Specific symbols are introduced to express the 6 basic rules of the Sudoku global solution, as the results, those Sudoku solving techniques are presented similar to the annotations in chess. Finnish mathematician Arto Inkala proposed 'the most difficult Sudoku puzzle' in 2007. Then, he designed another difficult Sudoku puzzle in 2012, named 'the thing Everest'. In the present book the solving process of those two difficult Sudoku puzzles are illustrated reliably by the specific symbols of the global solution step by step.

College students struggle with the switch from thinking of mathematics as a calculation based subject to a problem solving based subject. This book describes how the introduction to proofs course can be taught in a way that gently introduces students to this new way of thinking. This introduction utilizes recent research in neuroscience regarding how the brain learns best. Rather than jumping right into proofs, students are first taught how to change their mindset about learning, how to persevere through difficult problems, how to work successfully in a group, and how to reflect on their learning. With these tools in place, students then learn logic and problem solving as a further foundation.Next various proof techniques such as direct proofs, proof by contraposition, proof by contradiction, and mathematical induction are introduced. These proof techniques are introduced using the context of number theory. The last chapter uses Calculus as a way for students to apply the proof techniques they have learned.

College students struggle with the switch from thinking of mathematics as a calculation based subject to a problem solving based subject. This book describes how the introduction to proofs course can be taught in a way that gently introduces students to this new way of thinking. This introduction utilizes recent research in neuroscience regarding how the brain learns best. Rather than jumping right into proofs, students are first taught how to change their mindset about learning, how to persevere through difficult problems, how to work successfully in a group, and how to reflect on their learning. With these tools in place, students then learn logic and problem solving as a further foundation.Next various proof techniques such as direct proofs, proof by contraposition, proof by contradiction, and mathematical induction are introduced. These proof techniques are introduced using the context of number theory. The last chapter uses Calculus as a way for students to apply the proof techniques they have learned.

Originally published in 1934. This fourth edition originally published 1954., revised by C. W. K. Mundle. "It must be the desire of every reasonable person to know how to justify a contention which is of sufficient importance to be seriously questioned. The explicit formulation of the principles of sound reasoning is the concern of Logic". This book discusses the habit of sound reasoning which is acquired by consciously attending to the logical principles of sound reasoning, in order to apply them to test the soundness of arguments. It isn't an introduction to logic but it encourages the practice of logic, of deciding whether reasons in argument are sound or unsound. Stress is laid upon the importance of considering language, which is a key instrument of our thinking and is imperfect.

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.

Information Security and Optimization maintains a practical perspective while offering theoretical explanations. The book explores concepts that are essential for academics as well as organizations. It discusses aspects of techniques and tools-definitions, usage, and analysis-that are invaluable for scholars ranging from those just beginning in the field to established experts. What are the policy standards? What are vulnerabilities and how can one patch them? How can data be transmitted securely? How can data in the cloud or cryptocurrency in the blockchain be secured? How can algorithms be optimized? These are some of the possible queries that are answered here effectively using examples from real life and case studies. Features: A wide range of case studies and examples derived from real-life scenarios that map theoretical explanations with real incidents. Descriptions of security tools related to digital forensics with their unique features, and the working steps for acquiring hands-on experience. Novel contributions in designing organization security policies and lightweight cryptography. Presentation of real-world use of blockchain technology and biometrics in cryptocurrency and personalized authentication systems. Discussion and analysis of security in the cloud that is important because of extensive use of cloud services to meet organizational and research demands such as data storage and computing requirements. Information Security and Optimization is equally helpful for undergraduate and postgraduate students as well as for researchers working in the domain. It can be recommended as a reference or textbook for courses related to cybersecurity.

The crypto wars have raged for half a century. In the 1970s, digital privacy activists prophesied the emergence of an Orwellian State, made possible by computer-mediated mass surveillance. The antidote: digital encryption. The U.S. government warned encryption would not only prevent surveillance of law-abiding citizens, but of criminals, terrorists, and foreign spies, ushering in a rival dystopian future. Both parties fought to defend the citizenry from what they believed the most perilous threats. The government tried to control encryption to preserve its surveillance capabilities; privacy activists armed citizens with cryptographic tools and challenged encryption regulations in the courts. No clear victor has emerged from the crypto wars. Governments have failed to forge a framework to govern the, at times conflicting, civil liberties of privacy and security in the digital age—an age when such liberties have an outsized influence on the citizen–State power balance. Solving this problem is more urgent than ever. Digital privacy will be one of the most important factors in how we architect twenty-first century societies—its management is paramount to our stewardship of democracy for future generations. We must elevate the quality of debate on cryptography, on how we govern security and privacy in our technology-infused world. Failure to end the crypto wars will result in societies sleepwalking into a future where the citizen–State power balance is determined by a twentieth-century status quo unfit for this century, endangering both our privacy and security. This book provides a history of the crypto wars, with the hope its chronicling sets a foundation for peace.

The crypto wars have raged for half a century. In the 1970s, digital privacy activists prophesied the emergence of an Orwellian State, made possible by computer-mediated mass surveillance. The antidote: digital encryption. The U.S. government warned encryption would not only prevent surveillance of law-abiding citizens, but of criminals, terrorists, and foreign spies, ushering in a rival dystopian future. Both parties fought to defend the citizenry from what they believed the most perilous threats. The government tried to control encryption to preserve its surveillance capabilities; privacy activists armed citizens with cryptographic tools and challenged encryption regulations in the courts. No clear victor has emerged from the crypto wars. Governments have failed to forge a framework to govern the, at times conflicting, civil liberties of privacy and security in the digital age-an age when such liberties have an outsized influence on the citizen-State power balance. Solving this problem is more urgent than ever. Digital privacy will be one of the most important factors in how we architect twenty-first century societies-its management is paramount to our stewardship of democracy for future generations. We must elevate the quality of debate on cryptography, on how we govern security and privacy in our technology-infused world. Failure to end the crypto wars will result in societies sleepwalking into a future where the citizen-State power balance is determined by a twentieth-century status quo unfit for this century, endangering both our privacy and security. This book provides a history of the crypto wars, with the hope its chronicling sets a foundation for peace.

Proofs 101: An Introduction to Formal Mathematics serves as an introduction to proofs for mathematics majors who have completed the calculus sequence (at least Calculus I and II) and a first course in linear algebra. The book prepares students for the proofs they will need to analyze and write the axiomatic nature of mathematics and the rigors of upper-level mathematics courses. Basic number theory, relations, functions, cardinality, and set theory will provide the material for the proofs and lay the foundation for a deeper understanding of mathematics, which students will need to carry with them throughout their future studies. Features Designed to be teachable across a single semester Suitable as an undergraduate textbook for Introduction to Proofs or Transition to Advanced Mathematics courses Offers a balanced variety of easy, moderate, and difficult exercises

Proofs 101: An Introduction to Formal Mathematics serves as an introduction to proofs for mathematics majors who have completed the calculus sequence (at least Calculus I and II) and a first course in linear algebra. The book prepares students for the proofs they will need to analyze and write the axiomatic nature of mathematics and the rigors of upper-level mathematics courses. Basic number theory, relations, functions, cardinality, and set theory will provide the material for the proofs and lay the foundation for a deeper understanding of mathematics, which students will need to carry with them throughout their future studies. Features Designed to be teachable across a single semester Suitable as an undergraduate textbook for Introduction to Proofs or Transition to Advanced Mathematics courses Offers a balanced variety of easy, moderate, and difficult exercises

This book presents a set of historical recollections on the work of Martin Davis and his role in advancing our understanding of the connections between logic, computing, and unsolvability. The individual contributions touch on most of the core aspects of Davis' work and set it in a contemporary context. They analyse, discuss and develop many of the ideas and concepts that Davis put forward, including such issues as contemporary satisfiability solvers, essential unification, quantum computing and generalisations of Hilbert's tenth problem. The book starts out with a scientific autobiography by Davis, and ends with his responses to comments included in the contributions. In addition, it includes two previously unpublished original historical papers in which Davis and Putnam investigate the decidable and the undecidable side of Logic, as well as a full bibliography of Davis' work. As a whole, this book shows how Davis' scientific work lies at the intersection of computability, theoretical computer science, foundations of mathematics, and philosophy, and draws its unifying vision from his deep involvement in Logic.

There are thousands of books relating to poker, blackjack, roulette and baccarat, including strategy guides, statistical analysis, psychological studies, and much more. However, there are no books on Pell, Rouleno, Street Dice, and many other games that have had a short life in casinos! While this is understandable - most casino gamblers have not heard of these games, and no one is currently playing them - their absence from published works means that some interesting mathematics and gaming history are at risk of being lost forever. Table games other than baccarat, blackjack, craps, and roulette are called carnival games, as a nod to their origin in actual traveling or seasonal carnivals. Mathematics of Casino Carnival Games is a focused look at these games and the mathematics at their foundation. Features * Exercises, with solutions, are included for readers who wish to practice the ideas presented * Suitable for a general audience with an interest in the mathematics of gambling and games * Goes beyond providing practical 'tips' for gamblers, and explores the mathematical principles that underpin gambling games

This book is intended to be a thorough introduction to the subject of order and lattices, with an emphasis on the latter. It can be used for a course at the graduate or advanced undergraduate level or for independent study. Prerequisites are kept to a minimum, but an introductory course in abstract algebra is highly recommended, since many of the examples are drawn from this area. This is a book on pure mathematics: I do not discuss the applications of lattice theory to physics, computer science or other disciplines. Lattice theory began in the early 1890s, when Richard Dedekind wanted to know the answer to the following question: Given three subgroups EF , and G of an abelian group K, what is the largest number of distinct subgroups that can be formed using these subgroups and the operations of intersection and sum (join), as in E?FssDE?FN?GssE?DF?GN and so on? In lattice-theoretic terms, this is the number of elements in the relatively free modular lattice on three generators. Dedekind [15] answered this question (the answer is #)) and wrote two papers on the subject of lattice theory, but then the subject lay relatively dormant until Garrett Birkhoff, Oystein Ore and others picked it up in the 1930s. Since then, many noted mathematicians have contributed to the subject, including Garrett Birkhoff, Richard Dedekind, Israel Gelfand, George Gratzer, Aleksandr Kurosh, Anatoly Malcev, Oystein Ore, Gian-Carlo Rota, Alfred Tarski and Johnny von Neumann.

Merging logic and mathematics in deductive inference—an innovative, cutting-edge approach.

Optimization methods for logical inference? Absolutely, say Vijay Chandru and John Hooker, two major contributors to this rapidly expanding field. And even though "solving logical inference problems with optimization methods may seem a bit like eating sauerkraut with chopsticks. . . it is the mathematical structure of a problem that determines whether an optimization model can help solve it, not the context in which the problem occurs."

Presenting powerful, proven optimization techniques for logic inference problems, Chandru and Hooker show how optimization models can be used not only to solve problems in artificial intelligence and mathematical programming, but also have tremendous application in complex systems in general. They survey most of the recent research from the past decade in logic/optimization interfaces, incorporate some of their own results, and emphasize the types of logic most receptive to optimization methods—propositional logic, first order predicate logic, probabilistic and related logics, logics that combine evidence such as Dempster-Shafer theory, rule systems with confidence factors, and constraint logic programming systems.

Requiring no background in logic and clearly explaining all topics from the ground up, Optimization Methods for Logical Inference is an invaluable guide for scientists and students in diverse fields, including operations research, computer science, artificial intelligence, decision support systems, and engineering.

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.

The book is intended as an invitation to the topic of relations on a rather general basis. It fills the gap between the basic knowledge offered in countless introductory papers and books (usually comprising orders and equivalences) and the highly specialized monographs on mainly relation algebras, many-valued (fuzzy) relations, or graphs. This is done not only by presenting theoretical results but also by giving hints to some of the many interesting application areas (also including their respective theoretical basics).This book is a new - and the first of its kind - compilation of known results on binary relations. It offers relational concepts in both reasonable depth and broadness, and also provides insight into the vast diversity of theoretical results as well as application possibilities beyond the commonly known examples.This book is unique by the spectrum of the topics it handles. As indicated in its title these are:

In Mathematical Foundations of Public Key Cryptography, the authors integrate the results of more than 20 years of research and teaching experience to help students bridge the gap between math theory and crypto practice. The book provides a theoretical structure of fundamental number theory and algebra knowledge supporting public-key cryptography. Rather than simply combining number theory and modern algebra, this textbook features the interdisciplinary characteristics of cryptography-revealing the integrations of mathematical theories and public-key cryptographic applications. Incorporating the complexity theory of algorithms throughout, it introduces the basic number theoretic and algebraic algorithms and their complexities to provide a preliminary understanding of the applications of mathematical theories in cryptographic algorithms. Supplying a seamless integration of cryptography and mathematics, the book includes coverage of elementary number theory; algebraic structure and attributes of group, ring, and field; cryptography-related computing complexity and basic algorithms, as well as lattice and fundamental methods of lattice cryptanalysis. The text consists of 11 chapters. Basic theory and tools of elementary number theory, such as congruences, primitive roots, residue classes, and continued fractions, are covered in Chapters 1-6. The basic concepts of abstract algebra are introduced in Chapters 7-9, where three basic algebraic structures of groups, rings, and fields and their properties are explained. Chapter 10 is about computational complexities of several related mathematical algorithms, and hard problems such as integer factorization and discrete logarithm. Chapter 11 presents the basics of lattice theory and the lattice basis reduction algorithm-the LLL algorithm and its application in the cryptanalysis of the RSA algorithm. Containing a number of exercises on key algorithms, the book is suitable for use as a textbook for undergraduate students and first-year graduate students in information security programs. It is also an ideal reference book for cryptography professionals looking to master public-key cryptography.

This book brings together contributions by leading researchers in computational complexity theory written in honor of Somenath Biswas on the occasion of his sixtieth birthday. They discuss current trends and exciting developments in this flourishing area of research and offer fresh perspectives on various aspects of complexity theory. The topics covered include arithmetic circuit complexity, lower bounds and polynomial identity testing, the isomorphism conjecture, space-bounded computation, graph isomorphism, resolution and proof complexity, entropy and randomness. Several chapters have a tutorial flavor. The aim is to make recent research in these topics accessible to graduate students and senior undergraduates in computer science and mathematics. It can also be useful as a resource for teaching advanced level courses in computational complexity.

The Curry-Howard isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance,
minimal propositional logic corresponds to simply typed lambda-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, sequent calculus is related to explicit substitution, etc.
The isomorphism has many aspects, even at the syntactic level:
formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc.
But there is more to the isomorphism than this. For instance, it is an old idea---due to Brouwer, Kolmogorov, and Heyting---that a constructive proof of an implication is a procedure that transforms
proofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. The Curry-Howard isomorphism also provides theoretical foundations for many modern proof-assistant systems (e.g. Coq).
This book give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism. It can serve as an introduction to any or both of typed lambda-calculus and intuitionistic logic.

Key features
- The Curry-Howard Isomorphism treated as common theme
- Reader-friendly introduction to two complementary subjects: Lambda-calculus and constructive logics
- Thorough study of the connection between calculi and logics
- Elaborate study of classical logics and control operators
- Account of dialogue games for classical and intuitionistic logic
- Theoretical foundations of computer-assisted reasoning

. The Curry-Howard Isomorphism treated as the common theme.
. Reader-friendly introduction to two complementary subjects: lambda-calculus and constructive logics
. Thorough study of the connection between calculi and logics.
. Elaborate study of classical logics and control operators.
. Account of dialogue games for classical and intuitionistic logic.
. Theoretical foundations of computer-assisted reasoning"

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.

Over the last few decades the interest of logicians and mathematicians in constructive and computational aspects of their subjects has been steadily growing, and researchers from disparate areas realized that they can benefit enormously from the mutual exchange of techniques concerned with those aspects. A key figure in this exciting development is the logician and mathematician Helmut Schwichtenberg to whom this volume is dedicated on the occasion of his 70th birthday and his turning emeritus. The volume contains 20 articles from leading experts about recent developments in Constructive set theory, Provably recursive functions, Program extraction, Theories of truth, Constructive mathematics, Classical vs. intuitionistic logic, Inductive definitions, and Continuous functionals and domains.

This volume honours the life and work of Solomon Feferman, one of the most prominent mathematical logicians of the latter half of the 20th century. In the collection of essays presented here, researchers examine Feferman's work on mathematical as well as specific methodological and philosophical issues that tie into mathematics. Feferman's work was largely based in mathematical logic (namely model theory, set theory, proof theory and computability theory), but also branched out into methodological and philosophical issues, making it well known beyond the borders of the mathematics community. With regard to methodological issues, Feferman supported concrete projects. On the one hand, these projects calibrate the proof theoretic strength of subsystems of analysis and set theory and provide ways of overcoming the limitations imposed by Goedel's incompleteness theorems through appropriate conceptual expansions. On the other, they seek to identify novel axiomatic foundations for mathematical practice, truth theories, and category theory. In his philosophical research, Feferman explored questions such as "What is logic?" and proposed particular positions regarding the foundations of mathematics including, for example, his "conceptual structuralism." The contributing authors of the volume examine all of the above issues. Their papers are accompanied by an autobiography presented by Feferman that reflects on the evolution and intellectual contexts of his work. The contributing authors critically examine Feferman's work and, in part, actively expand on his concrete mathematical projects. The volume illuminates Feferman's distinctive work and, in the process, provides an enlightening perspective on the foundations of mathematics and logic.

Combinatory logic and lambda-calculus, originally devised in the 1920's, have since developed into linguistic tools, especially useful in programming languages. The authors' previous book served as the main reference for introductory courses on lambda-calculus for over 20 years: this long-awaited new version is thoroughly revised and offers a fully up-to-date account of the subject, with the same authoritative exposition. The grammar and basic properties of both combinatory logic and lambda-calculus are discussed, followed by an introduction to type-theory. Typed and untyped versions of the systems, and their differences, are covered. Lambda-calculus models, which lie behind much of the semantics of programming languages, are also explained in depth. The treatment is as non-technical as possible, with the main ideas emphasized and illustrated by examples. Many exercises are included, from routine to advanced, with solutions to most at the end of the book.

The series is devoted to the publication of high-level monographs on all areas of mathematical logic and its applications. It is addressed to advanced students and research mathematicians, and may also serve as a guide for lectures and for seminars at the graduate level.