Mathematical logic Books

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Book SynopsisThis book presents several recent advances in natural language semantics and explores the boundaries between syntax and semantics over the last two decades. It is based on some of the most recent theories in logic, such as linear logic and ludics, first created by Jean-Yves Girard, and it also provides some sharp analyses of computational semantical representations, explaining advanced theories in theoretical computer sciences, such as the lambda-mu and Lambek-Grishin calculi which were applied by Philippe de Groote and Michael Moortgat. The author also looks at Aarne Ranta's ‘proof as meaning’ approach, which was first based on Martin-Löf's Type Theory.Meaning, Logic and Ludics surveys the many solutions which have been proposed for the syntax-semantics interface, taking into account the specifications of linguistic signs (continuous or discontinuous) and the fundamental mechanisms developed by linguists and notable Generativists. This pioneering publication also presents ludics (in a chapter co-authored with Myriam Quatrini), a framework which allows us to characterize meaning as an invariant with regard to interaction between processes. It is an excellent book for advanced students, and academics alike, in the field of computational linguistics.Table of ContentsCompositional Semantics and Generative Grammar; Mathematics of Syntactic Structure; Elements of Proof Theory; Variants of the Lambek Calculus; Grammatical Reasoning; Minimalist Derivations; Continuations and Contexts; Towards Dynamics of Meaning; Interaction and Dialogue.

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Book SynopsisThe goal of this monograph is to give an accessible introduction to nonstandard methods and their applications, with an emphasis on combinatorics and Ramsey theory. It includes both new nonstandard proofs of classical results and recent developments initially obtained in the nonstandard setting. This makes it the first combinatorics-focused account of nonstandard methods to be aimed at a general (graduate-level) mathematical audience. This book will provide a natural starting point for researchers interested in approaching the rapidly growing literature on combinatorial results obtained via nonstandard methods. The primary audience consists of graduate students and specialists in logic and combinatorics who wish to pursue research at the interface between these areas.Table of Contents- Part I Preliminaries. - Ultrafilters. - Nonstandard Analysis. - Hyperfinite Generators of Ultrafilters. - Many Stars: Iterated Nonstandard Extensions. - LoebMeasure. - Part II Ramsey Theory. - Ramsey’s Theorem. - The Theorems of van der Waerden and Hales-Jewett. - From Hindman to Gowers. - Partition Regularity of Equations. - Part III Combinatorial Number Theory. - Densities and Structural Properties. - Working in the Remote Realm. - Jin’s Sumset Theorem. - Sumset Configurations in Sets of Positive Density. - Near Arithmetic Progressions in Sparse Sets. - The Interval Measure Property. - Part IV Other Topics. - Triangle Removal and Szemerédi Regularity. - Approximate Groups. - Foundations of Nonstandard Analysis.

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Book SynopsisThis volume explores A.P. Morse’s (1911-1984) development of a formal language for writing mathematics, his application of that language in set theory and mathematical analysis, and his unique perspective on mathematics. The editor brings together a variety of Morse’s works in this compilation, including Morse's book A Theory of Sets, Second Edition (1986), in addition to material from another of Morse’s publications, Web Derivatives, and notes for a course on analysis from the early 1950's. Because Morse provided very little in the way of explanation in his written works, the editor’s commentary serves to outline Morse’s goals, give informal explanations of Morse’s formal language, and compare Morse’s often unique approaches to more traditional approaches. Minor corrections to Morse’s previously published works have also been incorporated into the text, including some updated axioms, theorems, and definitions. The editor’s introduction thoroughly details the corrections and changes made and provides readers with valuable insight on Morse’s methods.A.P. Morse’s Set Theory and Analysis will appeal to graduate students and researchers interested in set theory and analysis who also have an interest in logic. Readers with a particular interest in Morse’s unique perspective and in the history of mathematics will also find this book to be of interest.Table of ContentsPreface.- Editor's Introduction.- Language and Inference.- Logic.- Set Theory.- Elementary Analysis.- Metrics.- Measure.- Linear Measure and Total Variation.- Integration.- Product Measures.- Web Derivatives.- Classical Differentiation.- The Construction of Definition.- The Consistency of the Axiom of Size.- Suggested Reading.- Publications of A.P. Morse.- Errata to A Theory of Sets, Second Edition.- Integration with Respect to Addor Functions.- The Henstock-Kurzweil Integral.

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Book SynopsisIn this book, Paulo Guilherme Santos studies diagonalization in formal mathematics from logical aspects to everyday mathematics. He starts with a study of the diagonalization lemma and its relation to the strong diagonalization lemma. After that, Yablo’s paradox is examined, and a self-referential interpretation is given. From that, a general structure of diagonalization with paradoxes is presented. Finally, the author studies a general theory of diagonalization with the help of examples from mathematics.Table of ContentsDiagonalization in Mathematics.- Diagonalization Lemma.- Fixed Point Theorems.- Paradoxes: Liar, Yablo’s Paradox, Curry’s Paradox.

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Book SynopsisIn diesem essential werden wesentliche Konzepte der Berechenbarkeitstheorie erörtert. Zunächst werden unterschiedliche Modelle der Berechenbarkeit eingeführt und ihre semantische Gleichwertigkeit gezeigt. Dieses Resultat steht in Einklang mit der Church-Turing-These, nach der jede intuitiv berechenbare Funktion partiell-rekursiv ist. Neben zentralen Instrumenten der Berechenbarkeit, wie etwa der Gödelisierung von berechenbaren Funktionen und der Existenz universeller berechenbarer Funktionen, stehen unentscheidbare Probleme im Fokus, wie etwa das Halteproblem sowie das Wortproblem für die Term-Ersetzung. Semi-entscheidbare Mengen werden beleuchtet und die zentralen Sätze von Rice und Rice-Shapiro werden skizziert. Table of ContentsBerechnungsmodelle.- Zentrale Konzepte.- Unentscheidbare Probleme.- Historie und Zusammenfassung

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Book SynopsisThis volume collects written versions of invited and contributed talks presented at the conference. It covers many areas of logic and the foundations of mathematics, as well as computer science.Table of ContentsSeminormal fine measures on Pk(lambda), Y. Abe; recursion theory and weak fragments of Peano arithmetic - a study of definable cuts, C.T. Chong and Y. Yang; lattice embedding into d-r.e. degrees preserving 0 and 1, D. Ding and L. Qian; on stationary reflection principles, Q. Feng; decidability and undecidability in the enumerable Turing degrees, S. Lempp; a note on weak segments of PFA, T. Miyamoto; linear set theory with strict comprehension, M. Shirahata; a solution to a problem of Marek and Truszcy ski, K. Su and H. Chen; DNK and natural reasoning, M. Yasugi and M. Nakata; Asian logic conferences, M. Yasugi; other papers.

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Book SynopsisAlthough there are some books dealing with algebraic theory of automata, their contents consist mainly of Krohn-Rhodes theory and related topics. The topics in the present book are rather different. For example, automorphism groups of automata and the partially ordered sets of automata are systematically discussed. Moreover, some operations on languages and special classes of regular languages associated with deterministic and nondeterministic directable automata are dealt with. The book is self-contained and hence does not require any knowledge of automata and formal languages.

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Book SynopsisThe 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.

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Book SynopsisThe 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.

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Book SynopsisCollege 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.

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Book SynopsisCollege 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.

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Book SynopsisMathematical 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.Table of ContentsEntering the Labyrinth; What? Why? How?; The Clue is in the Question; Thinking Outside the Box; "Precise Steps" Problems. Playing Preferans; Transformations of Plane; Geometrical Constructions with Restricted Elements; "Inventing" a Problem; Related Problems; Alternative Solutions Search; "Eureka";

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Book SynopsisMathematical 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.Table of ContentsEntering the Labyrinth; What? Why? How?; The Clue is in the Question; Thinking Outside the Box; "Precise Steps" Problems. Playing Preferans; Transformations of Plane; Geometrical Constructions with Restricted Elements; "Inventing" a Problem; Related Problems; Alternative Solutions Search; "Eureka";

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Book SynopsisInnovative Teaching: Best Practices from Business and Beyond for Mathematics Teachers provides educators with new and exciting ways to introduce material and methods to motivate and engage students by showing how some of the techniques commonly used in the business world — and beyond — are applicable to the world of education. It also offers educators practical advice with regard to the changing culture of education, keeping up with technology, navigating politics at work, interacting with colleagues, developing leadership skills, group behavior, and gender differences.Innovative Teaching demonstrates how the classroom environment is similar to the marketplace. Educators, like businesses, for example, must capture and hold the attention of their audience while competing with a constant stream of 'noise.' With the introduction of the Internet and the wide use of social media, promoters understand that they must not only engage their audience, but also incorporate audience feedback into the promotional work and product or service they offer. Innovative Teaching shows educators how to take the best practices from business — and beyond — and recombine these resources for appropriate use in the classroom.

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Book SynopsisInnovative Teaching: Best Practices from Business and Beyond for Mathematics Teachers provides educators with new and exciting ways to introduce material and methods to motivate and engage students by showing how some of the techniques commonly used in the business world — and beyond — are applicable to the world of education. It also offers educators practical advice with regard to the changing culture of education, keeping up with technology, navigating politics at work, interacting with colleagues, developing leadership skills, group behavior, and gender differences.Innovative Teaching demonstrates how the classroom environment is similar to the marketplace. Educators, like businesses, for example, must capture and hold the attention of their audience while competing with a constant stream of 'noise.' With the introduction of the Internet and the wide use of social media, promoters understand that they must not only engage their audience, but also incorporate audience feedback into the promotional work and product or service they offer. Innovative Teaching shows educators how to take the best practices from business — and beyond — and recombine these resources for appropriate use in the classroom.

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Book SynopsisThis book is for graduate students and researchers, introducing modern foundational research in mathematics, computer science, and philosophy from an interdisciplinary point of view. Its scope includes proof theory, constructive mathematics and type theory, univalent mathematics and point-free approaches to topology, extraction of certified programs from proofs, automated proofs in the automotive industry, as well as the philosophical and historical background of proof theory. By filling the gap between (under-)graduate level textbooks and advanced research papers, the book gives a scholarly account of recent developments and emerging branches of the aforementioned fields.

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Book SynopsisThe overall topic of the volume, Mathematics for Computation (M4C), is mathematics taking crucially into account the aspect of computation, investigating the interaction of mathematics with computation, bridging the gap between mathematics and computation wherever desirable and possible, and otherwise explaining why not.Recently, abstract mathematics has proved to have more computational content than ever expected. Indeed, the axiomatic method, originally intended to do away with concrete computations, seems to suit surprisingly well the programs-from-proofs paradigm, with abstraction helping not only clarity but also efficiency.Unlike computational mathematics, which rather focusses on objects of computational nature such as algorithms, the scope of M4C generally encompasses all the mathematics, including abstract concepts such as functions. The purpose of M4C actually is a strongly theory-based and therefore, is a more reliable and sustainable approach to actual computation, up to the systematic development of verified software.While M4C is situated within mathematical logic and the related area of theoretical computer science, in principle it involves all branches of mathematics, especially those which prompt computational considerations. In traditional terms, the topics of M4C include proof theory, constructive mathematics, complexity theory, reverse mathematics, type theory, category theory and domain theory.The aim of this volume is to provide a point of reference by presenting up-to-date contributions by some of the most active scholars in each field. A variety of approaches and techniques are represented to give as wide a view as possible and promote cross-fertilization between different styles and traditions.

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Book SynopsisLaws of Form is a seminal work in foundations of logic, mathematics and philosophy published by G Spencer-Brown in 1969. The book provides a new point of view on form and the role of distinction, markedness and the absence of distinction (the unmarked state) in the construction of any universe. A conference was held August 8-10, 2019 at the Old Library, Liverpool University, 19 Abercromby Square, L697ZN, UK to celebrate the 50th anniversary of the publication of Laws of Form and to remember George Spencer-Brown, its author. The book is a collection of papers introducing and extending Laws of Form written primarily by people who attended the conference in 2019.

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Book SynopsisCalculi of temporal logic are widely used in modern computer science. The temporal organization of information flows in the different architectures of laptops, the Internet, or supercomputers would not be possible without appropriate temporal calculi. In the age of digitalization and High-Tech applications, people are often not aware that temporal logic is deeply rooted in the philosophy of modalities. A deep understanding of these roots opens avenues to the modern calculi of temporal logic which have emerged by extension of modal logic with temporal operators. Computationally, temporal operators can be introduced in different formalisms with increasing complexity such as Basic Modal Logic (BML), Linear-Time Temporal Logic (LTL), Computation Tree Logic (CTL), and Full Computation Tree Logic (CTL*). Proof-theoretically, these formalisms of temporal logic can be interpreted by the sequent calculus of Gentzen, the tableau-based calculus, automata-based calculus, game-based calculus, and dialogue-based calculus with different advantages for different purposes, especially in computer science.The book culminates in an outlook on trendsetting applications of temporal logics in future technologies such as artificial intelligence and quantum technology. However, it will not be sufficient, as in traditional temporal logic, to start from the everyday understanding of time. Since the 20th century, physics has fundamentally changed the modern understanding of time, which now also determines technology. In temporal logic, we are only just beginning to grasp these differences in proof theory which needs interdisciplinary cooperation of proof theory, computer science, physics, technology, and philosophy.

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Book SynopsisThis textbook is aimed at transitioning high-school students who have already developed proficiency in mathematical problem solving from numerical-answer problems to proof-based mathematics. It serves to guide students on how to write and understand mathematical proofs. It covers proof techniques that are commonly used in several areas of mathematics, especially number theory, combinatorics, and analysis. In addition to just teaching the mechanics of proofs, this book showcases key materials in these areas, thus introducing readers to interesting mathematics along with proof techniques.

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Book SynopsisThis volume results from two programs that took place at the Institute for Mathematical Sciences at the National University of Singapore: Aspects of Computation — in Celebration of the Research Work of Professor Rod Downey (21 August to 15 September 2017) and Automata Theory and Applications: Games, Learning and Structures (20-24 September 2021).The first program was dedicated to the research work of Rodney G. Downey, in celebration of his 60th birthday. The second program covered automata theory whereby researchers investigate the other end of computation, namely the computation with finite automata, and the intermediate level of languages in the Chomsky hierarchy (like context-free and context-sensitive languages).This volume contains 17 contributions reflecting the current state-of-art in the fields of the two programs.

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Book SynopsisTopics covered in Discrete Mathematics have become essential tools in many areas of studies in recent years. This is primarily due to the revolution in technology, communications, and cyber security. The book treats major themes in a typical introductory modern Discrete Mathematics course: Propositional and predicate logic, proof techniques, set theory (including Boolean algebra, functions and relations), introduction to number theory, combinatorics and graph theory.An accessible, precise, and comprehensive approach is adopted in the treatment of each topic. The ability of abstract thinking and the art of writing valid arguments are emphasized through detailed proof of (almost) every result. Developing the ability to think abstractly and roguishly is key in any areas of science, information technology and engineering. Every result presented in the book is followed by examples and applications to consolidate its comprehension. The hope is that the reader ends up developing both the abstract reasoning as well as acquiring practical skills.All efforts are made to write the book at a level accessible to first-year students and to present each topic in a way that facilitates self-directed learning. Each chapter starts with basic concepts of the subject at hand and progresses gradually to cover more ground on the subject. Chapters are divided into sections and subsections to facilitate readings. Each section ends with its own carefully chosen set of practice exercises to reenforce comprehension and to challenge and stimulate readers.As an introduction to Discrete Mathematics, the book is written with the smallest set of prerequisites possible. Familiarity with basic mathematical concepts (usually acquired in high school) is sufficient for most chapters. However, some mathematical maturity comes in handy to grasp some harder concepts presented in the book.

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Book SynopsisThis book argues for neuromorphic systems as a technology of the future, which are oriented towards the energy efficiency of natural brains. Energy efficiency is a dramatic claim in times of environmental and climate challenges which should consider the sustainability goals of the United Nations (UN). Mathematically, neuromorphic computing is connected to analogue ('real') computing, which theoretically overcomes the limits of digital Turing computability. Therefore, the book also considers material sciences and engineering sciences which start to realize neuromorphic computing in hardware. Other mathematical formalisms such as quantum mechanics also open up new solutions (e.g., quantum computing) beyond the limits of digital Turing computability. These research fields are no longer merely of theoretical interest, they promise increasing innovation power of market interest. Nevertheless, neuromorphic computing is connected with deep logical, mathematical, and epistemic questions. Does it open new avenues to Artificial General Intelligence (AGI)? All these tendencies of research and innovation demonstrate that we need more integrated research in the foundations of logic, mathematics, physics, engineering sciences, cognitive science, and philosophy. The book is a plea for this kind of research.

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Book SynopsisThis book contains the material for a first course in pure model theory with applications to differentially closed fields. Topics covered in this book include saturated model criteria for model completeness and elimination of quantifiers; Morley rank and degree of element types; categoricity in power; two-cardinal theorems; existence and uniqueness of prime model extensions of substructures of models of totally transcendental theories; and homogeneity of models of ϖ1-categorical theories.Table of ContentsOrdinals and Diagrams; Similarity Types of Structures; Elementary Equivalence; Model Completeness; Skolemization of Structures; Saturated Structures; Omitting a Type; Homogeneous Structures; Inverse Systems of Compact Hausdorff Spaces; The Morley Derivative; Prime Model Extensions; Order Indiscernibles; The Baldwin-Lachlan Theorem; and other sections.

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Book SynopsisThe present monograph on matrix partial orders, the first on this topic, makes a unique presentation of many partial orders on matrices that have fascinated mathematicians for their beauty and applied scientists for their wide-ranging application potential. Except for the Löwner order, the partial orders considered are relatively new and came into being in the late 1970s. After a detailed introduction to generalized inverses and decompositions, the three basic partial orders — namely, the minus, the sharp and the star — and the corresponding one-sided orders are presented using various generalized inverses. The authors then give a unified theory of all these partial orders as well as study the parallel sums and shorted matrices, the latter being studied at great length. Partial orders of modified matrices are a new addition. Finally, applications are given in statistics and electrical network theory.∗DeceasedTable of ContentsIntroduction; Decompositions and Generalized Inverses; Minus Order; Sharp Order; Star Order; One-Sided Orders; Lowner Order and Majorization; Unified Theory of Matrix Partial Orders through Generalized Inverses; Parallel Sums; Schur Complements and Shorted Operators; Shorted Operators II; Supremum and Infimum for a Pair of Matrices; Partial Orders for Modified Matrices; Statistics; Electrical Network Theory.

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Book Synopsis'Points, questions, stories, and occasional rants introduce the 24 chapters of this engaging volume. With a focus on mathematics and peppered with a scattering of computer science settings, the entries range from lightly humorous to curiously thought-provoking. Each chapter includes sections and sub-sections that illustrate and supplement the point at hand. Most topics are self-contained within each chapter, and a solid high school mathematics background is all that is needed to enjoy the discussions. There certainly is much to enjoy here.'CHOICEEver notice how people sometimes use math words inaccurately? Or how sometimes you instinctively know a math statement is false (or not known)?Each chapter of this book makes a point like those above and then illustrates the point by doing some real mathematics through step-by-step mathematical techniques.This book gives readers valuable information about how mathematics and theoretical computer science work, while teaching them some actual mathematics and computer science through examples and exercises. Much of the mathematics could be understood by a bright high school student. The points made can be understood by anyone with an interest in math, from the bright high school student to a Field's medal winner.

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Book SynopsisThis book is about “diamond”, a logic of paradox. In diamond, a statement can be true yet false; an “imaginary” state, midway between being and non-being. Diamond's imaginary values solve many logical paradoxes unsolvable in two-valued Boolean logic. In this volume, paradoxes by Russell, Cantor, Berry and Zeno are all resolved. This book has three sections: Paradox Logic, which covers the classic paradoxes of mathematical logic, shows how they can be resolved in this new system; The Second Paradox, which relates diamond to Boolean logic and the Spencer-Brown “modulator”; and Metamathematical Dilemma, which relates diamond to Gödelian metamathematics and dilemma games.Table of ContentsParadox: Russell's Paradox; Santa Sentences; Antistrephon; Game Paradoxes; Paradox of the Boundary; Diamond: Diamond Values; Harmonic Functions; Diamond Circuits; Brownian Forms; Diamond Algebra: Bracket Algebra; Laws; Normal Forms; Completeness; Self-Reference: Re-entrance and Fixedpoints; Phase Order; The Outer Fixedpoints; Fixedpoint Lattices: Relative Lattices; Seeds and Spirals; Shared Fixed-Points; Limit Logic: Limit Fixedpoints; the Halting Theorem; Paradox Resolved: Russell's Paradox; Santa Sentences; Antistrephon; Game Paradoxes; The Continuum: Cantor's Paradox; Dedekind Splices; Zeno's Theorem; Fuzzy Chaos; Clique Theory: Cliques Defined; Clique Axioms; Graph Cliques; Clique Circuits; Orthogonal Logics: Analytic Functions; Star Logic; Harmonic Projection; Interferometry: Quadrature; Diffraction; Buzzers and Toggles; How to Count to Two: Brownian and Kauffman Modulators; Diffracting the Modulators; Rotors, Pumps and Tapes; the Ganglion; Metamathematics: Godelian Quanta; Meta-Logic; Dialectical Dilemma; Dilemma: Milo's Trick; Prisoner's Dilemma; Dilemma Diamond; Banker's Dilemma; The Unexpected Departure.

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Book SynopsisThis volume consists of papers selected from the presentations at the workshop and includes mainly recent developments in the fields of formal languages, automata theory and algebraic systems related to the theoretical computer science and informatics. It covers the areas such as automata and grammars, languages and codes, combinatorics on words, cryptosystems, logics and trees, Grobner bases, minimal clones, zero-divisor graphs, fine convergence of functions, and others.Table of ContentsAutomata and Grammars; Languages and Codes; Combinatorics on Words; Cryptosystems; Logics and Trees; Grobner Bases; Minimal Clones; Zero-Divisor Graphs; Fine Convergence of Functions; and other papers.

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Book SynopsisThis is a systematic and well-paced introduction to mathematical logic. Excellent as a course text, the book presupposes only elementary background and can be used also for self-study by more ambitious students.Starting with the basics of set theory, induction and computability, it covers propositional and first-order logic — their syntax, reasoning systems and semantics. Soundness and completeness results for Hilbert's and Gentzen's systems are presented, along with simple decidability arguments. The general applicability of various concepts and techniques is demonstrated by highlighting their consistent reuse in different contexts.Unlike in most comparable texts, presentation of syntactic reasoning systems precedes the semantic explanations. The simplicity of syntactic constructions and rules — of a high, though often neglected, pedagogical value — aids students in approaching more complex semantic issues. This order of presentation also brings forth the relative independence of syntax from the semantics, helping to appreciate the importance of the purely symbolic systems, like those underlying computers.An overview of the history of logic precedes the main text, while informal analogies precede introduction of most central concepts. These informal aspects are kept clearly apart from the technical ones. Together, they form a unique text which may be appreciated equally by lecturers and students occupied with mathematical precision, as well as those interested in the relations of logical formalisms to the problems of computability and the philosophy of logic.Table of ContentsHistory of Logic; Basic Set Theory and Induction; Turing Machines and Computability; Propositional Logic: Hilbert's and Gentzen's Proof Systems; Boolean and Set-Valued Semantics; DNF and CNF; Soundness and Completeness; First-Order Logic: Hilbert's and Gentzen's Proof Systems; Tarski's Semantics; Prenex NF; Term Models and Prolog; Soundness and Completeness; Exercises to Each Topic.

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Book SynopsisThis is a systematic and well-paced introduction to mathematical logic. Excellent as a course text, the book presupposes only elementary background and can be used also for self-study by more ambitious students.Starting with the basics of set theory, induction and computability, it covers propositional and first-order logic — their syntax, reasoning systems and semantics. Soundness and completeness results for Hilbert's and Gentzen's systems are presented, along with simple decidability arguments. The general applicability of various concepts and techniques is demonstrated by highlighting their consistent reuse in different contexts.Unlike in most comparable texts, presentation of syntactic reasoning systems precedes the semantic explanations. The simplicity of syntactic constructions and rules — of a high, though often neglected, pedagogical value — aids students in approaching more complex semantic issues. This order of presentation also brings forth the relative independence of syntax from the semantics, helping to appreciate the importance of the purely symbolic systems, like those underlying computers.An overview of the history of logic precedes the main text, while informal analogies precede introduction of most central concepts. These informal aspects are kept clearly apart from the technical ones. Together, they form a unique text which may be appreciated equally by lecturers and students occupied with mathematical precision, as well as those interested in the relations of logical formalisms to the problems of computability and the philosophy of logic.Table of ContentsHistory of Logic; Basic Set Theory and Induction; Turing Machines and Computability; Propositional Logic: Hilbert's and Gentzen's Proof Systems; Boolean and Set-Valued Semantics; DNF and CNF; Soundness and Completeness; First-Order Logic: Hilbert's and Gentzen's Proof Systems; Tarski's Semantics; Prenex NF; Term Models and Prolog; Soundness and Completeness; Exercises to Each Topic.

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Book SynopsisThe Asian Logic Conference is part of the series of logic conferences inaugurated in Singapore in 1981. It is normally held every three years and rotates among countries in the Asia-Pacific region. The 11th Asian Logic Conference was held at the National University of Singapore, in honor of Professor Chong Chitat on the occasion of his 60th birthday. The conference is on the broad area of logic, including theoretical computer science. It is considered a major event in this field and is regularly sponsored by the Association of Symbolic Logic. This volume contains papers from this meeting.Table of ContentsLimitwise Monotonic Functions and Their Applications (R G Downey et al.); Provably Δ02 and Weakly Descending Chains (T Arai); K-Trivials are NCR (G Barmpalias et al.); A Dichotomy for the Mackey Borel Structure (I Farah); Computable Dowd-Type Generic Oracles (M Kumabe & T Suzuki); Cappable CEA Sets and Ramsey's Theorem (A M Kach et al.); Amalgamation, Absoluteness, and Categoricity (J T Baldwin); Models of Long Sentences I (G Sacks); On Automatic Families (S Jain et al.); A Universally-Free Modal Logic (S C-M Yang).

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Book SynopsisThis penultimate volume contains numerous original, elegant, and surprising results in 1-dimensional cellular automata. Perhaps the most exciting, if not shocking, new result is the discovery that only 82 local rules, out of 256, suffice to predict the time evolution of any of the remaining 174 local rules from an arbitrary initial bit-string configuration. This is contrary to the well-known folklore that 256 local rules are necessary, leading to the new concept of quasi-global equivalence.Another surprising result is the introduction of a simple, yet explicit, infinite bit string called the super string S, which contains all random bit strings of finite length as sub-strings. As an illustration of the mathematical subtlety of this amazing discrete testing signal, the super string S is used to prove mathematically, in a trivial and transparent way, that rule 170 is as chaotic as a coin toss.Yet another unexpected new result, among many others, is the derivation of an explicit basin tree generation formula which provides an analytical relationship between the basin trees of globally-equivalent local rules. This formula allows the symbolic, rather than numerical, generation of the time evolution of any local rule corresponding to any initial bit-string configuration, from one of the 88 globally-equivalent local rules.But perhaps the most provocative idea is the proposal for adopting rule 137, over its three globally-equivalent siblings, including the heretofore more well-known rule 110, as the prototypical universal Turing machine.Table of ContentsPeriod-2 Rules; Period-3 Rules, Period-6 Rules, and Permutive Rules.

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Book SynopsisThe Asian Logic Conference is the most significant logic meeting outside of North America and Europe, and this volume represents work presented at, and arising from the 12th meeting. It collects a number of interesting papers from experts in the field. It covers many areas of logic.

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Book SynopsisThis book gives a detailed treatment of functional interpretations of arithmetic, analysis, and set theory. The subject goes back to Gödel's Dialectica interpretation of Heyting arithmetic which replaces nested quantification by higher type operations and thus reduces the consistency problem for arithmetic to the problem of computability of primitive recursive functionals of finite types. Regular functional interpretations, in particular the Dialectica interpretation and its generalization to finite types, the Diller-Nahm interpretation, are studied on Heyting as well as Peano arithmetic in finite types and extended to functional interpretations of constructive as well as classical systems of analysis and set theory. Kreisel's modified realization and Troelstra's hybrids of it are presented as interpretations of Heyting arithmetic and extended to constructive set theory, both in finite types. They serve as background for the construction of hybrids of the Diller-Nahm interpretation of Heyting arithmetic and constructive set theory, again in finite types. All these functional interpretations yield relative consistency results and closure under relevant rules of the theories in question as well as axiomatic characterizations of the functional translations.Table of ContentsArithmetic: Primitive Recursive Functionals; Λ- (Diller - Nahm) Interpretation of Heyting Arithmetic in Finite Types; The Dialectica Interpretation and Equality Functionals; Simultaneous Recursions in Linear Types; Computability, Consistency, Continuity; Modified Realization and its Hybrids; Hybrids of the Λ-Interpretation; N-Interpretations; Interpretations of Classical Arithmetic; Extensionality and Majorizability; Analysis: Bar Recursive Functionals; Λ- and Dialectica Interpretation of Bar Induction by Bar Recursion; Functional Interpretations of Classical Analysis; Computability of Bar Recursive Functionals; Set Theory: Constructive Set Functionals; Kripke - Platek Set Theory and Its Functional Interpretations; Constructive Set Theory and Its Λ-Interpretation; Modified Realizations of Constructive Set Theory; The Q-Hybrid of the Λ-Interpretation of Constructive Set Theory in Finite Types; Majorizability of Constructive Set Functionals.

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Book SynopsisEver since Paul Cohen's spectacular use of the forcing concept to prove the independence of the continuum hypothesis from the standard axioms of set theory, forcing has been seen by the general mathematical community as a subject of great intrinsic interest but one that is technically so forbidding that it is only accessible to specialists. In the past decade, a series of remarkable solutions to long-standing problems in C*-algebra using set-theoretic methods, many achieved by the author and his collaborators, have generated new interest in this subject. This is the first book aimed at explaining forcing to general mathematicians. It simultaneously makes the subject broadly accessible by explaining it in a clear, simple manner, and surveys advanced applications of set theory to mainstream topics.Table of ContentsPeano Arithmetic; Zermelo - Fraenkel Set Theory; Well-Ordered Sets; Ordinals; Cardinals; Relativization; Reflection; Forcing Posets; Generic Extensions; Forcing Equality; The Fundamental Theorem of Forcing; Forcing CH; Forcing not-CH; Families of Entire Functions; Self-Homeomorphisms of Beta N - N, I; Pure States on B(H); The Diamond Principle; Suslin's Problem, I; Naimark's problem; Product Forcing and Diamond^S; The Whitehead Problem, I; Two-Stage Iterated Forcing; Finite Support Iteration; Martin's Axiom; Suslin's Problem, II; The Whitehead Problem, II; The Open Coloring Axiom; Self-Homeomorphisms of Beta N - N, II; Automorphisms of the Calkin Algebra, I; Automorphisms of the Calkin Algebra, II; The Multiverse Interpretation.

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Book SynopsisThis 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.Table of ContentsSection I: Absoluteness, Truth, and Quotients (Ilijas Farah); A Multiverse Perspective on the Axiom of Constructiblity (Joel David Hamkins); Hilbert, Bourbaki and the Scorning of Logic (A R D Mathias); Toward Objectivity in Mathematics (Stephen G Simpson); Sort Logic and Foundations of Mathematics (Jouko Vaananen); Reasoning about Constructive Concepts (Nik Weaver); Perfect Infinites and Finite Approximation (Boris Zilber); Section II: An Objective Justification for Actual Infinity? (Stephen G Simpson); Oracle Questions (Theodore A Slaman and W Hugh Woodin).

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Book SynopsisIn the mathematical practice, the Baire category method is a tool for establishing the existence of a rich array of generic structures. However, in mathematics, the Baire category method is also behind a number of fundamental results such as the Open Mapping Theorem or the Banach-Steinhaus Boundedness Principle. This volume brings the Baire category method to another level of sophistication via the internal version of the set-theoretic forcing technique. It is the first systematic account of applications of the higher forcing axioms with the stress on the technique of building forcing notions rather than on the relationship between different forcing axioms or their consistency strengths.Table of ContentsThe Baire Category Theorem and the Baire Category Numbers; Coding into the Reals; Descriptive Set-Theoretic Consequences; Measure-Theoretic Consequences; Variations on the Souslin Hypothesis; The S- and L-Space Problems; The Side-Condition Method; Ideal Dichotomies; Coherent and Lipschitz Trees; Applications to the S-Space Problem and the Von Neumann Problem; Biorthogonal Systems; Structure of Compact Spaces; Ramsey Theory on Ordinals; Five Cofinal Types; Five Linear Orderings; mm and Cardinal Arithmetic; Reflection Principles; Appendices: Basic Notions; Preserving Stationary Sets; Historical and Other Comments.

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