An approach to complexity theory which offers a means of analysing algorithms in terms of their tractability. The authors consider the problem in terms of parameterized languages and taking "k-slices" of the language, thus introducing readers to new classes of algorithms which may be analysed more precisely than was the case until now. The book is as self-contained as possible and includes a great deal of background material. As a result, computer scientists, mathematicians, and graduate students interested in the design and analysis of algorithms will find much of interest.

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.

Poland has played an enormous role in the development of mathematical logic. Leading Polish logicians, like Lesniewski, Lukasiewicz and Tarski, produced several works related to philosophical logic, a field covering different topics relevant to philosophical foundations of logic itself, as well as various individual sciences. This collection presents contemporary Polish work in philosophical logic which in many respects continue the Polish way of doing philosophical logic. This book will be of interest to logicians, mathematicians, philosophers, and linguists.

This volume gathers selected papers presented at the Fourth Asian Workshop on Philosophical Logic, held in Beijing in October 2018. The contributions cover a wide variety of topics in modal logic (epistemic logic, temporal logic and dynamic logic), proof theory, algebraic logic, game logics, and philosophical foundations of logic. They also reflect the interdisciplinary nature of logic - a subject that has been studied in fields as diverse as philosophy, linguistics, mathematics, computer science and artificial intelligence. More specifically. The book also presents the latest developments in logic both in Asia and beyond.

This book is for researchers in computer science, mathematical logic, and philosophical logic. It shows the state of the art in current investigations of process calculi with mainly two major paradigms at work: linear logic and modal logic. The combination of approaches and pointers for further integration also suggests a grander vision for the field.

Self-contained, and collating for the first time material that has until now only been published in journals - often in Russian - this book will be of interest to functional analysts, especially those with interests in topological vector spaces, and to algebraists concerned with category theory. The closed graph theorem is one of the corner stones of functional analysis, both as a tool for applications and as an object for research. However, some of the spaces which arise in applications and for which one wants closed graph theorems are not of the type covered by the classical closed graph theorem of Banach or its immediate extensions. To remedy this, mathematicians such as Schwartz and De Wilde (in the West) and Rajkov (in the East) have introduced new ideas which have allowed them to establish closed graph theorems suitable for some of the desired applications. In this book, Professor Smirnov uses category theory to provide a very general framework, including the situations discussed by De Wilde, Rajkov and others. General properties of the spaces involved are discussed and applications are provided in measure theory, global analysis and differential equations.

Since their appearance in the late 19th century, the Cantor--Dedekind theory of real numbers and philosophy of the continuum have emerged as pillars of standard mathematical philosophy. On the other hand, this period also witnessed the emergence of a variety of alternative theories of real numbers and corresponding theories of continua, as well as non-Archimedean geometry, non-standard analysis, and a number of important generalizations of the system of real numbers, some of which have been described as arithmetic continua of one type or another. With the exception of E.W. Hobson's essay, which is concerned with the ideas of Cantor and Dedekind and their reception at the turn of the century, the papers in the present collection are either concerned with or are contributions to, the latter groups of studies. All the contributors are outstanding authorities in their respective fields, and the essays, which are directed to historians and philosophers of mathematics as well as to mathematicians who are concerned with the foundations of their subject, are preceded by a lengthy historical introduction.

This monograph provides a definitive overview of recent advances in the stability and oscillation of autonomous delay differential equations. Topics include linear and nonlinear delay and integrodifferential equations, which have potential applications to both biological and physical dynamic processes. Chapter 1 deals with an analysis of the dynamical characteristics of the delay logistic equation, and a number of techniques and results relating to stability, oscillation and comparison of scalar delay and integrodifferential equations are presented. Chapter 2 provides a tutorial-style introduction to the study of delay-induced Hopf bifurcation to periodicity and the related computations for the analysis of the stability of bifurcating periodic solutions. Chapter 3 is devoted to local analyses of nonlinear model systems and discusses many methods applicable to linear equations and their perturbations. Chapter 4 considers global convergence to equilibrium states of nonlinear systems, and includes oscillations of nonlinear systems about their equilibria. Qualitative analyses of both competitive and cooperative systems with time delays feature in both Chapters 3 and 4. Finally, Chapter 5 deals with recent developments in models of neutral differential equations and their applications to population dynamics. Each chapter concludes with a number of exercises and the overall exposition recommends this volume as a good supplementary text for graduate courses. For mathematicians whose work involves functional differential equations, and whose interest extends beyond the boundaries of linear stability analysis.

The twenty-six papers in this volume reflect the wide and still expanding range of Anil Nerode's work. A conference on Logical Methods was held in honor of Nerode's sixtieth birthday (4 June 1992) at the Mathematical Sciences Institute, Cornell University, 1-3 June 1992. Some of the conference papers are here, but others are from students, co-workers and other colleagues. The intention of the conference was to look forward, and to see the directions currently being pursued, in the development of work by, or with, Nerode. Here is a brief summary of the contents of this book. We give a retrospective view of Nerode's work. A number of specific areas are readily discerned: recursive equivalence types, recursive algebra and model theory, the theory of Turing degrees and r.e. sets, polynomial-time computability and computer science. Nerode began with automata theory and has also taken a keen interest in the history of mathematics. All these areas are represented. The one area missing is Nerode's applied mathematical work relating to the environment. Kozen's paper builds on Nerode's early work on automata. Recursive equivalence types are covered by Dekker and Barback, the latter using directly a fundamental metatheorem of Nerode. Recursive algebra is treated by Ge & Richards (group representations). Recursive model theory is the subject of papers by Hird, Moses, and Khoussainov & Dadajanov, while a combinatorial problem in recursive model theory is discussed in Cherlin & Martin's paper. Cenzer presents a paper on recursive dynamics.

The classical theory of computation has its origins in the work of Goedel, Turing, Church, and Kleene and has been an extraordinarily successful framework for theoretical computer science. The thesis of this book, however, is that it provides an inadequate foundation for modern scientific computation where most of the algorithms are real number algorithms. The goal of this book is to develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing. Along the way, the authors consider such fundamental problems as: * Is the Mandelbrot set decidable? * For simple quadratic maps, is the Julia set a halting set? * What is the real complexity of Newton's method? * Is there an algorithm for deciding the knapsack problem in a ploynomial number of steps? * Is the Hilbert Nullstellensatz intractable? * Is the problem of locating a real zero of a degree four polynomial intractable? * Is linear programming tractable over the reals? The book is divided into three parts: The first part provides an extensive introduction and then proves the fundamental NP-completeness theorems of Cook-Karp and their extensions to more general number fields as the real and complex numbers. The later parts of the book develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing.

Fuzzy Set Theory and Advanced Mathematical Applications contains contributions by many of the leading experts in the field, including coverage of the mathematical foundations of the theory, decision making and systems science, and recent developments in fuzzy neural control. The book supplies a readable, practical toolkit with a clear introduction to fuzzy set theory and its evolution in mathematics and new results on foundations of fuzzy set theory, decision making and systems science, and fuzzy control and neural systems. Each chapter is self-contained, providing up-to-date coverage of its subject. Audience: An important reference work for university students, and researchers and engineers working in both industrial and academic settings.

Fuzzy Sets, Logics and Reasoning about Knowledge reports recent results concerning the genuinely logical aspects of fuzzy sets in relation to algebraic considerations, knowledge representation and commonsense reasoning. It takes a state-of-the-art look at multiple-valued and fuzzy set-based logics, in an artificial intelligence perspective. The papers, all of which are written by leading contributors in their respective fields, are grouped into four sections. The first section presents a panorama of many-valued logics in connection with fuzzy sets. The second explores algebraic foundations, with an emphasis on MV algebras. The third is devoted to approximate reasoning methods and similarity-based reasoning. The fourth explores connections between fuzzy knowledge representation, especially possibilistic logic and prioritized knowledge bases. Readership: Scholars and graduate students in logic, algebra, knowledge representation, and formal aspects of artificial intelligence.

Thisvolume starts with the basicconceptsof FuzzyLogic: the membership function, the intersection and the union of fuzzy sets, fuzzy numbers, and the extension principle underlying the algorithmic operations. Several chapters are devoted to applications of FuzzyLogic in various branches of Operations Research: PERT planning with uncertain activity durations, SMART and the AHP for Multi-Criteria Decision Analysis (MCDA) with vague preferential statements, ELECTRE usingthe ideasof the AHP and SMART, and Multi-Objective Optimization (MOO) with weighted degrees of satisfaction. Finally, earlierstudiesof colour perception illustrate the attemptsto find a physiological basisfor the set-theoretical and the algorithmic operations in Fuzzy Logic. The last chapter also discusses somekey issues in linguistic categorization and the prospectsof FuzzyLogicas a multi-disciplinary research activity. I am greatly indebted to the Department of Mechanical Engineering and Applied Mechanics, College of Engineering, University of Michigan, Ann Arbor, for the splendid opportunity to start the actual work on this book during my sabbatical leavefrom Delft (1993 - 1994); to LAMSADE, Universite de Paris-Dauphine, where many ideas emerged duringtwo winter visits (1989, 1990); to the International Institute for Applied Systems Analysis, Laxenburg, Austria, whereI got further inspiration duringa number of summer visits (1992, 1995, and 1996); and to the NISSAN Foundation in The Netherlands who enabled me to visit several Japanese universities (June 1996). Moreover, I gratefully acknowledge the stimulating supportgiven by many colleagues inthe International Society on Multi-Criteria Decision Making and in the European Working Group "Aide Multicritere Ii la Decision."

This is the first book to present an up-to-date and self-contained account of Algebraic Complexity Theory that is both comprehensive and unified. Requiring of the reader only some basic algebra and offering over 350 exercises, it is well-suited as a textbook for beginners at graduate level. With its extensive bibliography covering about 500 research papers, this text is also an ideal reference book for the professional researcher. The subdivision of the contents into 21 more or less independent chapters enables readers to familiarize themselves quickly with a specific topic, and facilitates the use of this book as a basis for complementary courses in other areas such as computer algebra.

without a properly developed inconsistent calculus based on infinitesimals, then in consistent claims from the history of the calculus might well simply be symptoms of confusion. This is addressed in Chapter 5. It is further argued that mathematics has a certain primacy over logic, in that paraconsistent or relevant logics have to be based on inconsistent mathematics. If the latter turns out to be reasonably rich then paraconsistentism is vindicated; while if inconsistent mathematics has seri ous restriytions then the case for being interested in inconsistency-tolerant logics is weakened. (On such restrictions, see this chapter, section 3. ) It must be conceded that fault-tolerant computer programming (e. g. Chapter 8) finds a substantial and important use for paraconsistent logics, albeit with an epistemological motivation (see this chapter, section 3). But even here it should be noted that if inconsistent mathematics turned out to be functionally impoverished then so would inconsistent databases. 2. Summary In Chapter 2, Meyer's results on relevant arithmetic are set out, and his view that they have a bearing on G8del's incompleteness theorems is discussed. Model theory for nonclassical logics is also set out so as to be able to show that the inconsistency of inconsistent theories can be controlled or limited, but in this book model theory is kept in the background as much as possible. This is then used to study the functional properties of various equational number theories."

This book is devoted to some results from the classical Point Set Theory and their applications to certain problems in mathematical analysis of the real line. Notice that various topics from this theory are presented in several books and surveys. From among the most important works devoted to Point Set Theory, let us first of all mention the excellent book by Oxtoby [83] in which a deep analogy between measure and category is discussed in detail. Further, an interesting general approach to problems concerning measure and category is developed in the well-known monograph by Morgan [79] where a fundamental concept of a category base is introduced and investigated. We also wish to mention that the monograph by Cichon, W";glorz and the author [19] has recently been published. In that book, certain classes of subsets of the real line are studied and various cardinal valued functions (characteristics) closely connected with those classes are investigated. Obviously, the IT-ideal of all Lebesgue measure zero subsets of the real line and the IT-ideal of all first category subsets of the same line are extensively studied in [19], and several relatively new results concerning this topic are presented. Finally, it is reasonable to notice here that some special sets of points, the so-called singular spaces, are considered in the classi

This book gives a rigorous yet 'physics-focused' introduction to mathematical logic that is geared towards natural science majors. We present the science major with a robust introduction to logic, focusing on the specific knowledge and skills that will unavoidably be needed in calculus topics and natural science topics in general (rather than taking a philosophical math fundamental oriented approach that is commonly found in mathematical logic textbooks).

Reasoning under uncertainty is always based on a specified language or for malism, including its particular syntax and semantics, but also on its associated inference mechanism. In the present volume of the handbook the last aspect, the algorithmic aspects of uncertainty calculi are presented. Theory has suffi ciently advanced to unfold some generally applicable fundamental structures and methods. On the other hand, particular features of specific formalisms and ap proaches to uncertainty of course still influence strongly the computational meth ods to be used. Both general as well as specific methods are included in this volume. Broadly speaking, symbolic or logical approaches to uncertainty and nu merical approaches are often distinguished. Although this distinction is somewhat misleading, it is used as a means to structure the present volume. This is even to some degree reflected in the two first chapters, which treat fundamental, general methods of computation in systems designed to represent uncertainty. It has been noted early by Shenoy and Shafer, that computations in different domains have an underlying common structure. Essentially pieces of knowledge or information are to be combined together and then focused on some particular question or domain. This can be captured in an algebraic structure called valuation algebra which is described in the first chapter. Here the basic operations of combination and focus ing (marginalization) of knowledge and information is modeled abstractly subject to simple axioms."

Goal Directed Proof Theory presents a uniform and coherent methodology for automated deduction in non-classical logics, the relevance of which to computer science is now widely acknowledged. The methodology is based on goal-directed provability. It is a generalization of the logic programming style of deduction, and it is particularly favourable for proof search. The methodology is applied for the first time in a uniform way to a wide range of non-classical systems, covering intuitionistic, intermediate, modal and substructural logics. The book can also be used as an introduction to these logical systems form a procedural perspective. Readership: Computer scientists, mathematicians and philosophers, and anyone interested in the automation of reasoning based on non-classical logics. The book is suitable for self study, its only prerequisite being some elementary knowledge of logic and proof theory.

This book is the final version of a course on algorithmic information theory and the epistemology of mathematics and physics. It discusses Einstein and Goedel's views on the nature of mathematics in the light of information theory, and sustains the thesis that mathematics is quasi-empirical. There is a foreword by Cris Calude of the University of Auckland, and supplementary material is available at the author's web site. The special feature of this book is that it presents a new "hands on" didatic approach using LISP and Mathematica software. The reader will be able to derive an understanding of the close relationship between mathematics and physics. "The Limits of Mathematics is a very personal and idiosyncratic account of Greg Chaitin's entire career in developing algorithmic information theory. The combination of the edited transcripts of his three introductory lectures maintains all the energy and content of the oral presentations, while the material on AIT itself gives a full explanation of how to implement Greg's ideas on real computers for those who want to try their hand at furthering the theory." (John Casti, Santa Fe Institute)

Logic and Complexity looks at basic logic as it is used in Computer Science, and provides students with a logical approach to Complexity theory. With plenty of exercises, this book presents classical notions of mathematical logic, such as decidability, completeness and incompleteness, as well as new ideas brought by complexity theory such as NP-completeness, randomness and approximations, providing a better understanding for efficient algorithmic solutions to problems.

Divided into three parts, it covers:

- Model Theory and Recursive Functions - introducing the basic model theory of propositional, 1st order, inductive definitions and 2nd order logic. Recursive functions, Turing computability and decidability are also examined.

- Descriptive Complexity - looking at the relationship between definitions of problems, queries, properties of programs and their computational complexity.

- Approximation - explaining how some optimization problems and counting problems can be approximated according to their logical form.

Logic is important in Computer Science, particularly for verification problems and database query languages such as SQL. Students and researchers in this field will find this book of great interest.

This book has been designed to deal with the topics which are indispensable in the advanced age of computer science. The first three chapters cover mathematical logic, sets, relations and function. Next come the chapters on ordered sets, Boolean albegra and switching circuits and matrices. Finally there are individual chapters on combinatorics, discrete numeric functions, generating functinos, recurrence relations, algebraic structures and graph theory; Graphs are binary trees. The purpose of this book is to present principles and concepts of discrete structures as relevant to student learning. The matter has been presented in as simple and lucid manner as possible and a large number of solved examples to understand the concept and principle of the theory have been introduced.

Alfred Tarski was one of the two giants of the twentieth-century development of logic, along with Kurt Goedel. The four volumes of this collection contain all of Tarski's published papers and abstracts, as well as a comprehensive bibliography. Here will be found many of the works, spanning the period 1921 through 1979, which are the bedrock of contemporary areas of logic, whether in mathematics or philosophy. These areas include the theory of truth in formalized languages, decision methods and undecidable theories, foundations of geometry, set theory, and model theory, algebraic logic, and universal algebra.

Since 1990 the German Research Society (Deutsche Forschungsgemeinschaft, DFG) has been funding PhD courses (Graduiertenkollegs) at selected universi- ties in the Federal Republic of Germany. TU Berlin has been one of the first universities joining that new funding program of DFG. The PhD courses have been funded over aperiod of 9 years. The grant for the nine years sums up to approximately 5 million DM. Our Grnduiertenkolleg on Communication-based Systems has been assigned to the Computer Science Department of TU Berlin although it is a joined effort of all three universities in Berlin, Technische Uni- versitat (TU), Freie Universitat (FU), and Humboldt Universitat (HU). The Graduiertenkolleg has been started its program in October 1991. The professors responsible for the program are: Hartmut Ehrig (TU), Gunter Hommel (TU), Stefan Jahnichen (TU), Peter Lohr (FU), Miroslaw Malek (RU), Peter Pep- per (TU), Radu Popescu-Zeletin (TU), Herbert Weber (TU), and Adam Wolisz (TU). The Graduiertenkolleg is a PhD program for highly qualified persons in the field of computer science. Twenty scholarships have been granted to fellows of the Graduiertenkolleg for a maximal period of three years. During this time the fellows take part in a selected educational program and work on their PhD thesis.

2. The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3. Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ., . . . . 60 4. Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 A Simple Proof for a Result of Ollerenshaw on Steiner Trees . . . . . . . . . . 68 Xiufeng Du, Ding-Zhu Du, Biao Gao, and Lixue Qii 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2. In the Euclidean Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3. In the Rectilinear Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4. Discussion . . . . . . . . . . . . -. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Optimization Algorithms for the Satisfiability (SAT) Problem . . . . . . . . . 72 Jun Gu 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2. A Classification of SAT Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7:3 3. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV 4. Complete Algorithms and Incomplete Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5. Optimization: An Iterative Refinement Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6. Local Search Algorithms for SAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 7. Global Optimization Algorithms for SAT Problem . . . . . . . . . . . . . . . . . . . . . . . . 106 8. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9. Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 10. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Ergodic Convergence in Proximal Point Algorithms with Bregman Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Osman Guier 1. Introduction . . .: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 2. Convergence for Function Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 3. Convergence for Arbitrary Maximal Monotone Operators . . . . . . . . . . . .