This book constitutes the refereed proceedings of the 11th Conference on Computability in Europe, CiE 2015, held in Bucharest, Romania, in June/July 2015. The 26 revised papers presented were carefully reviewed and selected from 64 submissions and included together with 10 invited papers in this proceedings. The conference CiE 2015 has six special sessions: two sessions, Representing Streams and Reverse Mathematics, were introduced for the first time in the conference series. In addition to this, new developments in areas frequently covered in the CiE conference series were addressed in the further special sessions on Automata, Logic and Infinite Games; Bio-inspired Computation; Classical Computability Theory; as well as History and Philosophy of Computing.

Computable analysis is the modern theory of computability and complexity in analysis that arose out of Turing's seminal work in the 1930s. This was motivated by questions such as: which real numbers and real number functions are computable, and which mathematical tasks in analysis can be solved by algorithmic means? Nowadays this theory has many different facets that embrace topics from computability theory, algorithmic randomness, computational complexity, dynamical systems, fractals, and analog computers, up to logic, descriptive set theory, constructivism, and reverse mathematics. In recent decades computable analysis has invaded many branches of analysis, and researchers have studied computability and complexity questions arising from real and complex analysis, functional analysis, and the theory of differential equations, up to (geometric) measure theory and topology. This handbook represents the first coherent cross-section through most active research topics on the more theoretical side of the field. It contains 11 chapters grouped into parts on computability in analysis; complexity, dynamics, and randomness; and constructivity, logic, and descriptive complexity. All chapters are written by leading experts working at the cutting edge of the respective topic. Researchers and graduate students in the areas of theoretical computer science and mathematical logic will find systematic introductions into many branches of computable analysis, and a wealth of information and references that will help them to navigate the modern research literature in this field.

This book constitutes the proceedings of the 16th Conference on Computability in Europe, CiE 2020, which was planned to be held in Fisciano, Italy, during June 29 until July 3, 2020. The conference moved to a virtual format due to the coronavirus pandemic. The 30 full and 5 short papers presented in this volume were carefully reviewed and selected from 72 submissions. CiE promotes the development of computability-related science, ranging over mathematics, computer science and applications in various natural and engineering sciences, such as physics and biology, as well as related fields, such as philosophy and history of computing. CiE 2020 had as its motto Beyond the Horizon of Computability, reflecting the interest of CiE in research transgressing the traditional boundaries of computability theory.

Ordinal Computability discusses models of computation obtained by generalizing classical models, such as Turing machines or register machines, to transfinite working time and space. In particular, recognizability, randomness, and applications to other areas of mathematics are covered.

After a decade's development, evolutionary computation (EC) proves to be a powerful tool kit for economic analysis. While the demand for this equipment is increasing, there is no volume exclusively written for economists. This volume for the first time helps economists to get a quick grasp on how EC may support their research. A comprehensive coverage of the subject is given, that includes the following three areas: game theory, agent-based economic modelling and financial engineering. Twenty leading scholars from each of these areas contribute a chapter to the volume. The reader will find himself treading the path of the history of this research area, from the fledgling stage to the burgeoning era. The results on games, labour markets, pollution control, institution and productivity, financial markets, trading systems design and derivative pricing, are new and interesting for different target groups. The book also includes informations on web sites, conferences, and computer software.

This book constitutes the refereed proceedings of the 4th International Conference on Computability in Europe, CiE 2008, held in Athens, Greece, in June 2008. The 36 revised full papers presented together with 25 invited tutorials and lectures were carefully reviewed and selected from 108 submissions. Among them are papers of 6 special sessions entitled algorithms in the history of mathematics, formalising mathematics and extracting algorithms from proofs, higher-type recursion and applications, algorithmic game theory, quantum algorithms and complexity, and biology and computation.

This handbook offers a comprehensive treatise on Grammatical Evolution (GE), a grammar-based Evolutionary Algorithm that employs a function to map binary strings into higher-level structures such as programs. GE's simplicity and modular nature make it a very flexible tool. Since its introduction almost twenty years ago, researchers have applied it to a vast range of problem domains, including financial modelling, parallel programming and genetics. Similarly, much work has been conducted to exploit and understand the nature of its mapping scheme, triggering additional research on everything from different grammars to alternative mappers to initialization. The book first introduces GE to the novice, providing a thorough description of GE along with historical key advances. Two sections follow, each composed of chapters from international leading researchers in the field. The first section concentrates on analysis of GE and its operation, giving valuable insight into set up and deployment. The second section consists of seven chapters describing radically different applications of GE. The contributions in this volume are beneficial to both novices and experts alike, as they detail the results and researcher experiences of applying GE to large scale and difficult problems. Topics include: • Grammar design • Bias in GE • Mapping in GE • Theory of disruption in GE · Structured GE · Geometric semantic GE · GE and semantics · Multi- and Many-core heterogeneous parallel GE · Comparing methods to creating constants in GE · Financial modelling with GE · Synthesis of parallel programs on multi-cores · Design, architecture and engineering with GE · Computational creativity and GE · GE in the prediction of glucose for diabetes · GE approaches to bioinformatics and system genomics · GE with coevolutionary algorithms in cybersecurity · Evolving behaviour trees with GE for platform games · Business analytics and GE for the prediction of patient recruitment in multicentre clinical trials

This volume, with a foreword by Sir Roger Penrose, discusses the foundations of computation in relation to nature.It focuses on two main questions: What is computation? How does nature compute?The contributors are world-renowned experts who have helped shape a cutting-edge computational understanding of the universe. They discuss computation in the world from a variety of perspectives, ranging from foundational concepts to pragmatic models to ontological conceptions and philosophical implications.The volume provides a state-of-the-art collection of technical papers and non-technical essays, representing a field that assumes information and computation to be key in understanding and explaining the basic structure underpinning physical reality. It also includes a new edition of Konrad Zuse''s OC Calculating SpaceOCO (the MIT translation), and a panel discussion transcription on the topic, featuring worldwide experts in quantum mechanics, physics, cognition, computation and algorithmic complexity.The volume is dedicated to the memory of Alan M Turing OCo the inventor of universal computation, on the 100th anniversary of his birth, and is part of the Turing Centenary celebrations.

This collaborative book presents recent trends on the study of sequences, including combinatorics on words and symbolic dynamics, and new interdisciplinary links to group theory and number theory. Other chapters branch out from those areas into subfields of theoretical computer science, such as complexity theory and theory of automata. The book is built around four general themes: number theory and sequences, word combinatorics, normal numbers, and group theory. Those topics are rounded out by investigations into automatic and regular sequences, tilings and theory of computation, discrete dynamical systems, ergodic theory, numeration systems, automaton semigroups, and amenable groups. This volume is intended for use by graduate students or research mathematicians, as well as computer scientists who are working in automata theory and formal language theory. With its organization around unified themes, it would also be appropriate as a supplemental text for graduate level courses.

Automatic sequences are sequences over a finite alphabet generated by a finite-state machine. This book presents a novel viewpoint on automatic sequences, and more generally on combinatorics on words, by introducing a decision method through which many new results in combinatorics and number theory can be automatically proved or disproved with little or no human intervention. This approach to proving theorems is extremely powerful, allowing long and error-prone case-based arguments to be replaced by simple computations. Readers will learn how to phrase their desired results in first-order logic, using free software to automate the computation process. Results that normally require multipage proofs can emerge in milliseconds, allowing users to engage with mathematical questions that would otherwise be difficult to solve. With more than 150 exercises included, this text is an ideal resource for researchers, graduate students, and advanced undergraduates studying combinatorics, sequences, and number theory.