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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 comprehensive handbook contains 11 chapters grouped into parts on computability in analysis; complexity, dynamics, and randomness; and constructivity, logic, and descriptive complexity. Researchers and graduate students in the areas of theoretical computer science and mathematical logic will find systematic introductions into many branches of computable analysis, as well as a wealth of information and references that will help them to navigate the modern research literature in this field. Vasco Brattka is a professor for Theoretical Computer Science and Mathematical Logic at the Universität der Bundeswehr München. He is editor-in-chief of Computability, the journal of the association, Computability in Europe. Peter Hertling is a professor in the Institute for Theoretical Computer Science, Mathematics and Operations Research at UniBwM. He is an associate editor of Journal of Complexity.

Computers. --- Computer science—Mathematics. --- Mathematical logic. --- Theory of Computation. --- Mathematics of Computing. --- Mathematical Logic and Foundations. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Automatic computers --- Automatic data processors --- Computer hardware --- Computing machines (Computers) --- Electronic brains --- Electronic calculating-machines --- Electronic computers --- Hardware, Computer --- Computer systems --- Cybernetics --- Machine theory --- Calculators --- Cyberspace --- Computer science --- Mathematics. --- Computer mathematics --- Electronic data processing

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The workshop on Computability and Complexity in Analysis, CCA 2000, was hosted by the Department of Computer Science of the University of Wales Swansea, September 17{19, 2000. It was the fourth workshop in a successful series of workshops: CCA’95 in Hagen, Germany, CCA’96 in Trier, Germany, and CCA’98 in Brno, Czech Republic. About 40 participants from the countries United Kingdom, Germany, Japan, Italy, Russia, France, Denmark, Greece, and Ireland contributed to the success of this meeting. Altogether, 28 talkswere p- sented in Swansea. These proceedings include 23 papers which represent a cro- section through recent research on computability and complexity in analysis. The workshop succeeded in bringing together people interested in computability and complexity aspects of analysis and in exploring connections with nume- cal methods, physics and, of course, computer science. It was rounded o by a number of talks and papers on exact computer arithmetic and by a competition of v e implemented systems. A report on this competition has been included in these proceedings. We would like to thank the authors for their contributions and the referees for their careful work, and we hope for further inspiring and constructive meetings of the same kind. April 2001 Jens Blanck Vasco Brattka Peter Hertling Organization CCA2000was hosted by the Department of Computer Science of the University of Wales Swansea and took place on September 17{19, 2000.

Mathematical analysis --- Computable functions --- Computational complexity --- Applied Mathematics --- Engineering & Applied Sciences --- Computability theory --- Functions, Computable --- Partial recursive functions --- Recursive functions, Partial --- 517.1 Mathematical analysis --- Computer science. --- Computer programming. --- Computers. --- Algorithms. --- Mathematical logic. --- Database management. --- Computer Science. --- Computation by Abstract Devices. --- Database Management. --- Programming Techniques. --- Algorithm Analysis and Problem Complexity. --- Mathematical Logic and Formal Languages. --- Mathematical Logic and Foundations. --- Constructive mathematics --- Decidability (Mathematical logic) --- Computer software. --- Logic, Symbolic and mathematical. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Mathematics --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- Software, Computer --- Computer systems --- Data base management --- Data services (Database management) --- Database management services --- DBMS (Computer science) --- Generalized data management systems --- Services, Database management --- Systems, Database management --- Systems, Generalized database management --- Electronic data processing --- Informatics --- Science --- Algorism --- Algebra --- Arithmetic --- Computers --- Electronic computer programming --- Electronic digital computers --- Programming (Electronic computers) --- Coding theory --- Automatic computers --- Automatic data processors --- Computer hardware --- Computing machines (Computers) --- Electronic brains --- Electronic calculating-machines --- Electronic computers --- Hardware, Computer --- Cybernetics --- Machine theory --- Calculators --- Cyberspace --- Foundations --- Programming --- Mathematical analysis - Congresses --- Computable functions - Congresses --- Computational complexity - Congresses

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This book constitutes the refereed proceedings of the 9th Conference on Computability in Europe, CiE 2013, held in Milan, Italy, in July 2013. The 48 revised papers presented together with 1 invited lecture and 2 tutorials were carefully reviewed and selected with an acceptance rate of under 31,7%. Both the conference series and the association promote the development of computability-related science, ranging over mathematics, computer science and applications in various natural and engineering sciences such as physics and biology, and also including the promotion of related non-scientific fields such as philosophy and history of computing.

Mathematical logic --- Mathematical logic --- Logic --- Complex analysis --- Discrete mathematics --- Mathematics --- Computer science --- Computer architecture. Operating systems --- Computer. Automation --- complexiteit --- discrete wiskunde --- complexe analyse (wiskunde) --- bedrijfssoftware --- computers --- informatica --- ontwerpen --- programmeren (informatica) --- wiskunde --- logica --- logica --- computerkunde

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Mathematical logic --- Complex analysis --- Computer science --- Programming --- Information systems --- Artificial intelligence. Robotics. Simulation. Graphics --- Computer. Automation --- toegepaste informatica --- complexe analyse (wiskunde) --- informatica --- systeemontwikkeling (informatica) --- wiskunde --- KI (kunstmatige intelligentie) --- methodologieën --- logica --- robots

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Edited in collaboration with FoLLI, the Association of Logic, Language and Information, this volume constitutes a selection of papers presented at the Internatonal Conference on Infinity in Logic and Computation, ILC 2007, held in Cape Town, South Africa, in November 2007. The 7 revised papers presented together with 2 invited talks were carefully selected from 27 initial submissions during two rounds of reviewing and improvement. The papers address all aspects of infinity in automata theory, logic, computability and verification and focus on topics such as automata on infinite objects; combinatorics, cryptography and complexity; computability and complexity on the real numbers; infinite games and their connections to logic; logic, computability, and complexity in finitely presentable infinite structures; randomness and computability; transfinite computation; and verification of infinite state systems.

Mathematical logic --- Complex analysis --- Computer science --- Programming --- Information systems --- Artificial intelligence. Robotics. Simulation. Graphics --- Computer. Automation --- toegepaste informatica --- complexe analyse (wiskunde) --- informatica --- systeemontwikkeling (informatica) --- wiskunde --- KI (kunstmatige intelligentie) --- methodologieën --- logica --- robots