Search results for: computability-in-analysis-and-physics

Computability in Analysis and Physics

Author : Marian B. Pour-El
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Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the first publication in the Perspectives in Logic series, Pour-El and Richards present the first graduate-level treatment of computable analysis within the tradition of classical mathematical reasoning. The book focuses on the computability or noncomputability of standard processes in analysis and physics. Topics include classical analysis, Hilbert and Banach spaces, bounded and unbounded linear operators, eigenvalues, eigenvectors, and equations of mathematical physics. The work is self-contained, and although it is intended primarily for logicians and analysts, it should also be of interest to researchers and graduate students in physics and computer science.

Computability in analysis and physics

Author : Marian Boykan Pour-El
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New Computational Paradigms

Author : S.B. Cooper
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This superb exposition of a complex subject examines new developments in the theory and practice of computation from a mathematical perspective, with topics ranging from classical computability to complexity, from biocomputing to quantum computing. This book is suitable for researchers and graduate students in mathematics, philosophy, and computer science with a special interest in logic and foundational issues. Most useful to graduate students are the survey papers on computable analysis and biological computing. Logicians and theoretical physicists will also benefit from this book.

Computable Analysis

Author : Klaus Weihrauch
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Merging fundamental concepts of analysis and recursion theory to a new exciting theory, this book provides a solid fundament for studying various aspects of computability and complexity in analysis. It is the result of an introductory course given for several years and is written in a style suitable for graduate-level and senior students in computer science and mathematics. Many examples illustrate the new concepts while numerous exercises of varying difficulty extend the material and stimulate readers to work actively on the text.

Randomness and Undecidability in Physics

Author : K Svozil
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Recent findings in the computer sciences, discrete mathematics, formal logics and metamathematics have opened up a royal road for the investigation of undecidability and randomness in physics. A translation of these formal concepts yields a fresh look into diverse features of physical modelling such as quantum complementarity and the measurement problem, but also stipulates questions related to the necessity of the assumption of continua. Conversely, any computer may be perceived as a physical system: not only in the immediate sense of the physical properties of its hardware. Computers are a medium to virtual realities. The foreseeable importance of such virtual realities stimulates the investigation of an “inner description”, a “virtual physics” of these universes of computation. Indeed, one may consider our own universe as just one particular realisation of an enormous number of virtual realities, most of them awaiting discovery. One motive of this book is the recognition that what is often referred to as “randomness” in physics might actually be a signature of undecidability for systems whose evolution is computable on a step-by-step basis. To give a flavour of the type of questions envisaged: Consider an arbitrary algorithmic system which is computable on a step-by-step basis. Then it is in general impossible to specify a second algorithmic procedure, including itself, which, by experimental input-output analysis, is capable of finding the deterministic law of the first system. But even if such a law is specified beforehand, it is in general impossible to predict the system behaviour in the “distant future”. In other words: no “speedup” or “computational shortcut” is available. In this approach, classical paradoxes can be formally translated into no-go theorems concerning intrinsic physical perception. It is suggested that complementarity can be modelled by experiments on finite automata, where measurements of one observable of the automaton destroys the possibility to measure another observable of the same automaton and it vice versa. Besides undecidability, a great part of the book is dedicated to a formal definition of randomness and entropy measures based on algorithmic information theory. Contents:Algorithmic Physics: The Universe as a ComputerAlgorithmics and Recursive Function TheoryMechanism and DeterminismDiscrete PhysicsSource CodingLattice TheoryExtrinsic-Intrinsic ConceptAlgorithmic InformationComputational ComplexityUndecidabilityClassical ResultsComplementarityExtrinsic IndeterminismIntrinsic IndeterminismWeak Physical ChaosRandomnessRandomness in MathematicsRandom Fractals and 1/f NoiseChaotic Systems are Optimal Analogues of ThemselvesQuantum ChaosAlgorithmic EntropyEpilogue: Afterthoughts, Speculations and Metaphysics Readership: Physicists, computer scientists and mathematicians. keywords:Undecidability;Randomness;Algorithmic Information Theory;Automaton Logic;Quantum Logic;Recursive Function Theory;Discrete Physics;Lattice Theory;Intrinsic Perception;Halting ProblemCristian Calude

Computability and Complexity in Analysis

Author : Jens Blanck
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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.

New Computational Paradigms

Author : Barry S. Cooper
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This book constitutes the refereed proceedings of the first International Conference on Computability in Europe, CiE 2005, held in Amsterdam, The Netherlands in June 2005. The 68 revised full papers presented were carefully reviewed and selected from 144 submissions. Among them are papers corresponding to two tutorials, six plenary talks and papers of six special sessions involving mathematical logic and computer science at the same time as offering the methodological foundations for models of computation. The papers address many aspects of computability in Europe with a special focus on new computational paradigms. These include first of all connections between computation and physical systems (e.g., quantum and analog computation, neural nets, molecular computation), but also cover new perspectives on models of computation arising from basic research in mathematical logic and theoretical computer science.

PI 0 1 Classes in Computable Analysis and Topology

Author : Joseph Stephen Miller
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We explore aspects of 10 classes in Rn . These are the effective closed sets of computable analysis and natural analogs of the 10 classes in 2o, widely studied by computability theorists. In Chapter II, we characterize the fixable classes--the sets of fixed point of computable maps from the unit cube [0,1] n to itself--as the 10, classes which contain a nonempty, connected 10 subclass. This settles a question asked in [CJ00]. To prove that Brouwer's theorem is inconsistent with Russian constructivism, Orevkov gave a fixable class with no computable points [Ore63]. Our proof employs a generalization of Orevkov's construction, as well as the notion of topological degree . Homology theory is used in the definition and computation of the topological degree. Homology returns in Chapter III, where chains are used to take algorithmic advantage of the topological structure of a 10, class. We show that a 10 class homeomorphic to a sphere is located: the distance to the class is computable. Closed balls embedded as 10 classes are also studied. Chapter IV studies members of 10 classes which contain no computable points. These avoidable points were introduced by Kalantari and Welch [KW]. Avoidability is a type of effective non-computability; we introduce hyperavoidability, a stronger notion, and initiate the computability theoretic study of both classes, including their behavior in the Turing and weak truth-table degrees.

Models of Computation

Author : Maribel Fernandez
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A Concise Introduction to Computation Models and Computability Theory provides an introduction to the essential concepts in computability, using several models of computation, from the standard Turing Machines and Recursive Functions, to the modern computation models inspired by quantum physics. An in-depth analysis of the basic concepts underlying each model of computation is provided. Divided into two parts, the first highlights the traditional computation models used in the first studies on computability: - Automata and Turing Machines; - Recursive functions and the Lambda-Calculus; - Logic-based computation models. and the second part covers object-oriented and interaction-based models. There is also a chapter on concurrency, and a final chapter on emergent computation models inspired by quantum mechanics. At the end of each chapter there is a discussion on the use of computation models in the design of programming languages.

International mathematical news

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Proceedings

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Computability in Context

Author : S. Barry Cooper
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Addresses various ways computability and theoretical computer science enable scientists and philosophers to deal with mathematical and real-world issues. This book covers problems related to logic, mathematics, physical processes, real computation and learning theory.

Programs Proofs Processes

Author : Fernando Ferreira
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This book constitutes the refereed proceedings of the 6th Conference on Computability in Europe, CiE 2010, held in Ponta Delgada, Azores, Portugal, in June/July 2010. The 28 revised papers presented together with 20 invited lectures were carefully reviewed and selected from 90 submissions. The papers address not only the more established lines of research of computational complexity and the interplay between proofs and computation, but also novel views that rely on physical and biological processes and models to find new ways of tackling computations and improving their efficiency.

Computation and Logic in the Real World

Author : S. Barry Cooper
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CiE2007:ComputationandLogicintheRealWorld Siena,Italy,June18-23,2007 Computability in Europe (CiE) is an informal network of European scientists working on computability theory, including its foundations, technical devel- ment, and applications. Among the aims of the network is to advance our t- oretical understanding of what can and cannot be computed, by any means of computation. Its scienti'c vision is broad: computations may be performed with discrete or continuous data by all kinds of algorithms, programs, and - chines. Computations may be made by experimenting with any sort of physical system obeying the laws of a physical theory such as Newtonian mechanics, quantum theory, or relativity. Computations may be very general, depending upon the foundations of set theory; or very speci'c, using the combinatorics of ?nite structures. CiE also works on subjects intimately related to computation, especially theories of data and information, and methods for formal reasoning about computations. The sources of new ideas and methods include practical developments in areas such as neural networks, quantum computation, natural computation, molecular computation, computational learning. Applications are everywhere,especially, in algebra,analysisand geometry, or data types and p- gramming. Within CiE there is general recognition of the underlying relevance of computability to physics and a broad range of other sciences, providing as it does a basic analysis of the causal structure of dynamical systems.

Beyond the Horizon of Computability

Author : Marcella Anselmo
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Logic in Computer Science

Author : IEEE Computer Society. Technical Committee on Mathematical Foundations of Computing
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This text contains information on computational mathematics and complexity theory as presented at the 17th Annual IEEE Symposium on Logic in Computer Science.

Physics of Electric Propulsion

Author : Robert G. Jahn
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Geared toward advanced undergraduates and graduate students, this text develops the concepts of electrical acceleration of gases for propulsion, from primary physical principles to realistic space thruster designs. 1968 edition.

Reaction Diffusion Automata Phenomenology Localisations Computation

Author : Andrew Adamatzky
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Reaction-diffusion and excitable media are amongst most intriguing substrates. Despite apparent simplicity of the physical processes involved the media exhibit a wide range of amazing patterns: from target and spiral waves to travelling localisations and stationary breathing patterns. These media are at the heart of most natural processes, including morphogenesis of living beings, geological formations, nervous and muscular activity, and socio-economic developments. This book explores a minimalist paradigm of studying reaction-diffusion and excitable media using locally-connected networks of finite-state machines: cellular automata and automata on proximity graphs. Cellular automata are marvellous objects per se because they show us how to generate and manage complexity using very simple rules of dynamical transitions. When combined with the reaction-diffusion paradigm the cellular automata become an essential user-friendly tool for modelling natural systems and designing future and emergent computing architectures. The book brings together hot topics of non-linear sciences, complexity, and future and emergent computing. It shows how to discover propagating localisation and perform computation with them in very simple two-dimensional automaton models. Paradigms, models and implementations presented in the book strengthen the theoretical foundations in the area for future and emergent computing and lay key stones towards physical embodied information processing systems.

Logic and Theory of Algorithms

Author : Arnold Beckmann
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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.

Mathematical Theory and Computational Practice

Author : Klaus Ambos-Spies
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This book constitutes the proceedings of the 5th Conference on Computability in Europe, CiE 2009, held in Heidelberg, Germany, during July 19-24, 2009. The 34 papers presented together with 17 invited lectures were carefully reviewed and selected from 100 submissions. The aims of the conference is to advance our theoretical understanding of what can and cannot be computed, by any means of computation. It is the largest international meeting focused on computability theoretic issues.