Author: E.R. Griffor
Publisher: Elsevier
ISBN: 9780080533049
Category : Mathematics
Languages : en
Pages : 724
Book Description
The chapters of this volume all have their own level of presentation. The topics have been chosen based on the active research interest associated with them. Since the interest in some topics is older than that in others, some presentations contain fundamental definitions and basic results while others relate very little of the elementary theory behind them and aim directly toward an exposition of advanced results. Presentations of the latter sort are in some cases restricted to a short survey of recent results (due to the complexity of the methods and proofs themselves). Hence the variation in level of presentation from chapter to chapter only reflects the conceptual situation itself. One example of this is the collective efforts to develop an acceptable theory of computation on the real numbers. The last two decades has seen at least two new definitions of effective operations on the real numbers.
Handbook of Computability Theory
Author: E.R. Griffor
Publisher: Elsevier
ISBN: 9780080533049
Category : Mathematics
Languages : en
Pages : 724
Book Description
The chapters of this volume all have their own level of presentation. The topics have been chosen based on the active research interest associated with them. Since the interest in some topics is older than that in others, some presentations contain fundamental definitions and basic results while others relate very little of the elementary theory behind them and aim directly toward an exposition of advanced results. Presentations of the latter sort are in some cases restricted to a short survey of recent results (due to the complexity of the methods and proofs themselves). Hence the variation in level of presentation from chapter to chapter only reflects the conceptual situation itself. One example of this is the collective efforts to develop an acceptable theory of computation on the real numbers. The last two decades has seen at least two new definitions of effective operations on the real numbers.
Publisher: Elsevier
ISBN: 9780080533049
Category : Mathematics
Languages : en
Pages : 724
Book Description
The chapters of this volume all have their own level of presentation. The topics have been chosen based on the active research interest associated with them. Since the interest in some topics is older than that in others, some presentations contain fundamental definitions and basic results while others relate very little of the elementary theory behind them and aim directly toward an exposition of advanced results. Presentations of the latter sort are in some cases restricted to a short survey of recent results (due to the complexity of the methods and proofs themselves). Hence the variation in level of presentation from chapter to chapter only reflects the conceptual situation itself. One example of this is the collective efforts to develop an acceptable theory of computation on the real numbers. The last two decades has seen at least two new definitions of effective operations on the real numbers.
Undecidable Theories
Author: Alfred Tarski
Publisher: Elsevier
ISBN: 0444533788
Category : Decidability (Mathematical logic)
Languages : en
Pages : 109
Book Description
Publisher: Elsevier
ISBN: 0444533788
Category : Decidability (Mathematical logic)
Languages : en
Pages : 109
Book Description
Recursive Model Theory
Author:
Publisher: Elsevier
ISBN: 9780080533698
Category : Computers
Languages : en
Pages : 619
Book Description
Recursive Model Theory
Publisher: Elsevier
ISBN: 9780080533698
Category : Computers
Languages : en
Pages : 619
Book Description
Recursive Model Theory
Structure of Decidable Locally Finite Varieties
Author: Ralph McKenzie
Publisher: Springer Science & Business Media
ISBN: 1461245524
Category : Mathematics
Languages : en
Pages : 209
Book Description
A mathematically precise definition of the intuitive notion of "algorithm" was implicit in Kurt Godel's [1931] paper on formally undecidable propo sitions of arithmetic. During the 1930s, in the work of such mathemati cians as Alonzo Church, Stephen Kleene, Barkley Rosser and Alfred Tarski, Godel's idea evolved into the concept of a recursive function. Church pro posed the thesis, generally accepted today, that an effective algorithm is the same thing as a procedure whose output is a recursive function of the input (suitably coded as an integer). With these concepts, it became possible to prove that many familiar theories are undecidable (or non-recursive)-i. e. , that there does not exist an effective algorithm (recursive function) which would allow one to determine which sentences belong to the theory. It was clear from the beginning that any theory with a rich enough mathematical content must be undecidable. On the other hand, some theories with a substantial content are decidable. Examples of such decidabLe theories are the theory of Boolean algebras (Tarski [1949]), the theory of Abelian groups (Szmiele~ [1955]), and the theories of elementary arithmetic and geometry (Tarski [1951]' but Tarski discovered these results around 1930). The de termination of precise lines of division between the classes of decidable and undecidable theories became an important goal of research in this area. algebra we mean simply any structure (A, h(i E I)} consisting of By an a nonvoid set A and a system of finitary operations Ii over A.
Publisher: Springer Science & Business Media
ISBN: 1461245524
Category : Mathematics
Languages : en
Pages : 209
Book Description
A mathematically precise definition of the intuitive notion of "algorithm" was implicit in Kurt Godel's [1931] paper on formally undecidable propo sitions of arithmetic. During the 1930s, in the work of such mathemati cians as Alonzo Church, Stephen Kleene, Barkley Rosser and Alfred Tarski, Godel's idea evolved into the concept of a recursive function. Church pro posed the thesis, generally accepted today, that an effective algorithm is the same thing as a procedure whose output is a recursive function of the input (suitably coded as an integer). With these concepts, it became possible to prove that many familiar theories are undecidable (or non-recursive)-i. e. , that there does not exist an effective algorithm (recursive function) which would allow one to determine which sentences belong to the theory. It was clear from the beginning that any theory with a rich enough mathematical content must be undecidable. On the other hand, some theories with a substantial content are decidable. Examples of such decidabLe theories are the theory of Boolean algebras (Tarski [1949]), the theory of Abelian groups (Szmiele~ [1955]), and the theories of elementary arithmetic and geometry (Tarski [1951]' but Tarski discovered these results around 1930). The de termination of precise lines of division between the classes of decidable and undecidable theories became an important goal of research in this area. algebra we mean simply any structure (A, h(i E I)} consisting of By an a nonvoid set A and a system of finitary operations Ii over A.
Computability Theory
Author: S. Barry Cooper
Publisher: CRC Press
ISBN: 1420057561
Category : Mathematics
Languages : en
Pages : 420
Book Description
Computability theory originated with the seminal work of Gödel, Church, Turing, Kleene and Post in the 1930s. This theory includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, and subrecursive hierarchy classifications. Recent work in computability theory has focused on Turing definability and promises to have far-reaching mathematical, scientific, and philosophical consequences. Written by a leading researcher, Computability Theory provides a concise, comprehensive, and authoritative introduction to contemporary computability theory, techniques, and results. The basic concepts and techniques of computability theory are placed in their historical, philosophical and logical context. This presentation is characterized by an unusual breadth of coverage and the inclusion of advanced topics not to be found elsewhere in the literature at this level. The book includes both the standard material for a first course in computability and more advanced looks at degree structures, forcing, priority methods, and determinacy. The final chapter explores a variety of computability applications to mathematics and science. Computability Theory is an invaluable text, reference, and guide to the direction of current research in the field. Nowhere else will you find the techniques and results of this beautiful and basic subject brought alive in such an approachable and lively way.
Publisher: CRC Press
ISBN: 1420057561
Category : Mathematics
Languages : en
Pages : 420
Book Description
Computability theory originated with the seminal work of Gödel, Church, Turing, Kleene and Post in the 1930s. This theory includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, and subrecursive hierarchy classifications. Recent work in computability theory has focused on Turing definability and promises to have far-reaching mathematical, scientific, and philosophical consequences. Written by a leading researcher, Computability Theory provides a concise, comprehensive, and authoritative introduction to contemporary computability theory, techniques, and results. The basic concepts and techniques of computability theory are placed in their historical, philosophical and logical context. This presentation is characterized by an unusual breadth of coverage and the inclusion of advanced topics not to be found elsewhere in the literature at this level. The book includes both the standard material for a first course in computability and more advanced looks at degree structures, forcing, priority methods, and determinacy. The final chapter explores a variety of computability applications to mathematics and science. Computability Theory is an invaluable text, reference, and guide to the direction of current research in the field. Nowhere else will you find the techniques and results of this beautiful and basic subject brought alive in such an approachable and lively way.
Decidable Theories
Author: J.R. Büchi
Publisher: Springer
ISBN: 9783540063452
Category : Mathematics
Languages : en
Pages : 232
Book Description
Publisher: Springer
ISBN: 9783540063452
Category : Mathematics
Languages : en
Pages : 232
Book Description
Decision Problems for Equational Theories of Relation Algebras
Author: H. Andréka
Publisher: American Mathematical Soc.
ISBN: 0821805959
Category : Mathematics
Languages : en
Pages : 126
Book Description
This work presents a systematic study of decision problems for equational theories of algebras of binary relations (relation algebras). For example, an easily applicable but deep method, based on von Neumann's coordinatization theorem, is developed for establishing undecidability results. The method is used to solve several outstanding problems posed by Tarski. In addition, the complexity of intervals of equational theories of relation algebras with respect to questions of decidability is investigated. Using ideas that go back to Jonsson and Lyndon, the authors show that such intervals can have the same complexity as the lattice of subsets of the set of the natural numbers. Finally, some new and quite interesting examples of decidable equational theories are given. The methods developed in the monograph show promise of broad applicability. They provide researchers in algebra and logic with a new arsenal of techniques for resolving decision questions in various domains of algebraic logic.
Publisher: American Mathematical Soc.
ISBN: 0821805959
Category : Mathematics
Languages : en
Pages : 126
Book Description
This work presents a systematic study of decision problems for equational theories of algebras of binary relations (relation algebras). For example, an easily applicable but deep method, based on von Neumann's coordinatization theorem, is developed for establishing undecidability results. The method is used to solve several outstanding problems posed by Tarski. In addition, the complexity of intervals of equational theories of relation algebras with respect to questions of decidability is investigated. Using ideas that go back to Jonsson and Lyndon, the authors show that such intervals can have the same complexity as the lattice of subsets of the set of the natural numbers. Finally, some new and quite interesting examples of decidable equational theories are given. The methods developed in the monograph show promise of broad applicability. They provide researchers in algebra and logic with a new arsenal of techniques for resolving decision questions in various domains of algebraic logic.
Theory Reasoning in Connection Calculi
Author: Peter Baumgartner
Publisher: Springer
ISBN: 3540492100
Category : Technology & Engineering
Languages : en
Pages : 290
Book Description
The ability to draw inferences is a central operation in any artificial intelligence system. Automated reasoning is therefore among the traditional disciplines in AI. Theory reasoning is about techniques for combining automated reasoning systems with specialized and efficient modules for handling domain knowledge called background reasoners. Connection methods have proved to be a good choice for implementing high-speed automated reasoning systems. They are the starting point in this monograph,in which several theory reasoning versions are defined and related to each other. A major contribution of the book is a new technique of linear completion allowing for the automatic construction of background reasoners from a wide range of axiomatically given theories. The emphasis is on theoretical investigations, but implementation techniques based on Prolog are also covered.
Publisher: Springer
ISBN: 3540492100
Category : Technology & Engineering
Languages : en
Pages : 290
Book Description
The ability to draw inferences is a central operation in any artificial intelligence system. Automated reasoning is therefore among the traditional disciplines in AI. Theory reasoning is about techniques for combining automated reasoning systems with specialized and efficient modules for handling domain knowledge called background reasoners. Connection methods have proved to be a good choice for implementing high-speed automated reasoning systems. They are the starting point in this monograph,in which several theory reasoning versions are defined and related to each other. A major contribution of the book is a new technique of linear completion allowing for the automatic construction of background reasoners from a wide range of axiomatically given theories. The emphasis is on theoretical investigations, but implementation techniques based on Prolog are also covered.
Models and Computability
Author: S. Barry Cooper
Publisher: Cambridge University Press
ISBN: 0521635500
Category : Computers
Languages : en
Pages : 433
Book Description
Second of two volumes providing a comprehensive guide to the current state of mathematical logic.
Publisher: Cambridge University Press
ISBN: 0521635500
Category : Computers
Languages : en
Pages : 433
Book Description
Second of two volumes providing a comprehensive guide to the current state of mathematical logic.
Theory of Computation
Author: D. P. Acharjya
Publisher: MJP Publisher
ISBN:
Category : Science
Languages : en
Pages : 416
Book Description
Theory of computation is the scientific discipline concerned with the study of general properties of computation and studies the inherent possibilities and limitations of efficient computation that makes machines more intelligent and enables them to carry out intellectual processes. This book deals with all those concepts by developing the standard mathematical models of computational devices, and by investigating the cognitive and generative capabilities of such machines. The book emphasizes on mathematical reasoning and problem-solving techniques that penetrate computer science. Each chapter gives a clear statement of definition and thoroughly discusses the concepts, principles and theorems with illustrative and other descriptive materials.
Publisher: MJP Publisher
ISBN:
Category : Science
Languages : en
Pages : 416
Book Description
Theory of computation is the scientific discipline concerned with the study of general properties of computation and studies the inherent possibilities and limitations of efficient computation that makes machines more intelligent and enables them to carry out intellectual processes. This book deals with all those concepts by developing the standard mathematical models of computational devices, and by investigating the cognitive and generative capabilities of such machines. The book emphasizes on mathematical reasoning and problem-solving techniques that penetrate computer science. Each chapter gives a clear statement of definition and thoroughly discusses the concepts, principles and theorems with illustrative and other descriptive materials.