Computability Complexity Logic

Likewise the concept of M-enumerability or M-generability can be based directly on that of comput ability by declaring of a set x that X = { f(n); n < N), for an M-comput able function f, in other words X is the set of elements f(OX, ...

Author: E. Börger

Publisher: Elsevier

ISBN: 008088704X

Category: Mathematics

Page: 591

View: 331


The theme of this book is formed by a pair of concepts: the concept of formal language as carrier of the precise expression of meaning, facts and problems, and the concept of algorithm or calculus, i.e. a formally operating procedure for the solution of precisely described questions and problems. The book is a unified introduction to the modern theory of these concepts, to the way in which they developed first in mathematical logic and computability theory and later in automata theory, and to the theory of formal languages and complexity theory. Apart from considering the fundamental themes and classical aspects of these areas, the subject matter has been selected to give priority throughout to the new aspects of traditional questions, results and methods which have developed from the needs or knowledge of computer science and particularly of complexity theory. It is both a textbook for introductory courses in the above-mentioned disciplines as well as a monograph in which further results of new research are systematically presented and where an attempt is made to make explicit the connections and analogies between a variety of concepts and constructions.

Logical Foundations of Computer Science

Generalising slightly the notions of a strict computability model and of a simulation between them, which were elaborated by Longley and Normann in [9], we define canonical computability models over certain categories and appropriate ...

Author: Sergei Artemov

Publisher: Springer Nature

ISBN: 9783030931001

Category: Mathematics

Page: 377

View: 666


This book constitutes the refereed proceedings of the International Symposium on Logical Foundations of Computer Science, LFCS 2022, held in Deerfield Beach, FL, USA, in January 2022. The 23 revised full papers were carefully reviewed and selected from 35 submissions. The scope of the Symposium is broad and includes constructive mathematics and type theory; homotopy type theory; logic, automata, and automatic structures; computability and randomness; logical foundations of programming; logical aspects of computational complexity; parameterized complexity; logic programming and constraints; automated deduction and interactive theorem proving; logical methods in protocol and program verification; logical methods in program specification and extraction; domain theory logics; logical foundations of database theory; equational logic and term rewriting; lambda and combinatory calculi; categorical logic and topological semantics; linear logic; epistemic and temporal logics; intelligent and multiple-agent system logics; logics of proof and justification; non-monotonic reasoning; logic in game theory and social software; logic of hybrid systems; distributed system logics; mathematical fuzzy logic; system design logics; other logics in computer science.

Reverse Mathematics

One of our primary tools for studying the difficulty of producing a solution to a problem will be computability. The theory of computability is a field in its own right, of which we will need only certain pieces.

Author: Damir D. Dzhafarov

Publisher: Springer Nature

ISBN: 9783031113673

Category: Computers

Page: 498

View: 175


Reverse mathematics studies the complexity of proving mathematical theorems and solving mathematical problems. Typical questions include: Can we prove this result without first proving that one? Can a computer solve this problem? A highly active part of mathematical logic and computability theory, the subject offers beautiful results as well as significant foundational insights. This text provides a modern treatment of reverse mathematics that combines computability theoretic reductions and proofs in formal arithmetic to measure the complexity of theorems and problems from all areas of mathematics. It includes detailed introductions to techniques from computable mathematics, Weihrauch style analysis, and other parts of computability that have become integral to research in the field. Topics and features: Provides a complete introduction to reverse mathematics, including necessary background from computability theory, second order arithmetic, forcing, induction, and model construction Offers a comprehensive treatment of the reverse mathematics of combinatorics, including Ramsey's theorem, Hindman's theorem, and many other results Provides central results and methods from the past two decades, appearing in book form for the first time and including preservation techniques and applications of probabilistic arguments Includes a large number of exercises of varying levels of difficulty, supplementing each chapter The text will be accessible to students with a standard first year course in mathematical logic. It will also be a useful reference for researchers in reverse mathematics, computability theory, proof theory, and related areas. Damir D. Dzhafarov is an Associate Professor of Mathematics at the University of Connecticut, CT, USA. Carl Mummert is a Professor of Computer and Information Technology at Marshall University, WV, USA.

Sociative Logics and Their Applications Essays by the Late Richard Sylvan

Quantum computability, for example, diverges somewhat (disappointingly little sofar) from classical computability. It was eventhought, for a short time, that quantum computability might “solve the halting problem', a problem that in ...

Author: Dominic Hyde

Publisher: Routledge

ISBN: 9781351723725

Category: Social Science

Page: 441

View: 997


This title was first published in 2003. Richard Sylvan died in 1996, he had made contributions to many areas of philosophy, such as, relevant and paraconsistent logic, Meinongianism and metaphysics and environmental ethics. One of his "trademarks" was the taking up of unpopular views and defending them. To Richard Sylvan ideas were important, wether they were his or not. This is a book of ideas, based on a collection of work found after his death, a chance for readers to see his vision of his projects. This collected works represents material drafted between 1982 and 1996, and the theme is that a small band of logics, namely pararelevant logics, offer solutions to many problems, puzzles and paradoxes in the philosophy of science.

Logic and Scientific Methods

Computability via enumerations . Skordev's Combinatory spaces . Abstract Church Thesis . Computability in abstract structures of arbitrary power . Computability in multiple - valued structures . • Petio Petkov — Constructive mathematics ...

Author: Maria Luisa Dalla Chiara

Publisher: Springer Science & Business Media

ISBN: 0792343832

Category: Science

Page: 564

View: 705


This is the first of two volumes comprising the papers submitted for publication by the invited participants to the Tenth International Congress of Logic, Methodology and Philosophy of Science, held in Florence, August 1995. The Congress was held under the auspices of the International Union of History and Philosophy of Science, Division of Logic, Methodology and Philosophy of Science. The invited lectures published in the two volumes demonstrate much of what goes on in the fields of the Congress and give the state of the art of current research. The two volumes cover the traditional subdisciplines of mathematical logic and philosophical logic, as well as their interfaces with computer science, linguistics and philosophy. Philosophy of science is broadly represented, too, including general issues of natural sciences, social sciences and humanities. The papers in Volume One are concerned with logic, mathematical logic, the philosophy of logic and mathematics, and computer science.

Slicing the Truth

What is computability theory about? There is likely no single answer that will satisfy everyone working in the field, and many might not even consider questions at this level of generality and abstraction to be interesting, ...

Author: Denis R Hirschfeldt

Publisher: World Scientific

ISBN: 9789814612630

Category: Mathematics

Page: 232

View: 174


This book is a brief and focused introduction to the reverse mathematics and computability theory of combinatorial principles, an area of research which has seen a particular surge of activity in the last few years. It provides an overview of some fundamental ideas and techniques, and enough context to make it possible for students with at least a basic knowledge of computability theory and proof theory to appreciate the exciting advances currently happening in the area, and perhaps make contributions of their own. It adopts a case-study approach, using the study of versions of Ramsey's Theorem (for colorings of tuples of natural numbers) and related principles as illustrations of various aspects of computability theoretic and reverse mathematical analysis. This book contains many exercises and open questions. Contents:Setting Off: An IntroductionGathering Our Tools: Basic Concepts and NotationFinding Our Path: König's Lemma and ComputabilityGauging Our Strength: Reverse MathematicsIn Defense of DisarrayAchieving Consensus: Ramsey's TheoremPreserving Our Power: ConservativityDrawing a Map: Five DiagramsExploring Our Surroundings: The World Below RT22Charging Ahead: Further TopicsLagniappe: A Proof of Liu's Theorem Readership: Graduates and researchers in mathematical logic. Key Features:This book assumes minimal background in mathematical logic and takes the reader all the way to current research in a highly active areaIt is the first detailed introduction to this particular approach to this area of researchThe combination of fully worked out arguments and exercises make this book well suited to self-study by graduate students and other researchers unfamiliar with the areaKeywords:Reverse Mathematics;Computability Theory;Computable Mathematics;Computable Combinatorics

Theory and Applications of Models of Computation

4th International Conference, TAMC 2007, Shanghai, China, May 22-25, 2007, Proceedings Jin-Yi Cai, Barry S. Cooper, Hong Zhu. Computability on Subsets of Locally Compact Spaces Yatao Xu1 and Tanja Grubba2 1 Nanjing University, ...

Author: Jin-Yi Cai

Publisher: Springer

ISBN: 9783540725046

Category: Computers

Page: 772

View: 573


This book constitutes the refereed proceedings of the 4th International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007. It addresses all major areas in computer science; mathematics, especially logic; and the physical sciences, particularly with regard to computation and computability theory. The papers particularly focus on algorithms, complexity and computability theory.

Philosophy of Mathematics

The question whether the Church-Turing Thesis is a neo-Pythagoeanistic claim, ultimately depends on the definition of computability. If our conception about effective computability is merely limited to Turing mechanical principles, ...

Author: Ahmet Cevik

Publisher: CRC Press

ISBN: 9781000468809

Category: Mathematics

Page: 352

View: 358


The philosophy of mathematics is an exciting subject. Philosophy of Mathematics: Classic and Contemporary Studies explores the foundations of mathematical thought. The aim of this book is to encourage young mathematicians to think about the philosophical issues behind fundamental concepts and about different views on mathematical objects and mathematical knowledge. With this new approach, the author rekindles an interest in philosophical subjects surrounding the foundations of mathematics. He offers the mathematical motivations behind the topics under debate. He introduces various philosophical positions ranging from the classic views to more contemporary ones, including subjects which are more engaged with mathematical logic. Most books on philosophy of mathematics have little to no focus on the effects of philosophical views on mathematical practice, and no concern on giving crucial mathematical results and their philosophical relevance, consequences, reasons, etc. This book fills this gap. The book can be used as a textbook for a one-semester or even one-year course on philosophy of mathematics. "Other textbooks on the philosophy of mathematics are aimed at philosophers. This book is aimed at mathematicians. Since the author is a mathematician, it is a valuable addition to the literature." - Mark Balaguer, California State University, Los Angeles "There are not many such texts available for mathematics students. I applaud efforts to foster the dialogue between mathematics and philosophy." - Michele Friend, George Washington University and CNRS, Lille, France

From Sets and Types to Topology and Analysis

21 COMPUTABILITY ON NON-SEPARABLE BANACH SPACES AND LANDAU'S THEOREM VASCO BRATTKA Abstract While there is a well-established concept of a computable normed space in the separable case, one can prove that there is no way to represent ...

Author: Laura Crosilla

Publisher: Oxford University Press on Demand

ISBN: 9780198566519

Category: Mathematics

Page: 371

View: 679


Bridging the foundations and practice of constructive mathematics, this text focusses on the contrast between the theoretical developments - which have been most useful for computer science - and more specific efforts on constructive analysis, algebra and topology.

Introduction to the Philosophy of Science

The proposal is that , in view of the coherence of several conceptually very different approaches to computability in characterizing one and the same class functions , in view of the evidence Turing's work provides that the functions in ...

Author: Merrilee H. Salmon

Publisher: Hackett Publishing

ISBN: 0872204502

Category: Science

Page: 474

View: 192


Originally published: Englewood Cliffs, N.J.: Prentice Hall, c1992.