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Author: Pavel Pudlák
Publisher: Springer Science & Business Media
ISBN: 3319001191
Category : Mathematics
Languages : en
Pages : 699
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Book Description
The two main themes of this book, logic and complexity, are both essential for understanding the main problems about the foundations of mathematics. Logical Foundations of Mathematics and Computational Complexity covers a broad spectrum of results in logic and set theory that are relevant to the foundations, as well as the results in computational complexity and the interdisciplinary area of proof complexity. The author presents his ideas on how these areas are connected, what are the most fundamental problems and how they should be approached. In particular, he argues that complexity is as important for foundations as are the more traditional concepts of computability and provability. Emphasis is on explaining the essence of concepts and the ideas of proofs, rather than presenting precise formal statements and full proofs. Each section starts with concepts and results easily explained, and gradually proceeds to more difficult ones. The notes after each section present some formal definitions, theorems and proofs. Logical Foundations of Mathematics and Computational Complexity is aimed at graduate students of all fields of mathematics who are interested in logic, complexity and foundations. It will also be of interest for both physicists and philosophers who are curious to learn the basics of logic and complexity theory.
Author: Pavel Pudlák
Publisher: Springer Science & Business Media
ISBN: 3319001191
Category : Mathematics
Languages : en
Pages : 699
Get Book Here
Book Description
The two main themes of this book, logic and complexity, are both essential for understanding the main problems about the foundations of mathematics. Logical Foundations of Mathematics and Computational Complexity covers a broad spectrum of results in logic and set theory that are relevant to the foundations, as well as the results in computational complexity and the interdisciplinary area of proof complexity. The author presents his ideas on how these areas are connected, what are the most fundamental problems and how they should be approached. In particular, he argues that complexity is as important for foundations as are the more traditional concepts of computability and provability. Emphasis is on explaining the essence of concepts and the ideas of proofs, rather than presenting precise formal statements and full proofs. Each section starts with concepts and results easily explained, and gradually proceeds to more difficult ones. The notes after each section present some formal definitions, theorems and proofs. Logical Foundations of Mathematics and Computational Complexity is aimed at graduate students of all fields of mathematics who are interested in logic, complexity and foundations. It will also be of interest for both physicists and philosophers who are curious to learn the basics of logic and complexity theory.
Author: Jan Krajicek
Publisher: Cambridge University Press
ISBN: 0521452058
Category : Computers
Languages : en
Pages : 361
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Book Description
Discusses the deep connections between logic and complexity theory, and lists a number of intriguing open problems.
Author: Neil Immerman
Publisher: Springer Science & Business Media
ISBN: 1461205395
Category : Computers
Languages : en
Pages : 275
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Book Description
By virtue of the close relationship between logic and relational databases, it turns out that complexity has important applications to databases such as analyzing the parallel time needed to compute a query, and the analysis of nondeterministic classes. This book is a relatively self-contained introduction to the subject, which includes the necessary background material, as well as numerous examples and exercises.
Author: Howard Straubing
Publisher: Springer Science & Business Media
ISBN: 1461202892
Category : Computers
Languages : en
Pages : 235
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Book Description
The study of the connections between mathematical automata and for mal logic is as old as theoretical computer science itself. In the founding paper of the subject, published in 1936, Turing showed how to describe the behavior of a universal computing machine with a formula of first order predicate logic, and thereby concluded that there is no algorithm for deciding the validity of sentences in this logic. Research on the log ical aspects of the theory of finite-state automata, which is the subject of this book, began in the early 1960's with the work of J. Richard Biichi on monadic second-order logic. Biichi's investigations were extended in several directions. One of these, explored by McNaughton and Papert in their 1971 monograph Counter-free Automata, was the characterization of automata that admit first-order behavioral descriptions, in terms of the semigroup theoretic approach to automata that had recently been developed in the work of Krohn and Rhodes and of Schiitzenberger. In the more than twenty years that have passed since the appearance of McNaughton and Papert's book, the underlying semigroup theory has grown enor mously, permitting a considerable extension of their results. During the same period, however, fundamental investigations in the theory of finite automata by and large fell out of fashion in the theoretical com puter science community, which moved to other concerns.
Author: Stephen Cook
Publisher: Cambridge University Press
ISBN: 9781107694118
Category : Mathematics
Languages : en
Pages : 0
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Book Description
This book treats bounded arithmetic and propositional proof complexity from the point of view of computational complexity. The first seven chapters include the necessary logical background for the material and are suitable for a graduate course. Associated with each of many complexity classes are both a two-sorted predicate calculus theory, with induction restricted to concepts in the class, and a propositional proof system. The result is a uniform treatment of many systems in the literature, including Buss's theories for the polynomial hierarchy and many disparate systems for complexity classes such as AC0, AC0(m), TC0, NC1, L, NL, NC, and P.
Author: E. Börger
Publisher: Elsevier
ISBN: 008088704X
Category : Computers
Languages : en
Pages : 618
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Book Description
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.
Author: Jean-Yves Girard
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 516
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Book Description
"This long awaited book ... fills essential gaps in monographic literature on proof theory and prepares readers for volume 2 (to be published soon) containing an exposition of the author's new approach to proof theory for higher order logic. Even in traditional topics, like Gödel's completeness and incompleteness theorems, and cut elemination, accents are different compared to books by Kleene, Schütte, or Takeuti, which are strongly influenced by Hilbert's aim: to make mathematical theories (number theory, analysis etc.) more reliable by transformations of formalized proofs. The author is much closer to the approach of G. Kreisel (to whom this book is dedicated): Hilbert's program needs drastic rethinking and one of the main tasks is in finding mathematical applications of the results obtained in proof theory. Possibly, it is not a pure chance that the system of second order functionals developed by the author in his normalization proof for second order logic (was rediscovered and) became a tool in computer science. The book under review presents not only this material, but also other results by the author which became a part of modern proof theory including analysis of cut-free provability in terms of 3-valued logic. The material which was not previously covered (at least in such detail) in proof-theoretic monographs includes strong normalizability proofs (after Tait and Gandy), applications of reflection principles, recursive ordinals, operations on local correct (but not necessarily well-founded) omega-derivations, no-counterexample interpretation, using proof theory to extract combinatory estimates with a detailed treatment of van der Waerden's theorem. This is a difficult, but rewarding postgraduate-level textbook. The author does not avoid philosophical questions, and such discussion supported by theorems is certainly fruitful, although the reviewer would not agree with all author's conclusions"-- description of volume 1.
Author: Ivo Düntsch
Publisher: Springer Nature
ISBN: 3030714306
Category : Philosophy
Languages : en
Pages : 591
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Book Description
This book is dedicated to the work of Alasdair Urquhart. The book starts out with an introduction to and an overview of Urquhart’s work, and an autobiographical essay by Urquhart. This introductory section is followed by papers on algebraic logic and lattice theory, papers on the complexity of proofs, and papers on philosophical logic and history of logic. The final section of the book contains a response to the papers by Urquhart. Alasdair Urquhart has made extremely important contributions to a variety of fields in logic. He produced some of the earliest work on the semantics of relevant logic. He provided the undecidability of the logics R (of relevant implication) and E (of relevant entailment), as well as some of their close neighbors. He proved that interpolation fails in some of those systems. Urquhart has done very important work in complexity theory, both about the complexity of proofs in classical and some nonclassical logics. In pure algebra, he has produced a representation theorem for lattices and some rather beautiful duality theorems. In addition, he has done important work in the history of logic, especially on Bertrand Russell, including editing Volume four of Russell’s Collected Papers.
Author: Sanjeev Arora
Publisher: Cambridge University Press
ISBN: 0521424267
Category : Computers
Languages : en
Pages : 609
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Book Description
New and classical results in computational complexity, including interactive proofs, PCP, derandomization, and quantum computation. Ideal for graduate students.
Author: Richmond H. Thomason
Publisher: Springer
ISBN: 0792304152
Category : Philosophy
Languages : en
Pages : 222
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Book Description
cians concerned with using logical tools in philosophy have been keenly aware of the limitations that arise from the original con centration of symbolic logic on the idiom of mathematics, and many of them have worked to create extensions of the received logical theories that would make them more generally applicable in philosophy. Carnap's Testability and Meaning, published in 1936 and 1937, was a good early example of this sort of research, motivated by the inadequacy of first-order formalizations of dis 'This sugar cube is soluble in water'. positional sentences like And in fact there is a continuous history of work on this topic, extending from Carnap's paper to Shoham's contribution to the present volume . . Much of the work in philosophical logic, and much of what has appeared in The Journal of Philosophical Logic, was mo tivated by similar considerations: work in modal logic (includ ing tense, deontic, and epistemic logic), intensional logics, non declaratives, presuppositions, and many other topics. In this sort of research, sin.ce the main point is to devise new formalisms, the technical development tends to be rather shallow in comparison with mathematical logic, though it is sel dom absent: theorems need to be proved in order to justify the formalisms, and sometimes these are nontrivial. On the other hand, much effort has to go into motivating a logical innovation.
