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Proof Theory And Logical Complexity
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Book Synopsis Proof Theory and Logical Complexity by : Jean-Yves Girard
Download or read book Proof Theory and Logical Complexity written by Jean-Yves Girard and published by . This book was released on 1987 with total page 516 pages. Available in PDF, EPUB and Kindle. Book excerpt: "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.
Book Synopsis Proof Theory and Logical Complexity by : Jean-Yves Girard
Download or read book Proof Theory and Logical Complexity written by Jean-Yves Girard and published by Elsevier Science & Technology. This book was released on 1987 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: "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.
Book Synopsis Proof Theory and Logical Complexity by : Jean-Yves Girard
Download or read book Proof Theory and Logical Complexity written by Jean-Yves Girard and published by . This book was released on 1987 with total page 503 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Arithmetic, Proof Theory, and Computational Complexity by : Peter Clote
Download or read book Arithmetic, Proof Theory, and Computational Complexity written by Peter Clote and published by Clarendon Press. This book was released on 1993-05-06 with total page 442 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book principally concerns the rapidly growing area of "Logical Complexity Theory", the study of bounded arithmetic, propositional proof systems, length of proof, etc and relations to computational complexity theory. Additional features of the book include (1) the transcription and translation of a recently discovered 1956 letter from K Godel to J von Neumann, asking about a polynomial time algorithm for the proof in k-symbols of predicate calculus formulas (equivalent to the P-NP question), (2) an OPEN PROBLEM LIST consisting of 7 fundamental and 39 technical questions contributed by many researchers, together with a bibliography of relevant references.
Book Synopsis Logical Foundations of Mathematics and Computational Complexity by : Pavel Pudlák
Download or read book Logical Foundations of Mathematics and Computational Complexity written by Pavel Pudlák and published by Springer Science & Business Media. This book was released on 2013-04-22 with total page 699 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Logical Foundations of Proof Complexity by : Stephen Cook
Download or read book Logical Foundations of Proof Complexity written by Stephen Cook and published by Cambridge University Press. This book was released on 2014-03-06 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Proof Complexity written by Jan Krajíček and published by Cambridge University Press. This book was released on 2019-03-28 with total page 533 pages. Available in PDF, EPUB and Kindle. Book excerpt: Proof complexity is a rich subject drawing on methods from logic, combinatorics, algebra and computer science. This self-contained book presents the basic concepts, classical results, current state of the art and possible future directions in the field. It stresses a view of proof complexity as a whole entity rather than a collection of various topics held together loosely by a few notions, and it favors more generalizable statements. Lower bounds for lengths of proofs, often regarded as the key issue in proof complexity, are of course covered in detail. However, upper bounds are not neglected: this book also explores the relations between bounded arithmetic theories and proof systems and how they can be used to prove upper bounds on lengths of proofs and simulations among proof systems. It goes on to discuss topics that transcend specific proof systems, allowing for deeper understanding of the fundamental problems of the subject.
Book Synopsis Proof theory and logical complexity. 1 vol by : Jean-Yves GIRARD
Download or read book Proof theory and logical complexity. 1 vol written by Jean-Yves GIRARD and published by . This book was released on 1987 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Semantics and Proof Theory of the Logic of Bunched Implications by : David J. Pym
Download or read book The Semantics and Proof Theory of the Logic of Bunched Implications written by David J. Pym and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 323 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a monograph about logic. Specifically, it presents the mathe matical theory of the logic of bunched implications, BI: I consider Bl's proof theory, model theory and computation theory. However, the mono graph is also about informatics in a sense which I explain. Specifically, it is about mathematical models of resources and logics for reasoning about resources. I begin with an introduction which presents my (background) view of logic from the point of view of informatics, paying particular attention to three logical topics which have arisen from the development of logic within informatics: • Resources as a basis for semantics; • Proof-search as a basis for reasoning; and • The theory of representation of object-logics in a meta-logic. The ensuing development represents a logical theory which draws upon the mathematical, philosophical and computational aspects of logic. Part I presents the logical theory of propositional BI, together with a computational interpretation. Part II presents a corresponding devel opment for predicate BI. In both parts, I develop proof-, model- and type-theoretic analyses. I also provide semantically-motivated compu tational perspectives, so beginning a mathematical theory of resources. I have not included any analysis, beyond conjecture, of properties such as decidability, finite models, games or complexity. I prefer to leave these matters to other occasions, perhaps in broader contexts.
Book Synopsis Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs by : Ivo Düntsch
Download or read book Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs written by Ivo Düntsch and published by Springer Nature. This book was released on 2021-09-24 with total page 591 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Proof Complexity and Feasible Arithmetics by : Paul W. Beame
Download or read book Proof Complexity and Feasible Arithmetics written by Paul W. Beame and published by American Mathematical Soc.. This book was released on 1998 with total page 335 pages. Available in PDF, EPUB and Kindle. Book excerpt: The 16 papers reflect some of the breakthroughs over the past dozen years in understanding whether or not logical inferences can be made in certain situations and what resources are necessary to make such inferences, questions that play a large role in computer science and artificial intelligence. They discuss such aspects as lower bounds in proof complexity, witnessing theorems and proof systems for feasible arithmetic, algebraic and combinatorial proof systems, and the relationship between proof complexity and Boolean circuit complexity. No index. Member prices are $47 for institutions and $35 for individuals. Annotation copyrighted by Book News, Inc., Portland, OR.
Book Synopsis Basic Proof Theory by : A. S. Troelstra
Download or read book Basic Proof Theory written by A. S. Troelstra and published by Cambridge University Press. This book was released on 2000-07-27 with total page 436 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and then apply them elsewhere in much more complex settings. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to selected exercises have been included.
Author :Jean Goubault-Larrecq Publisher :Springer Science & Business Media ISBN 13 :9781402003684 Total Pages :448 pages Book Rating :4.0/5 (36 download)
Book Synopsis Proof Theory and Automated Deduction by : Jean Goubault-Larrecq
Download or read book Proof Theory and Automated Deduction written by Jean Goubault-Larrecq and published by Springer Science & Business Media. This book was released on 2001-11-30 with total page 448 pages. Available in PDF, EPUB and Kindle. Book excerpt: Interest in computer applications has led to a new attitude to applied logic in which researchers tailor a logic in the same way they define a computer language. In response to this attitude, this text for undergraduate and graduate students discusses major algorithmic methodologies, and tableaux and resolution methods. The authors focus on first-order logic, the use of proof theory, and the computer application of automated searches for proofs of mathematical propositions. Annotation copyrighted by Book News, Inc., Portland, OR
Book Synopsis Bounded Arithmetic, Propositional Logic and Complexity Theory by : Jan Krajicek
Download or read book Bounded Arithmetic, Propositional Logic and Complexity Theory written by Jan Krajicek and published by Cambridge University Press. This book was released on 1995-11-24 with total page 361 pages. Available in PDF, EPUB and Kindle. Book excerpt: Discusses the deep connections between logic and complexity theory, and lists a number of intriguing open problems.
Book Synopsis Proof Theory in Computer Science by : Reinhard Kahle
Download or read book Proof Theory in Computer Science written by Reinhard Kahle and published by Springer Science & Business Media. This book was released on 2001-09-28 with total page 249 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the International Seminar on Proof Theory in Computer Science, PTCS 2001, held in Dagstuhl Castle, Germany, in October 2001. The 13 thoroughly revised full papers were carefully reviewed and selected for inclusion in the book. Among the topics addressed are higher type recursion, lambda calculus, complexity theory, transfinite induction, categories, induction-recursion, post-Turing analysis, natural deduction, implicit characterization, iterate logic, and Java programming.
Book Synopsis A First Course in Logic by : Shawn Hedman
Download or read book A First Course in Logic written by Shawn Hedman and published by OUP Oxford. This book was released on 2004-07-08 with total page 452 pages. Available in PDF, EPUB and Kindle. Book excerpt: The ability to reason and think in a logical manner forms the basis of learning for most mathematics, computer science, philosophy and logic students. Based on the author's teaching notes at the University of Maryland and aimed at a broad audience, this text covers the fundamental topics in classical logic in an extremely clear, thorough and accurate style that is accessible to all the above. Covering propositional logic, first-order logic, and second-order logic, as well as proof theory, computability theory, and model theory, the text also contains numerous carefully graded exercises and is ideal for a first or refresher course.
Book Synopsis A First Course in Logic by : Shawn Hedman
Download or read book A First Course in Logic written by Shawn Hedman and published by Oxford University Press on Demand. This book was released on 2004 with total page 431 pages. Available in PDF, EPUB and Kindle. Book excerpt: "The ability to reason and think in a logical manner forms the basis of learning for most mathematics, computer science, philosophy and logic students. Based on the author's teaching notes at the University of Maryland and aimed at a broad audience, thistext covers the fundamental topics in classical logic in a clear, thorough and accurate style that is accessible to all the above. Covering propositional logic, first-order logic, and second-order logic, as well as proof theory, computability theory, andmodel theory, the text also contains numerous carefully graded exercises and is ideal for a first or refresher course."--BOOK JACKET.
