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Logical Foundations Of Proof Complexity
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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 2010-01-25 with total page 496 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. The result is a uniform treatment of many systems in the literature.
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.
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: Offers a self-contained work presenting basic ideas, classical results, current state of the art and possible future directions in proof complexity.
Book Synopsis Principia Mathematica by : Alfred North Whitehead
Download or read book Principia Mathematica written by Alfred North Whitehead and published by . This book was released on 1910 with total page 688 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Logic for Computer Science by : Jean H. Gallier
Download or read book Logic for Computer Science written by Jean H. Gallier and published by Courier Dover Publications. This book was released on 2015-06-18 with total page 532 pages. Available in PDF, EPUB and Kindle. Book excerpt: This advanced text for undergraduate and graduate students introduces mathematical logic with an emphasis on proof theory and procedures for algorithmic construction of formal proofs. The self-contained treatment is also useful for computer scientists and mathematically inclined readers interested in the formalization of proofs and basics of automatic theorem proving. Topics include propositional logic and its resolution, first-order logic, Gentzen's cut elimination theorem and applications, and Gentzen's sharpened Hauptsatz and Herbrand's theorem. Additional subjects include resolution in first-order logic; SLD-resolution, logic programming, and the foundations of PROLOG; and many-sorted first-order logic. Numerous problems appear throughout the book, and two Appendixes provide practical background information.
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 Computational Complexity by : Sanjeev Arora
Download or read book Computational Complexity written by Sanjeev Arora and published by Cambridge University Press. This book was released on 2009-04-20 with total page 609 pages. Available in PDF, EPUB and Kindle. Book excerpt: New and classical results in computational complexity, including interactive proofs, PCP, derandomization, and quantum computation. Ideal for graduate students.
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 2010-01-25 with total page 496 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.
Book Synopsis Logical Foundations of Cyber-Physical Systems by : André Platzer
Download or read book Logical Foundations of Cyber-Physical Systems written by André Platzer and published by Springer. This book was released on 2018-08-31 with total page 639 pages. Available in PDF, EPUB and Kindle. Book excerpt: Cyber-physical systems (CPSs) combine cyber capabilities, such as computation or communication, with physical capabilities, such as motion or other physical processes. Cars, aircraft, and robots are prime examples, because they move physically in space in a way that is determined by discrete computerized control algorithms. Designing these algorithms is challenging due to their tight coupling with physical behavior, while it is vital that these algorithms be correct because we rely on them for safety-critical tasks. This textbook teaches undergraduate students the core principles behind CPSs. It shows them how to develop models and controls; identify safety specifications and critical properties; reason rigorously about CPS models; leverage multi-dynamical systems compositionality to tame CPS complexity; identify required control constraints; verify CPS models of appropriate scale in logic; and develop an intuition for operational effects. The book is supported with homework exercises, lecture videos, and slides.
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 Kurt Gödel and the Foundations of Mathematics by : Matthias Baaz
Download or read book Kurt Gödel and the Foundations of Mathematics written by Matthias Baaz and published by Cambridge University Press. This book was released on 2011-06-06 with total page 541 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume commemorates the life, work and foundational views of Kurt Gödel (1906–78), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency - with the other widely accepted axioms of set theory - of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances and ideas for future directions not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Gödel personally, providing some interesting insights into his life. By putting his ideas and life's work into the context of current thinking and perceptions, this book will extend the impact of Gödel's fundamental work in mathematics, logic, philosophy and other disciplines for future generations of researchers.
Book Synopsis Logical Foundations of Probability by : Rudolf Carnap
Download or read book Logical Foundations of Probability written by Rudolf Carnap and published by . This book was released on 1951 with total page 636 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Proofs from THE BOOK by : Martin Aigner
Download or read book Proofs from THE BOOK written by Martin Aigner and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 194 pages. Available in PDF, EPUB and Kindle. Book excerpt: According to the great mathematician Paul Erdös, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such "perfect proofs," those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, analysis, combinatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics.
Book Synopsis Practical Foundations of Mathematics by : Paul Taylor
Download or read book Practical Foundations of Mathematics written by Paul Taylor and published by Cambridge University Press. This book was released on 1999-05-13 with total page 590 pages. Available in PDF, EPUB and Kindle. Book excerpt: Practical Foundations collects the methods of construction of the objects of twentieth-century mathematics. Although it is mainly concerned with a framework essentially equivalent to intuitionistic Zermelo-Fraenkel logic, the book looks forward to more subtle bases in categorical type theory and the machine representation of mathematics. Each idea is illustrated by wide-ranging examples, and followed critically along its natural path, transcending disciplinary boundaries between universal algebra, type theory, category theory, set theory, sheaf theory, topology and programming. Students and teachers of computing, mathematics and philosophy will find this book both readable and of lasting value as a reference work.
Book Synopsis Perspectives in Logic by : Stephen Cook
Download or read book Perspectives in Logic written by Stephen Cook and published by . This book was released on 2010 with total page 479 pages. Available in PDF, EPUB and Kindle. Book excerpt: A treatise on bounded arithmetic and propositional proof complexity by the leader in the field.
Book Synopsis Justification As Ignorance by : Sven Rosenkranz
Download or read book Justification As Ignorance written by Sven Rosenkranz and published by Oxford University Press, USA. This book was released on 2021-03 with total page 301 pages. Available in PDF, EPUB and Kindle. Book excerpt: Justification as Ignorance offers an original account of epistemic justification as both non-factive and luminous, vindicating core internalist intuitions without construing justification as an internal condition knowable by reflection alone. Sven Rosenkranz conceives of justification, in its doxastic and propositional varieties, as a kind of epistemic possibility of knowing and of being in a position to know. His account contrasts with recent alternative views that characterize justification in terms of the metaphysical possibility of knowing. Instead, he develops a suitable non-normal multi-modal epistemic logic for knowledge and being in a position to know that respects the finding that these notions create hyperintensional contexts. He also defends his conception of justification against well-known anti-luminosity arguments, shows that the account allows for fruitful applications and principled solutions to the lottery and preface paradoxes, and provides a metaphysics of justification and its varying degrees of strength that is compatible with core assumptions of the knowledge-first approach and disjunctivist conceptions of mental states.
Book Synopsis Arithmetic Circuits by : Amir Shpilka
Download or read book Arithmetic Circuits written by Amir Shpilka and published by Now Publishers Inc. This book was released on 2010 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions. The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we discuss the recent lower bounds for multilinear circuits and formulas, the advances in the question of deterministically checking polynomial identities, and the results regarding reconstruction of arithmetic circuits. We do, however, also cover part of the classical works on arithmetic circuits. In order to keep this monograph at a reasonable length, we do not give full proofs of most theorems, but rather try to convey the main ideas behind each proof and demonstrate it, where possible, by proving some special cases.
