• Computers

Basic Proof Theory


Author: A. S. Troelstra,H. Schwichtenberg
Publisher: Cambridge University Press
ISBN: 9780521779111
Category: Computers
Page: 417
View: 7901
Introduction to proof theory and its applications in mathematical logic, theoretical computer science and artificial intelligence.

    • Computers

Basic Simple Type Theory


Author: J. Roger Hindley
Publisher: Cambridge University Press
ISBN: 9780521465182
Category: Computers
Page: 186
View: 6027
Type theory is one of the most important tools in the design of higher-level programming languages, such as ML. This book introduces and teaches its techniques by focusing on one particularly neat system and studying it in detail. By concentrating on the principles that make the theory work in practice, the author covers all the key ideas without getting involved in the complications of more advanced systems. This book takes a type-assignment approach to type theory, and the system considered is the simplest polymorphic one. The author covers all the basic ideas, including the system's relation to propositional logic, and gives a careful treatment of the type-checking algorithm that lies at the heart of every such system. Also featured are two other interesting algorithms that until now have been buried in inaccessible technical literature. The mathematical presentation is rigorous but clear, making it the first book at this level that can be used as an introduction to type theory for computer scientists.

    • Computers

Topology Via Logic


Author: Steven Vickers
Publisher: Cambridge University Press
ISBN: 9780521576512
Category: Computers
Page: 200
View: 2493
This is an advanced textbook on topology for computer scientists. It is based on a course given by the author to postgraduate students of computer science at Imperial College.

    • Computers

Proofs and Types


Author: Jean-Yves Girard,Yves Lafont,Paul Taylor
Publisher: Cambridge University Press
ISBN: 9780521371810
Category: Computers
Page: 192
View: 2342
This text is an outgrowth of notes prepared by J. Y. Girard for a course at the University of Paris VII. It deals with the mathematical background of the application to computer science of aspects of logic (namely the correspondence between proposition & types). Combined with the conceptual perspectives of Girard's ideas, this sheds light on both the traditional logic material & its prospective applications to computer science. The book covers a very active & exciting research area, & it will be essential reading for all those working in logic & computer science.

    • Computers

Extensions of First-Order Logic


Author: Maria Manzano
Publisher: Cambridge University Press
ISBN: 9780521354356
Category: Computers
Page: 388
View: 7885
This book introduces some extensions of classical first-order logic and applies them to reasoning about computer programs. The extensions considered are: second-order logic, many-sorted logic, w-logic, modal logic type theory and dynamic logic. These have wide applications in various areas of computer science, philosophy, natural language processing and artificial intelligence. Researchers in these areas will find this book a useful introduction and comparative treatment.

    • Computers

Modal Logic

Graph. Darst
Author: Patrick Blackburn,Maarten de Rijke,Yde Venema
Publisher: Cambridge University Press
ISBN: 9780521527149
Category: Computers
Page: 554
View: 845
This is an advanced 2001 textbook on modal logic, a field which caught the attention of computer scientists in the late 1970s. Researchers in areas ranging from economics to computational linguistics have since realised its worth. The book is for novices and for more experienced readers, with two distinct tracks clearly signposted at the start of each chapter. The development is mathematical; prior acquaintance with first-order logic and its semantics is assumed, and familiarity with the basic mathematical notions of set theory is required. The authors focus on the use of modal languages as tools to analyze the properties of relational structures, including their algorithmic and algebraic aspects, and applications to issues in logic and computer science such as completeness, computability and complexity are considered. Three appendices supply basic background information and numerous exercises are provided. Ideal for anyone wanting to learn modern modal logic.

    • Philosophy

Concepts of Proof in Mathematics, Philosophy, and Computer Science


Author: Dieter Probst,Peter Schuster
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 1501502646
Category: Philosophy
Page: 384
View: 990
This book provides the reader with research arising from the Humboldt-Kolleg ‘Proof’ held in Bern in fall 2013, which gathered leading experts actively involved with the concept ‘proof’ in philosophy, mathematics and computer science. This volume aims to do justice to the breadth and depth of the subject and presents relevant current conceptions and technical advances featuring ‘proof’ in those fields.

    • Philosophy

Advances in Natural Deduction

A Celebration of Dag Prawitz's Work
Author: Luiz Carlos Pereira,Edward Hermann Haeusler,Valeria de Paiva
Publisher: Springer
ISBN: 9400775482
Category: Philosophy
Page: 279
View: 5085
This collection of papers, celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science. The range of contributions includes material on the extension of natural deduction with higher-order rules, as opposed to higher-order connectives, and a paper discussing the application of natural deduction rules to dealing with equality in predicate calculus. The volume continues with a key chapter summarizing work on the extension of the Curry-Howard isomorphism (itself a by-product of the work on natural deduction), via methods of category theory that have been successfully applied to linear logic, as well as many other contributions from highly regarded authorities. With an illustrious group of contributors addressing a wealth of topics and applications, this volume is a valuable addition to the libraries of academics in the multiple disciplines whose development has been given added scope by the methodologies supplied by natural deduction. The volume is representative of the rich and varied directions that Prawitz work has inspired in the area of natural deduction.

    • Mathematics

Typed Lambda Calculi and Applications

9th International Conference, TLCA 2009, Brasilia, Brazil, July 1-3, 2009, Proceedings
Author: Pierre-Louis Curien
Publisher: Springer
ISBN: 3642022731
Category: Mathematics
Page: 417
View: 6199
This book constitutes the refereed proceedings of the 9th International Conference on Typed Lambda Calculi and Applications, TLCA 2009, held in Brasilia, Brazil in July 2008 in conjunction with RTA 2007, the 19th International Conference on Rewriting Techniques and Applications as part of RDP 2009, the 5th International Conference on Rewriting, Deduction, and Programming. The 27 revised full papers presented together with 2 invited talks were carefully reviewed and selected from 53 submissions. The papers present original research results that are broadly relevant to the theory and applications of typed calculi and address a wide variety of topics such as proof-theory, semantics, implementation, types, and programming.

    • Philosophy

An Introduction to Substructural Logics


Author: Greg Restall
Publisher: Routledge
ISBN: 1135111316
Category: Philosophy
Page: 400
View: 6263
This book introduces an important group of logics that have come to be known under the umbrella term 'susbstructural'. Substructural logics have independently led to significant developments in philosophy, computing and linguistics. An Introduction to Substrucural Logics is the first book to systematically survey the new results and the significant impact that this class of logics has had on a wide range of fields.The following topics are covered: * Proof Theory * Propositional Structures * Frames * Decidability * Coda Both students and professors of philosophy, computing, linguistics, and mathematics will find this to be an important addition to their reading.

    • Mathematics

Logic and Structure


Author: Dirk van Dalen
Publisher: Springer Science & Business Media
ISBN: 3662029626
Category: Mathematics
Page: 220
View: 2976
New corrected printing of a well-established text on logic at the introductory level.

    • Computers

Lambda-calculus, Combinators and Functional Programming


Author: G. E. Revesz
Publisher: Cambridge University Press
ISBN: 0521345898
Category: Computers
Page: 181
View: 2641
Provides computer science students and researchers with a firm background in lambda-calculus and combinators.

    • Mathematics

First-Order Logic and Automated Theorem Proving


Author: Melvin Fitting
Publisher: Springer Science & Business Media
ISBN: 1468403575
Category: Mathematics
Page: 242
View: 7816
There are many kinds of books on formal logic. Some have philosophers as their intended audience, some mathematicians, some computer scientists. Although there is a common core to all such books they will be very dif ferent in emphasis, methods, and even appearance. This book is intended for computer scientists. But even this is not precise. Within computer sci ence formal logic turns up in a number of areas, from program verification to logic programming to artificial intelligence. This book is intended for computer scientists interested in automated theorem proving in classical logic. To be more precise yet, it is essentially a theoretical treatment, not a how-to book, although how-to issues are not neglected. This does not mean, of course, that the book will be of no interest to philosophers or mathematicians. It does contain a thorough presentation of formal logic and many proof techniques, and as such it contains all the material one would expect to find in a course in formal logic covering completeness but not incompleteness issues. The first item to be addressed is, what are we talking about and why are we interested in it. We are primarily talking about truth as used in mathematical discourse, and our interest in it is, or should be, self-evident. Truth is a semantic concept, so we begin with models and their properties. These are used to define our subject.

    • Computers

Term Rewriting and All That


Author: Franz Baader,Tobias Nipkow
Publisher: Cambridge University Press
ISBN: 9780521779203
Category: Computers
Page: 316
View: 4217
Unified and self-contained introduction to term-rewriting; suited for students or professionals.

    • Mathematics

Applied Proof Theory: Proof Interpretations and their Use in Mathematics


Author: Ulrich Kohlenbach
Publisher: Springer Science & Business Media
ISBN: 3540775331
Category: Mathematics
Page: 536
View: 3444
This is the first treatment in book format of proof-theoretic transformations - known as proof interpretations - that focuses on applications to ordinary mathematics. It covers both the necessary logical machinery behind the proof interpretations that are used in recent applications as well as – via extended case studies – carrying out some of these applications in full detail. This subject has historical roots in the 1950s. This book for the first time tells the whole story.

    • Philosophy

What Logics Mean

From Proof Theory to Model-Theoretic Semantics
Author: James W. Garson
Publisher: Cambridge University Press
ISBN: 1107471001
Category: Philosophy
Page: 260
View: 1301
What do the rules of logic say about the meanings of the symbols they govern? In this book, James W. Garson examines the inferential behaviour of logical connectives (such as 'and', 'or', 'not' and 'if ... then'), whose behaviour is defined by strict rules, and proves definitive results concerning exactly what those rules express about connective truth conditions. He explores the ways in which, depending on circumstances, a system of rules may provide no interpretation of a connective at all, or the interpretation we ordinarily expect for it, or an unfamiliar or novel interpretation. He also shows how the novel interpretations thus generated may be used to help analyse philosophical problems such as vagueness and the open future. His book will be valuable for graduates and specialists in logic, philosophy of logic, and philosophy of language.

    • Mathematics

Lectures on the Curry-Howard Isomorphism


Author: Morten Heine Sørensen,Pawel Urzyczyn
Publisher: Elsevier
ISBN: 9780080478920
Category: Mathematics
Page: 456
View: 3019
The Curry-Howard isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance, minimal propositional logic corresponds to simply typed lambda-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, sequent calculus is related to explicit substitution, etc. The isomorphism has many aspects, even at the syntactic level: formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc. But there is more to the isomorphism than this. For instance, it is an old idea---due to Brouwer, Kolmogorov, and Heyting---that a constructive proof of an implication is a procedure that transforms proofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. The Curry-Howard isomorphism also provides theoretical foundations for many modern proof-assistant systems (e.g. Coq). This book give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism. It can serve as an introduction to any or both of typed lambda-calculus and intuitionistic logic. Key features - The Curry-Howard Isomorphism treated as common theme - Reader-friendly introduction to two complementary subjects: Lambda-calculus and constructive logics - Thorough study of the connection between calculi and logics - Elaborate study of classical logics and control operators - Account of dialogue games for classical and intuitionistic logic - Theoretical foundations of computer-assisted reasoning · The Curry-Howard Isomorphism treated as the common theme. · Reader-friendly introduction to two complementary subjects: lambda-calculus and constructive logics · Thorough study of the connection between calculi and logics. · Elaborate study of classical logics and control operators. · Account of dialogue games for classical and intuitionistic logic. · Theoretical foundations of computer-assisted reasoning

    • Computers

Data Refinement

Model-Oriented Proof Methods and Their Comparison
Author: Willem-Paul de Roever,Kai Engelhardt,Karl-Heinz Buth
Publisher: Cambridge University Press
ISBN: 0521641705
Category: Computers
Page: 423
View: 8102
The goal of this book is to provide a comprehensive and systematic introduction to the important and highly applicable method of data refinement and the simulation methods used for proving its correctness. The authors concentrate in the first part on the general principles needed to prove data refinement correct. They begin with an explanation of the fundamental notions, showing that data refinement proofs reduce to proving simulation. The book's second part contains a detailed survey of important methods in this field, which are carefully analysed, and shown to be either incomplete, with counterexamples to their application, or to be always applicable whenever data refinement holds. This is shown by proving, for the first time, that all these methods can be described and analysed in terms of two simple notions: forward and backward simulation. The book is self-contained, going from advanced undergraduate level and taking the reader to the state of the art in methods for proving simulation.

    • Mathematics

Proof Theory

Sequent Calculi and Related Formalisms
Author: Katalin Bimbo
Publisher: CRC Press
ISBN: 1466564660
Category: Mathematics
Page: 386
View: 9843
Although sequent calculi constitute an important category of proof systems, they are not as well known as axiomatic and natural deduction systems. Addressing this deficiency, Proof Theory: Sequent Calculi and Related Formalisms presents a comprehensive treatment of sequent calculi, including a wide range of variations. It focuses on sequent calculi for various non-classical logics, from intuitionistic logic to relevance logic, linear logic, and modal logic. In the first chapters, the author emphasizes classical logic and a variety of different sequent calculi for classical and intuitionistic logics. She then presents other non-classical logics and meta-logical results, including decidability results obtained specifically using sequent calculus formalizations of logics. The book is suitable for a wide audience and can be used in advanced undergraduate or graduate courses. Computer scientists will discover intriguing connections between sequent calculi and resolution as well as between sequent calculi and typed systems. Those interested in the constructive approach will find formalizations of intuitionistic logic and two calculi for linear logic. Mathematicians and philosophers will welcome the treatment of a range of variations on calculi for classical logic. Philosophical logicians will be interested in the calculi for relevance logics while linguists will appreciate the detailed presentation of Lambek calculi and their extensions.

    • Mathematics

Algebraic Curves Over Finite Fields


Author: Carlos Moreno
Publisher: Cambridge University Press
ISBN: 9780521459013
Category: Mathematics
Page: 260
View: 4414
Develops the theory of algebraic curves over finite fields, their zeta and L-functions and the theory of algebraic geometric Goppa codes.