Computation is revolutionizing our world, even the inner world of the 'pure' mathematician. Mathematical methods - especially the notion of proof - that have their roots in classical antiquity have seen a radical transformation since the 1970s, as successive advances have challenged the priority of reason over computation. Like many revolutions, this one comes from within. Computation, calculation, algorithms - all have played an important role in mathematical progress from the beginning - but behind the scenes, their contribution was obscured in the enduring mathematical literature. To understand the future of mathematics, this fascinating book returns to its past, tracing the hidden history that follows the thread of computation. Along the way it invites us to reconsider the dialog between mathematics and the natural sciences, as well as the relationship between mathematics and computer science. It also sheds new light on philosophical concepts, such as the notions of analytic and synthetic judgment. Finally, it brings us to the brink of the new age, in which machine intelligence offers new ways of solving mathematical problems previously inaccessible. This book is the 2007 winner of the Grand Prix de Philosophie de l'Academie Francaise.

Logic is a branch of philosophy, mathematics and computer science. It studies the required methods to determine whether a statement is true, such as reasoning and computation.
"Proofs and Algorithms: Introduction to Logic and Computability "is an introduction to the fundamental concepts of contemporary logic - those of a proof, a computable function, a model and a set. It presents a series of results, both positive and negative, - Church's undecidability theorem, G del 's incompleteness theorem, the theorem asserting the semi-decidability of provability - that have profoundly changed our vision of reasoning, computation, and finally truth itself.
Designed for undergraduate students, this book presents all that philosophers, mathematicians and computer scientists should know about logic.

ThisvolumecontainsthepaperspresentedatIJCAR2008, the4thInternational Joint Conference on Automated Reasoning, held August 12-15, 2008, in S- ney (Australia). The IJCAR conference series is aimed at unifying the di?erent research principles within automated reasoning. IJCAR 2008 was the fusion of several major international events: -CADE: The International Conference on Automated Deduction -FroCoS: The Symposium on Frontiers of Combining Systems -FTP: The Workshop on First-Order Theorem Proving - TABLEAUX: The Conference on Analytic Tableaux and Related Methods Previous versions of IJCAR were held in Seattle (USA) in 2006, Cork (Ireland) in 2004, and Siena (Italy) in 2001. These proceedings comprise 4 contributions by invited speakers, 26 research papers, and 13 system descriptions. The volume also includes a short overview of the CASC-J4 competition for automated theorem proving systems that was conductedduringIJCAR2008.TheinvitedspeakerswereHubertComon-Lundh, NachumDershowitz, AartiGupta, andCarstenLutz.Theirtalkscoveredabroad spectrum of automated reasoning themes, viz., veri?cation of security protocols, prooftheoreticalframeworksfor ?rst-orderlogic, automateddecisionprocedures and software veri?cation, and description logics. The contributed papers were selected from 80 research paper submissions and 17 system description submissions. Each submission was reviewed by at least three reviewers, and decisions were reached after two weeks of discussion through an electronic Program Committee meeting. The submissions, reviews, and discussion were coordinated using the EasyChair conference management system. The accepted papers spanned a wide spectrum of researchin automated reasoning, including saturation, equationalreasoninganduni?cation, automa- based methods, description logics and related logics, sati?ability modulo theory, decidable logics, reasoning about programs, and higher-order l

This book contains the proceedings of the 12th International Conference on TheoremProvinginHigherOrderLogics(TPHOLs 99), whichwasheldinNice at the University of Nice-Sophia Antipolis, September 14{17, 1999. Thirty- ve papers were submitted as completed research, and each of them was refereed by at least three reviewers appointed by the program committee. Twenty papers were selected for publication in this volume. Followingawell-establishedtraditioninthisseriesofconferences, anumberof researchers also came to discuss work in progress, using short talks and displays at a poster session. These papers are included in a supplementary proceedings volume. These supplementary proceedings take the form of a book published by INRIA in its series of research reports, under the following title: Theorem ProvinginHigherOrderLogics: EmergingTrends1999. The organizers were pleased that Dominique Bolignano, Arjeh Cohen, and Thomas Kropf accepted invitations to be guest speakers for TPHOLs 99. For several years, D. Bolignano has been the leader of the VIP team in the Dyade consortium between INRIA and Bull and is now at the head of a company Trusted Logic. His team has been concentrating on the use of formal methods for the e ective veri cationof securityproperties for protocols used in electronic commerce. A. Cohen has had a key in?uence on the development of computer algebra in The Netherlands and his contribution has been of particular imp- tance to researchersinterested in combining the severalknown methods of using computers to perform mathematical investigations. T. Kropf is an important actor in the Europe-wide project PROSPER, which aims to deliver the be- ts of mechanized formal analysis to system builders in industry."

This book presents a collection of revised refereed papers selected from the presentations accepted for the Second International Workshop on Higher-Order Algebra, Logic, and Term Rewriting, HOA '95, held in Paderborn, Germany, in September 1995.
The 14 research papers included, together with an invited paper by Jan Willem Klop, report state-of-the-art results; the relevant theoretical aspects are addressed, and in addition existing proof systems and term rewriting systems are discussed.

Computation is revolutionizing our world, even the inner world of the 'pure' mathematician. Mathematical methods - especially the notion of proof - that have their roots in classical antiquity have seen a radical transformation since the 1970s, as successive advances have challenged the priority of reason over computation. Like many revolutions, this one comes from within. Computation, calculation, algorithms - all have played an important role in mathematical progress from the beginning - but behind the scenes, their contribution was obscured in the enduring mathematical literature. To understand the future of mathematics, this fascinating book returns to its past, tracing the hidden history that follows the thread of computation. Along the way it invites us to reconsider the dialog between mathematics and the natural sciences, as well as the relationship between mathematics and computer science. It also sheds new light on philosophical concepts, such as the notions of analytic and synthetic judgment. Finally, it brings us to the brink of the new age, in which machine intelligence offers new ways of solving mathematical problems previously inaccessible. This book is the 2007 winner of the Grand Prix de Philosophie de l'Academie Francaise.

We've known about algorithms for millennia, but we've only been writing c- puter programs for a few decades. A big di?erence between the Euclidean or Eratosthenes age and ours is that since the middle of the twentieth century, we express the algorithms we conceive using formal languages: programming languages. Computer scientists are not the only ones who use formal languages. - tometrists, for example, prescribe eyeglasses using very technical expressions, ? ? such as "OD: -1.25 (-0.50) 180 OS: -1.00 (-0.25) 180," in which the parent- ses are essential. Many such formal languages have been created throughout history: musical notation, algebraic notation, etc. In particular, such languages have long been used to control machines, such as looms and cathedral chimes. However, until the appearance of programming languages, those languages were only of limited importance: they were restricted to specialised ?elds with only a few specialists and written texts of those languages remained relatively scarce. This situation has changed with the appearance of programming l- guages, which have a wider range of applications than the prescription of e- glassesorthecontrolofaloom, areusedbylargecommunities, andhaveallowed the creation of programs of many hundreds of thousands of lines.

The design and implementation of programming languages, from Fortran and Cobol to Caml and Java, has been one of the key developments in the management of ever more complex computerized systems. Introduction to the Theory of Programming Languages gives the reader the means to discover the tools to think, design, and implement these languages. It proposes a unified vision of the different formalisms that permit definition of a programming language: small steps operational semantics, big steps operational semantics, and denotational semantics, emphasising that all seek to define a relation between three objects: a program, an input value, and an output value. These formalisms are illustrated by presenting the semantics of some typical features of programming languages: functions, recursivity, assignments, records, objects, ... showing that the study of programming languages does not consist of studying languages one after another, but is organized around the features that are present in these various languages. The study of these features leads to the development of evaluators, interpreters and compilers, and also type inference algorithms, for small languages.

This book constitutes the refereed proceedings of the Joint 25th International Conference on Rewriting Techniques and Applications, RTA 2014, and 12th International Conference on Typed Lambda-Calculi and Applications, TLCA 2014, held as part of the Vienna Summer of Logic, VSL 2014, in Vienna, Austria, in July 2014. The 28 revised full papers and 3 short papers presented were carefully reviewed and selected from 87 submissions. The papers provide research results on all aspects of rewriting and typed lambda calculi, ranging from theoretical and methodological issues to applications in various contexts. They address a wide variety of topics such as algorithmic aspects, implementation, logic, types, semantics, and programming.

Algorithms are probably the most sophisticated tools that people have had at their disposal since the beginnings of human history. They have transformed science, industry, society. They upset the concepts of work, property, government, private life, even humanity. Going easily from one extreme to the other, we rejoice that they make life easier for us, but fear that they will enslave us. To get beyond this vision of good vs evil, this book takes a new look at our time, the age of algorithms. Creations of the human spirit, algorithms are what we made them. And they will be what we want them to be: it's up to us to choose the world we want to live in.

Algorithms are probably the most sophisticated tools that people have had at their disposal since the beginnings of human history. They have transformed science, industry, society. They upset the concepts of work, property, government, private life, even humanity. Going easily from one extreme to the other, we rejoice that they make life easier for us, but fear that they will enslave us. To get beyond this vision of good vs evil, this book takes a new look at our time, the age of algorithms. Creations of the human spirit, algorithms are what we made them. And they will be what we want them to be: it's up to us to choose the world we want to live in.