This textbook offers a detailed introduction to the methodology and applications of sequent calculi in propositional logic. Unlike other texts concerned with proof theory, emphasis is placed on illustrating how to use sequent calculi to prove a wide range of metatheoretical results. The presentation is elementary and self-contained, with all technical details both formally stated and also informally explained. Numerous proofs are worked through to demonstrate methods of proving important results, such as the cut-elimination theorem, completeness, decidability, and interpolation. Other proofs are presented with portions left as exercises for readers, allowing them to practice techniques of sequent calculus. After a brief introduction to classical propositional logic, the text explores three variants of sequent calculus and their features and applications. The remaining chapters then show how sequent calculi can be extended, modified, and applied to non-classical logics, including modal, intuitionistic, substructural, and many-valued logics. Sequents and Trees is suitable for graduate and advanced undergraduate students in logic taking courses on proof theory and its application to non-classical logics. It will also be of interest to researchers in computer science and philosophers.

Proof Theory of Modal Logic is devoted to a thorough study of proof systems for modal logics, that is, logics of necessity, possibility, knowledge, belief, time, computations etc. It contains many new technical results and presentations of novel proof procedures. The volume is of immense importance for the interdisciplinary fields of logic, knowledge representation, and automated deduction.

This volume contains the papers selected for presentation at the conference and two abstracts from invited speakers. The programme committee selected these 25 papers from 12 countries out of 65 submissions from 17 countries. The rst JELIA meeting was in Rosco , France, ten years ago. Afterwards, it took place in the Netherlands, Germany, United Kingdom, Portugal, and now again in Germany. The proceedings of the last four meetings appeared in the Springer-Verlag LNCS series, and a selected series of papers of the English and the Portuguese meeting appeared as special issues in the Journal of Applied Non-Classical Logics and in the Journal of Automated Reasoning, respectively. The aim of JELIA was and still is to provide a forum for the exchange of ideas and results in the domain of foundations of AI, focusing on rigorous descriptions of some aspects of intelligence. These descriptions are promoted by applications, and produced by logical tools and methods. The papers contained in this volume cover the following topics: 1. Logic programming 2. Epistemic logics 3. Theorem proving 4. Non-monotonic reasoning 5. Non-standard logics 6. Knowledge representation 7. Higher order logics We would like to warmly thank the authors, the invited speakers, the m- bers of the program committee, and the additional reviewers listed below. They all have made these proceedings possible and ensured their quality.

A First Course in Logic is an introduction to first-order logic suitable for first and second year mathematicians and computer scientists. There are three components to this course: propositional logic; Boolean algebras; and predicate/first-order, logic. Logic is the basis of proofs in mathematics — how do we know what we say is true? — and also of computer science — how do I know this program will do what I think it will? Surprisingly little mathematics is needed to learn and understand logic (this course doesn't involve any calculus). The real mathematical prerequisite is an ability to manipulate symbols: in other words, basic algebra. Anyone who can write programs should have this ability.

Logic has acquired a reputation for difficulty, perhaps because many of the approaches adopted have been more suitable for mathematicians than computer scientists. This book shows that the subject is not inherently difficult and that the connections between logic and declarative language are straightforward. Many exercises have been included in the hope that these will lead to a much greater confidence in manual proofs, therefore leading to a greater confidence in automated proofs.

Providing an in-depth introduction to fundamental classical and non-classical logics, this textbook offers a comprehensive survey of logics for computer scientists. Logics for Computer Science contains intuitive introductory chapters explaining the need for logical investigations, motivations for different types of logics and some of their history. They are followed by strict formal approach chapters. All chapters contain many detailed examples explaining each of the introduced notions and definitions, well chosen sets of exercises with carefully written solutions, and sets of homework. While many logic books are available, they were written by logicians for logicians, not for computer scientists. They usually choose one particular way of presenting the material and use a specialized language. Logics for Computer Science discusses Gentzen as well as Hilbert formalizations, first order theories, the Hilbert Program, Godel's first and second incompleteness theorems and their proofs. It also introduces and discusses some many valued logics, modal logics and introduces algebraic models for classical, intuitionistic, and modal S4 and S5 logics. The theory of computation is based on concepts defined by logicians and mathematicians. Logic plays a fundamental role in computer science, and this book explains the basic theorems, as well as different techniques of proving them in classical and some non-classical logics. Important applications derived from concepts of logic for computer technology include Artificial Intelligence and Software Engineering. In addition to Computer Science, this book may also find an audience in mathematics and philosophy courses, and some of the chapters are also useful for a course in Artificial Intelligence.

Logic is a foundational mathematical discipline for Computer Science. This unique compendium provides the main ideas and techniques originating from logic. It is divided into two volumes — propositional logic and predicate logic. The volume presents some of the most important concepts starting with a variety of logic formalisms — Hilbert/Frege systems, tableaux, sequents, and natural deduction in both propositional and first-order logic, as well as transformations between these formalisms. Topics like circuit design, resolution, cutting planes, Hintikka sets, paramodulation, and program verification, which do not appear frequently in logic books are discussed in detail.The useful reference text has close to 800 exercises and supplements to deepen understanding of the subject. It emphasizes proofs and overcomes technical difficulties by providing detailed arguments. Computer scientists and mathematicians will benefit from this volume.

Handbook of Automated Reasoning

This book constitutes the refereed proceedings of the 22nd International Conference on Automated Deduction, CADE-22, held in Montreal, Canada, in August 2009. The 27 revised full papers and 5 system descriptions presented were carefully reviewed and selected from 77 submissions. Furthermore, three invited lectures by distinguished experts in the area were included. The papers are organized in topical sections on combinations and extensions, minimal unsatisfiability and automated reasoning support, system descriptions, interpolation and predicate abstraction, resolution-based systems for non-classical logics, termination analysis and constraint solving, rewriting, termination and productivity, models, modal tableaux with global caching, arithmetic.

Felty PuzzleTool:AnExampleofProgrammingComputationandDeduction . . 214 MichaelJ. C. Gordon AFormalApproachtoProbabilisticTermination. ... ... 230 JoeHurd UsingTheoremProvingforNumericalAnalysis. ... ... . 246 MicaelaMayero QuotientTypes:AModularApproach. ... ... ... 263 AlekseyNogin SequentSchemaforDerivedRules ... ... ... . 281 AlekseyNogin, JasonHickey AlgebraicStructuresandDependentRecords ... ... . 298 VirgilePrevosto, DamienDoligez, Thþ er` eseHardin ProvingtheEquivalenceofMicrostepandMacrostepSemantics. ... 314 KlausSchneider WeakestPreconditionforGeneralRecursiveProgramsFormalizedinCoq.