This treatise presents an integrated perspective on the interplay of set theory and graph theory, providing an extensive selection of examples that highlight how methods from one theory can be used to better solve problems originated in the other. Features: explores the interrelationships between sets and graphs and their applications to finite combinatorics; introduces the fundamental graph-theoretical notions from the standpoint of both set theory and dyadic logic, and presents a discussion on set universes; explains how sets can conveniently model graphs, discussing set graphs and set-theoretic representations of claw-free graphs; investigates when it is convenient to represent sets by graphs, covering counting and encoding problems, the random generation of sets, and the analysis of infinite sets; presents excerpts of formal proofs concerning graphs, whose correctness was verified by means of an automated proof-assistant; contains numerous exercises, examples, definitions, problems and insight panels.

This book presents a set of historical recollections on the work of Martin Davis and his role in advancing our understanding of the connections between logic, computing, and unsolvability. The individual contributions touch on most of the core aspects of Davis' work and set it in a contemporary context. They analyse, discuss and develop many of the ideas and concepts that Davis put forward, including such issues as contemporary satisfiability solvers, essential unification, quantum computing and generalisations of Hilbert's tenth problem. The book starts out with a scientific autobiography by Davis, and ends with his responses to comments included in the contributions. In addition, it includes two previously unpublished original historical papers in which Davis and Putnam investigate the decidable and the undecidable side of Logic, as well as a full bibliography of Davis' work. As a whole, this book shows how Davis' scientific work lies at the intersection of computability, theoretical computer science, foundations of mathematics, and philosophy, and draws its unifying vision from his deep involvement in Logic.

This treatise presents an integrated perspective on the interplay of set theory and graph theory, providing an extensive selection of examples that highlight how methods from one theory can be used to better solve problems originated in the other. Features: explores the interrelationships between sets and graphs and their applications to finite combinatorics; introduces the fundamental graph-theoretical notions from the standpoint of both set theory and dyadic logic, and presents a discussion on set universes; explains how sets can conveniently model graphs, discussing set graphs and set-theoretic representations of claw-free graphs; investigates when it is convenient to represent sets by graphs, covering counting and encoding problems, the random generation of sets, and the analysis of infinite sets; presents excerpts of formal proofs concerning graphs, whose correctness was verified by means of an automated proof-assistant; contains numerous exercises, examples, definitions, problems and insight panels.

This book presents a set of historical recollections on the work of Martin Davis and his role in advancing our understanding of the connections between logic, computing, and unsolvability. The individual contributions touch on most of the core aspects of Davis' work and set it in a contemporary context. They analyse, discuss and develop many of the ideas and concepts that Davis put forward, including such issues as contemporary satisfiability solvers, essential unification, quantum computing and generalisations of Hilbert's tenth problem. The book starts out with a scientific autobiography by Davis, and ends with his responses to comments included in the contributions. In addition, it includes two previously unpublished original historical papers in which Davis and Putnam investigate the decidable and the undecidable side of Logic, as well as a full bibliography of Davis' work. As a whole, this book shows how Davis' scientific work lies at the intersection of computability, theoretical computer science, foundations of mathematics, and philosophy, and draws its unifying vision from his deep involvement in Logic.

This must-read text presents the pioneering work of the late Professor Jacob (Jack) T. Schwartz on computational logic and set theory and its application to proof verification techniques, culminating in the AEtnaNova system, a prototype computer program designed to verify the correctness of mathematical proofs presented in the language of set theory. Topics and features: describes in depth how a specific first-order theory can be exploited to model and carry out reasoning in branches of computer science and mathematics; presents an unique system for automated proof verification in large-scale software systems; integrates important proof-engineering issues, reflecting the goals of large-scale verifiers; includes an appendix showing formalized proofs of ordinals, of various properties of the transitive closure operation, of finite and transfinite induction principles, and of Zorn's lemma.

This must-read text presents the pioneering work of the late Professor Jacob (Jack) T. Schwartz on computational logic and set theory and its application to proof verification techniques, culminating in the AEtnaNova system, a prototype computer program designed to verify the correctness of mathematical proofs presented in the language of set theory. Topics and features: describes in depth how a specific first-order theory can be exploited to model and carry out reasoning in branches of computer science and mathematics; presents an unique system for automated proof verification in large-scale software systems; integrates important proof-engineering issues, reflecting the goals of large-scale verifiers; includes an appendix showing formalized proofs of ordinals, of various properties of the transitive closure operation, of finite and transfinite induction principles, and of Zorn's lemma."