'Satisfiability (SAT) related topics have attracted researchers from various disciplines: logic, applied areas such as planning, scheduling, operations research and combinatorial optimization, but also theoretical issues on the theme of complexity and much more, they all are connected through SAT. My personal interest in SAT stems from actual solving: The increase in power of modern SAT solvers over the past 15 years has been phenomenal. It has become the key enabling technology in automated verification of both computer hardware and software' - Edmund M. Clarke (FORE Systems University Professor of Computer Science and Professor of Electrical and Computer Engineering at Carnegie Mellon University). 'Bounded Model Checking (BMC) of computer hardware is now probably the most widely used model checking technique. The counterexamples that it finds are just satisfying instances of a Boolean formula obtained by unwinding to some fixed depth a sequential circuit and its specification in linear temporal logic. Extending model checking to software verification is a much more difficult problem on the frontier of current research. One promising approach for languages like C with finite word-length integers is to use the same idea as in BMC but with a decision procedure for the theory of bit-vectors instead of SAT. All decision procedures for bit-vectors that I am familiar with ultimately make use of a fast SAT solver to handle complex formulas' - Edmund M. Clarke (FORE Systems University Professor of Computer Science and Professor of Electrical and Computer Engineering at Carnegie Mellon University). 'Decision procedures for more complicated theories, like linear real and integer arithmetic, are also used in program verification. Most of them use powerful SAT solvers in an essential way. Clearly, efficient SAT solving is a key technology for 21st century computer science. I expect this collection of papers on all theoretical and practical aspects of SAT solving will be extremely useful to both students and researchers and will lead to many further advances in the field' - Edmund M. Clarke (FORE Systems University Professor of Computer Science and Professor of Electrical and Computer Engineering at Carnegie Mellon University).

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Handbook of Satisfiability

ion and general ideas.” Compare Meinong, Über Möglichkeit und Wahrscheinlichkeit, p. 178: “Besonders geeignet sind vielmehr Begriffsgegenstände, wie uns deren etwa durch Definitionen gegeben werden. Das Dreick z.B., darin hatte der sicher nicht überrationalistische Locke gegen Berkeley und gegen viele Spätere) am Ende doch recht, ist als solches weder gleichseitig noch gleichschenklig, weder rechtwinklig noch schiefwinklig, noch das Gegenteil davon: es ist in diesen Hinsichten und noch in vielen anderen eben unbestimmt. Gegenstände dieser Art stehen in deutlichen Gegensatz zu solchen, die, wie wir deren oben zuerst betrachtet haben, in bezug auf alle wie immer gearteten Gegenstände bestimmt sind. Man kann solche Gegenstände mit Recht vollständig bestimmte nennen, Blaues, Dreieck und ihresgleichen dagegen unvollständig bestimmte.” 15Hume, A Treatise of Human Nature, [1739-40], edited by Selby-Bigge, second edition revised by Nidditch (Oxford: The Clarendon Press, 1978), p. 17.

Handbook of the history of logic

Radial visualization, or the practice of displaying data in a circular or elliptical pattern, is an increasingly common technique in information visualization research. In spite of its prevalence, little work has been done to study this visualization paradigm as a methodology in its own right. We provide a historical review of radial visualization, tracing it to its roots in centuries-old statistical graphics. We then identify the types of problem domains to which modern radial visualization techniques have been applied. A taxonomy for radial visualization is proposed in the form of seven design patterns encompassing nearly all recent works in this area. From an analysis of these patterns, we distill a series of design considerations that system builders can use to create new visualizations that address aspects of the design space that have not yet been explored. It is hoped that our taxonomy will provide a framework for facilitating discourse among researchers and stimulate the development of additional theories and systems involving radial visualization as a distinct design metaphor.

A Survey of Radial Methods for Information Visualization

Apart from explicit node-link representations, implicit visualizations and especially the Treemap as their frontrunner have acquired a solid position among the available techniques to visualize hierarchies. Their advantage is a highly space-efficient graphical representation that does not require explicit drawing of edges. In this paper, we survey the design space for this class of visualization techniques. We establish the design space along the four axes of dimensionality, edge representation, node representation, and layout by examining existing implicit hierarchy visualization techniques. The survey is completed by casting some light into regions of the design space that have not yet been explored. Our design space is not a mere theoretical construct, but a practically usable tool for rapid visualization development. To that end, we discuss a software implementation of the introduced design space.

The Design Space of Implicit Hierarchy Visualization: A Survey

A variety of data analysis problems can be modelled by defining multiple sets over a collection of elements and analyzing the relations between these sets. Despite their simple concept, visualizing sets is a non-trivial problem due to the large number of possible relations between them. We provide a systematic overview of state-of-the-art techniques for visualizing different kinds of set relations. We classify these techniques into 7 main categories according to the visual representations they use and the tasks they support. We compare the categories to provide guidance for choosing an appropriate technique for a given problem. Finally, we identify challenges in this area that need further research and propose possible directions to address with these challenges.

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Visualizing Sets and Set-typed Data: State-of-the-Art and Future Challenges

Lessons on sets have become commonplace in schools today. Venn diagrams proliferate and, even in primary schools, children can be seen sorting and classifying objects by size, colour and shape and placing them in spaces marked out on the floor by chalk outlines or wooden hoops. Older children learn that such diagrams are named after the English logician, John Venn, and that through them we can represent the relations of membership and inclusion and the operations of union, intersection and complementation. A rectangle is drawn to represent the universe U: subsets of U are represented by the interiors of circles, or other closed curves within U, i.e. subspaces of the rectangle. The elements of U are represented by points within the rectangle, the elements of a subset A by points within the corresponding subspace of the rectangle and the elements of A′ by points within the rectangle but outside the region representing A.

A Note on the Historical Development of Logic Diagrams: Leibniz, Euler and Venn

The Concept of Number. By Christoph J. Scriba. Pp. 216. 16s. 1968. (Vol. 825/825a of Hochschulskripten, Bibliographisohes Institut, Mannheim and Zurich.)

Margaret E. Baron

Papers

A Note on the Historical Development of Logic Diagrams: Leibniz, Euler and Venn

The Concept of Number. By Christoph J. Scriba. Pp. 216. 16s. 1968. (Vol. 825/825a of Hochschulskripten, Bibliographisohes Institut, Mannheim and Zurich.)

A Note on Robert Recorde and the Dienes Blocks