Computers in Ramsey Theory; Testing, Constructions and Nonexistence

Computers in Ramsey Theory Ramsey theory is often regarded as the study of how order emerges from randomness. While originated in mathematical logic, it has applications in geometry, number theory, game theory, information theory, approximation algorithms, and other areas of mathematics and theoretical computer science. Ramsey theory studies the conditions of when a combinatorial object necessarily contains some smaller given objects. The central concept in Ramsey theory is that of arrowing, which in the case of graphs describes when colorings of larger graphs necessarily contain monochromatic copies of given smaller graphs. The role of Ramsey numbers is to quantify some of the general existential theorems in Ramsey theory, always involving arrowing. The determination of whether this arrowing holds is notoriously difficult, and thus it leads to numerous computational challenges concerning various types of Ramsey numbers and closely related Folkman numbers. This talk will overview how computers are increasingly used to study the bounds on Ramsey and Folkman numbers, and properties of Ramsey arrowing in general. This is happening in the area where traditional approaches typically call for classical computer-free proofs. It is evident that now we understand Ramsey theory much better than a few decades ago, increasingly due to computations. Further, more such progress and new insights based on computations should be anticipated

The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi

A Science of Reasoning

This paper addresses the question of how we can understand reasoning in general and mathematical proofs in particular. It argues the need for a high-level understanding of proofs to complement the low-level understanding provided by Logic. It proposes a role for computation in providing this high-level understanding, namely by the association of proof plans with proofs. Proof plans are defined and examples are given for two families of proofs. Criteria are given for assessing the association of a proof plan with a proof. 1 Motivation: the understanding of mathematical proofs The understanding of reasoning has interested researchers since, at least, Aristotle. Logic has been proposed by Aristotle, Boole, Frege and others as a way of formalising arguments and understanding their structure. There have also been psychological studies of how people and animals actually do reason. The work on Logic has been especially influential in the automation of reasoning. For instance, resolution..

Mathematical practice, crowdsourcing, and social machines

The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend. Mathematical practice is an emerging interdisciplinary field which draws on philosophy and social science to understand how mathematics is produced. Online mathematical activity provides a novel and rich source of data for empirical investigation of mathematical practice - for example the community question answering system {\it mathoverflow} contains around 40,000 mathematical conversations, and {\it polymath} collaborations provide transcripts of the process of discovering proofs. Our preliminary investigations have demonstrated the importance of "soft" aspects such as analogy and creativity, alongside deduction and proof, in the production of mathematics, and have given us new ways to think about the roles of people and machines in creating new mathematical knowledge. We discuss further investigation of these resources and what it might reveal. Crowdsourced mathematical activity is an example of a "social machine", a new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity, and the subject of major international research endeavours. We outline a future research agenda for mathematics social machines, a combination of people, computers, and mathematical archives to create and apply mathematics, with the potential to change the way people do mathematics, and to transform the reach, pace, and impact of mathematics research.Comment: To appear, Springer LNCS, Proceedings of Conferences on Intelligent Computer Mathematics, CICM 2013, July 2013 Bath, U

The Information-Flow Approach to Ontology-Based Semantic Integration

In this article we argue for the lack of formal foundations for ontology-based semantic alignment. We analyse and formalise the basic notions of semantic matching and alignment and we situate them in the context of ontology-based alignment in open-ended and distributed environments, like the Web. We then use the mathematical notion of information flow in a distributed system to ground three hypotheses that enable semantic alignment. We draw our exemplar applications of this work from a variety of interoperability scenarios including ontology mapping, theory of semantic interoperability, progressive ontology alignment, and situated semantic alignment