Fingerprint databases for theorems

We discuss the advantages of searchable, collaborative, language-independent databases of mathematical results, indexed by "fingerprints" of small and canonical data. Our motivating example is Neil Sloane's massively influential On-Line Encyclopedia of Integer Sequences. We hope to encourage the greater mathematical community to search for the appropriate fingerprints within each discipline, and to compile fingerprint databases of results wherever possible. The benefits of these databases are broad - advancing the state of knowledge, enhancing experimental mathematics, enabling researchers to discover unexpected connections between areas, and even improving the refereeing process for journal publication.Comment: to appear in Notices of the AM

The use of data-mining for the automatic formation of tactics

This paper discusses the usse of data-mining for the automatic formation of tactics. It was presented at the Workshop on Computer-Supported Mathematical Theory Development held at IJCAR in 2004. The aim of this project is to evaluate the applicability of data-mining techniques to the automatic formation of tactics from large corpuses of proofs. We data-mine information from large proof corpuses to find commonly occurring patterns. These patterns are then evolved into tactics using genetic programming techniques

Impossibility of Gathering, a Certification

Recent advances in Distributed Computing highlight models and algorithms for autonomous swarms of mobile robots that self-organise and cooperate to solve global objectives. The overwhelming majority of works so far considers handmade algorithms and proofs of correctness. This paper builds upon a previously proposed formal framework to certify the correctness of impossibility results regarding distributed algorithms that are dedicated to autonomous mobile robots evolving in a continuous space. As a case study, we consider the problem of gathering all robots at a particular location, not known beforehand. A fundamental (but not yet formally certified) result, due to Suzuki and Yamashita, states that this simple task is impossible for two robots executing deterministic code and initially located at distinct positions. Not only do we obtain a certified proof of the original impossibility result, we also get the more general impossibility of gathering with an even number of robots, when any two robots are possibly initially at the same exact location.Comment: 10

Machine learning discovers invariants of braids and flat braids

We use machine learning to classify examples of braids (or flat braids) as trivial or non-trivial. Our ML takes form of supervised learning using neural networks (multilayer perceptrons). When they achieve good results in classification, we are able to interpret their structure as mathematical conjectures and then prove these conjectures as theorems. As a result, we find new convenient invariants of braids, including a complete invariant of flat braids.Comment: 24 page