Galois Connections, T-CUBES, & P2P Data Mining Witold Litwin CERIA, Université Paris Dauphine, Pl. du Mal. de Lattre, 75016 Paris, France mailto:Witold.Litwin@dauphine.fr Abstract. Galois connections are bread and butter of the formal concept analy- sis. They concern objects with properties, represented typically as a single- valued (.true or null) binary attributes. The closed sets and Galois lattices are the most studied connections. We generalize them to the relational database universe with the multi-valued domains. We show interesting queries that ap- pear from, hard or impossible with SQL at present. As remedy, we generalize CUBE to a new operator we call θ-CUBE, writing T-CUBE It calculates the groups according to all the values of the θ operator popular with the relational joins. We show also the utility of new aggregate functions LIST and T- GROUP. In this context We finally discuss scalable distributed algorithms in P2P or grid environment for T-CUBE queries Our proposals should benefit to both: the data mining and the concept analysis over many objects. 1. Introduction The formal concept analysis studies the relationship between objects and the proper- ties in a space of objects and properties. The Galois connection in this universe is a re- lationship among some objects and some properties. Whether an object has a property is typically indicated by a binary attribute of the object. Probably the most studied Ga- lois connection is among a set O of all the objects sharing some set P of properties such that there is no property beyond P that would be also shared by all the objects in O. One qualifies (O, P) as closed set. The closed sets over the subsets of a set of ob- jects sharing a set of properties can be ordered by inclusion over P or O. A popular result is a Galois lattice. Finding a closed set let us conclude about the maximal common set of properties. The set may be then abstracted into a concept. The lattice calculus let us see possible abstractions among the concepts. For instance, the objects may be the students for some diploma. A property may be the final “pass” grade at a course, Fig. 1. A closed set would be any couple (S, P) such that S contains all the students who passed all the courses in P, and, for any other course, at least one of the students in S failed. It could be for example students 1,3,5 who were the only and all to pass courses a,b,d,e,g. The Galois lattice would show the inclusion connections. It could show for instance for our closed set that students 1,3 form also a close set over more courses as they share also d. The set of students pass- ing all the exams (one extremity of the lattice) may or may not be empty. Likewise, there may or may not be the course that all students pass (the other extremity). The dean may obviously be interested in mining the resulting (Galois) connections, i.e.,