Concept Lattices and Logical Inference
Galois closure is a way of discovering maximal sets of
objects with a common set of properties.
Rudolf Wille exploited this characterization in his development of
concept lattices.
We have extended this technique to extract
all logically
consistent, universally quantified, assertions that can be made given
a collection of existentially quantified facts.
This has also been called discrete, deterministic data mining, or DDDM,
because it extracts necessary, not just statistically probable,
associations from the input data.

[PT02] J.L. Pfaltz, C. M. Taylor
Closed Set Mining of Biological Data,
BIOKDD 2002, Workshop on Data Mining in Bioinformatics,
(at KDD 2002, Edmonton, Alberta,
July 2002, 4348.

[PT02] J.L. Pfaltz, C. M. Taylor
Scientific Discovery through Iterative Transformations
of Concept Lattices,
Workshop on Discrete Mathematics and Data Mining,
(at 2nd SIAM Conf. on Data Mining, Arlington VA)
April 2002, 6574.

[P08] J.L. Pfaltz,
Establishing Logical Rules from Empirical Data
Intern. Journal on Artificial Intelligence Tools
,
Vol. 17, no. 5 (2008) 9851001
Finally, our attention has been focused on the transformation
of closure systems.
The ability to smoothly transform one closure system into another lies at the
heart of successful DDDM as described in [P01].
But, it is expected that since we have proven that closed, complete
transformations, or morphisms, induce a welldefined, cartesian closed,
category of discrete closure systems [P04a, PS12] we will uncover many more
applications of interest to computer science.
For example, separation in social networks behaves differently under
continuous transformations than one would expect [P12].

[P01] J.L. Pfaltz,
Transformations of Concept
Graphs: An Approach to Empirical Induction ,
2nd Intern'l Workshop
on Graph Transformation and Visual Modeling Techniques, GTVM 2001,
Satellite Workshop of ICALP 2001, Crete, Greece July 2001, 320326.

[P04a] J.L. Pfaltz,
A Category of Discrete Closure Systems,
Spatial representation:Discrete vs. Continuous Computational Models ,
Dagstuhl Seminar 04351, August 2004.

[PS12] J.L. Pfaltz and Josef Slapal,
Transformations of discrete closure systems
Acta Math. Hungar.
,
(to appear), 2012.

[P11] J.L. Pfaltz,
Mathmetical Continuity in Dynamic Social Networks,
3rd International Conf. on Social Informatics, SOCINFO 2011 ,
October 2011,
LNCS #6984, 3650.

[P12] J.L. Pfaltz,
Entropy in Social Networks,
4th International Conf. on Social Informatics, SOCINFO 2012 ,
December 2012,
(in review).