Applications of Closure Concepts

Although our research has been largely theoretical, we have explored several applications of closure concepts which appear to be very promising.

One of these has been the formulation of 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.

Another application occurs in the field of discrete digital topology. Here, we have been able to successfuly formulate a Jordan Surface Theorem for arbitrarily tesselated n-dimensional discrete geometries.

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 well-defined, cartesian closed, category of discrete closure systems [P04a] we will uncover many more applications of interest to computer science. For example, this category has some striking parallels to the Scott topology of continuous domains which forms the theoretical justification of computation in general.

Demonstration that convex closure is the key to creating a category of partially ordered sets, or acyclic graphs [P04b], opens the door to better research into parallel computation.