Spatial Representation: Discrete vs. Continuous Computational Models

About the seminar

Topological notions and methods have been successfully applied in various areas of computer science. Programming language semantics and computing with exact real numbers are two important examples. Computerized geometrical constructions have many applications in engineering. The seminar will concentrate on an important approach which is basic to these applications, i.e. spatial representation.

Due to the digital nature of most applications, the structures used in computer science are different from the mathematical structures that are classically used in engineering and that are based on the continuum. Typical features of these digital structures are asymmetry and partiality. Whereas classical spaces contain only the ideal elements that are the result of a computation (approximation) process, spaces that also allow reasoning on such processes in a formal way must as well contain the partial (and finite) objects appearing during a computation. Only they can be observed in finite time.

The seminar was devoted to the study of several topological structures. The leading example of such is the domain (in Scott's sense), and it is closely related to locales. Here, the finitely observable properties of a process are the primary objects of study. The ideal entities, which are the first class citizens of classical mathematical structures, are obtained as derived objects. These have given rise to a constructive treatment of topological spaces, Formal Topology.

More about the seminar contents and talks in the Online Seminar Proceedings