Информационная система "Конференции"

Abstracts

Applications of logic in computer science

The purpose of this paper is to present recent developments in Conceptual Knowledge Processing, a branch of Knowledge Representation which is based on the mathematical theory of Formal Concept Analysis (FCA). This theory has its origins in the logical, algebraical, and geometrical roots which gave rise to the development of lattice theory by Garrett Birkhoff. The mathematical notions of closure systems, Galois connections, and complete lattices are combined in FCA (as developed since 1979 in the Research Group Concept Analysis at Darmstadt University of Technology) with basic mathematical tools for the description of practically relevant parts of the world, as for example data tables and rules. For that purpose one has to handle theoretically the concepts which are used in theory and practice. That is done by the introduction of the notion of emph{formal concepts} of emph{formal contexts} as defined by Rudolf Wille (1982). That leads to a emph{Contextual Logic} which does not start with assertions and their truth values, but with emph{concepts} which are then connected to emph{judgments} from which emph{conclusions} can be drawn. More precisely, we are using conceptual structures, namely formal contexts and their concept lattices, as conceptual semantics for the representation of knowledge frames. These conceptual semantics and their concept lattices cover all kinds of granularity, the usual continuous scales (for space and time in physics) as well as discrete or even hybrid scales. The formal concepts of the chosen semantics are then used to build judgments in the form of relational expressions which are usually represented in the rows of a database table. The experiences with the conceptual representation of database tables (or many-valued contexts) using emph{Conceptual Scaling} led us to introduce emph{Conceptual Semantic Systems (CSS)} in which the basic judgments connecting formal concepts of the chosen conceptual semantics are taken as formal objects of the underlying many-valued context. It is shown in this paper that Conceptual Semantic Systems yield a simple mathematical framework for discussing the connections between the ordinal oriented FCA and the nominal oriented relational database theory. Moreover, the previously introduced emph{Conceptual Time Systems with Actual Objects and a Time Relation (CTSOT)} which allow for a clear mathematical notion of states, situations, transitions, and life tracks are special cases of Conceptual Semantic Systems. It was shown by the author that automata and Turing machines can be represented in a natural way as special CTSOTs. The more general notion of CSSs was developed for investigating the emph{particle-wave-problem in physics}. Indeed, that problem is solved now in the following sense: for spatiotemporal CSSs one can define particles and waves as general objects such that classical particles and waves in physics can be represented as particles and waves respectively of suitably constructed spatiotemporal CSS. That also allows for a mathematical understanding of wave packets in physics as special emph{distributed objects} in a CSS. The definition of emph{aspects} of a distributed object in a CSS with respect to some view Q yields a mathematical understanding of the famous ``probability distribution of a quantum mechanical system". Applications of CSSs in industrial and clinical practice are presented.

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