Relations between fuzzy time intervals

Time intervals like "tonight", which are usually not very precise, can be modeled as fuzzy sets. But this causes the problem that the relations between points and intervals and between two intervals, which are usually very trivial, become very complex when the intervals are fuzzy sets. Moreover, there are many different possibilities to define such relations. In this paper a very flexible operator-based approach to point-interval and interval-interval relations is proposed, where the intervals are fuzzy time intervals over the real numbers. The relations yield non-trivial fuzzy values even if the intervals are crisp. As an example for an application, consider a database with, say, a cinema timetable, and you query the timetable give me all performances ending before midnight. The usual "before" relation will exclude the performances ending a second after midnight. With the fuzzy before relation you can get instead of a sharp drop to 0 at midnight decreasing fuzzy values after midnight, and these can be used to order the results of the query. The intervals and relations are implemented in the Fu-TIRe library (Fuzzy Time intervals and Relations). FuTIRe is an open source C++ library.