A New Framework for Reasoning about Points, Intervals and Durations

We present a new framework for reasoning about points , intervals and dura t ions-Poin t Interval Dura t i on Network ( P I D N ) . The P I D N adequately handles bo th qual i ta t ive and quant i ta ive tempora l in fo rmat ion . We show that I n terval Algebra, Point Algebra, TCSP, P D N and A P D N become special cases of P I D N . The under ly ing algebraic structure of P I D N is closed under composi t ion and intersection. Determ i n i g consistency of P I D N is N P l l a r d . However, we identify some tractable subclasses of P I D N . We show tha t path consistency is not sufficient to ensure global consistency o[ the tractable subclasses of P I D N . We identify a subclass for which enforcing 4-consistency suffices to ensure the global consistency, and prove tha t this subclass is m a x i m a l for qual i ta t ive constraints. Our approach is based on the geometric in terpreta t ion of the domains of temporal objects. Interestingly, the classical Helly\s Theorem of 1923 is used to prove the complexi ty for the tractable subclass.