Pseudo-Permutations II: Geometry and Representation Theory

Representation and reasoning about time-dependent information is an active and important research area in Artificial Intelligence and in Computer Science. Vilain and Kautz’s point algebra (see [VK86]) is one of the most influential models used for representing and reasoning with qualitative time information, because the reasoning tasks are solvable in polynomial time in that model, contrary to more powerful models, as Allen’s Interval Algebra (see [All81]) in which the reasoning tasks are computationally intractable in general. The problem is the following: one considers a set of events which happen at certain dates, and wants to use this information to solve a problem, take a decision. However, it is often not meaningful when the events occur, while the relevant informations are the temporal relations between events: did event i happened before, during or after event j ? In this context, it is natural to represent the temporal relations between n events by an ordered sequence of nonempty parts of 1 n such that each integer appears exactly once. If i is in a part of 1 n which appears in the sequence before the part which contains j, then the event i happened before the event j. If they appear in the same part, they occured simultaneously. For example, the sequence 1 3 4 2 means that event 1 occured first, event 2 occured last, and events 3 and 4 occured at the same time. In the following, we will call such a sequence a pseudopermutation of order n (in the example, n 4), we will use parentheses instead of braces, and we will remove unnecessary comas. Therefore, we will write the example as 1 34 2 . Having defined the pseudo-permutations, one can study their combinatorial properties. They appear to have a very rich structure and many interesting properties that were first presented in [KLNPS00]. Among