Minimal Periodical Representations of Calendar Expressions

The formalization of time granularity, the study of computational methods to manipulate granularities and their applications have been the subject of much research in the recent years. This paper aims at filling the gap between symbolic representations defined thrugh calendar algebras and mathematical representations defined through periodical sets of instants. Considering a specific calendar algebra, the paper shows how each algebraic operator changes the periodical structure of the granularities given as operands. This leads to a conversion procedure between calendar expressions and periodical representations of granularities. From a practical point of view, our results allow users to specify granularities using the algebra (which is more natural) and to process them as periodical sets (as most current computational methods do). The paper also shows experimental evidence that the provided mapping is in most cases optimal and presents an efficient post-processing method for those cases where optimality must be always guaranteed.