Supporting temporal reasoning by mapping calendar expressions to minimal periodic sets

In the recent years several research efforts have focused on the concept of time granularity and its applications. A first stream of research investigated the mathematical models behind the notion of granularity and the algorithms to manage temporal data based on those models. A second stream of research investigated symbolic formalisms providing a set of algebraic operators to define granularities in a compact and compositional way. However, only very limited manipulation algorithms have been proposed to operate directly on the algebraic representation making it unsuitable to use the symbolic formalisms in applications that need manipulation of granularities. This paper aims at filling the gap between the results from these two streams of research, by providing an efficient conversion from the algebraic representation to the equivalent low-level representation based on the mathematical models. In addition, the conversion returns a minimal representation in terms of period length. Our results have a major practical impact: users can more easily define arbitrary granularities in terms of algebraic operators, and then access granularity reasoning and other services operating efficiently on the equivalent, minimal low-level representation. As an example, we illustrate the application to temporal constraint reasoning with multiple granularities. From a technical point of view, we propose an hybrid algorithm that interleaves the conversion of calendar subexpressions into periodical sets with the minimization of the period length. The algorithm returns set-based granularity representations having minimal period length, which is the most relevant parameter for the performance of the considered reasoning services. Extensive experimental work supports the techniques used in the algorithm, and shows the efficiency and effectiveness of the algorithm.