The granularity of given temporal information is the level of abstraction at which information is expressed. Different units of measure allow one to represent different granularities. Indeterminacy is often present in temporal information given at different granularities: temporal indeterminacy is related to incomplete knowledge of when the considered fact happened. Focusing on temporal databases, different granularities and indeterminacy have to be considered in expressing valid time, i.e., the time at which the information is true in the modeled reality. In this paper, we propose HMAP (The term is the transliteration of an ancient Greek poetical word meaning “day”.), a temporal data model extending the capability of defining valid times with different granularity and/or with indeterminacy. In HMAP, absolute intervals are explicitly represented by their start,end, and duration: in this way, we can represent valid times as “in December 1998 for five hours”, “from July 1995, for 15 days”, “from March 1997 to October 15, 1997, between 6 and 6:30 p.m.”. HMAP is based on a three-valued logic, for managing uncertainty in temporal relationships. Formulas involving different temporal relationships between intervals, instants, and durations can be defined, allowing one to query the database with different granularities, not necessarily related to that of data. In this paper, we also discuss the complexity of algorithms, allowing us to evaluate HMAP formulas, and show that the formulas can be expressed as constraint networks falling into the class of simple temporal problems, which can be solved in polynomial time.
