Time Units and Calendars

We are investigating a formal representation of time units and calendars, as restricted temporal entities for reasoning about activities. We examine characteristics of time units, and provide a categorization of the hierarchical relations among them. Hence we define an abstract hierarchical unit structure (a calendar structure) that expresses specific relations and properties among the units that compose it. Calendar structures subsume systems that can be based on discrete units together with a repetitive containment relation. The motivation for this work is to (ultimately) be able to reason about schedulable, repeated activities, specified using calendars. Examples of such activities include going to a specific class every Tuesday and Thursday during a semester, attending a seminar every first day of a month, and going to swim every other day. Defining a precise representation and developing or adapting known efficient algorithms to this domain would provide a valuable framework for scheduling systems. In a representation scheme for such activities, one would ideally be able to determine consistency among several repeated activities, find a minimal set of potential ways repeated activities may interact, convert between time units, etc. Such a structure would be a restriction of, for example, the general algebra of intervals (Allen 1983); hence one might hope that the resulting restricted structure would have good computational properties. Work has been done around repeated activities, for example (Poesio & Brachman 1991), where a main concern is the implementation of variants of constraint propagation algorithms to detect overlapping repeated activities. We search for a different (more general and formalized) representation of the temporal entities. We explore further the date concept, used in their work as a reference interval, building a structure that formalizes dates in calendars. Human, schedulable activities are based on conventional systems called calendars. Examples of calendars include the Gregorian calendar as well as uni1436 Student Abstracts versity calendars and a business calendar, where the latter calendars are defined in terms of the Gregorian. Calendars are repetitive, cyclic temporal objects. We define an abstract structure that formalizes calendars as composed of time units which are related by a decomposition relation, which is a containment relation involving repetition and other specific characteristics. Time units decompose into contiguous sequences of other time units in various ways. For example, the expressions year k month, with Cons(year,month,l2) and Alig(year,month) indicate that the time unit year decomposes into month in a constant and aligned way, with a repetition factor of 12. We have studied transitivity properties of constancy and alignment in the decomposition relation. The decomposition relation is defined as a partial order on the set of time units. A calendar structure is a hierarchical structure based on the decomposition of time units and expresses relationships that hold between them in several calendars. At this point in the research, the formalism is under systematic study, particularly regarding its mathematical properties. Representation of specific temporal objects based on the formalism are under study also. Still to be addressed are considerations about algorithms that would best fit with this formalism,. so that we may obtain efficient inferences when reasoning about single and repeated activities. Algorithms developed for temporal constraint satisfaction problems, or variations, will be considered in this matter. We believe also that the hierarchy defined represents a generic approach, appropriate to represent any measurement system based on discrete units that relate with a repetitive contaimnent relation, such as the Metric or Imperial measurement systems.