Abstract
The key notion for modelling calendar systems and many other periodic temporal notion is the mathematical concept of a partitioning of the real numbers. A partitioning of R splits the time axis into a sequence of intervals. Basic time units like seconds, minutes, hours, days, weeks, months, years etc. can all be represented as partitionings of R with finite partitions. Besides the basic time units in calendar systems, there are a lot of other temporal notions which can be modelled as partitions: the seasons, the ecclesiastical calendars, financial years, semesters at universities, the sequence of sunrises and sunsets, the sequence of the tides, the sequence of school holidays etc. Almost all systems for modelling periodic temporal notions developed so far identify partitions, granules or whatever they are called, by labels or coordinates which are essentially sequences of integers. In this paper it is show how these integer coordinates as identifiers for partitions can be generalised to 'partition access specifiers’. An example for a non-trivial partition access specifier is a path in a tree which represents hierarchically nested partitions. A bus timetable, for example, can be specified this way: ’(in very winter, in every week, (in day 0-4, hour 5, minute 20, bus B1, hour 6, minute 20 bus B2 ...), (in day 5-6, hour 8, minute 20 bus B1, ...)), (in every spring ...)...’. A particular 'partition access specifiers’ is then a sequence of integers ’10/1/2/..’. The first integer represents an absolute coordinate (season 10). The other integers represent shifts: 1 week after the start of season 10, 2 days after the start of this week etc. The main data structures and algorithms for these 'tree partitionings’ are presented in this paper.

URL:
http://rewerse.net/publications/rewerse-publications.html#REWERSE-TR-2006-07

BibTeX:

@techreport{REWERSE-TR-2006-07,
	author = {Hans J\"urgen Ohlbach},
	title = {Periodic temporal notions as ’tree partitionings’},
	institution = {Institute for Informatics, University of Munich},
	year = {2006},
	type = {{research report, REWERSE-TR-2006-07}},
	number = {REWERSE-TR-2006-07},
	url = {http://rewerse.net/publications/rewerse-publications.html#REWERSE-TR-2006-07}
}