On Completeness of a Positional Interval Logic with Equality, Overlap and Subinterval Relations

Abstract

This paper presents an interval-based positional logic and proves its completeness. The proposed logic combines positional operators, defined by time intervals, with modal operators used to express the subinterval relationship. Moreover, the logic can be easily extended either by including more predicates for absolute temporal references or by adding other operators for relative and periodical temporal references. The positional logic can be used to express and reason about time-stamped temporal information, particularly stored in temporal databases. The underlying time structure is motivated by the common-sense calendar-clock style time, where multiple time granularities to any precision, often involved in temporal information, can be very naturally supported. Although each positional/modal operator is normal, the completeness proof of the proposed positional logic is more complicated than the standard completeness proof of normal modal logics.