Interval Temporal Description Logics

In this paper, we construct a combination HS-Litehorn of the Halpern-Shoham interval temporal logic HS [15] with the description logic DL-Litehorn [12, 1], which is a Horn extension of the standard language OWL 2 QL. The temporal operators of HS are of the form 〈R〉 (‘diamond’) and [R] (‘box’), where R is one of Allen’s interval relations After, Begins, Ends, During, Later, Overlaps and their inverses (Ā, B̄, Ē, D̄, L̄, Ō). The propositional variables of HS are interpreted by sets of closed intervals [i, j] of some flow of time (e.g., Z, R), and a formula 〈R〉φ ([R]φ) is regarded to be true in [i, j] iff φ is true in some (respectively, all) interval(s) [i′, j′] such that [i, j]R[i′, j′] in Allen’s interval algebra. In HS-Litehorn, we represent temporal data by means of assertions such as SummerSchool(RW, t1, t2) and teaches(US,DL, s1, s2), which say that RW is a summer school that takes place in the time interval [t1, t2] and US teaches DL in the time interval [s1, s2]. Note that temporal databases store data in a similar format [17]. Temporal concept and role inclusions are used to impose constraints on the data and introduce new concepts and roles. For example, AdvCourseu〈D̄〉MorningSession v ⊥ says that advanced courses are not given in the morning sessions described by 〈B̄〉LectureDayu〈A〉Lunch v MorningSession; teaches v [D]teaches claims that the role teaches is downward hereditary (or stative) in the sense that if it holds in some interval then it also holds in all of its sub-intervals; [D](〈O〉teaches t 〈D̄〉teaches) u 〈B〉teaches u 〈E〉teaches v teaches, on the contrary, states that teaches is coalesced (or upward hereditary). The inclusions teaches v [D]teaches and [D](〈O〉teaches t 〈D̄〉teaches) v teaches ensure that teaches is both upward and downward hereditary. On the other hand, ‘rising stock market’ and ‘high average speed’ are typical examples of concepts that are not downward hereditary; for a discussion of these notions see [6, 21, 18]. Although the complexity of full HS-Litehorn remains unknown, in this paper we define two fragments, HS-LiteH/flat horn and HS-Lite H[G] horn , where satisfiability and instance checking are P-complete for both combined and data complexity. Our interest in tractable description logics with interval temporal operators is motivated by possible applications in ontology-based data access (OBDA) [12] to temporal databases. In this context, we naturally require reasonably expressive yet tractable ontology and query languages with temporal constructs (although