A Compatible Approach to Temporal Description Logics

Temporal description logics (TDLs) have been studied by many researchers (see e.g., [1, 10] for surveys and [4, 2, 15] for recent or important results). These TDLs are, however, not compatible in the following sense: these are not embeddable into the standard (non-temporal) description logics (DLs), and hence the existing algorithms for testing satisfiability in the standard DLs are not available for these TDLs. Such a compatibility issue is important for obtaining reusable and practical algorithms for temporal reasoning in ontologies. In this paper, two compatible TDLs, XALC and BALCl, are introduced by combining and modifying the description logic ALC [14] and Prior’s tomorrow tense logic [12, 13]. XALC has the next-time operator, and BALCl has some restricted versions of the next-time, any-time and some-time operators, in which the time domain is bounded by a positive integer l. Semantical embedding theorems of XALC and BALCl into ALC are shown. By using these embedding theorems, the concept satisfiability problems for XALC and BALCl are shown to be decidable. The complexities of the decision procedures for XALC and BALCl are also shown to be the same complexity as that for ALC. Next, tableau calculi, T XALC (for XALC) and T BALCl (for BALCl), are introduced, and syntactical embedding theorems of these calculi into a tableau calculus, T ALC (for ALC), are proved. The completeness theorems for T XALC and T BALCl are proved by combining both the semantical and syntactical embedding theorems. Prior’s tomorrow tense logic, which is a base logic of XALC and BALCl, is regarded as the next-time fragment of linear-time temporal logic (LTL) [11], and hence XALC and BALCl may also be familiar with many users of the existing LTL-based TDLs. The bounded temporal operators in BALCl are, indeed, regarded as restricted versions of the corresponding LTL-operators. Although the standard temporal operators of LTL have an infinite (unbounded) time domain, i.e., the set ω of natural numbers, the bounded operators which are presented in this paper have a bounded time domain which is restricted by a fixed positive integer l, i.e., the set ωl := {x ∈ ω | x ≤ l}. To restrict the time domain of temporal operators is not a new idea. Such an idea has been discussed [5–9]. It is known that to restrict the time domain is a technique to obtain a decidable or efficient fragment of first-order LTL [8]. Proc. 23rd Int. Workshop on Description Logics (DL2010), CEUR-WS 573, Waterloo, Canada, 2010.