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Temporal Description Logics Over Finite Traces Temporal Description Logics over Finite Traces Alessandro Artale, Andrea Mazzullo, Ana Ozaki KRDB Research Centre, Free University of Bozen-Bolzano, Italy [email protected] Abstract. Linear temporal logic interpreted over finite traces (LTLf ) has been used as a formalism for temporal specification in automated planning, process modelling and (runtime) verification. In this paper, we lift some results from the propositional case to a decidable fragment of first-order temporal logic obtained by combining the description logic ALC f with LTLf , denoted as TU ALC. We show that the satisfiability problem f in TU ALC is ExpSpace-complete, as already known for the infinite trace case, while it decreases to NExpTime when we consider finite traces with a fixed number of instants, and even to ExpTime when additionally restricting to globally interpreted TBoxes. Moreover, we investigate two f model-theoretical notions for TU ALC: we show that, differently from the infinite trace case, it enjoys the bounded model property; we also discuss the notion of insensitivity to infiniteness, providing characterisations and sufficient conditions to preserve it. 1 Introduction Since the introduction of linear temporal logic (LTL) by Pnueli [33], several (propositional and first-order) LTL-based formalisms have been developed for applications such as automated planning [11, 12, 15–17], process modelling [1, 31] and (runtime) verification of programs [14, 24, 32–35]. Related research in knowledge representation has focussed on decidable fragments of first-order LTL [28], many of them obtained by combining LTL with description logics (DLs), and for this reason called temporal description logics (TDLs) [3, 4, 30, 36]. In this latter setting, the complexity of satisfiability problems can range from ExpSpace-complete, for the full-fledged combination of the description logic ALC with LTL [23, 27], down to ExpTime- or PSpace-complete, by restricting the application of temporal operators [10, 23], or even in NP or NLogSpace, by weakening both the temporal and the description logic components, for some TDLs used in temporal conceptual data modelling [5, 6]. Besides the usual LTL semantics defined over the natural numbers, rececently attention has been devoted to study LTL semantics based on finite traces, which are temporal structures having only a finite number of time points, with the result- ing logic sometimes known as linear temporal logic over finite traces (LTLf ) [13, 18–22]. Indeed, for several of the already mentioned applications, the finiteness of the time dimension is a fairly natural restriction, since the processes involved are usually required to terminate within a finite amount of time. Motivated by similar applications, some recent work has considered TDLs in the context of runtime verification [8] and business process modelling [2]. Nonetheless, these attempts are still based on the infinite trace semantics for temporal operators (sometimes allowed only over axioms [8]). On the way to overcome such limitations, in this work we combine DLs f with LTLf . In particular, we consider the logic TU ALC defined as the DL ALC extended with temporal operators on both concepts and formulas (but without temporalised or rigid roles), and interpreted over models with a finite time f dimension. We show that the complexity of the satisfiability problem in TU ALC is ExpSpace-complete, as for the infinite case, and lowers down to NExpTime if we restrict to traces with a bound k on the number of instants, and even to ExpTime for a fragment with only globally interpreted TBoxes. We then consider f two model-theoretical notions. Firstly, we show that TU ALC has the bounded model property, with the bound being double exponential in the size of the formula. We then study the notion of a formula being insensitive to infiniteness, considered in [19, 8, 13, 14] for dealing with the possibility to preserve the satisfiability of formulas from the finite to the infinite case and thus to reuse TDL algorithms developed for the infinite case. This paper is organised as follows. In Section 2, we introduce TU ALC and we provide the semantics both over infinite and over finite traces. In Section 3 f we study the complexity of the satisfiability problem for TU ALC formulas, while Section 4 refines these results for traces with fixed k instants. Section 5 investigates f model-theoretical notions of TU ALC formulas. Finally, in Section 6, we conclude with some open problems and possible directions for future work. 2 Temporal Description Logics In the following, we first define the temporal language TU ALC as a temporal extension of the description logic ALC. The language TU ALC is obtained by extending ALC with the binary temporal operator until (U). We then define the semantics based on traces, which are temporal structures having either a f finite or an infinite number of time points. We denote by TU ALC the language TU ALC interpreted over finite traces. We further add the upperscript k and write k TU ALC to indicate that the formulas are interpreted over finite traces with at most k 2 N; k > 0; time points. Syntax. Let NC, NR and NI be countably infinite and pairwise disjoint sets of concept names, role names, and individual names, respectively. A TU ALC concept is an expression of the form: C; D ::= A j :C j C u D j 9R:C j C U D; where A 2 NC and R 2 NR. For instance, the expression A U (B u 9R::A) is a TU ALC concept. A TU ALC axiom is either a concept inclusion (CI) of the form C v D, or an assertion of the form A(a) or R(a; b), where C; D are TU ALC concepts, A 2 NC, R 2 NR, and a; b 2 NI. TU ALC formulas are of the form: '; ::= α j :' j ' ^ j ' U ; where α is a TU ALC axiom. I Semantics. A (TDL) interpretation is a structure I = (∆ ; (In)n2I ), where I is an interval of the form [0; 1) or [0; l], with l 2 N, and each In is a classical DL interpretation with domain ∆I: we have AIn ⊆ ∆I, RIn ⊆ ∆I×∆I, and aIi = Ij I a 2 ∆ for all a 2 NI and i; j 2 N, i.e., the interpretation of individual names is fixed (in the following denoted as aI ). We refer to a temporal interpretation I I = (∆ ; (In)n2I ) as a finite trace if I = [0; l], and as an infinite trace if I = [0; 1). The stipulation that all time points share the same domain ∆I is called the constant domain assumption (meaning that objects are not created or destroyed over time), and it is the most general choice in the sense that increasing, decreasing, and varying domains can all be reduced to it [23]. We now define f the semantics of TU ALC considering I = [0; 1). The semantics of TU ALC is defined in the same way except that I is of the form [0; l], for some l 2 N. For k a fixed k 2 N; k > 0, we define the semantics of TU ALC considering I = [0; l] with l ≤ k − 1. The interpretation of concepts at instant n of interpretation I I = (∆ ; (In)n2I ) is defined as follows: (:C)In = ∆I n CIn ; (C u D)In = CIn \ DIn ; (9R:C)In = fd 2 ∆I j 9d0 2 CIn :(d; d0) 2 RIn g; (C U D)In = fd 2 ∆I j 9m 2 I; m > n: d 2 DIm ^ 8i 2 (n; m): d 2 CIi g: We say that a concept C is satisfied in I if CI0 6= ;. Satisfaction of a formula ' in I at time point n 2 I (written I; n j= ') is inductively defined as follows: I; n j= C v D iff CIn ⊆ DIn ; I; n j= :φ iff not I; n j= φ, I; n j= A(a) iff aI 2 AIn ; I; n j= φ ^ iff I; n j= φ and I; n j= ; I; n j= R(a; b) iff (aI ; bI ) 2 RIn ; I; n j= φ U iff 9m 2 I; m > n: I; m j= ; and 8i 2 (n; m): I; i j= φ We say that ' is satisfied in I, writing I j= ', if I; 0 j= '. A formula ' logically implies a formula if every I that satisfies ' satisfies also , and we write ' j= . f Moreover, ' is said to be TU ALC satisfiable (or satisfiable over finite traces) if it is satisfied in some finite trace. Given a concept C and a formula ', we say that C is satisfiable over finite traces with respect to ' if there is a finite trace I such that ' is satisfied in I and C is satisfied in I, in symbols ' 6j= C v ?. Concept satisfiability w.r.t. ' can be reduced to formula satisfiability by checking whether the following formula is satisfiable: ' ^ C(a), for a fresh a 2 NI. We will use the following standard equivalences for concepts: ? ≡ A u :A, > ≡ :?; 8R:C ≡ :9R::C; (C t D) ≡ :(:C u :D); C ≡ 1C ≡ ? U C; n n−1 C ≡ C, with n > 1; C ≡ >U C; C ≡ : :C#. Similar# abbreviations F #also hold## for formulas. We often3 write F =2 (∆ ; (Fn3)n2[0;l]) to indicate that a I trace is finite and I = (∆ ; (In)n2I ) when it is infinite. 3 Satisfiability over Finite Traces In the following we show how to reduce the formula satisfiability problem over finite traces to the same problem over infinite traces. Similar to the encoding proposed in [20] for propositional LTL, we introduce a fresh concept name E to capture the finiteness of the temporal dimension.
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