Proceedings, Eleventh International Conference on Principles of Knowledge Representation and Reasoning (2008)

LTL over Description Axioms

Franz Baader∗ Silvio Ghilardi Carsten Lutz TU Dresden, Germany Universit`adegli Studi di Milano, Italy TU Dresden, Germany [email protected] [email protected] [email protected]

Abstract tion of DLs with Halpern and Shoham’s logic of inter- vals (Schmiedel 1990), formalisms inspired by action log- Most of the research on temporalized Description Log- ics (Artale & Franconi 1998), the treatment of time points ics (DLs) has concentrated on the case where tempo- ral operators can occur within DL concept descriptions. and intervals as a concrete domains (Lutz 2001), and the In this setting, reasoning usually becomes quite hard combination of standard DLs with standard (propositional) if rigid roles, i.e., roles whose interpretation does not temporal into logics with a two-dimensional seman- change over time, are available. In this paper, we con- tics, where one dimension is for time and the other for the sider the case where temporal operators are allowed to DL domain (Schild 1993; Wolter & Zakharyaschev 1999; occur only in front of DL axioms (i.e., ABox assertions Gabbay et al. 2003). In this paper, we follow the last ap- and general concept inclusion axioms), but not inside proach, where we use the basic DL ALC (Schmidt-Schauß of concepts descriptions. As the temporal component, & Smolka 1991) in the DL component and linear tempo- we use (LTL) and in the DL com- ral logic (LTL) (Pnueli 1977) in the temporal component. ponent we consider the basic DL ALC. We show that However, even after the DL and the temporal logic to be reasoning in the presence of rigid roles becomes con- siderably simpler in this setting. combined have been fixed, there remain several degrees of freedom when defining the resulting temporalized DL. On the one hand, one must decide which pieces of syntax Introduction temporal operators can be applied to. Temporal operators In many applications of Description Logics (DLs) (Baader may be allowed to be used as concept constructors, as re- et al. 2003), such as the use of DLs as ontology lan- quired by the above example of a concussion with no loss guages or conceptual modeling languages, being able to rep- of consciousness, which could be defined using the until- resent dynamic aspects of the application domain would be operator U of LTL as follows: quite useful. This is, for instance, the case if one wants ∃finding.Concussion ⊓ (1) to use DLs as conceptual modeling languages for temporal Conscious U ∃procedure.Examination. databases (Artale et al. 2002). Another example are medi- cal ontologies, where the faithful representation of concepts Alternatively or in addition, temporal operators may be would often require the description of temporal patterns. As applied to TBox axioms (i.e., general concept inclusions, a simple example, consider the concept “Concussion with GCIs) and/or to ABox assertions. For example, the tem- no loss of consciousness,” which is modeled as a primitive poralized TBox axiom (i.e., not further defined) concept in the medical ontology 32 UScitizen insured by HealthInsurer SNOMED CT.1 As argued in (Schulz, Mark´o, & Suntis- ( ⊑ ∃ . ) rivaraporn 2006), a correct representation of this concept says that there is a future time point from which on US citi- should actually say that, after the concussion, the patient re- zens will always have health insurance, and the formula Ψ: mained conscious until the examination. 3 finding Concussion BOB Since the expressiveness of pure DLs is not sufficient (∃ . )( ) ∧ (2) to describe such temporal patterns, a plethora of tempo- Conscious(BOB)U(∃procedure.Examination)(BOB) ral extensions of DLs have been investigated in the litera- says that, sometime in the future, Bob will have a concussion ture.2 These include approaches as diverse as the combina- with no loss of consciousness between the concussion and ∗Supported by NICTA, Canberra Research Lab. the examination. Copyright c 2008, Association for the Advancement of Artificial On the other hand, one must decide whether one wants Intelligence (www.aaai.org). All rights reserved. to have rigid concepts and/or roles, i.e., concepts/roles 1 see http://www.ihtsdo.org/our-standards/ whose interpretation does not vary over time. For exam- 2 For a more thorough survey of the literature on temporalized ple, the concept Human and the role has father should DLs, see the technical report accompanying this paper (Baader, Ghilardi, & Lutz 2008) and the survey papers (Artale & Franconi 2000; 2001; Lutz, Wolter, & Zakharyaschev 2008).

684 probably be rigid since a human being will stay a hu- rem 14.15 on page 605 there). However, also in (Gabbay et man being and have the same father over his/her life-time, al. 2003), the setting where temporal operators are allowed whereas Conscious should be a flexible concept (i.e., not to occur only in front of axioms is considered only in the rigid) since someone that is conscious at the moment need absence of rigid symbols. not always by conscious. Similarly, insured by should Obviously, the temporalized DLs we investigate in this be modeled as a flexible role. Using a logic that can- paper cannot be used to define temporal concepts such as not enforce rigidity of concepts/roles may result in un- (1) for concussion with no loss of consciousness. However, intended models, and thus prevent certain useful infer- they are nevertheless useful in ontology-based applications ences to be drawn. For example, the concept description since they can be used to reason about a temporal sequence ∃has father.Human ⊓ 3 (∀has father .¬Human) is only un- of ABoxes w.r.t. a (global or temporalized) TBox. For exam- satisfiable if both has father and Human are rigid. ple, in an emergency ward, the vital parameters of a patient The combination of (extensions of) ALC and LTL in are monitored in short intervals (sometimes not longer than which temporal operators can be applied to concept descrip- 10 minutes), and additional information is available from tions, TBox axioms, and ABox assertions has been studied the patient record and added by doctors and nurses. Using by Wolter, Zakharyaschev, and others (see, e.g., (Wolter & concepts defined in a medical ontology like SNOMED CT, Zakharyaschev 1999; Gabbay et al. 2003)). In particular, a high-level view of the medical status of the patient at a it is known that the basic reasoning problems such as satis- given time point can be given by an ABox. Obviously, the fiability are EXPSPACE-complete. In this setting, rigid con- sequence of ABoxes obtained this way can be described us- cepts can be defined, but rigid roles cannot. In fact, as shown ing temporalized ABox assertions. Critical situations, which in (Gabbay et al. 2003), the addition of rigid roles causes un- require the intervention of a doctor, can then be described by decidability even w.r.t. a global TBox (i.e., where the same a formula in our temporalized DL, and recognized using the TBox axioms must hold at all time points). Decidability can reasoning procedures developed in this paper. For example, be regained by dropping TBoxes altogether, but the deci- given a formula φ encoding a sequence of ABoxes describ- sion problem is still hard for non-elementary time. Decid- ing the medical status of Bob, starting at some time point able combinations of DLs and temporal logics that allow for t0, and the formula ψ defined in (2), we can check whether rigid roles can be obtained by strongly restricting either the Bob sometime after t0 had a concussion with no loss of con- temporal component (Artale, Lutz, & Toman 2007) or the sciousness by testing φ ∧ ¬ψ for unsatisfiability. DL component (Artale et al. 2007). In this paper, we follow a different approach for regain- Basic definitions ing decidability in the presence of rigid roles: temporal op- erators are allowed to occur only in front of axioms (i.e., The temporalized DL ALC-LTL introduced in this paper ABox assertions and TBox axioms), but not inside concept combines the basic DL ALC with linear temporal logic descriptions. We show that reasoning becomes simpler in (LTL). We start by recalling the relevant definitions for this setting: with rigid roles, satisfiability is decidable (more ALC. precisely: 2-EXPTIME-complete); without rigid roles, the Definition 1. Let NC, NR, and NI respectively be disjoint complexity decreases to NEXPTIME-complete; and with- sets of concept names, role names, and individual names. out any rigid symbols, it decreases further to EXPTIME- The set of ALC-concept descriptions is the smallest set such complete (i.e., the same complexity as reasoning in ALC that alone). We also consider two other ways of decreasing the • all concept names are ALC-concept descriptions; complexity of satisfiability to EXPTIME. On the one hand, satisfiability without rigid roles (but with rigid concepts) be- • if C,D are ALC-concept descriptions and r ∈ NR, then comes EXPTIME-complete if GCIs can occur only as global ¬C, C ⊔ D, C ⊓ D, ∃r.C, and ∀r.C are ALC-concept axioms that must hold in every temporal world. Note that, in descriptions. this case, ABox assertions are not assumed to be global, i.e., A general concept inclusion axiom (GCI) is of the form the valid ABox assertions may vary over time. On the other C ⊑ D, where C,D are ALC-concept descriptions, and hand, satisfiability with rigid concepts and roles becomes an assertion is of the form a : C or (a, b): r where C is EXPTIME-complete if the temporal component is restricted an ALC-concept description, r is a role name, and a, b are appropriately by replacing the temporal operators until (U) individual names. We call both GCIs and assertions - 3 ALC and next (X) of LTL with diamond ( ), which expresses axioms. A Boolean combination of ALC-axioms is called a “sometime in the future.” Boolean ALC-knowledge base, i.e., The situation we concentrate on in this paper (i.e., where temporal operators are allowed to occur only in front of ax- • every ALC-axiom is a Boolean ALC-knowledge base; ioms) has been considered before only for the case where • if B1 and B2 are Boolean ALC-knowledge bases, then so there are no rigid concepts or roles. The combination ap- are B1 ∧ B2, B1 ∨ B2, and ¬B1. proach introduced in (Finger & Gabbay 1992) yields a de- An -TBox is a conjunctionof GCIs, and an -ABox cision procedure for this case, whose worst-case complex- ALC ALC is a conjunction of assertions. ity is, however, non-optimal. Our EXPTIME upper bound for this case actually also follows from more general results According to this definition, TBoxes and ABoxes are spe- in (Gabbay et al. 2003) (see the remark following Theo- cial kinds of Boolean knowledge bases. However, note that

685 they are often written as sets of axioms rather than as con- time point i ∈ {0, 1, 2,...}, validity of φ in I at time i (writ- junctions of these axioms. The semantics of ALC is defined ten I, i |= φ) is defined inductively: through the notion of an interpretation. I, i |= C ⊑ D iff CIi ⊆ DIi Definition 2. An interpretation is a pair I I where I = (∆ , · ) I, i |= a : C iff aIi ∈ CIi the domain ∆I is a non-empty set, and ·I is a function that I, i |= (a, b): r iff (aIi , bIi ) ∈ rIi assigns to every concept name A a set AI ⊆ ∆I, to every I, i |= φ ∧ ψ iff I, i |= φ and I, i |= ψ role name r a rI ⊆ ∆I × ∆I , and to every I, i |= φ ∨ ψ iff I, i |= φ or I, i |= ψ individual name an element I I . This function is a a ∈ ∆ I, i |= ¬φ iff not I, i |= φ extended to ALC-concept descriptions as follows: I, i |= Xφ iff I, i + 1 |= φ • (C ⊓ D)I = CI ∩ DI, (C ⊔ D)I = CI ∪ DI, (¬C)I = I, i |= φUψ iff there is k ≥ i such that I, k |= ψ ∆I \ CI ; and I, j |= φ for all j, i ≤ j < k I I there is a I with I • (∃r.C) = {x ∈ ∆ | y ∈ ∆ (x, y) ∈ r As mentioned above, for some concepts and roles it is not and y ∈ CI }; desirable that their interpretation changes over time. Thus, I I I I • (∀r.C) = {x ∈ ∆ | for all y ∈ ∆ , (x, y) ∈ r we will sometimes assume that a subset of the set of concept I implies y ∈ C }. and role names can be designated as being rigid. We will The interpretation I is a model of the ALC-axioms C ⊑ D, call the elements of this subset rigid concept names and rigid a : C, and (a, b): r iff it respectively satisfies CI ⊆ DI , role names. Concept and role names that do not belong to aI ∈ CI , and (aI , bI) ∈ rI . The notion of a model is this subset will be called flexible. extended to Boolean ALC-knowledge bases as follows: Definition 5. We say that the ALC-LTL structure I = •I is a model of B1 ∧ B2 iff it is a model of B1 and B2; (Ii)i=0,1,... respects rigid concept names (role names) iff AIi = AIj (rIi = rIj ) holds for all i, j ∈ {0, 1, 2,...} •I is a model of B1 ∨ B2 iff it is a model of B1 or B2; and all rigid concept names A (rigid role names r). •I is a model of ¬B1 iff it is not a model of B1. We say that the Boolean ALC-knowledge base B is consis- The satisfiability problem in ALC-LTL tent iff it has a model. The concept description C is sat- Dependingon whether rigid conceptand role names are con- isfiable w.r.t. the GCI D ⊑ D iff there is a model I of 1 2 sidered or not, we obtain different variants of the satisfiabil- D ⊑ D with CI 6= ∅. 1 2 ity problem. In Description Logics, it is often assumed that the inter- Definition 6. Let φ be an ALC-LTL formula and assume pretations satisfy the unique name assumption (UNA), i.e., that a subset of the set of concept and role names has been different individual names are interpreted by different ele- designated as being rigid. ments of the domain. For LTL, we use the variant with a non-strict until (U) and • We say that φ is satisfiable w.r.t. rigid names iff there is an a next (X) operator. Instead of first introducing the proposi- ALC-LTL structure I respecting rigid concept and role tional temporal logic LTL, we directly define our new tem- names such that I, 0 |= φ. poralized DL, called ALC-LTL. The difference to LTL is • We say that φ is satisfiable w.r.t. rigid concepts iff there is that ALC-axioms replace propositional letters. an ALC-LTL structure I respecting rigid concept names I Definition 3. ALC-LTL formulae are defined by induction: such that , 0 |= φ. • We say that φ is satisfiable without rigid names (or simply • if α is an ALC-axiom, then α is an ALC-LTL formula; satisfiable) iff there is an ALC-LTL structure I such that • if φ, ψ are ALC-LTL formulae, then so are φ ∧ ψ, φ ∨ ψ, I, 0 |= φ. ¬φ, φUψ, and Xφ. In this paper, we show that the complexity of the satisfia- As usual, we use true as an abbreviation for A(a) ∨ bility problem for ALC-LTL strongly depends on which of ¬A(a), 3φ as an abbreviation for trueUφ (diamond, which the above cases one considers. Note that it does not really should be read as “sometime in the future”), and 2φ as an make sense to consider satisfiability w.r.t. rigid role names, abbreviation for ¬3¬φ (box, which should be read as “al- but without rigid concept names, as a separate case when in- ways in the future”). The semantics of ALC-LTL is based vestigating the complexity of the satisfiability problem. In on ALC-LTL structures, which are sequences of ALC- fact, rigid concepts can be simulated by rigid roles: just interpretations over the same non-empty domain ∆ (con- introduce a new rigid role name rA for each rigid concept stant domain assumption). We assume that every individ- name A, and then replace A by ∃rA.⊤. ual name stands for a unique element of ∆ (rigid individual Another dimension that influences the complexity of the names), and we make the unique name assumption. satisfiability problem is whether GCIs occur globally or lo- Definition 4. An ALC-LTL structure is a sequence I = cally in the formula. Intuitively, a GCI occurs globally if it Ii must hold in every world of the -LTL structure. (Ii)i=0,1,... of ALC-interpretations Ii = (∆, · ) obeying ALC the UNA (called worlds) such that aIi = aIj for all individ- Definition 7. We say that φ is an ALC-LTL formula with ual names a and all i, j ∈ {0, 1, 2,...}. Given an ALC-LTL global GCIs iff it is of the form φ = 2B ∧ ϕ where B is a formula φ, an ALC-LTL structure I = (Ii)i=0,1,..., and a conjunction of ALC-axioms and ϕ is an ALC-LTL formula

686 that doesnot containGCIs. We denote the fragment of ALC- rigid names. We build its propositional abstraction φ by re- LTL that contains only ALC-LTL formulae with global GCIs placing each ALC-axiom by a propositional variable such b by ALC-LTL|gGCI. that there is a 1–1 relationship between the ALC-axioms Note that saying, in the above definition, that B is a con- α1, . . . , αn occurring in φ and the propositional variables junction of ALC-axioms just means that B is a TBox to- p1, . . . , pn used for the abstraction. We assume in the fol- gether with an ABox. We could have restricted B to being lowing that pi was used to replace αi (i = 1, . . . , n). a conjunction of GCIs (i.e., a TBox) since assertions α in B Consider a set S ⊆ P({p1, . . . , pn}), i.e., a set of subsets could be moved as conjuncts 2α to ϕ.3 However, it turns out of {p1, . . . , pn}. Such a set induces the following (proposi- to be moreconvenientto allow also ABox assertions to occur tional) LTL formula: in the “global part” 2B of φ. Also note that it is important to restrict B to being a conjunction of ALC-axioms rather φS := φ ∧ 2   p ∧ ¬p than an arbitrary Boolean ALC-knowledge base. In fact, X_∈S p^∈X p^6∈X the lower complexity for the case of satisfiability w.r.t. rigid b b    concepts obtained in this case (see Table 1 below) would not If φ is satisfiable in an ALC-LTL structure I = (Ii)i=0,1,..., hold without this restriction (see Corollary 6.8 in (Baader, then there is an S ⊆ P({p1, . . . , pn}) such that φS is sat- Ghilardi, & Lutz 2008)). isfiable in a propositional LTL structure. In fact, for each b Instead of restricting to ALC-LTL formulae with global ALC-interpretation Ii of I, we define the set GCIs, we can also restrict the temporal component, by con- Xi := {pj | 1 ≤ j ≤ n and Ii satisfies αj}, sidering the fragment ALC-LTL|3 of ALC-LTL in which 3 is the only temporal operator. In this fragment, neither U nor and then take S = {Xi | i = 0, 1,...}. We say that X is definable. S is induced by the ALC-LTL structure I = (Ii)i=0,1,.... The fact that I satisfies φ implies that its propositional ab- Definition 8. ALC-LTL|3 formulae are defined by induc- tion: straction satisfies φS , where the propositional abstraction I = (wi)i=0,1,... of I is defined such that world wi makes • if α is an ALC-axiom, then α is an ALC-LTL|3 formula; b variable pj true iff Ii satisfies αj. However, guessing such if are -LTL 3 formulae, then so are , , b • φ, ψ ALC | φ∧ψ φ∨ψ a set S ⊆ P({p1, . . . , pn}) and then testing whether the ¬φ, and 3φ. induced propositional LTL formula φS is satisfiable is not The semantics of ALC-LTL|3 formulae is defined as in sufficient for checking satisfiability w.r.t. rigid names of the the case of ALC-LTL. In particular, the interpretation of the ALC-LTL formula φ. We must alsob check whether the diamond operator is defined as guessed set S can indeed be induced by some ALC-LTL structure that respects the rigid concept and role names. I, i |= 3φ iff there is k ≥ i such that I, k |= φ. To this purpose, assume that a set S = {X1,...,Xk} ⊆ Table 1 summarizes the results of our investigation of the P({p1, . . . , pn}) is given. For every i, 1 ≤ i ≤ k, and every complexity of the satisfiability problem in ALC-LTL and its flexible concept name A (flexible role name r) occurring in (i) (i) (i) fragments. α1, . . . , αn, we introduce a copy A (r ). We call A (i) (i) (r ) the ith copy of A (r). The ALC-axiom αj is obtained Reasoning with rigid names from αj by replacing every occurrence of a flexible name by its ith copy. The sets X (1 ≤ i ≤ k) induce the following In this section, we investigate the complexity of the satis- i Boolean ALC-knowledge bases: fiability problem in ALC-LTL and ALC-LTL|gGCI if rigid concepts and roles are available. (i) (i) Bi := αj ∧ ¬αj Theorem 9. Satisfiability w.r.t. rigid names is 2-EXPTIME- pj^∈Xi pj^6∈Xi complete both in ALC-LTL and in ALC-LTL|gGCI. Lemma 10. The ALC-LTL formula φ is satisfiable w.r.t. 2-EXPTIME-hardness for satisfiability w.r.t. rigid names rigid names iff there is a set S = {X1,...,Xk} ⊆ and with global GCIs (i.e., in ALC-LTL|gGCI) can be shown P({p1, . . . , pn}) such that the propositional LTL formula by a (quite intricate) reduction of the word problem for ex- φS is satisfiable and the Boolean ALC-knowledge base ponentially space bounded alternating Turing machines (see B := 1≤i≤k Bi is consistent. the Appendix). Obviously, this also yields 2-EXPTIME- b Proof.V For the “only if” direction, recall that we have hardness for the more general case with arbitrary GCIs (i.e., already seen how an ALC-LTL structure I = (I ) in ALC-LTL). ι ι=0,1,... satisfying canbeusedtodefineaset In the following, we prove the complexity upper bound φ S ⊆ P({p1, . . . , pn}) for ALC-LTL. Obviously, this also establishes the same up- such that φS is satisfiable. Let S = {X1,...,Xk}. For each ι = 0, 1,... there is an index iι ∈ {1, . . . , k} such that Iι per bound for the restricted case of ALC-LTL|gGCI. Thus, let b φ be an ALC-LTL formula to be tested for satisfiability w.r.t. induces the set Xiι , i.e., Xi = {pj | 1 ≤ j ≤ n and Iι satisfies αj}, 3This is the reason why we talk about ALC-LTL formulae with ι global GCIs in this case, rather than about ALC-LTL formulae and, conversely, for each i ∈ {1, . . . , k} thereis an index ι ∈ with global axioms. {0, 1, 2,...} such that i = iι. Let ι1, . . . , ιk ∈ {0, 1, 2,...}

687 W.r.t. rigid names W.r.t. rigid concepts Without rigid names ALC -LTL 2-EP X TIME-complete NEP X TIME-complete EP X TIME-complete ALC -LTL|gGCI 2-EP X TIME-complete EP X TIME-complete EP X TIME-complete ALC -LTL|3 EP X TIME-complete EP X TIME-complete EP X TIME-complete

Table 1: Complexity of the satisfiability problem in ALC-LTL and its fragments.

be such that iι1 = 1, . . . , iιk = k. The ALC-interpretation The Boolean ALC-knowledge base B is a conjunction of n Ji is obtained from Iιi by interpreting the ith copy of each k ≤ 2 Boolean ALC-knowledge bases Bi, where the size flexible name like the original flexible name, and by for- of each Bi is polynomial in the size of φ. The consistency getting about the interpretations of the flexible names. By problem for Boolean ALC-knowledge base is EXPTIME- our construction of Ji and our definition of the Boolean complete (see, e.g., Theorem 2.27 in (Gabbay et al. 2003)). ALC-knowledge base Bi, we have that Ji is a model of Consequently, consistency of B can also be tested in double-

Bi. Recall that the interpretations Iι1 ,..., Iιk (and thus also exponential time in the size of the input formula φ. J1,..., Jk) all have the same domain. In addition, the in- Overall, we thus have double-exponentially many tests, terpretations of the rigid names coincide in Iι1 ,..., Iιk (and where each test takes double-exponential time. This pro- thus also in J1,..., Jk) and the flexible symbols have been vides us with a double-exponential bound for testing satisfi- renamed. Thus, the union J of J1,..., Jk is a well-defined ability in ALC-LTL w.r.t. rigid names based on Lemma 10. ALC-interpretation, and it is easy to see that it is a model of The hardness proof given in the Appendix shows that this B = 1≤i≤k Bi. double-exponential upper bound is indeed optimal. How- ToV show the “if” direction, assume that there is a set ever, to get an intuition for what actually makes the prob- S = {X1,...,Xk} ⊆ P({p1, . . . , pn}) such that φS is lem so hard, let us analyze in more detail where the al- satisfiable and B := B is consistent. Let I = gorithm sketched above needs to spend double-exponential 1≤i≤k i b time. The algorithm needs (w ) be a propositionalV LTL structure satisfying φ , ι ι=0,1,... b S 2n and let J be an ALC-interpretation satisfying B. By the def- 1. to consider 2 many subsets S = {X1,...,Xk} of b P({p1, . . . , pn}); inition of φS , for every world wι there is exactly one index iι ∈ {1, . . . , k} such that wι satisfies 2. for each such subset test for satisfiability; b φS p ∧ ¬p. 3. test B = 1≤i≤k Bi for consistency.b V p∈^Xiι p6∈^Xiι The main culprit is 3. Intuitively, the presence of rigid roles enforces that the consistency test for the conjunction For i ∈ {1, . . . , k}, we use the ALC-interpretation J satis- B needs to be done for the whole conjunction, fying B to define an ALC-interpretation J as follows: J 1≤i≤k i i i and cannot be reduced to separate test for the conjuncts (or interprets the rigid names like J , and it interprets the flexi- V some enriched versions of the conjuncts). Thus, the Boolean ble names just as J interprets the ith copies of them. Note ALC-knowledge base to be tested for consistency is expo- that the interpretations J are over the same domain and i nentially large. Since the consistency problem for Boolean respect the rigid symbols, i.e., they interpret them identi- ALC-knowledge bases is EXPTIME-complete, testing this cally. We can now define an ALC-LTL structure respecting knowledge base for consistency requires in the worst-case rigid symbols and satisfying φ as follows: I := (I ) ι ι=0,1,... double-exponentialtime. In contrast, 2. is actually harmless. where I := J . ❏ ι iι According to what we have said above, it needs exponential space, but we will see in the next section that this can even It remains to show that this lemma provides us with a de- be reduced to exponential time. Finally, 1. also does not re- cision procedure for satisfiability in ALC-LTL w.r.t. rigid ally require double-exponential time since one could guess names that runs in deterministic double-exponential time. XP IME n an appropriate set S within NE T . First, note that there are 22 many subsets S of P({p1, . . . , pn}) to be tested, where n is of course lin- Reasoning with rigid concepts early bounded by the size of φ. For each of these subsets S = {X1,...,Xk}, whose cardinality k is bounded by In this section, we consider the case where rigid concept 2n, we need to check satisfiability of φ and consistency names are available, but not rigid role names. First, note S that, in contrast to temporal DLs where temporal opera- of B = Bi. 1≤i≤k b tors may occur inside of concept descriptions, rigid con- The sizeV of φS is at most exponential in the size of φ, cept names cannot easily be expressed within the logic with- and the complexity of the satisfiability problem in proposi- out rigid concept names. In fact, the GCIs A ⊑ 2A and b tional LTL is in PSPACE, and thusin particularin EXPTIME. ¬A ⊑ 2¬A express that A must be interpreted in a rigid Consequently, satisfiability of φS can be tested in double- way. However, they are not allowed by the syntax of ALC- exponential time in the size of φ. LTL since the box is applied directly to a concept, and not to b

688 an axiom. We will show below that, for ALC-LTL, the pres- occurring in φ. We guess a set T ⊆ P({A1,...,Ar}) and ence of rigid concept names indeed increases the complex- a mapping t : {a1, . . . , as} → T . Again, this guess can ity of the satisfiability problem, unless GCIs are restricted to clearly be done within NEXPTIME. being global. First, we treat the case of arbitrary GCIs, and Given T and t, we extend the knowledge bases Bi then the special case of global GCIs. to knowledge bases Bi(T , t) as follows. For Y ⊆ {A1,...,Ar}, let CY be the concept description CY := Theorem 11. Satisfiability in ALC-LTL w.r.t. rigid concepts b is NEXPTIME-complete. A ⊓ ¬A. We define Bi(T , t) as A⊓∈Y A⊓6∈Y A detailed proof of the lower bound, by reduction of the b n+1 2 -bounded domino problem (B¨orger, Gr¨adel, & Gure- Bi ∧ a : CY ∧ ⊤ ⊑ CY ∧ ¬(⊤ ⊑ ¬CY ) Y⊔∈T vich 1997; Tobies 1999), can be found in (Baader, Ghi- t(a^)=Y Y^∈T lardi, & Lutz 2008). In the proof of the upper bound, we want to reuse Lemma 10. Obviously, we can guess a set Since we consider the case where only concept names can S = {X1,...,Xk} ⊆ P({p1, . . . , pn}), within NEXP- be rigid, we know that the Boolean ALC-knowledge bases TIME. However, there are two obstacles on our way to a Bi are built over disjoint sets of role names. The only names NEXPTIME decision procedure. shared among the knowledge bases Bi are the rigid concept names. First, the propositional LTL formula φS is of size expo- nential in the size of φ. Thus, a direct application of the Lemma 12. The Boolean ALC-knowledge base B := PACE b PS decision procedure for satisfiability in propositional 1≤i≤k Bi is consistent iff there is a set T ⊆ LTL would only yield an EXPSPACE upper bound, which is VP({A1,...,Ar}) and a mapping t : {a1, . . . , as} → T not good enough. However, note that the only effect of the such that the Boolean knowledge bases Bi(T , t) for i = b box-formula in φS is to restrict the worlds w in a proposi- 1, . . . , k are separately consistent. b tional LTL structureb satisfying φ to being induced by one Proof. For the “only if” direction, assume that B = of the elements of S. One way of deciding satisfiability of a I b 1≤i≤k Bi has a model I = (∆, · ). Let T consist of those propositional LTL formula φ is to construct a B¨uchi automa- b Vsets Y ⊆ {A1,...,Ar} such that there is a d ∈ ∆ with ton Ab that accepts the propositional LTL structures satisfy- d ∈ (C )I , and let t be the mapping satisfying t(a) = Y iff φ b Y aI ∈ (C )I . It is easy to see that, with this choice of T and ing φ. To be more precise, let Σ := P({p1, . . . , pn}). Then Y t, all the knowledge bases B (T , t) for i = 1, . . . , k have I a given propositional LTL structure I = (wι)ι=0,1,... can be i b as model. represented by an infinite word X0X1 ... over Σ, where Xι b To show the “if” direction,b assume that there is a set consists of the propositional variables that wι makes true. T ⊆ P({A1,...,Ar}) and a mapping t : {a1, . . . , as} → T The B¨uchi automaton Aφb is built such that it accepts exactly those infinite words over Σ that represent propositional LTL such that the Boolean knowledge bases Bi(T , t) for i = Ii structures satisfying φ. Consequently, φ is satisfiable iff the 1, . . . , k have models Ii = (∆i, · ). We can assume with- out loss of generality4 that b language accepted by Aφb is non-empty. The size of Aφb is b b the domains are countably infinite, and exponential in the size of φ, and the emptiness test for B¨uchi • ∆i automata is polynomialin the size of the automaton. The au- • in each model Ii, the sets Y ∈ T are realized by count- b tomaton Aφb can now easily be modified into one accepting ably infinitely many individuals, i.e., there are countably I exactly the words representing propositional LTL structures infinitely many elements d ∈ ∆i such that d ∈ (CY ) i . b satisfying φS . In fact, we just need to remove all transitions Consequently, the domains ∆i are partitioned into the count- ably infinite sets ∆i(Y ) (for Y ∈ T ), which are defined as that use ab letter from Σ \ S. Obviously, this modification can be done in time polynomial in the size of Ab, and thus follows: b φ Ii in time exponential in the size of φ. The size of the resulting ∆i(Y ) := {d ∈ ∆i | d ∈ (CY ) } automaton is obviously still only exponential in the size of b In addition, for each individual name a ∈ {a , . . . , a } we , and thus its emptiness can be tested in time exponential 1 s φ have in the size of φ (and hence of φ). Ii b a ∈ ∆i(t(a)) The second obstacle is the fact that B = B is of b 1≤i≤k i We are now ready to define the model I of . As exponential size, and thus testing it directly for consistency I = (∆, · ) B V the domain of I we take the domain of I , i.e., ∆ := ∆ . using the known EXPTIME decision procedure for satisfia- 1 1 Accordingly, we define ∆(Y ) := ∆ (Y ) for all Y ∈ T . bility of Boolean ALC-knowledge bases would provide us 1 Because of the properties stated above, there exist bijections with a double-exponentialtime bound. Instead of testing the π : ∆ → ∆ such that consistency of B directly we reduce this test to k separate i i consistency tests, each requiring time exponential in the size 4This is an easy consequence of the fact that Boolean ALC- of φ. Before we can do this, we need another guessing step. knowledge bases always have a finite model and that the countably Assume that A1,...,Ar are all the rigid concept names oc- infinite disjoint union of a model of a Boolean ALC-knowledge curring in φ, and that a1, . . . , as are all the individual names base is again a model of this knowledge base.

689 • the restriction of πi to ∆i(Y ) is a bijection between of T is at most exponential in n, and the size of each Y ∈ T ∆i(Y ) and ∆(Y ); is linear in n. We say that B is consistent w.r.t. T iff it has • π respects individual names, i.e., π (aIi ) = aI1 holds a model that is also a model of (3). The following lemma i i b for all a ∈ {a1, . . . , as}. (Note that we have the unique can be shown by an adaptation of the proof of Theorem 2.27 name assumption for individual names.) in (Gabbay et al. 2003), which shows that the consistency problem for Boolean ALC-knowledge bases is in EXPTIME We us these bijections to define the interpretation function (see (Baader, Ghilardi, & Lutz 2008) for details). ·I of I as follows: • If A is a flexible concept name, then B contains its copies Lemma 13. Let B be a Boolean ALC-knowledge base of (i) size n, A ,...,A concept names occurring in B, and T ⊆ A for i = 1, . . . , k. Their interpretation is defined as 1 rb P({A ,...,A }). Then, consistency of B w.r.t. T can be follows: 1 r b (A(i))I := {π(d) | d ∈ AIi }. decided in time exponential in n. b • All role names r are flexible, and B contains their copies Overall, this completes the proof of Theorem 11. In fact, r(i) for i = 1, . . . , k. Their interpretation is defined as after two NEXPTIME guesses, all we have to do are k (i.e., follows: exponentially many) EXPTIME consistency tests. (r(i))I := {(π(d), π(e)) | (d, e) ∈ rIi }. Restricting GCIs to global ones decreases the complexity of the satisfiability problem. • If A is a rigid concept names, then we define Theorem 14. Satisfiability in ALC-LTL|gGCI w.r.t. rigid AI := AI1 concepts is EXPTIME-complete. EXPTIME-hardness is an easy consequence of the well- • If a is an individual name, then we define known fact that concept satisfiability in ALC w.r.t. a single I I1 a := a GCI is EXPTIME-complete: C is satisfiable w.r.t. D1 ⊑ D2 iff a : C ∧ 2(D1 ⊑ D2) is satisfiable. To prove the lemma, it remains to show that I is a model of To prove the EXPTIME upper bound, we consider an all the knowledge bases Bi (i = 1, . . . , k). This is an imme- ALC-LTL formula φ = 2B ∧ ϕ, where B is a conjunction diate consequence of the fact that πi is an isomorphism be- of ALC-axioms and ϕ is an ALC-LTL formula that does not tween Ii and I w.r.t. the concept and role names occurring contain GCIs. We say that B is φ-exhaustive if, for every in- in Bi. The isomorphism condition is satisfied for flexible dividual name a and every rigid concept name A occurring I concepts and roles by our definition of · , and for individ- in φ, either a : A or a : ¬A occurs as a conjunct in B. We ual names by our assumptions on πi. Now, let A be a rigid can assume without loss of generality that B is φ-exhaustive. I I concept name. We must show that d ∈ A i iff πi(d) ∈ A In fact, given an arbitrary Boolean ALC-knowledge base B, holds for all d ∈ ∆i. Since ∆i is partitioned into the sets we can build all the φ-exhaustive knowledge bases B′ that ∆i(Y ) for Y ∈ T , we know that there is a Y ∈ T such that are obtained from B by conjoining to it, for every individ- I d ∈ ∆i(Y ), i.e., d ∈ (CY ) i . In addition, we know that ual name a and every rigid concept name A occurring in φ, I1 I πi(d) ∈ ∆(Y ), i.e., πi(d) ∈ (CY ) = (CY ) . This im- either a : A or a : ¬A. Obviously, φ = 2B ∧ ϕ is sat- I I plies that d ∈ A i iff A ∈ Y iff πi(d) ∈ A , which finishes isfiable w.r.t. rigid concepts iff 2B′ ∧ ϕ is satisfiable w.r.t. the proof that πi is an isomorphism between Ii and I. Con- rigid concepts for one of the extension B′ of B obtained this sequently, I is a model of B = 1≤i≤k Bi, which in turn way. Since the size of each such an extension is polyno- finishes the proof of the lemma. V ❏ mial and there are only exponentially many such extensions, it is sufficient to show that testing satisfiability of 2B′ ∧ ϕ To complete the proof of Theorem 11, we must show that w.r.t. rigid concepts for φ-exhaustive knowledge bases B′ is the consistency of Bi(T , t) can be decided in time exponen- in EXPTIME. tial in the size of the input formula φ. Note that this is not Following the approach used in the proof of Theorem 9, b we abstract every ABox assertion α occurring in ϕ by a trivial. In fact, while the size of Bi ∧ t(a)=Y a : CY is i polynomial in the size of φ, the cardinalityV of T , and thus propositional variable pi, thus building the propositional the size of LTL-formula ϕ. Next, we compute the set S, which consists of those X ⊆ {p1, . . . , pn} for which the Boolean ALC- ⊤ ⊑ CY ∧ ¬(⊤ ⊑ ¬CY ), (3) knowledge baseb b Y⊔∈T Y^∈T B := B ∧ α ∧ ¬α can be exponential in the size of φ. Decidability of the con- X j j pj^∈X pj^6∈X sistency of Bi(T , t) in time exponential in the size of φ is, however, an immediate consequence of the next lemma. To is consistent. This computation can be done in exponential formulate thisb lemma, we need to introduce one more no- time since it requires exponentially many EXPTIME consis- tation. Let B be a Boolean ALC-knowledge base of size tency tests. n, A ,...,A concept names occurring in B, and T ⊆ Lemma 15. Let φ = 2B∧ϕ be such that B is a φ-exhaustive 1 br P({A1,...,Ar}). Note that this implies that the cardinality conjunction of ALC-axioms and ϕ is an ALC-LTL formula b

690 not containing GCIs. Then φ is satisfiable w.r.t. rigid con- such that the propositional LTL|3 formula φS is satisfiable cepts iff the propositional LTL formula and the Boolean ALC-knowledge base B := Bi is b 1≤i≤k consistent. V b 2 ϕS := ϕ ∧ ( ( pj ∧ ¬pj)) b Proof. The “if” direction of this lemma is an immediate X_∈S pj^∈X pj^6∈X b b consequence of Lemma 10. For the “only if” direction, we is satisfiable. can use the proof of the “only if” direction Lemma 10. The only difference is that we start with an -LTL structure The proof of this lemma can again be found in (Baader, ALC I respecting rigid concept and role names of weight at most Ghilardi, & Lutz 2008). Note that this actually completes such that J . The existence of such a structure the proof of Theorem 14. In fact, as shown in the proof of |φ| + 2 , 0 |= φ is guaranteed by Lemma 17. It is easy to see that then the b Theorem 11, satisfiability of ϕS can be decided in exponen- set S = {X1,...,Xk} induced by this structure is indeed tial time. ❏ b of cardinality k ≤ |φ| + 2. Reasoning without rigid names In this section, we consider the case where we have no rigid It remains to show that this lemma provides us with a de- names at all. cision procedure for satisfiability in ALC-LTL|3 w.r.t. rigid names that runs in deterministic exponential time. There Theorem 16. Satisfiability without rigid names in ALC- m(m+2) are ≤ 2 subsets S ⊆ P({p1, . . . , pn}) of cardinality LTL and in ALC-LTL|gGCI is EXPTIME-complete. ≤ m + 2 to be considered, and the size of each such sub- EXPTIME-hardness is again an easy consequence of the set S = {X1,...,Xk} is polynomial in m. Thus, the size fact that concept satisfiability in ALC w.r.t. a single GCI is of both φS and B = 1≤i≤k Bi is polynomial in m. Since EXPTIME-complete. As already mentioned in the introduc- satisfiability in propositionalV LTL is in PSPACE and the con- tion, the EXPTIME upper bound follows from more general sistencyb problems for Boolean ALC-knowledge bases is in results proved in Chapter 11 of (Gabbay et al. 2003) (see EXPTIME, this shows that satisfiability in ALC-LTL|3 w.r.t. the remark following Theorem 14.14 on page 605 of (Gab- rigid names is in EXPTIME. bay et al. 2003)). A direct proof of the upper bound, which EXPTIME-hardness of satisfiability in ALC-LTL|3 (even is similar to the proof of Theorem 14, is given in (Baader, without any rigid names) is again an easy consequence of Ghilardi, & Lutz 2008). the fact that concept satisfiability in ALC w.r.t. a single GCI is EXPTIME-complete. Restricting the temporal component Theorem 19. Satisfiability in ALC-LTL|3 w.r.t. rigid names In this section, we consider the fragment ALC-LTL|3 of is an EXPTIME-complete problem. The same is true for sat- ALC-LTL, in which 3 is the only temporal operator. Our isfiability w.r.t. rigid concepts and without rigid names. aim is to show that satisfiability in ALC-LTL|3 w.r.t. rigid names is in EXPTIME. The main reason for this is that we can restrict the attention to ALC-LTL structures respecting Conclusion rigid concept and role names that consist of at most poly- The faithful modelling of dynamically changing environ- nomially many distinct ALC-interpretations. The weight of ments with a temporalized DL often requires the availability the ALC-LTL structure I = (Ii)i=0,1,... is defined to be the 5 of rigid concepts and roles. We have shown that decidability cardinality of the set {Ii | i = 0, 1,...}. and an elementary complexity upper bound can be achieved Lemma 17. Let φ be an ALC-LTL|3 formula of size m. also in the presence of rigid roles by restricting the appli- If φ is satisfiable w.r.t. rigid names, then there is an ALC- cation of temporal operators to DL axioms. This is a big LTL structure J respecting rigid concept and role names of advance over the case where temporal operators can occur weight at most |φ| + 2 such that J, 0 |= φ. inside concept descriptions, in which rigid roles cause unde- The proof of this fact, which can be found in (Baader, cidability in the presence of a TBox and hardness for non- Ghilardi, & Lutz 2008), is a straightforward generalization elementary time even without a TBox. The decision procedures we have described in this paper to ALC-LTL|3 of a verysimilar proof for LTL|3, the restric- tion of propositional LTL to its diamond fragment (see, e.g., were developed for the purpose of showing worst-case com- Lemma 6.40 in (Blackburn, de Rijke, & Venema 2001)). plexity upper bounds. The major topic for future work is to Based on this lemma, the following modified version of optimize them such that they can be used in practice, where Lemma 10 can be shown. we will first concentrate on the application scenario sketched in the introduction. Lemma 18. Let φ be an ALC-LTL|3 formula of size m. Then, φ is satisfiable w.r.t. rigid names iff there is a set S = References {X1,...,Xk} ⊆ P({p1, . . . , pn}) of cardinality k ≤ m+2

5 Artale, A., and Franconi, E. 1998. A temporal description Recall that all the ALC-interpretations within one ALC-LTL logic for reasoning about actions and plans. JAIR 9. structure have the same domain. For this reason, we can use exact equality of interpretations rather than equality up to isomorphism Artale, A., and Franconi, E. 2000. A survey of temporal when defining the weight of an ALC-LTL structure. extensions of description logics. AMAI 30.

691 Artale, A., and Franconi, E. 2001. Temporal description Appendix logics. In Handbook of Time and Temporal Reasoning in In this appendix, we show the hardness part of Theorem 9. AI. The MIT Press. Note that the proof of the hardness part of Theorem 11 is Artale, A.; Franconi, E.; Wolter, F.; and Zakharyaschev, simpler than the proof given below. It can be found in M. 2002. A temporal for reasoning over (Baader, Ghilardi, & Lutz 2008). conceptual schemas and queries. In Proc. JELIA’2002, Springer LNCS 2424. Lemma 20. Satisfiability in ALC-LTL|gGCI w.r.t. rigid names is 2-EXPTIME-hard. Artale, A.; Kontchakov, R.; Lutz, C.; Wolter, F.; and Za- kharyaschev, M. 2007. Temporalising tractable description Proof. The proof is by reduction of the word problem for logics. In Proc. TIME’07, IEEE Press. exponentially space bounded alternating Turing machines (ATMs). An ATM is of the form M = (Q, Σ, Γ, q0, Θ), Artale, A.; Lutz, C.; and Toman, D. 2007. A description where Q = Q ⊎ Q ⊎ {q , q } is a finite set of states, logic of change. In Proc. IJCAI’07, AAAI Press. ∃ ∀ a r partitioned into existential states from Q∃, universal states Baader, F.; Calvanese, D.; McGuinness, D.; Nardi, D.; from Q∀, an accepting state qa, and a rejecting state qr; Σ and Patel-Schneider, P. F., eds. 2003. The Description is the input alphabet and Γ ⊇ Σ the work alphabet con- Logic Handbook: Theory, Implementation, and Applica- taining a blank symbol B/∈ Σ; q0 ∈ Q∃ ∪ Q∀ is the tions. Cambridge University Press. initial state; and the transition relation Θ is of the form Baader, F.; Ghilardi, S.; and Lutz, C. 2008. LTL over de- Θ ⊆ Q × Γ × Q × Γ × {L,R}. We write Θ(q, σ) for ′ ′ scription logic axioms. LTCS-Report 08-01, TU Dresden, {(q , θ, M) | (q, σ, q , b, M) ∈ Θ}. Germany. A configuration of an ATM is a word wqw′ with w, w′ ∈ See http://lat.inf.tu-dresden.de/research/reports.html. Γ∗ and q ∈ Q. The intended meaning is that the (one-sided infinite) tape contains the word ww′ with only blanks behind Blackburn, P.; de Rijke, M.; and Venema, Y. 2001. Modal it, the machine is in state q, and the head is on the left-most Logic. Cambridge University Press. symbol of w′. The successor configurations of a configura- B¨orger, E.; Gr¨adel, E.; and Gurevich, Y. 1997. The Classi- tion wqw′ are defined in the usual way in terms of the transi- cal Decision Problem. Springer-Verlag. tion relation Θ.A halting configuration is of the form wqw′ Chandra, A.; Kozen, D.; and Stockmeyer, L. 1981. Alter- with q ∈ {qa, qr}. We may assume w.l.o.g. that any con- nation. JACM 28. figuration other than a halting configuration has at least one successor configuration. A computation of an ATM on a Finger, M., and Gabbay, D. 1992. Adding a temporal di- M word is a (finite or infinite) sequence of successive config- mension to a logic system. JoLLI 2. w urations K1,K2,... . The ATMs considered here have only Gabbay, D.; Kurusz, A.; Wolter, F.; and Zakharyaschev, finite computations on any input. Since this case is simpler M. 2003. Many-dimensional Modal Logics: Theory and than the general one, we only define acceptance for ATMs Applications. Elsevier. with finite computations and refer to (Chandra, Kozen, & Lutz, C.; Wolter, F.; and Zakharyaschev, M. 2008. Tempo- Stockmeyer 1981) for the full definition. Let M be such ral description logics: A survey. In Proc. TIME’08, IEEE an ATM. A halting configuration is accepting iff it is of the ′ ′ Press. form wqaw . For other configurations K = wqw , the ac- Lutz, C. 2001. Interval-based temporal reasoning with ceptance behaviour depends on q: if q ∈ Q∃, then K is ac- general TBoxes. In Proc. IJCAI’01, AAAI Press. cepting iff at least one successor configuration is accepting; if q ∈ Q∀, then K is accepting iff all successor configura- Pnueli, A. 1977. The temporal logic of programs. In Proc. tions are accepting. Finally, the ATM M with initial state FOCS’77. q0 accepts the input w iff the initial configuration q0w is ac- Schild, K. 1993. Combining terminological logics with cepting. We use L(M) to denote the language accepted by tense logic. In Proc. EPIA’93, Springer LNCS 727. M, i.e., Schmidt-Schauß, M., and Smolka, G. 1991. Attributive L(M) = {w ∈ Σ∗ | M accepts w}. concept descriptions with complements. AIJ 48. Schmiedel, A. 1990. A temporal terminological logic. In The word problem for M is the following decision problem: ∗ Proc. AAAI’90, AAAI Press. given a word w ∈ Σ , does w ∈ L(M) hold or not? There exists an exponentially space bounded ATM Schulz, S.; Mark´o, K.; and Suntisrivaraporn, B. 2006. M = (Q, Σ, Γ, q , Θ) whose word problem is 2-EXPTIME- Complex occurrents in clinical terminologies and their rep- 0 hard (Chandra, Kozen, & Stockmeyer 1981). Our aim is to resentation in a formal language. In Proc. 1st European reduce the word problem for this ATM M to satisfiability in Conference on SNOMED CT (SMCS’06). ALC-LTL|gGCI w.r.t. rigid names. We may assume that the Tobies, S. 1999. A NEXPTIME-completedescription logic length of every computation of on k is boundedby 2 M w ∈ Σ strictly contained in C . In Proc. CSL’99, Springer LNCS k 2 , and all the configurations ′ in such computations 1683. 2 wqw satisfy |ww′| ≤ 2k. We may also assume w.l.o.g. that M Wolter, F., and Zakharyaschev, M. 1999. Temporalizing never attempts to move to the left when it is on the left-most description logics. In Proc. FroCoS’98, Wiley. tape cell.

692 ∗ Let w = σ0 · ·· σk−1 ∈ Σ be an input to M. We alternating Turing machine, it is not enough to consider one construct an ALC-LTL|gGCI formula φM,w such that w ∈ sequence of configurations. For a configuration with a uni- L(M) iff φM,w is satisfiable w.r.t. rigid names. In an ALC- versal state, we must consider all successor configurations. LTL structure satisfying φM,w, each domain element from Thus, we do not consider a single sequence of r-successors, ∆ describes a single tape cell of a configuration of M. The but rather a tree of r-successors. formula φM,w, that will be defined below is actually not The main problem to solve when defining the reduction of the syntactic form 2B ∧ ϕ (where B is a conjunction of is to ensure that each configuration following a given con- ALC-axioms and ϕ is an ALC-LTL formula that does not figuration in the tree of r-successors is actually a successor contain GCIs) required for ALC-LTL formulae with global configuration, i.e., tape cells that are not immediately to the GCIs. Instead, φM,w is a conjunction of formulae of the left or right of the head remain unchanged, and the other form tape cells are changed according to the transition relation. • 2α where α is an ALC-axiom, For the first type of cells this means that, given a cell num- bered i in the current configuration, the next cell with the • ψ where ψ is an ALC-LTL formula not containing GCIs. same number should carry the same symbol. However, we Since 2 distributes over conjunction, it is obvious that such cannot remember the value i of the A-counter when going a formula is equivalent to an ALC-LTL formula with global down along the sequence of r-successors since this counter k GCIs. In the definition of φM,w, we use the following sym- is incremented (modulo 2 ) when going to an r-successor. bols: This is where the temporal dimension comes into play. Here, we realize an A′-counter, using the (flexible) concept names • a single individual name a that identifies the first tape cell ′ ′ of the first configuration; A0,...,Ak−1, whose value does not change along the r- dimension, but is incremented (modulo 2k) along the tem- • a single rigid role name r to represent “going to the next poral dimension. This additional counter, together with the tape cell in the same configuration” and “going from the flexible copies of the symbols from Q and Γ, can be used to last tape cell in a configuration to the first tape cell in a transfer a symbol from a tape cell in a given configurationto successor configuration”; the corresponding tape cell in a successor configuration (see • the elements of Q and Γ are viewed as rigid concept below). names; In the following, we use φ → ψ as an abbreviation for , as an abbreviation for , and • rigid concept names A0,...,Ak−1 are the bits of a binary ¬φ∨ψ C ⇒ D ¬C ⊔D C ⇔ D counter that numbers the tape cells in each configuration; as an abbreviation for (C ⇒ D) ⊓ (D ⇒ C). The reduction formula φM,w is the conjunctionof the fol- • auxiliary rigid concept name I and H; I indicates the ini- lowing formulae: tial configuration and H indicates that, in the current con- We start by setting up I, H, r, and the A-counter: figuration, the head is to the left of the current tape cell; • I behaves as described, i.e., it marks the initial configura- • auxiliary rigid concept names Tq,σ,M for all q ∈ Q, σ ∈ tion, whose first tape cell is represented by the individual Γ, and M ∈ {L,R}; intuitively, Tq,σ,M is true if, in the a: current configuration, the head is on the left neighboring 2 (a : I) cell and the machine executes transition (q, σ, M); k 2 I ⊓ ¬(CA = 2 − 1) ⊑ ∀r.I • for each element of Q and Γ, a flexible concept name • H behaves as described, i.e., it marks the tape cells that which is distinguished from its rigid version by a prime; are to the right of the head, where the head position is ′ ′ • (flexible) concept names A0,...,Ak−1 to realize an or- indicated by having a state concept at this cell: thogonalcounter (in the sense that it countsalong the tem- 2 k poral dimension instead of along r). (H ⊔ q) ⊓ ¬(CA = 2 − 1) ⊑ ∀r.H  q⊔∈Q  Before giving the formal reduction, let us explain the un- • there is always an r-successor, except when we meet the derlying intuition. As said above, a single configuration is head in a halting configuration: described as a sequence of r-successors of length 2k of the individual representing its first tape cell. The tape cells of 2 (¬(q ⊔ q ) ⊑ ∃r.⊤) a configuration are numbered from 0 to 2k − 1, using the a r counter realized through the concept names A ,...,Ak . • the counter realized by A0,...,Ak−1 has value 0 at a, 0 −1 k We denote the concept that expresses that the counter has and it is incremented along r (modulo 2 ): k value i, 0 ≤ i < 2 , by (CA = i); i.e., (CA = 0) 2 (a :(CA = 0)) denotes ¬A0 ⊓ ¬A1 ⊓ ... ⊓ ¬Ak−1, (CA = 1) denotes k 2 A ⊓ ¬A ⊓ ... ⊓ ¬Ak , ..., (CA = 2 − 1) denotes ⊤ ⊑ Aj ⇒ 0 1 −1 ⊓i

693 Some properties of runs of ATMs can be formalized without plus 1 (modulo 2k), which can be expressed by a recasting using the temporal dimension: of the incrementation concept given already twice above:

• The initial configuration is the one induced by the input ′ ′ ′ Aj ⇒ (Ai ⇒ ¬Ai) ⊓ (¬Ai ⇒ Ai) ⊓ w = σ0 . . . σk−1: ⊓i

694