A Description Logic of Change

A Description Logic of Change

Alessandro Artale University of Bolzano, Italy artale@inf.unibz.it Carsten Lutz University of Dresden, Germany clu@tcs.inf.tu-dresden.de David Toman University of Waterloo, Canada david@uwaterloo.ca

Abstract

We combine the modal logic S5 with the de- scription logic (DL) ALCQI. The resulting multi-dimensional DLS5_ALCQI supports reason- ing about change by allowing to express that con- cepts and roles change over time. It cannot, how- ever, discriminate between changes in the past and in the future. Our main technical result is that satisfiability ofS5ALCQI concepts with respect to general TBoxes (including GCIs) is decidable and 2-EXPTIME-complete. In contrast, reasoning in temporal DLs thatareable to discriminate between past and future is inherently undecidable. We argue that our logic is sufficient for reasoning about tem- poral conceptual models with time-stamping con- straints.

1 Introduction

An important application of Temporal Description Logics (TDLs) is the representation of and reasoning about tem- poral conceptual models [Artale, 2004; Artaleet al., 2003;

2002]. Knowledge captured by such models is translated into a TDL TBox and reasoning algorithms for TDL are then used to detect inconsistencies and implicit IS-A rela- tions in the temporal model [Artale and Franconi, 1999;

Artaleet al., 2002; Calvaneseet al., 1998]. A serious obstacle for putting this general idea to work is posed by the fact that for many natural temporal conceptual formalisms and their associated TDLs, reasoning turns out to be undecidable.

The most prominent witnesses of this problem are the var- ious temporal entity-relationship (TER) models used to de- sign temporal databases [Chomicki and Toman, 2005]. TERs are classical ER data models extended with two additional classes of constraints that model the temporal evolution of data in an application domain [Spaccapietra et al., 1998].

First,timestamping constraintsare used to distinguish tempo- ral and atemporal components of a TER model. Timestamp- ing is usually implemented by marking entities (i.e., classes), relationships and attributes assnapshot,temporary, orunre- stricted. The idea behind such a classification is to express

∗The research was supported by EU IST-2005-7603 FET Project TONESand by NSERC.

that object membership in entities, relationships, and attribute values cannot or must change in time; this is achieved by snapshot and temporary marks in the diagram, respectively.

Second, evolution constraints govern object migration be- tween entities and can state, for example, that every instance of the entityChildwill eventually become an instance of the entityAdult.

TER models with both timestamping and evolution con- straints can be translated into the TDL DLR_{U S} [Artale et al., 2002]. Unfortunately, reasoning in this logic is undecid- able. Moreover, the computational problems are not due to the translation to TDLs: even direct reasoning in the gener- ally less powerful TER models is undecidable [Artale, 2004].

There are two principal ways around this problem. The first approach restricts the application of timestamping: it allows arbitrary timestamping of entities, but gives up timestamp- ing of relationships and attributes (i.e., all relationships and attributes are unrestricted). This re-establishes decidability of TER models with restricted timestamping and evolution constraints [Artaleet al., 2002]. The second approach to re- gaining decidability allows for full use of timestamping, but prohibits the use of evolution constraints.

This second alternative is pursued in the current paper. We devise a multi-dimensional description logic S5_ALCQI that is obtained by combining the modal logic S5with the stan- dard DL ALCQI. TheS5modality can be applied to both concepts and roles; axioms in the TBox are, however, inter- preted globally. This logic can be viewed as a description logic of change: it can express that concept and role mem- berships change in time, but does not permit discriminating between changes in the past or future. We show that TER models with full timestamping (i.e., timestamping on entities, relationships, and attributes) but without evolution constraints can be captured byS5_ALCQI TBoxes.

The main contribution of this paper is to show that rea- soning in S5_ALCQI is decidable. We also pinpoint the ex- act computational complexity by showing 2-EXPTIMEcom- pleteness. Thus, adding theS5change modalitypushes the complexity ofALCQI, which is EXPTIME-complete, by one exponential. Our upper bound can be viewed as an extension of the decidability result for a simpler multi-dimensional DL, S5_ALC, [Gabbayet al., 2003] which is not capable of cap- turing TER models. However, we had to develop completely new proof techniques as the decidability proof forS5ALCre-

lies on the ability to duplicate points in a model, which is im- possible in the presence of S5_ALCQI’s number restrictions.

Our lower bound applies also toS5_ALC, hence we show that this logic is also 2-EXPTIME-complete.

The paper is organized as follows. Section 2 introduces the logicS5_ALCQI. Section 3 shows that reasoning inS5_ALCQI is decidable in 2-EXPTIMEby first providing a tree abstrac- tion of S5ALCQI interpretations and then presenting a 2- EXPTIMEprocedure that checks for the existence of such a tree abstraction. Section 4 illustrates howS5ALCQI is able to capture conceptual models with timestamping constraints.

Full proofs and the 2-EXPTIMElower bound can be found in the full version of this paper [Artaleet al., 2006].

2 The Logic S5

The logic S5_ALCQI combines the modal logic S5 and the description logic ALCQI in a spirit similar to the multi- dimensional DLs [Gabbay et al., 2003; Wolter and Za- kharyaschev, 1999]. The syntax of S5_ALCQI is defined as follows. LetNC and NR be disjoint and countably infinite sets ofconcept namesandrole names. We assume thatNR

is partitioned into two countably infinite setsNgloandNlocof global role namesandlocal role names. The setROLofroles is defined as{r, r⁻,3r,3r⁻,2r,2r⁻|r∈N_R}. The set of conceptsCONis defined inductively as follows:NC⊆CON;

ifC, D∈CON,r∈ROL, andn∈N, then the following are also inCON: ¬C,CuD,(>n r C), and3C. ATBoxis a finite set ofgeneral concept inclusions (GCIs)CvDwith C, D∈CON.

An S5_ALCQI-interpretation I is a pair (W,I) with W a non-empty set of worlds and I a function assigning an ALCQI-interpretation I(w) = (∆,·^I,w)to eachw ∈ W, with∆a fixed non-empty set called thedomainand·^I,w a function mapping eachA ∈ NCto a subsetA^I,w ⊆ ∆and eachr∈N_Rto a relationr^I,w ⊆∆×∆; forr∈N_glowe ad- ditionally requirer^I,w =r^I,vfor allw, v ∈W. We extend the mapping·^I,wto complex roles and concepts as follows:

(r⁻)^I,w:={(y, x)∈∆×∆|(x, y)∈r^I,w}

(3r)^I,w:={(x, y)∈∆×∆| ∃v∈W : (x, y)∈r^I,v} (2r)^I,w:={(x, y)∈∆×∆| ∀v∈W : (x, y)∈r^I,v} (¬C)^I,w:= ∆\C^I,w

(CuD)^I,w:=C^I,w∩D^I,w

(>n r C)^I,w:={x∈∆|]{y∈∆|(x, y)∈r^I,w

andy∈C^I,w} ≥n}

(3C)^I,w:={x∈∆| ∃v∈W:x∈C^I,v}

AnS5_ALCQI-interpretationI= (W,I)is amodelof a TBox T iff it satisfiesC^I,w ⊆D^I,w for allC vD ∈ T andw ∈ W. It is amodelof a conceptCifC^I,w 6=∅for somew ∈ W. A conceptCissatisfiablew.r.t. a TBoxT if there exists a common model ofC andT. Note that whenS5ALCQI is considered a temporal description logic, the elements ofW correspond to time points. We do not distinguish between global and localconcept names because a concept nameA can easily be enforced to be global using the GCIAv2A.

The concept constructorsCtD,∃r.C,∀r.C,(6n r C), (=n r C),2C,>, and⊥are defined as abbreviations in the

usual way. For roles, we allow only single applications ofS5 modalities and inverse. It is easily seen that roles obtained by nesting modal operators and inverse in an arbitrary way can be converted into an equivalent role in our simple form: mul- tiple temporal operators are absorbed and inverses commute with temporal operators.

The fragmentS5_ALC ofS5_ALCQI is obtained by allowing only roles of the formr,3r, and2r, and replacing the con- cept constructor (> n r C)by∃r.C. We note that neither S5ALC norS5ALCQI enjoys the finite model property: there are concepts and TBoxes that are only satisfiable in models with both an infinite set of worldsW and an infinite domain

∆. An example of this phenomenon is the concept¬Cand the TBox{¬C v3C, C v ∃r.¬C, ¬C v ∀r.¬C}, with r∈Nglo.

3 Reasoning in S5

We show that inS5ALCQI, satisfiability w.r.t. TBoxes is de- cidable in 2-EXPTIME. For simplicity, throughout this sec- tion we assume that only local role names are used. This can be done w.l.o.g. as global role names can be simulated by2r, forra fresh local role name. LetC₀andT be a concept and a TBox whose satisfiability is to be decided. We introduce the following notation. For rolesr, we useInv(r)to denote r⁻ifr ∈NR,sifr=s⁻,3Inv(s)ifr=3s, and2Inv(s) ifr=2s. We userol(C₀,T)to denote the smallest set that contains all sub-roles used inC0andT and is closed under Inv. We usecl(C0,T)to denote the smallest set containing all sub-concepts appearing inC0andT that is closed under negation: ifC∈cl(C0,T)and “¬” is not the top level oper- ator inC, then¬C∈cl(C₀,T).

In the rest of this section we devise tree abstractions of models ofC₀andT which we call(C₀,T)-trees. In the sub- sequent section, we then show how to construct looping tree automata that accept the(C₀,T)-trees and thus reduce satis- fiability inS5ALCQIto the emptiness problem of looping tree automata, yielding decidability ofS5_ALCQI.

3.1 Tree Abstractions ofS5ALCQImodels

Intuitively, for a(C₀,T)-treeτthat abstracts a modelIofC₀ andT, the root node ofτcorresponds to an objectxinIthat realizesC₀. Successors of the root inτcorrespond to objects inIthat can be reached fromxby traversing a role in some S5world. Similarly, further nodes inτcorrespond to objects ofI reachable fromxby traversing multiple roles. To de- scribe the concept and role interpretations ofIin its abstrac- tionτ, we decorate the nodes ofτwithextended quasistates as introduced in Definition 3 below. Extended quasistates are defined in terms of types and quasistates, which we introduce first. Intuitively, a type describes the concept memberships of a domain elementx∈∆in a singleS5world.

Definition 1 (Type). A type t for C0,T is a subset of cl(C₀,T)such that

¬C∈tiffC6∈t for¬C∈cl(C0,T) CuD∈tiffC∈tandD∈t forCuD∈cl(C₀,T)

D∈t if C∈t forCvD∈ T

We usetp(C0,T)to denote the set of all types forC0andT. To describe the concept memberships of a domain element in allS5worlds, we use quasistates:

Definition 2 (Quasistate). Let W be a set and f : W → tp(C0,T)a function such that for allw∈W we have:

3C∈f(w)iffC∈f(v)for somev∈W.

We call the pair (W, f) a quasistate witness and the set {f(v)|v∈W}aquasistate.

To check whether a set of types{t₁, . . . , t_n} forms a valid quasistate, we can simply check whether the pair(W, f), with W ={t1, . . . , tn}andfthe identity function, is a quasistate witness. Note, however, that each quasistate has many wit- nesses.

Quasistates only abstract concept membership of a partic- ular object in all worlds. To capture the role structure relating objects adjacent insomeS5world in a givenS5_ALCQImodel, we develop the notion of aextended quasistate. Ultimately, in the desired tree abstraction these two objects turn into a par- ent and a child nodes; the child is then labeled by the extended quasistate in question. Note the similarity to handling inverse roles using thedouble blockingtechnique used in tableau al- gorithms forALCQI[Horrockset al., 1999].

Definition 3(Extended Quasistate). LetW be a set,(W, f) and(W, g)quasistate witnesses, andh:W →rol(C₀,T)∪ {2ε}forε6∈rol(C0,T)such that, for everyr∈NR∪ {s⁻| s∈N_R}:

1. ifh(w) = 3r, for somew ∈ W, thenh(v) = r, for somev∈W;

2. ifh(w) =r, for somew ∈W, then eitherh(v) = 3r orh(v) =r, for allv∈W;

3. it is not the case thath(w) =rfor allw∈W;

4. ifh(w) = 2r, for somew ∈ W, thenh(v) =2r, for allv∈W.

We call (W, f, g, h)anextended quasistate witnessand the set of triplesQ(W, f, g, h) ={(f(v), g(v), h(v))|v ∈W} anextended quasistate. Elements ofQ(W, f, g, h)are called extended typesandetp(C0,T)denotes the set of all extended types for C₀ and T. We say that Q(W, f, g, h) realizes a concept C if C ∈ f(w) for some w ∈ W; we say that Q(W, f, g, h)is rootifh(w) =2εfor allw∈W.

Intuitively, given a node labeled with the extended quasistate witness(W, f, g, h), the quasistate witness(W, f)describes the node which is labeled with the extended quasistate,(W, g) describes the predecessor of this node, andhdescribes the role connections between the two nodes. Conditions 1 to 4 ensure that the mappinghassigns roles in a way that respects the semantics of modal operators. To fully understand these conditions, note that we assume an ordering3r ≤ r ≤2r between roles which allows us to use a single role in the ex- tended quasistate to capture all theimpliedroles. The dummy roleεis intended only for use with the root object, which does not have a predecessor.

We now introduce the concept of amatching successor.

The main difficulty is to properly capture the effects ofqual- ified number restrictions(>n r C)which constrain the pos- sible combinations of extended quasistates in(C0,T)-trees:

the extended quasistates assigned to children nodes must sat- isfy the qualified number restrictions of the parent node.

Definition 4 (Matching Successor). Let W and Γ be sets, x /∈Γ, and letebe a function mapping eachy∈Γ∪{x}to an extended quasistate witness(W, fy, gy, hy)such thatgy=fx

for all y ∈ Γ. We call (W,Γ, x, e) a matching successor witnessif for allw∈W:

1. if(> n r C) ∈ fx(w)andC 6∈ gx(w) orInv(r) 6≤

hx(w)then|{y∈Γ|r≤hy(w), C∈fy(w)}| ≥n, 2. if(>n r C)∈fx(w), then|{y ∈Γ|r≤hy(w), C ∈

f_y(w)}| ≥n−1,

3. if(> n r C) ∈ cl(T, C0),C ∈ gx(w), andInv(r) ≤ hx(w), and|{y∈Γ|r≤hy(w), C∈fy(w)}| ≥n−1 then(>n r C)∈f_x(w),

4. if (> n r C) ∈ cl(T, C0) and |{y ∈ Γ | r ≤ h_y(w), C∈f_y(w)}| ≥nthen(>n r C)∈f_x(w).

The pair(Q(W, f_x, g_x, h_x),{Q(W, fy, g_y, h_y) | y ∈ Γ})is called amatching successor,.

We say that two matching successor witnesses areequiva- lentif they define the same matching successor.

The intuition behind this definition is as follows: the object xstands for a parent node (described by f_x) and the set of objectsΓfor all its children (described byfy). The extended quasistates are chosen in a consistent way w.r.t. the informa- tion that is represented twice: the parent part of the extended quasistates labeling the children matches the quasistate at- tached to the parent itself (i.e.,gy = fx for ally ∈ Γ). A matching successor witnessis then a witness such that the ex- tended quasistates attached toxand to all elements ofΓcan be used to build a part of a model ofC0andT without violat- ing any qualifying number restrictions. Also, the domain of such a model is eventually built from the objects{x} ∪Γ. As already mentioned, matching successors are the most crucial ingredient to the definition of(C0,T)-trees.

Definition 5((C0,T)-tree). Letτ = (N, E, G, n0)be a tu- ple such that (N, E)is a tree with root n₀ ∈ N andGa mapping ofτ’s nodes to extended quasistates. Then τ is a (C0,T)-treeif:

1. G(n₀)realizesC₀; 2. G(n0)is root;

3. for alln∈N, the pair(G(n),{G(m)|(n, m)∈E})is a matching successor.

Note that the matching successor witnessesinduced by the matching successors consisting of the extended quasistates associated with a node and its children in the(C0,T)-tree do not necessarily share the same setW. This poses a difficulty when showing that(C₀,T)-trees are proper abstractions of models ofC0 andT: when we want to convert such a tree into an interpretation, we need to decide on a common set of worldsW. This difficulty can be overcome using the follow- ing lemma, which shows that we can assume that all extended quasistate witnesses are based on sets of worldWof identical cardinality.

Lemma 6(Compatible Matching Successor). There exists an infinite cardinalαsuch that the following holds: for every

x y1 y2 · · · ylyl+1 · · · y_m−1ym · · · yn yn+1 · · · ∈Γ w ν_x ν₁ ν₂ ν_l ν_l+1 ν_m−1 ν ν ν_n+1 · · ·



y worldwis replaced by worlds{v₁, . . . , v_l} v1 νx ν1 ν ν νl+1 ν_m−1

v2 νx ν ν2 ν ν νl+1 ν_m−1

... ... ...

vl νx ν ν νl νl+1 ν_m−1

Figure 1: Reducing the size of a Matching Successor Witness.

matching successor witness(W,Γ, x, e), there is a matching successor witness(W⁰,Γ, x, e⁰)such that

• (W,Γ, x, e)and(W⁰,Γ, x, e⁰)define the same matching successor and;

• for all y ∈ Γ ∪ {x} and all extended types ν ∈ Q(W⁰, f_y⁰, g⁰_y, h⁰_y), we have:

|{w∈W⁰|(f_y⁰(w), g_y⁰(w), h⁰_y(w)) =ν}|=α.

Intuitively, the Lemma is proved by replicating elements of W a sufficient number of times. When all extended quasis- tate witnesses for a(C₀,T)-tree are based on a set of worlds of identical cardinality, we can connect these extended qua- sistate witnesses to a model by simply permuting the setW in an appropriate way. In this way, we can prove the difficult right-to-left direction of the following theorem.

Theorem 7. C₀is satisfiable w.r.t.T iff a(C₀,T)-tree exists.

3.2 Decidability and Complexity ofS5ALCQI

We now develop an effective procedure to check whether there exists a(C0,T)-tree for a given conceptC0and TBox T. We also show that the procedure runs in 2-EXPTIME. Since it is easy to see that the number of all possiblematch- ing successors (q, Q) is 3-exponential, we cannot simply generate all of them and check whether they give rise to a (C0,T)-tree. Instead, we start by showing that if a(C0,T)- tree exists, then there is one withslim matching successors witnesses only, i.e., all matching successors in this tree are witnessed by matching successor witnesses whose size is at most exponential. In the rest of the paper, letmaxC₀,T = P

(>m r C)∈cl(C0,T)m, andn=|cl(C₀,T)|.

Lemma 8. Let(W,Γ, x, e)be a matching successor witness for a matching successor(q, Q). Then there is aQ⁰⊆Qand a matching successor witness(W⁰,Γ⁰, x, e⁰)for(q, Q⁰)such that:|Γ⁰| ≤n·2²ⁿ·(max_C0,T + 1), and|W⁰| ≤(1 +|Γ⁰|)· n·2²ⁿ.

We call a matching successor witnessslimif it satisfies the cardinality bounds given in the above Lemma. We call a matching successorslimif it has a slim matching successor witness.

To prove Lemma 8, we need to construct the required slim matching successor witness(W⁰,Γ⁰, x, e⁰)from(W,Γ, x, e).

To this end, we choose a setΓ⁰ ⊆ Γ and a functionµ that associates an extended typeµ(y)with everyy∈Γ⁰such that

• µ(y)∈Q(W, fy, gy, hy)and

• if ν ∈ Q(W, fz, gz, hz)for some z ∈ Γ\Γ⁰ then we have|{y∈Γ⁰|µ(y) =ν}|=max_C0,T + 1.

The functionµtells us which objects inΓ⁰ can, in every par- ticular world w, be used to fulfill number restrictions that have been originally fulfilled by (extended types of) objects inΓ\Γ⁰.

The setΓ⁰, in turn, is the basis to constructing aslim wit- nessas it can always be chosen in a way such that|Γ⁰| ≤

|etp(C0,T)|·(maxC₀,T+1)≤n·2ⁿ·(maxC₀,T+1). Finally, for a witness in which|Γ|is bounded as above, we can simply eliminatesuperfluous worldsofW to obtain a slim witness.

This can be done by keeping at most|etp(C₀,T)| ≤n·2²ⁿ worlds for each element of {x} ∪Γ⁰; those worlds can be chosen fromWindependently.

The crucial step of the actual construction is illustrated in Figure 1: consider a particular world w. In the origi- nal witness the number restrictions in the parent object are fulfilled, e.g., by the objectsy_l+1, . . . , y_n with y_m, . . . , y_n falling outside of the set Γ⁰. Assume first, for simplicity, that the objectsy_m, . . . , y_nhave been assigned a common ex- tended typeν. We then pick objectsy1, . . . , yl∈Γsuch that µ(y1) = · · · = µ(yl) = ν. Sincen−m+ 1 ≤ maxC₀,T

we can always find l ≤ max_C0,T + 1of such objects inΓ⁰ such thatl =n−m+ 2; follows from the definition ofΓ⁰. Thus we can transform the old witness to a new one as de- picted in Figure 1. Whenever more than one extended type is associated withy_m, . . . , y_nin the original witness, we simply pick objects inΓ⁰with an appropriate matchingµvalue and proceed similarly to the above example. To construct a slim matching successor witness we fix the setΓ⁰ and apply this transformation to allw ∈ W independently. Note that the transformation preservesquasistates associated with all ob- jects inΓ⁰(hence allS5modalities are preserved) and that all number restrictions are met.

Lemma 9. There is a procedure that runs in 2-EXPTIMEto

generate all slim matching successors.

We simply use a brute-force approach to enumerate all can- didates for slim matching successor witnesses up to exponen- tially sizedΓandW and test for satisfaction of the conditions in Definition 4.

As the next step, we show that whenever a(C0,T)-tree ex- ists, then there is a(C₀,T)-tree constructed solely from slim matching successors, i.e.,(G(n),{G(m)| (n, m)∈ E})is a slim matching successor for alln ∈ N. We call such a (C0,T)-treeslim.

Lemma 10. For anyC₀ and T, a slim(C₀,T)-tree exists whenever a(C0,T)-tree exists.

Since the children in a slim matching successor are a subset of the children in the original matching successor, it is easy to convert an arbitrary(C₀,T)-tree into a slim one.

Finally, to check whether a slim(C0,T)-tree exists we de- fine a looping tree automatonA_C0,T that accepts exactly the slim(C0,T)-trees. To check satisfiability ofC0 w.r.t.T, it then suffices to check whether this looping automaton accepts at least one such tree. This observation yields a 2-EXPTIME

decision procedure for satisfiability inS5_ALCQIas the empti- ness problem for looping tree automata is decidable in time linear in the size of the automaton [Vardi and Wolper, 1986].

We use extended quasistates as states of the automaton AC₀,T andslim matching successorsto define the transition relation. SinceC0,T-trees do not have a constant branch- ing degree, we useamorphous looping automata which are similar to the automata model introduced in [Kupferman and Vardi, 2001] except that in our case the branching degree is bounded and thus the transition relation can be represented finitely in a trivial way.

Definition 11(Looping Tree Automaton). Alooping tree au- tomatonA = (Q, M, I, δ)for anM-labeled tree is defined by a setQof states, an alphabet M, a setI ⊆ Qof initial states, and a transition relationδ⊆Q×M ×2^Q.

LetT = (N, E, `, r)be a tree with rootr ∈ N and label- ing function` : N → M. A runofAonTis a mapping γ : N → Qsuch that γ(r) ∈ I and(γ(α), `(α),{γ(β) | (α, β) ∈ E}) ∈ δfor allα ∈ N. A looping automatonA acceptsthose labeled treesTfor which there exists a run of AonT.

We construct an automaton forC0andT as follows.

Definition 12. Let C₀ be a concept and T an S5_ALCQI TBox. The looping automatonAC₀,T = (Q, M, I, δ)is de- fined by setting M = Q = etp(C₀,T), I := {q ∈ Q | qrealizesC0andqis root}, andδ to the set of those tuples (q, q, Q) such thatQ ∈ 2^Q and(q, Q)is a slim matching successor forC0andT.

The following Lemma states that the obtained looping au- tomata behaves as expected.

Lemma 13. τ is a slim (C₀,T)-tree iff τ is accepted by AC₀,T.

It is easily seen that there are at most 2-EXPmany extended quasistates and thusAC₀,T has at most 2-EXP many states.

To construct the transition function of the automaton, we need to construct all slim matching successors which can be

done in 2-EXPTIME by Lemma 9. Since emptiness of loop- ing automata can be checked in polynomial time, the overall decision procedure for satisfiability inS5_ALCQI runs in 2- EXPTIME. This holds regardless of whether numbers inside number restrictions are coded in unary or in binary.

Theorem 14. Satisfiability inS5_ALCQI w.r.t TBoxes is de- cidable and 2-EXPTIME-complete.

The lower bound in Theorem 14 is obtained by reducing the word problem of exponentially space-bounded, alternating Turing machines. Since the reduction does not use inverse role and qualifying number restrictions, we also obtain a 2- EXPTIMElower bound for satisfiability onS5_ALC.

Corollary 15. Satisfiability inS5_ALC w.r.t TBoxes is decid- able and 2-EXPTIME-complete.

4 Capturing Conceptual Schemata

It is known that the TDLALCQI_{U S} is able to capture the temporal conceptual modelERV T, a TER model that sup- ports timestamping and evolution constraints, IS-A links, dis- jointness, covering, and participation constraints [Artaleet al., 2003]. In ERV T, timestamping is implemented using a marking approach as sketched in the introduction. The translation of atemporal constructs is similar to the one us- ingALCQI_{U S}; see [Artaleet al., 2003] for full details and examples. In the following we briefly recall the translation of atemporal constructs and then show thatS5_ALCQIis suffi- cient to capture the fragment ofERV T that has timestamping as the only temporal construct.

When translating ERV T to TDLs, entities E—denoting sets of abstract objects—are mapped into concept namesA_E and attributesP—denoting functions associating mandatory concrete properties of entities—are mapped into roles names r_P interpreted as total functions, which is enforced by the GCI> v(= 1rP >). InS5_ALCQI, unrestricted entities and attributes need no special treatment. Properties of snapshot or temporary entities and attributes are captured as follows:

A_E v2A_E snapshot entity AE v3¬AE temporary entity A_E v ∃2r_P.> snapshot attribute AE v ∀2rP.⊥ temporary attribute

Relationships—n-ary relations between abstract objects—are translated using reification: each n-ary relationship R is translated into a concept nameA_Rwithnglobalrole names r₁, . . . , r_n. Intuitively, for each instancex∈A^I,w_R , the tuple (y1, . . . , yn)with(x, yi)∈r^I,w_i is a tuple in the relationship Rat a time pointw. To ensure that every instance ofARgives rise to a unique tuple inR, we use GCIs> v (= 1 r_i >), for1 ≤i≤n. To capturesnapshot relationships, we assert A_R v 2A_R, while for temporary relationships, we assert ARv3¬ARin the TBox.

Note that the latter GCIs do not fully capture tempo- rary relationships. As an example, consider the interpreta- tion I = ({w₁, w₂},I), with ∆ = {a, a⁰, b, c}, A^I,w_R ¹ = {a}, A^I,w_R ² = {a⁰}, r^I,w₁ ¹ = {(a, b)}, r^I,w₂ ¹ = {(a, c)}, r₁^I,w2 ={(a⁰, b)}, andr₂^I,w2 ={(a⁰, c)}. Although the GCI

AR v 3¬AR(expressing temporary relationships) is satis- fied,(b, c)is constantly in the temporary relationshipR. This is due to a mismatch between the models of anERV Tschema and the models of its translation intoS5_ALCQI. In particu- lar, in models ofERV T, tuples belonging to relationships are unique while in models of the reified translation there may be twodistinctobjects connected through the global rolesr_i to the sameobjects, thus representing the same tuple (e.g.

the objectsa, a⁰in the above interpretation). Then,S5_ALCQI models in which the above situation occurs do not directly correspond to anyERV T model. Similarly to [Calvaneseet al., 1999], however, it is possible to show that:(i)there are so calledsafe models ofS5_ALCQI that are in one-to-one cor- respondence with ERV T models, and (ii) every satisfiable S5_ALCQI concept is also satisfied in a safe model. When reasoning aboutERV T schemas, we can thus safely ignore non-safe models. AnS5ALCQI interpretationI= (W,I)is safefor anERV T schema if, for everyn-ary relationshipR reified with the global functional rolesri, and everyw∈W, we have the following:

∀x, y, x1, . . . , xn∈∆ :¬((x, x1)∈r₁^I,w∧(y, x1)∈r^I,w₁ ∧. . .∧ (x, xn)∈r^I,w_n ∧(y, xn)∈r_n^I,w).

It is not hard to see that:(1)the model in the example above is not safe, and(2)given a safe model, the above GCIs correctly capture the temporal behavior of relationships.

5 Conclusions

This work introduces the modal description logicS5_ALCQI as a logic for representing and reasoning in temporal con- ceptual models with timestamping constraints. A novel technique is used to show decidability and 2-EXPTIME- completeness forS5ALCQI. This is also the first decidability result that allows reasoning in temporal conceptual models with timestamping for entities, relationships, and attributes.

Furthermore, reasoning on the less expressive logicS5_ALC is also shown to be 2-EXPTIME-complete.

This paper leaves several interesting open problems for further investigation. The fine line separating the decidable TDLs from the undecidable ones is not fully explored: we plan to investigate further extensions ofS5_ALCQI that still enjoy decidability. Two natural candidates areS5_ALCQI that allows, in addition toS5modalities, an irreflexive2(thus en- abling statements abouteverywhere else) andS5_ALCQI with temporalized axioms (enabling TBox statements to appear in scope ofS5operators). Another open issue concerns decid- ability and complexity ofS5_ALCQI in finite models.

On the knowledge representation side, we believe that a converse translation—from TER with full timestamping to S5_ALCQI—is also possible; this result would allow to fully characterize the complexity of reasoning in TER with times- tamping. The limits of the expressive power of S5_ALCQI w.r.t. various constraints that have appeared in literature on temporal models other than timestamping also remain to be determined.

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