Temporalising Tractable Description Logics

Temporalising Tractable Description Logics

A. Artale R. Kontchakov C. Lutz

Faculty of Computer Science School of CS and IS Department of Computer Science Free University of Bozen-Bolzano Birkbeck College Dresden University of Technology

I-39100 Bozen-Bolzano, Italy London WC1E 7HX, U.K. 01062 Dresden, Germany artale@inf.unibz.it roman@dcs.bbk.ac.uk lutz@tcs.inf.tu-dresden.de

F. Wolter M. Zakharyaschev

Department of Computer Science School of Comp. Sci. and Inf. Syst.

University of Liverpool Birkbeck College Liverpool L69 3BX, U.K. London WC1E 7HX, U.K.

frank@csc.liv.ac.uk michael@dcs.bbk.ac.uk

Abstract

It is known that for temporal languages, such as first- orderLT L, reasoning about constant (time-independent) relations is almost always undecidable. This applies to tem- poral description logics as well: constant binary relations together with general concept subsumptions in combina- tions ofLT Land the basic description logicALC cause undecidability. In this paper, we explore temporal exten- sions of two recently introduced families of ‘weak’ descrip- tion logics known as DL-Lite and EL. Our results are twofold: temporalisations of even rather expressive vari- ants of DL-Lite turn out to be decidable, while the tem- poralisation ofELwith general concept subsumptions and constant relations is undecidable.

1. Introduction

Over the last 15 years, many temporalised versions of description logics (DLs) have been suggested and investi- gated. We refer the reader to the survey papers and mono- graph [6, 14, 4] where the history of the development of both interval and point-based temporal extensions of DLs is discussed in full detail. Our main concern in this pa- per are extensions of DLs by point-based temporal logics, in particular the standard linear time temporal logicLT L (see [13] and references therein). The current state of the art in this field can be summarised as follows: it is gen- erally agreed that the semantics of combined temporal de- scription logics should be based on the Cartesian products

of the flow of time (the natural numbersNforLT L) and the domains of the DL interpretations. Thus, a model for the combined language consists of a flow of snapshots that rep- resent the domains of interest at various time points. This semantics corresponds to the semantics of first-order tem- poral logics (more precisely, to first-order temporal mod- els withconstant domains; varying and expanding domains have been considered as well in temporalised DLs, but they are not within the scope of this paper). In fact, the transla- tion of standard DLs into first-order logic can be extended to a translation of temporalised DLs into first-order tempo- ral logics. For this semantics, the expressivity and com- putational complexity of combinations of LT Land DLs extending the standard Boolean DLALC have been com- pletely classified [14, 4]. Instead of trying to summarise all the available results here, we only point out one of the main insights from this investigation:

• combinations ofLT LandALC, which allow general concept inclusions(GCIs)C₁vC₂, are decidable(in fact, usually EXPSPACE-complete)if, and only if, the temporal operators are not applied to binary relations (roles)and, more generally, no constraints are imposed on the binary relations.

In other words, as long as one only wants to reason about the temporal behaviour of axioms (corresponding to closed for- mulas) and concepts (corresponding to unary predicates), the resulting combination is likely to be decidable; but as soon as the combination allows reasoning about the tem- poral behaviour of binary relations it becomes undecidable.

This phenomenon is well understood and reflected in the de- finition of, e.g., the monodic fragments of first-order tempo-

ral logics [17, 12]. In particular, the undecidability results hold for the most important temporal constraint on binary relations, namely, that a role isconstant over time: even a single constant role results in an undecidable combina- tion ofALC andLT Lwith GCIs. Without GCIs, tempo- ral description logics may be decidable even with constant roles [14].

Unfortunately, many applications of temporal descrip- tion logics (say, temporal data modelling, which will be briefly discussed in Section 3, or dynamic ontologies) re- quire both GCIs and temporal constraints on roles, in par- ticular constant roles. It was this problem that motivated the research which resulted in this paper. More precisely, our main aim was to find out whether it is possible to de- sign useful combinations ofLT Land DLs with GCIs and constant roles that are still decidable.

Recent developments in description logic have opened a new path to follow in designing such languages. First, the recognition of the importance oftractablereasoning and, in particular, query answering over DL ontologies with GCIs has given rise to the investigation of the newDL-Lite family of DLs [10, 11, 2]. And second, the use of huge DL-based ontologies with GCIs in bio- and medical informatics has led to the introduction and investigation of ‘weak’ DLs (re- flecting the expressive power of existing ontologies) with tractable subsumption algorithms, namely, the EL-family of DLs [5, 7, 8]. Both families of DLs lack some of the expressive power ofALCbut have nevertheless proved ex- pressive enough for a number of applications. In this paper, we explore to which extent these new families of DLs can provide basis for useful and still decidable combinations of LT Land DLs with GCIs and constant roles.

The obtained results are twofold. On the one hand, we prove in Section 4 that the combination of one of the most expressive versionsDL-Litebool ofDL-LitewithLT Lis in- deed decidable (in EXPSPACE), even with GCIs and con- stant roles. Moreover, its Krom fragment turns out to be decidable in PSPACE. The proofs are based on an embed- ding into the one-variable fragment of first-order temporal logic. This means, in particular, that reasoning in temporal DL-Lite can be supported by available temporal provers;

see, e.g., [12]. On the other hand, we show in Section 7 that the corresponding combination ofELandLT Lis undecid- able. The meaning of these results is analysed in Section 8.

2. Temporal extension of DL-Lite

We begin by introducing the temporal extension TDL-Litebool of one of the most expressive description log- icsDL-Litebool of theDL-Litefamily [2]. It combines the temporal operatorsofLT L,(‘at the next moment’) and U (‘until’), with the language ofDL-Lite_bool in a straight- forward manner by applying them to concepts and Boolean

combinations of GCIs and ABox assertions. Moreover, we will distinguish between local and global role names.

Thus,TDL-Litebool containsobject namesa₀, a₁, . . .,con- cept namesA₀, A₁, . . ., local role namesP₀, P₁, . . ., and global role namesT₀, T₁, . . .. RolesR,basic conceptsB andconceptsCofTDL-Lite_bool are defined as follows:

R ::= Pi | P_i⁻ | Ti | T_i⁻, B ::= ⊥ | Ai | ≥q R, C ::= B | ¬C | C₁uC₂ |

C | C1UC2,

whereq ≥ 1 is a natural number (note that the results of this paper do not depend on whetherqis given in unary or in binary). TDL-Lite_bool formulasare built from atoms of the form

C1vC2, C(ai), R(ai, aj)

with the help of the Boolean connectives (say,¬and∧) and the temporal operatorsandU. The atomsC1 vC2are often called general concept inclusions (GCIs), while the atomsC(ai)andR(ai, aj)are calledABox assertions.

ATDL-Litebool interpretationIis a function

I(n) = ∆, a^I(n)₀ , . . . , A^I(n)₀ , . . . , P₀^I(n), . . . , T₀^I(n), . . . ,

where∆is a nonempty set,n∈N,a^I(n)_i ∈∆,A^I(n)_i ⊆∆, P_i^I(n) ⊆ ∆×∆,T_i^I(n) ⊆ ∆×∆, witha^I(n)_i = a^I(m)_i andT_i^I(n) =T_i^I(m), for alln, m∈N, anda^I(n)_i 6=a^I(n)_j , for all i 6= j and all n ∈ N(the last condition means the unique name assumption, which standard in DL). The role and concept formation constructors are interpreted inI as follows (whereR_iis either a local or global role name):

(R⁻_i )^I(n) =

(y, x)|(x, y)∈R^I(n)_i ,

⊥^I(n) = ∅, (≥q R)^I(n) =

x∈∆|]{y|(x, y)∈R^I(n)} ≥q , (¬C)^I(n) = ∆\C^I(n),

(C1uC2)^I(n) = C₁^I(n)∩C₂^I(n), (C)^I(n) = C^I(n+1),

(C1U C2)^I(n) = [

k>n

C₂^I(k)∩ \

n<m<k

C₁^I(m) .

The standard abbreviations > ≡ ¬⊥, ∃R ≡ (≥1R), C₁ t C₂ ≡ ¬(¬C1 u ¬C2), ≤q R ≡ ¬(≥q+ 1R), (=q R) ≡ (≤qR)u(≥q R), 3FC ≡ > U C (‘some time in the future’) and2FC ≡ ¬3F¬C (‘always in the future’) we need in what follows are self-explanatory and correspond to the intended semantics.

Thesatisfaction relation(I, n)|=ϕ, for aTDL-Litebool

formulaϕ, is defined inductively:

(I, n)|=C1vC2 iff C₁^I(n)⊆C₂^I(n), (I, n)|=C(ai) iff a^I(n)_i ∈C^I(n),

(I, n)|=R(ai, aj) iff (a^I(n)_i , a^I(n)_j )∈R^I(n), (I, n)|=¬ϕ iff (I, n)6|=ϕ,

(I, n)|=ϕ1∧ϕ2 iff (I, n)|=ϕ1and(I, n)|=ϕ2, (I, n)|=ϕ iff (I, n+ 1)|=ϕ,

(I, n)|=ϕ1Uϕ2 iff there isk > nwith(I, k)|=ϕ2

and(I, m)|=ϕ1for alln < m < k.

We will also freely use the Booleans→and∨and the tem- poral operators2F and3F for formulas. A formulaϕis satisfiableif there is an interpretationIand a time pointn such that(I, n)|=ϕ.

Observe that the interpretation of object names and global role names is time-independent, while the interpre- tation of local role names and concepts is allowed to vary over time. Time-independent concepts can be introduced by means of the axioms2⁺F AvA

and2⁺F AvA , where2⁺Fϕ≡ϕ∧2Fϕ.

At first sight one might think that the satisfiability prob- lem for this logic is undecidable because using a single global functional role T (functionality can be ensured by the axiom≥2T v ⊥) with functional T⁻ one can easily enforce the existence of aN×Ngrid, which could possibly be used to encode the undecidableN×Ntiling problem.

However, the language is not capable of expressing the re- quirements on colour matching in the domain ‘dimension,’

i.e., that if(x, y)∈T^I(n)then the colours of tiles covering xandymatch (which can be easily expressed with the qual- ified existential quantifier ∃T.C). In fact, as we shall see in the next section, TDL-Litebool can be embedded in the one-variable fragment of first-order temporal logic, which is known to be decidable, actually, EXPSPACE-complete;

see, e.g., [14]. Note that satisfiability inDL-Litebool is NP- complete [2].

3. Temporal data modelling with TDL-Lite

Here we briefly discuss howTDL-Litebool can be used for temporal data modelling. It was argued in [10] that the underlying DL DL-Litebool can represent atemporal con- ceptual data models like UML class diagrams and Entity- Relationship models. For example, one maps entities E, denoting sets of abstract objects, into concept names A_E. Then one can represent the subclass relation (ISA) and disjointness between E₁ and E₂ by A_E1 v A_E2 and A_E1 v ¬A_E2, respectively, and to express thatE iscov- eredbyE1, . . . , En one can useAE v AE₁t · · · tAE_n

andAE1 vAE, . . . , AEn vAE. To capture ann-ary re- lationshipRover entitiesE1, . . . , En, onereifiesthe rela- tionship. First, take a concept nameA_Randnrole names R₁, . . . , R_n. The GCIsA_R v(= 1R_i)ensure that every instance ofA_Rgives rise to a unique tuple inR; the GCIs

∃R⁻_i v A_Ei guarantee that only instances of E₁, . . . , E_n may be connected by R. Participation constraints are captured by cardinality restrictions AE_i v (≥k R⁻_i )and AE_i v(≤m R⁻_i ). An attributeP of an entityE, associat- ing values of a concrete domainDto instances ofE, is con- sidered as a binary relationship linkingEwithD: this can be captured by a conceptAP and a pair of functional roles P1andP2with the GCIsAP v(= 1P1),AP v(= 1P2),

∃P₁⁻ vAEand∃P₂⁻vD.

In the temporal context, we can express all those con- straints using 2⁺F(C₁ v C₂) instead of the atemporal C₁ v C₂. Below we writeC₁ v^∗ C₂for2⁺_F(C₁ v C₂).

However, even at this basic level, global roles are already required: when reifying relationships, to ensure that every instance ofARrepresents the same tuple at different times, the rolesRi should be global; similarly, the rolesP1 and P2 introduced for an attributeP should be global. More- over, concrete domains should be constant and disjoint: this is captured by(D v^∗ D)∧(D v^∗ D), for allD, and Dv^∗¬D⁰, for all distinct concrete domainsD, D⁰.

In addition, the temporal constructors of TDL-Litebool

are able to represent dynamic aspects of conceptual mod- els. Timestampingis the basic temporal constraint used to model the temporal behaviour of entities, relationships and attributes [18, 3]. It is implemented either by marking enti- ties, relationships and attributes assnapshotortemporary, or leaving them unmarked. An object belongs to a snap- shot entity either never or at all times, no object may belong to a temporary entity at all times, and there are no tempo- ral assumptions about instances of unmarked entities. The meaning of timestamps for relationships and attributes is analogous. In TDL-Litebool timestamps are expressed by the following formulas: (AE v^∗ AE)∧(AE v^∗ AE) for asnapshot/global entityand(> v3⁺F¬AE)for atem- porary entity. Timestamping formulas for a relationshipR involve the concept nameA_R that reifies the relationship;

then we need (A_R v^∗ A_R)∧(A_R v^∗ A_R)for the snapshot/global relationship, and(> v 3⁺_F¬A_R)for the temporary relationship. Attributes are treated similarly.

Finally, TDL-Litebool is capable of capturing dynamic transitionsbetween entities where objects of a source entity, E1, migrate to a target entity,E2, with the help of the GCI AE₁ v^∗3FAE₂.

It was observed in [1] that temporal conceptual models with timestamping and evolution constraints can be trans- lated into the DLDLR_{U S}and that reasoning with temporal models with both timestamping and dynamic constraints is undecidable. The main difference here is thatTDL-Litebool

lacks the ability to represent sub-relationships which is an essential part in the undecidability proof.

4. TDL-Lite

is E

S

-complete

This result is proved by providing a satisfiability pre- serving translation ofTDL-Litebool formulas into theone- variable fragmentQT L¹of first-order temporal logic with- out function symbols and equality. To define the syntax of QT L¹, fix one variablex. Then the formulas ofQT L¹are constructed from unary predicatesP(x)andP(a_i)(where a_i is a constant) and propositional variables pusing the standard connectives of first-order logic (with quantifiers

∀xand∃x) and the temporal operatorsandU. QT L¹- models and the satisfaction relation between formulas and time points are defined in the obvious way by modifying the definition ofTDL-Liteboolinterpretations (however, there is no unique name assumption in this case); for details we re- fer the reader to [14], where the following is also shown:

Theorem 1. The satisfiability problem forQT L¹-formulas isEXPSPACE-complete.

Now we define a translation·^† ofTDL-Litebool formu- las into QT L¹. Let ϕbe a TDL-Litebool formula. De- note byrole(ϕ)the set of both local and global role names occurring in ϕ, by g-role(ϕ)the set of global role names in ϕ, and by ob(ϕ) the set of object names in ϕ. Let role^±(ϕ) = {R, R⁻ | R ∈ role(ϕ)} andg-role^±(ϕ) = {T, T⁻ | T ∈ g-role(ϕ)}. Denote byq_ϕ the maximum numerical parameter inϕ.

With every object namea_i ∈ob(ϕ)we associate the in- dividual constanta_iofQT L¹and with every concept name Aithe unary predicateAi(x)from the signature ofQT L¹. For eachR ∈ role^±(ϕ), we also introduceqϕfresh unary predicatesEqR(x), for1 ≤q ≤qϕ. Intuitively, for each n,E1R(x)andE1R⁻(x)represent the domain and range of R at momentn (i.e.,E1R(x)andE1R⁻(x)are inter- preted by the sets of points withat least oneR-successor andat least oneR-predecessor at momentn, respectively), while EqR(x)and EqR⁻(x) represent the sets of points with at least q distinct R-successors and at least q dis- tinctR-predecessors at moment n. Additionally, for each paira_i, a_j ∈ ob(ϕ)and each roleR ∈ role^±(ϕ), we take a freshpropositional variableRa_ia_j of QT L¹ to encode R(a_i, a_j).

By induction on the construction of aTDL-Lite_bool con- ceptCwe define theQT L¹- formulaC^∗:

⊥^∗=⊥,

(A)^∗=A(x), (≥q R)^∗=EqR(x), (¬C)^∗=¬C^∗(x), (C₁uC₂)^∗=C₁^∗(x)∧C₂^∗(x), (C)^∗=C^∗(x), (C1UC2)^∗=C₁^∗(x)UC₂^∗(x),

whereAis a concept name andRa role. Next, we extend this translation toTDL-Litebool-formulas:

(C₁vC₂)^∗=∀x(C₁^∗(x)→C₂^∗(x)),

(C(ai))^∗=C^∗(ai), (R(ai, aj))^∗=Raiaj, (¬ψ)^∗=¬ψ^∗, (ψ₁∧ψ₂)^∗=ψ₁^∗∧ψ^∗₂, (ψ)^∗=ψ^∗, (ψ1Uψ2)^∗=ψ₁^∗Uψ^∗₂,

where C, C1, C2 are concepts, R is a role and ai, aj are object names.

The following formulas express some natural properties of the role domains and ranges. For everyR ∈ role^±(ϕ), we need twoQT L¹-sentences:

ε(R) = ∃x E1R(x) → ∃xinv(E₁R)(x), (1) δ(R) =

q_ϕ−1

^

q=1

∀x Eq+1R(x)→E_qR(x) , (2)

where inv(E1R) is the predicate E1R⁻_k if R = Rk and E1Rk if R = R⁻_k. Sentence (1) says that if the domain ofRis not empty then its range is not empty either.

We also need formulas representing the relation of the Ra_ia_j with the unary predicates for the role domain and range. For a roleR∈role^±(ϕ), let

ω(R) =

qϕ

^

q=1

^

a∈ob(ϕ) aj1,...,a_jq∈ob(ϕ)

j_i6=j_i0fori6=i⁰

^^q

i=1

Raaj_i→EqR(a) , (3)

ι(R) = ^

ai,aj∈ob(ϕ)

Raiaj →inv(R)ajai

, (4)

whereinv(R)a_ja_i is the propositional variable R⁻_ka_ja_i if R=R_kandR_ka_ja_iifR=R⁻_k.

For every global roleT ∈g-role^±(ϕ)we need two addi- tional sentences:

γ₁(T) =

q_ϕ

^

q=1

∀x E_qT(x)↔E_qT(x)

, (5) γ2(T) = ^

ai,aj∈ob(ϕ)

(T aiaj ↔T aiaj). (6)

Finally, we set ϕ^† = ϕ^∗ ∧ ^

R∈role^±(ϕ)

2⁺F ε(R)∧δ(R)∧ω(R)∧ι(R)

∧ ^

T∈g-role^±(ϕ)

2⁺F γ1(T)∧γ2(T) .

Theorem 2. ATDL-Litebool formulaϕis satisfiable iffϕ^† is satisfiable.

Proof. (⇐)LetMbe a first-order temporal model with a countabledomainDand let(M,0) |=ϕ^†. We denote the interpretation of unary predicatesP and propositional vari- ablespinMat momentnbyP^M,nandp^M,n. The inter- pretation of constantsainMis denoted bya^M. Let

W₀=

a^M |a∈ob(ϕ) ⊆D.

Without loss of generality we may assume that all thea^M are distinct.

We are going to construct aTDL-Litebool interpretation Isatisfyingϕthat is based on the domain

∆ =W0 ∪ (D×N).

The interpretations of object names inIare given by their interpretations inM: a^I(n) = a^M ∈ W0. The interpreta- tionsA^I(n)of concept namesAinIare defined by taking

A^I(n)=

w∈∆|(M, n)|=A^∗[cp(w)] , (7) where the functioncp: ∆→Dis defined as follows:

cp(w) =

(w, ifw∈W0,

d, ifw= (d, n)∈D×N. (8) We will callwacopyofcp(w). Now, for eachR∈role(ϕ) and eachn∈N, we introduce inductively the interpretation R^I(n). (For globalRthis can be done for some fixedn, say 0, and then copied for all othern.)R^I(n)will be defined as the union

R^I(n)=

∞

[

m=0

R^n,m,

where, for allm≥0,R^n,m⊆W_R^n,m×W_R^n,m, W_R^n,m⊆W_R^n,m+1 and

∞

[

m=0

W_R^n,m= ∆.

We start withW_R^n,0 =W₀. The setW_R^n,m\W_R^n,m−1, for m ≥0, will be denoted byV_R^n,m; for convenience, we let W_R^n,−1=∅, so thatV_R^n,0=W0.

First we define therequiredR-rankrⁿ(R, d)ofd∈ D at momentn:

rⁿ(R, d) =











0, if(M, n)|=¬E1R[d],

q, if(M, n)|=EqR∧ ¬Eq+1R[d], for1≤q < q_ϕ, qϕ, if(M, n)|=Eq_ϕR[d].

It follows from (2) that rⁿ(R, d)is a function and that if d∈Dandrⁿ(R, d) =qthen(M, n)|=E_q⁰R[d]whenever 1 ≤ q⁰ ≤ q, and(M, n) |= ¬Eq⁰R[d]forq < q⁰ ≤ qϕ.

We also define theactual R-rankrⁿ_m(R, w)ofw ∈ ∆ at momentnand stepmby taking

rⁿ_m(R, w) =







q, ifw∈ ≥q R^n,m.∆\≥q+ 1R^n,m.∆, for0≤q < q_ϕ, qϕ, ifw∈ ≥qϕR^n,m.∆,

where≥q S.∆ =

x∈∆|]{y|(x, y)∈S} ≥q , for a binary relationS.

For the basis of induction we set R^n,0=

(a^M_i , a^M_j )∈W0×W0|(M, n)|=Raiaj . (9) By (3) and (4), for bothRandR⁻(whereR⁻⁻ =R) and allw∈W₀,

r₀ⁿ(R, w) ≤ rⁿ(R,cp(w)). (10) Suppose that the W_R^n,mand R^n,m have already been de- fined form ≥ 0. If we had rⁿ_m(R, w) = rⁿ(R,cp(w)), for bothRandR⁻and allw∈W_R^n,m, then the interpreta- tionR^n,mwe need forR^I(n)would have been constructed.

However, in general this is not the case because there may be some ‘defects’ in the sense that the actual rank of some points is smaller than the required rank. Consider the fol- lowing two sets of defects inR^n,m:

Λ^n,m_R =

w∈V_R^n,m|rⁿ_m(R, w)< rⁿ(R,cp(w)) , Λ^n,m_R− =

w∈V_R^n,m|rⁿ_m(R⁻, w)< rⁿ(R⁻,cp(w)) . The purpose of, say, Λ^n,m_R is to identify those ‘defective’

pointsw ∈ V_R^n,mfrom which preciselyrⁿ(R,cp(w))dis- tinctR-arrows should start (according toM), but some ar- rows are still missing (onlyr_mⁿ(R, w)many arrows exist).

To ‘cure’ these defects, we extendW_R^n,mtoW_R^n,m+1 and R^n,mtoR^n,m+1according to the following rules:

(Λ^n,m_R ) Letw ∈Λ^n,m_R . Denoteq=rⁿ(R, d)−r_mⁿ(R, w) andd = cp(w). Then(M, n) |= Eq⁰R[d]for some q⁰ ≥ q > 0. By (2), (M, n) |= E1R[d]and, by (1), there is d⁰ ∈ D such that(M, n) |= E1R⁻[d⁰]. In this case we take q fresh copies w⁰₁, . . . , w⁰_q of d⁰, i.e., w₁⁰, . . . , w⁰_q ∈ ({d⁰} ×N)\W_R^n,m, add them to W_R^n,m+1 and add the pairs(w, w_i⁰), 1 ≤ i ≤ q, to R^n,m+1.

(Λ^n,m_R−) Letw∈Λ^n,m−_R− . Denoteq=rⁿ(R⁻, d)−r_mⁿ(R⁻, w) andd =cp(w). Then(M, n)|=E_q⁰R⁻[d]for some q⁰ ≥ q > 0. By (2), (M, n) |= E₁R⁻[d] and, by (1), there is d⁰ ∈ D with (M, n) |= E₁R[d⁰].

In this case we takeqfresh copiesw⁰₁, . . . , w⁰_q of d⁰, i.e., w₁⁰, . . . , w⁰_q ∈ ({d⁰} ×N)\W_R^n,m, add them to W_R^n,m+1 and add the pairs(w⁰_i, w), 1 ≤ i ≤ q, to R^n,m+1.

(Ω) Finally, if all defects for R in W_R^n,m have already been cured we take, for every d ∈ D, a fresh copy (d, l)∈({d} ×N)\W_R^n,mwith minimalland add it toW_R^n,m+1.

It should be clear that the rule (Ω) guarantees that S∞

m=0W_R^n,m = ∆. Now we observe the following im- portant property of the construction: for allm₀ ≥ 0 and w∈V_R^n,m0,

r_mⁿ(R, w) =











0, ifm < m₀,

q, ifm=m0, for some

q≤rⁿ(R,cp(w)), rⁿ(R,cp(w)), ifm > m0.

(11)

To prove this property, consider all possible cases. If m < m₀ then w has not been added to W_R^n,m yet, i.e., w /∈ W_R^n,m, and so rⁿ_m(R, w) = 0. If m = m₀ and m₀= 0thenr_mⁿ(R, w)≤rⁿ(R,cp(w))follows from (10).

If m = m0 andm0 > 0 thenw was added at step m0

either to cure a defect of some point w⁰ ∈ W_R^n,m0−1 or by(Ω). In the latter case we clearly haverⁿ_m(R, w) = 0, and sorⁿ_m(R, w)≤ rⁿ(R,cp(w)). In the former case this means that either(w⁰, w) ∈ R^n,m0 andw⁰ ∈ Λ^n,m_R ⁰⁻¹or (w, w⁰)∈R^n,m0andw⁰∈Λ^n,m_R−⁰⁻¹. In the first case

(M, n)|=E1R⁻[cp(w)]. (12) Sincefresh witnessesware picked up every time the rule (Λ^n,m_R ⁰⁻¹) is applied and those witnesses satisfy (12), we obtain r_mⁿ₀(R, w) = 0, rⁿ_m0(R⁻, w) = 1 and rⁿ(R⁻,cp(w)) ≥ 1. The second case is similar. If m=m0+ 1then all defects ofware cured at stepm0+ 1 by applying the rules (Λ^n,m_R ⁰) and (Λ^n,m_R−⁰). Therefore, rⁿ_m0+1(R, w) = rⁿ(R,cp(w)). Ifm > m0+ 1then (11) follows from the observation that new arrows involvingw can only be added at stepm₀+ 1, that is, for allm≥0,

R^n,m+1\R^n,m ⊆

V_R^n,m×V_R^n,m+1 ∪ V_R^n,m+1×V_R^n,m. (13) Finally, recall that if R is global then, by (5) and (6), the above inductive procedure does not depend onn and R^I(n)=R^I(m), for alln, m∈N.

It follows that, for allR∈role^±(ϕ),1≤q≤q_ϕ,n∈N andw∈∆,

(M, n)|=E_qR[cp(w)] iff w∈ ≥q R^I(n).∆. (14) Indeed, if (M, n) |= E_qR[cp(w)] then, by definition, rⁿ(R,cp(w)) ≥ q. Let w ∈ V_R^n,m0. Then, by (11), rⁿ_m(R, w) = rⁿ(R,cp(w)) ≥ q, for all m > m₀. It follows from the definition of rⁿ_m(R, w) and R^I(n) that w ∈ ≥q R^I(n).∆. Conversely, letw ∈ ≥q R^I(n).∆and

w ∈ V_R^n,m0. Then, by (11), we have q ≤ r_mⁿ(R, w) = rⁿ(R,cp(w)), for all m > m0. So, by the definition of rⁿ(R,cp(w))and (2), we have(M, n)|=E_qR[cp(w)].

Now we show by induction on the construction of con- ceptsCinϕthat, for alln∈Nandw∈∆,

(M, n)|=C^∗[cp(w)] iff w∈C^I(n). (15) The basis of induction is trivial for B = ⊥ and follows from (7) ifB =Aiand (14) ifB =≥q R. The induction step for the Booleans (C=¬C1andC=C1uC2) and the temporal operators (C =C1andC =C1U C2) follows from the induction hypothesis.

Finally, we show that for each subformulaψofϕ, (M, n)|=ψ^∗ iff (I, n)|=ψ. (16) Forψ=C1 vC2andψ=C(ai), this follows from (15).

For ψ = Rk(ai, aj), (a^I_i, a^I_j) ∈ R^I(n)_k iff, by (13), (a^I_i, a^I_j) ∈ R^n,0_k iff, by (9),(M, n) |= Rkaiaj. The case ψ = R_k⁻(a_i, a_j) is similar. The induction step for the Booleans (ψ = ¬ψ1 andψ = ψ1∧ψ2) and the tempo- ral operators (ψ=ψ1andψ=ψ1Uψ2) follows from the induction hypothesis.

Thus, we obtain (I,0) |= ϕ. The implication(⇒) is straightforward.

The translationϕ^†ofϕis obviously too lengthy to pro- vide us with reasonably low complexity results. However, it follows from the proof above that in fact a lot of information in this translation is redundant and can be safely omitted.

We define now a more concise translation ofϕintoQT L¹. ForR ∈ role^±(ϕ), let Q^R_ϕ be the set of natural numbers containing1and all the numerical parametersqfor which

≥q Roccurs inϕ. Then we set ϕ^[=ϕ^∗ ∧ ^

R∈role^±(ϕ)

2⁺F ε(R)∧δ^[(R)∧ω^[(R)∧ι(R)

∧ ^

T∈g-role^±(ϕ)

2⁺F γ₁^[(T)∧γ2(T) ,

whereε(R),ι(R)andγ₂(T)are as before (see (1), (4) and (6), respectively),

δ^[(R) = ^

q,q⁰∈Q^R_ϕ, q⁰>q q⁰>q⁰⁰>qfor noq⁰⁰∈Q^R_ϕ

∀x Eq⁰R(x)→EqR(x)

, (17)

ω^[(R) = ^

q∈Q^R_ϕ

^

a∈ob(ϕ) a_j1,...,a_jq∈ob(ϕ)

ji6=j_i0fori6=i⁰

q

^

i=1

Raa_ji →E_qR(a) , (18)

γ^[₁(T) = ^

q∈Q^T_ϕ

∀x EqT(x)↔E_qT(x)

. (19)

Corollary 3. ATDL-Litebool formulaϕis satisfiable iff the QT L¹-sentenceϕ^[is satisfiable.

Proof. Follows from the fact thatϕ^†is satisfiable iffϕ^[ is satisfiable. Indeed, if (M,0) |= ϕ^† then (M,0) |= ϕ^[. Conversely, if(M,0) |= ϕ^[then one can construct a new modelM⁰based on the same domainDasMby taking

• A^M0,n=A^M,n, for all concept namesAandn∈N;

• EqR^M0,n=Eq⁰R^M,n, forR∈role^±(ϕ),1≤q≤qϕ

andn∈N, whereq⁰is the maximum number fromQ^R_ϕ withq⁰≤q;

• Raiaj to be true inM⁰ atniff(M, n)|=Raiaj, for allR∈role^±(ϕ), allai, aj∈ob(ϕ)and alln∈N;

• a^M0 =a^M, for alla∈ob(ϕ).

It follows immediately from the definition that we have (M⁰,0) |=ϕ^†. (For example,(M⁰,0) |=ϕ^∗ follows from the fact that for every concept (≥q R) from ϕ we have EqR^M0,n=EqR^M,n, for alln∈N.)

This observation makes it possible to prove the following result:

Theorem 4. The satisfiability problem forTDL-Litebool is EXPSPACE-complete.

Proof. As we know, satisfiability forQT L¹is EXPSPACE- complete. However, we cannot use this result directly be- cause the size of ϕ^[ is exponential in the number of ob- ject names (in fact, double exponential, if qϕ is given in binary): |ϕ^[| ≤const· |ϕ|+|ob(ϕ)|^qϕ+1. Instead, we use the EXPSPACEalgorithm presented in [14, Theorem 11.30]

(see also [16]) which, given aQT L¹-sentenceψ, decides whetherψis satisfiable or not by guessing an ultimately pe- riodical quasimodel such that the lengths of its prefix and its period are bounded by some numbersl1andl2, respec- tively. In general, bothl1andl2are double exponential in the length|ψ|of ψ. Hence, the algorithm requires single exponential space to write down the two numbers. The al- gorithm also requires exponential space to store at most 3 state candidates. Clearly, every realisable state candidateC forϕ^[is uniquely determined by the following parameters:

• the set of propositional variables and the set of closed subformulas of the form ∀x χ(x) that belong to the types ofC;

• for every type inC, the set of all open subformulas that belong to this type.

It is easy to compute thatϕ^[contains|role^±(ϕ)| · |ob(ϕ)|² propositional variables and|ϕ|+3·|role^±(ϕ)|closed subfor- mulas of the form∀x χ(x). Therefore, the ‘propositional’

part of a state candidate can be stored in space bounded

byp1(|ϕ|), wherep1is a polynomial. Next, for each type forϕ^[, the number of open subformulas that belong to this type is bounded by |ϕ|, and the number of types in every state candidate is bounded by 2^|ϕ|. Therefore, the ‘type’

part of a state candidate can be stored in space bounded by 2^|ϕ|· |ϕ|, and so the overall space required to store a state candidate forϕ^[is bounded by2^p2(|ϕ|), for some polyno- mialp2. Now, [14, Theorem 11.26] provides more precise upper bounds onl1andl2:

l₁≤](ϕ^[) and l₂≤k_ϕ[·](ϕ^[)·[²(ϕ^[) +](ϕ^[), where](ϕ^[)is the number of distinct state candidates,[(ϕ^[) the number of distinct types, andk_ϕ[ the number of ‘even- tualities,’ i.e., subformulas ofϕ^[ of the formχ₁U χ₂. It follows from the above argument that](ϕ^[)≤2^{2p2 (|ϕ|)}and [(ϕ^[) ≤ 2^p1(|ϕ|)·2^|ϕ|(every type for ϕ^[ is uniquely de- termined by its ‘propositional’ part and the subset of open subformulas that belong to it). Finally, the numberk_ϕ[ of

‘eventualities’ is bounded by|ϕ|+2·|role^±(ϕ)|. This shows that although the length ofϕ^[is (double) exponential in|ϕ|, the numbers l1 andl2 are only double exponential in |ϕ|

(not triple exponential as one would expect). Therefore, the algorithm of [14, Theorem 11.30] runs in EXPSPACE.

The EXPSPACE lower bound follows from the fact that there is a satisfiability preserving polynomial translation from QT L¹ toTDL-Litebool. First, by introducing new unary predicates one can transform, in a satisfiability pre- serving way, each QT L¹-formula into aQT L¹-sentence containing neither ∃xnor nested ∀x. Such a sentence ϕ can be translated intoTDL-Liteboolby first associating with every unary predicateP(x)a concept name(P(x))^‡=A_P. For every subformulaψofϕwith freex, we obtain a con- ceptψ^‡ by distributing the translation·^‡ over the connec- tives,U,¬and∧, e.g.,(ψ1∧ψ2)^‡ =ψ^‡₁uψ₂^‡. For each subformula of the form ∀x ψ, set(∀x ψ)^‡ = (> v ψ^‡).

Now, forQT L¹-sentences, the translation·^‡ again distrib- utes over the connectives,U,¬and∧. It is easily seen thatϕis satisfiable iffϕ^‡is satisfiable.

The same lower bound follows also from Theorem 10 below.

5. TDL-Lite

is PS

-complete

Consider now the Krom fragment TDL-Litekrom of TDL-Litebool with atomic formulas of the form

D1vD2, ¬D1vD2, D1v ¬D2, D(a_i), R(a_i, a_j),

where conceptsD1, D2are formed from basic conceptsB by means ofonly:

D ::= B | D. (20)

We can still apply all temporal operators and the Booleans to formulas. (Note that spatio-temporal logics of a similar kind were considered in [15] and [9]. Note also that satisfi- ability for the underlying DLDL-Litekromis NLOGSPACE- complete [2].)

It is readily seen that the·^[-translations ofTDL-Lite_krom formulas can be transformed in a satisfiability preserving way (by introducing abbreviations for nestedoperators) to formulas of the following fragmentQT L¹_kromofQT L¹:

Q(x) ::= Pi(x) | ¬Pi(x), L(x) ::= Q(x) | Q(x),

ϕ ::= ∀x L1(x)∨L₂(x)

| L(a_j) |

¬ϕ | ϕ1∧ϕ2 | ϕ | ϕ1Uϕ2,

where the Pi are unary predicate symbols and the aj are constants. PredicatesPi(x)and their negations¬Pi(x)will be calledliterals; literalsQ(x)and-prefixed literals will be calledtemporal literals.

In this section we establish (using the quasimodel tech- nique from [14]) a PSPACEupper bound for satisfiability of QT L¹_kromformulas from which we obtain the following re- sult (using Lemma 8 and an argument similar to the proof of Theorem 4):

Theorem 5. The satisfiability problem forTDL-Litekrom

formulas isPSPACE-complete.

We denote by¬L(x)˙ the formula equivalent to¬L(x) in the above restricted syntax, e.g., ¬˙Pi(x)is¬Pi(x) and¬˙¬Pi(x)isPi(x). For every formula of the form

Q(x), we reserve a unary predicateQ⁰(x)called thesur- rogateof Q(x). Note that we introduce surrogates only for temporal literals (unlike ‘standard’ quasimodels, here we do not need to explicitly introduce surrogates for other temporal subformulas). Given a formulaψ, denote byψthe result of replacing all subformulas ofψof the formQ(x) by their surrogates.

For a QT L¹_krom sentence ϕ, let clϕ be the union of sub0ϕ,Σ_ϕand theΞ^a_ϕ, fora∈conϕ, wheresub0ϕis the set ofclosedsubformulas ofϕ,conϕthe set of all constants inϕ, and

Λϕ=

Pi(x), ¬Pi(x), Pi(x), ¬Pi(x) |

Pi(x)a predicate inϕ , Σϕ=

∀x(L1(x)∨L2(x)) | L1(x), L2(x)∈Λϕ , Ξ^a_ϕ=

L(a) | L(x)∈Λϕ , fora∈conϕ.

Astate candidateCforϕis any subset ofclϕsatisfying the properties

(qs⁰_K) χC=V

ψ∈C∩Σϕψ is satisfiable;

(qs¹_K) for everyψ∈Σϕ, if χ_C|=ψ then ψ∈C;

(qs⁰_c) for everyL(a)∈Ξ^a_ϕ, ¬L(a)˙ ∈C iff L(a)∈/C;

(qs¹_c) for everyL₁(a), L₂(a)∈Ξ^a_ϕ,

ifL₁(a), L₂(a)∈Cthen∀x( ˙¬L₁(x)∨¬L˙ ₂(x))6∈C;

(qs^¬) for every¬ψ∈sub0ϕ, ¬ψ∈C iff ψ /∈C;

(qs^∧) for everyψ1∧ψ2∈sub0ϕ,

ψ1∧ψ2∈C iff ψ1, ψ2∈C.

Let q be a map associating with everyw ∈ Na state candidateq(w)forϕ. We callqaquasimodelforϕif the following conditions hold:

(qm0) ϕ∈q(w0), for somew0≥0;

(qm1) for every∀x(Q1(x)∨Q2(x))∈Σϕ,

∀x(Q₁(x)∨Q₂(x))∈q(w)

iff ∀x(Q₁(x)∨Q₂(x))∈q(w+ 1);

(qm2) for everyQ(a)∈Ξ^a_ϕ,

Q(a)∈q(w) iff Q(a)∈q(w+ 1);

(qm3) for everyψ∈sub0ϕ,

ψ∈q(w) iff ψ∈q(w+ 1);

(qm4) for every ψ₁ U ψ₂ ∈ sub0ϕ, ψ₁U ψ₂ ∈ q(w) iff there is k > 0 such that ψ₂ ∈ q(w+k) and ψ₁∈q(w+n), for all0< n < k.

Lemma 6. AQT L¹_kromsentenceϕis satisfiable iff there is a quasimodel forϕ.

Proof. Suppose(M, w0)|=ϕ. Then q(w) =

ψ∈clϕ|(M, w)|=ψ

defines a quasimodel forϕ. Conversely, suppose thatqis a quasimodel forϕ.

Claim 7. If {L₁(x), . . . , L_k(x)} ⊆ Λ_ϕ and C is a state candidate forϕ, then

χ_C ∧ ∃x L₁(x)∧ · · · ∧L_k(x)

(21) is satisfiable iff there are no 1 ≤ i, j ≤ k such that

∀x( ˙¬Li(x)∨¬L˙ j(x))∈C.

Proof of claim. As formula (21) is a conjunction of the form∀x χ1(x)∧ ∃x χ2(x), it is satisfiable iff the formula χ1[a]∧χ2[a]is satisfiable, whereais a constant symbol.

Now, if χ1[a]∧χ2[a] is satisfiable then there are no i, j such that∀x( ˙¬Li(x)∨¬L˙ j(x))∈C. Conversely, suppose that there are no suchi, j, butχ₁[a]∧χ₂[a]is not satisfi- able. Thenχ₁[a]|=¬χ2[a]. By(qs⁰_K),χ₁[a]is satisfiable.

Moreover as it is a 2-CNF,

χ₁[a]|=¬L₁[a]∨ · · · ∨ ¬L_k[a]

implies that there are i, j withχ₁[a] |= ¬L_i[a]∨ ¬L_j[a].

It follows from(qs¹_K)that¬L_i[a]∨ ¬L_j[a]is a conjunct of χ1[a], contrary to our assumption.

Say thatt⊆Λϕis atypefor a state candidateCif

• L(x)∈t iff ¬L(x)˙ ∈/ t, for everyL(x)∈Λϕ;

• ifL₁(x), L₂(x)∈tthen∀x( ˙¬L1(x)∨¬L˙ 2(x))6∈C, for everyL₁(x), L₂(x)∈Λ_ϕ.

By Claim 7, iftis a type forCthenχC∧∃xV

tis satisfiable.

Denote byTwthe set of all types forq(w). A pair of types (t,t⁰)is calledsuitableifQ(x)∈tiffQ(x)∈t⁰. Then the following two properties hold:

(succ) for eacht∈T_wthere ist⁰ ∈T_w+1such that(t,t⁰) is a suitable pair;

(pred) for eacht⁰ ∈T_w+1there ist∈T_wsuch that(t,t⁰) is a suitable pair.

To show (succ), suppose that t ∈ Tw, but there is no t⁰ ∈ Tw+1 such that (t,t⁰) is a suitable pair. Let

Q1(x), . . . ,Qk(x)be all temporal literals of the form

Q(x)int. Then

χ_q(w+1)∧ ∃x Q1(x)∧ · · · ∧Qk(x)

is not satisfiable. By Claim 7, there are i, j such that

∀x( ˙¬Q_i(x)∨ ¬Q˙ _j(x)) ∈ q(w+ 1). Then, by (qm₁),

∀x( ˙¬Q_i(x)∨¬˙Q_j(x)) ∈ q(w), and so, by Claim 7, the formulaχ_q(w)∧ ∃x Q1(x)∧ · · · ∧Qk(x)

is not satisfiable, contrary to our assumption. Property(pred)is proved analogously.

Now we define a setRof ‘runs’ throughqby taking all r∈Q

w∈NT_w

such that(r(w), r(w+ 1))is a suitable pair for everyw. By (succ)and(pred), for everywand every typet∈Twthere isr∈Rsuch thatr(w) =t.

Fora∈conϕandw∈N, let t^w_a =

L(x)∈Λϕ|L(a)∈q(w) .

It follows from(qs⁰_c)and(qs¹_c)that thet^w_a are types. More- over, by(qm₂),(t^w_a,t^w+1_a )is a suitable pair for every w.

Thus, there isr_a∈Rwithr_a(w) =t^w_a, for everyw.

Consider the model M = R, a^M₀ , . . . , P₀^M,w, . . . , wherea^M_j = r_aj andP_i^M,w =

r ∈ R | P_i ∈ r(w) . It is readily checked that(M, w0)|=ϕ.

Lemma 8. A QT L¹_krom-sentence ϕis satisfiable iff there is an ultimately periodical quasimodel q for ϕsuch that q(l1+w) =q(l1+l2+w), for everyw ∈ Nand some l1, l2withl1≤](ϕ)andl2≤kϕ·](ϕ) +](ϕ), where](ϕ) is the number of distinct state candidates forϕandk_ϕthe number of eventualities inϕ.

Proof. Similar to the proof of [14, Theorem 11.26].

Theorem 9. The satisfiability problem for QT L¹_krom is PSPACE-complete.

Proof. The upper bound follows from Lemma 8 using an algorithm that first guessesl1 andl2and then tries to con- struct an ultimately periodical quasimodel (see [14, The- orem 11.30]). The lower bound follows from PSPACE- hardness ofLT L(which is a fragment ofQT L¹_krom).

6. TDL-Lite

is E

S

-complete

Consider the Horn fragment TDL-Litehorn of TDL-Litebool whose atomic formulas are of the form

D1u · · · uDkvD, D(ai), R(ai, aj), whereD, D1, . . . , Dkare formed from basic conceptsBby means ofonly as in (20). Again we can apply all tempo- ral operators and the Booleans to formulas. (Note that sat- isfiability for the underlying DLDL-Litehornis P-complete [2].)

Theorem 10. The satisfiability problem forTDL-Litehorn

isEXPSPACE-complete.

Proof. The upper bound follows from Theorem 4. The lower one is proved by reduction of theN×2ⁿ corridor tiling problem that is known to be EXPSPACE-complete (for details see, e.g., [19, 16]): given an instance(T, τ0, n), whereTis a finite set of tile types,τ0∈Tis a tile type, and n∈Nis given in unary, decide whetherTtiles theN×2ⁿ- corridor{(x, y)|x∈N, 0≤y <2ⁿ}in such a way that τ₀ is placed at(0,0)and the top and bottom sides of the corridor are of some fixed colour, say,white. We construct aTDL-Litehorn formulaϕ_{T ,τ0,n} such that (i) its length is polynomial in|T|andn, and (ii)Ttiles theN×2ⁿcorridor (withτ₀on(0,0)and with white top and bottom sides) iff ϕ_{T ,τ0,n}is satisfiable.

The formulaϕT ,τ₀,nwill be constructed in a number of steps. To explain the meaning of its subformulas, let us fix some interpretationIwith some domain∆.

LetSτ, forτ ∈ T, be role names and suppose that the following formula holds inIat 0:

2⁺_F _

τ∈T

> v ∃S_τ

∧ ^

τ6=τ⁰

2⁺_F ∃S_τu ∃S_τ⁰ v ⊥ . (22) Then there is a uniquely determined sequenceτ₀, τ₁, . . . of tile types such that∃S^I(m)τm = ∆and(∃S_τ⁻_m)^I(m)6=∅, for everym∈N; see Fig. 1.

Suppose also that the following formulas hold inIat 0:

^

τ∈T

2⁺F ∃S_τ⁻ v lⁿ

j=1Q_j u N

, (23)

^

τ∈T

2⁺F ∃S_τ⁻uNv ⊥

, (24)

2⁺F NvN

. (25)