Query Inseparability by Games

Query Inseparability by Games

E. Botoeva¹, R. Kontchakov², V. Ryzhikov¹, F. Wolter³, and M. Zakharyaschev²

1 KRDB Research Centre, Free University of Bozen-Bolzano, Italy {botoeva,ryzhikov}@inf.unibz.it

2 Department of Computer Science and Inf. Systems, Birkbeck, University of London, U.K.

{roman,michael}@dcs.bbk.ac.uk

3 Department of Computer Science, University of Liverpool, U.K.

wolter@liverpool.ac.uk

Abstract. We investigate conjunctive query inseparability of description logic knowledge bases (KBs) with respect to a given signature, a fundamental prob- lem for KB versioning, module extraction, forgetting and knowledge exchange.

We develop a game-theoretic technique for checking query inseparability of KBs expressed in fragments of Horn-ALCHI, and show a number of complexity re- sults ranging from P to EXPTIMEand 2EXPTIME. We also employ our results to resolve two major open problems forOWL 2 QLby showing that TBox query in- separability and the membership problem for universal UCQ-solutions in knowl- edge exchange are both EXPTIME-complete for combined complexity.

Introduction

A description logic (DL) knowledge base (KB) consists of a terminological box (TBox), storing conceptual knowledge, and an assertion box (ABox), storing data. Typical ap- plications of KBs involve answering queries over incomplete data sources (ABoxes) augmented by ontologies (TBoxes) that provide additional information about the do- main of interest as well as a convenient vocabulary for user queries. The standard query language in such applications, which balances expressiveness and computational com- plexity, is the language of conjunctive queries (CQs).

With typically large data, often tangled ontologies, and the hard problem of answer- ing CQs over ontologies, various transformation and comparison tasks are becoming indispensable for KB engineering and maintenance. For example, to make answering certain CQs more efficient, one may want to extract from a given KB a smaller module returning the same answers to those CQs as the original KB; to provide the user with a more convenient query vocabulary, one may want to reformulate the KB in a new lan- guage. These tasks are known as module extraction [19] and knowledge exchange [2];

other relevant tasks include versioning, revision and forgetting [10, 20, 15].

In this paper, we investigate the following relationship between KBs that is funda- mental for all such tasks. LetΣbe a signature consisting of concept and role names. We call KBsK1andK2Σ-query inseparableand writeK1≡Σ K2if any CQ formulated inΣhas the same answers overK1andK2.The relativisation to (smaller) signatures is crucial to support the tasks mentioned above:

(versioning) When comparing twoversionsK₁andK₂of a KB with respect to their answers to CQs in a relevant signatureΣ, the task is to check whetherK₁≡_ΣK₂. (modularisation) AΣ-module of a KBKis a KBK⁰⊆ Ksuch thatK⁰ ≡Σ K. If we are only interested in answering CQs inΣoverK, then we can achieve our aim by querying anyΣ-module ofKinstead ofKitself.

(knowledge exchange) In knowledge exchange, we want to transform a KBK1 in a signatureΣ1 to a new KBK2 in a disjoint signature Σ2 connected toΣ1 via a declarative mapping specification given by a TBoxT12. Thus, the target KB K2

should satisfy the conditionK1∪ T12≡Σ2 K2, in which case it is called auniversal UCQ-solution(CQ and UCQ inseparabilities coincide for Horn DLs).

(forgetting) A KBK⁰ is the result offorgettinga signatureΣ in a KBK ifK⁰ does not useΣand gives the same answers to CQs without symbols inΣasK: that is, sig(K⁰)⊆sig(K)\ΣandK⁰ ≡_sig(K)\ΣK.

We study the data and combined complexity of decidingΣ-query inseparability for KBs expressed in various fragments of the DL Horn-ALCHI [14], which include DL-Lite^H_core [7] and EL[3] underlying the W3C profilesOWL 2 QL andOWL 2 EL.

To establish upper complexity bounds, we develop a novel game-theoretic technique for checking finiteΣ-homomorphic embeddability between (possibly infinite) materi- alisations of KBs. For all of the considered DLs,Σ-query inseparability turns out to be P-complete for data complexity, which matches the data complexity of CQ evalua- tion for all of our DLs lying outside theDL-Litefamily. For combined complexity, the obtained tight complexity results are summarised in the diagram below.

Horn-ALCHI Horn-ALCI Horn-ALCH

Horn-ALC ELH

EL

DL-Lite^H_horn

DL-Lite_horn

DL-Lite^H_core

DL-Lite_core

P Thms. 12, 24

EXPTIME Thms. 12, 25

EXPTIME Thms. 23, 25

P Thms. 16, 24 2EXPTIME

Thms. 23, 25

forward strategy

arbitrary strategy

backward+forward strategy

Most interesting are EXPTIME-completeness ofDL-Lite^H_coreand 2EXPTIME-complete- ness ofHorn-ALCI, which contrast with NP-completeness and EXPTIME-complete- ness of CQ evaluation for those logics. ForDL-Litewithout role inclusions andELH, Σ-query inseparability is P-complete, while CQ evaluation is NP-complete. In general, it is the combined presence of inverse roles and qualified existential restrictions (or role inclusions) that makesΣ-query inseparability hard.

We apply our results to resolve two important open problems. First, we show that the membership problem for universal UCQ-solutions in knowledge exchange for KBs inDL-Lite^H_core is EXPTIME-complete for combined complexity, which settles an open question of Arenaset al.[1], where only PSPACE-hardness was established. We also show thatΣ-query inseparability ofDL-Lite^H_coreTBoxes is EXPTIME-complete, which closes the PSPACE–EXPTIMEgap that was left open by Konevet al.[12].

Recall that TBoxesT₁ andT₂ are Σ-query inseparable if, for all Σ-ABoxes A (which only use concept and role names fromΣ), the KBs(T1,A)and(T2,A)areΣ- query inseparable. TBox and KB inseparabilities have different applications. The for- mer supports ontology engineering when data is not known or changes frequently: one

can equivalently replace one TBox with another only if they return the same answers to queries foreveryΣ-ABox. In contrast, KB inseparability is useful in applications where data is stable—such as knowledge exchange or variants of module extraction and for- getting with fixed data—in order to use the KB in a new application or as a compilation step to make CQ answering more efficient. As we show below, TBox and KBΣ-query inseparabilities also have different computational properties.

All the omitted proofs can be found in the full version of the paper [6].

Preliminaries

All the DLs for which we investigate KBΣ-query inseparability are Horn fragments of ALCHI. To define these DLs, we fix sequences ofindividual namesai,concept namesAi, androle namesPi, wherei < ω. Aroleis either a role namePior aninverse role P_i⁻; we assume that(P_i⁻)⁻ = P_i. Concepts in the DLsALCI,ALC, andEL are defined as usual.DL-Litehorn-conceptsare constructed from concept names using the constructors>,⊥,u, and∃R.>andDL-Litecore-concepts areDL-Litehorn-concepts withoutu; in other words, they arebasic conceptsof the form⊥,>,A_ior∃R.>.

For a DLL, anL-concept inclusion(CI) takes the formC v D, whereCandD areL-concepts. AnL-TBox,T, is a finite set ofL-CIs. AnALCHI,DL-Lite^H_hornand DL-Lite^H_core TBox can also contain a finite number ofrole inclusions(RIs)R1 vR2, where theR_i are roles. InELHTBoxes, RIs have no inverse roles.DL-LiteTBoxes may also containdisjointness constraintsB₁uB₂ v ⊥andR₁uR₂ v ⊥, for basic conceptsB_iand rolesR_i. To introduce the Horn fragments of these DLs, we require the standard recursive definition [9, 11] of positive occurrences of a concept. A TBox T isHornif no concept of the formCtDoccurs positively inT, and no concept of the form¬Cor∀R.Coccurs negatively inT. In the DLHorn-LonlyHorn-L-TBoxes are allowed. AnABox,A, is a finite set ofassertionsof the formAk(ai)orPk(ai, aj).

AnL-TBoxT and an ABoxAtogether form anLknowledge base(KB)K= (T,A).

The set of individual names inKis denoted byind(K).

The semantics for the DLs is defined in the usual way based on interpretations I = (∆^I,·^I)that comply with theunique name assumption:a^I_i 6=a^I_j fori 6=j [4].

We writeI |= αin case an inclusion or assertionαis true in I. If I |= α, for all α∈ T ∪ A, thenIis amodelof a KBK= (T,A); in symbols:I |=K.Kisconsistent if it has a model.K |=αmeans thatI |=αfor allI |=K.

Aconjunctive query(CQ)q(x)is a formula∃yϕ(x,y), whereϕis a conjunction of atoms of the formA_k(z₁)orP_k(z₁, z₂)withz_i∈x∪y. A tupleainind(K)(of the same length asx) is acertain answertoq(x)overKifI |=q(a)for allI |=K; in this case we writeK |=q(a).

Asignature,Σ, is a set of concept and role names. By aΣ-concept,Σ-role,Σ-CQ, etc. we understand any concept, role, CQ, etc. constructed using the names fromΣ.

Definition 1. LetK₁andK₂be KBs andΣa signature. ThenK₁Σ-query entailsK₂ ifK2|=q(a)impliesK1|=q(a)forallΣ-CQsq(x)and all tuplesainind(K2). And K1andK2areΣ-query inseparableif theyΣ-query entail each other; in which case we writeK1≡ΣK2.

SinceΣ-query inseparability can be reduced to twoΣ-query entailment checks, we can prove complexity upper bounds for entailment. Conversely, for most languages we have a semantically transparent reduction ofΣ-query entailment toΣ-query inseparability:

Theorem 2. For any of our DLsLcontainingELor having role inclusions,Σ-query entailment forL-KBs isLOGSPACE-reducible toΣ-query inseparability forL-KBs.

We now consider the relationship between inseparability and universal UCQ-solu- tions in knowledge exchange. SupposeK1andK2are KBs in disjoint signaturesΣ1and Σ2. LetT12be a mapping consisting of inclusions of the formS1vS2, where theSiare concept (or role) names inΣi. ThenK2is a universal UCQ-solution for(K1,T12, Σ2) ifK1∪ T12≡Σ2 K2. Deciding the latter is called themembership problem for universal UCQ-solutions. For DLsLwith role inclusions, the problem whetherK1∪T12≡Σ2 K2

is aΣ₂-query inseparability problem inL. Conversely, we have:

Theorem 3. Σ-query entailment for any of our DLsLisLOGSPACE-reducible to the membership problem for universal UCQ-solutions inL.

Semantic Characterisation

In this section, we give a semantic characterisation ofΣ-query entailment based on an abstract notion of materialisation and finite homomorphisms between such structures.

LetKbe a KB. An interpretationI is called amaterialisationofKifK |=q(a) just in caseI |= q(a), for all CQs q(x)and tuplesa inind(K). We say thatK is materialisableif it has a materialisation. Materialisations can be used to characterise KBΣ-query entailment by means ofΣ-homomorphisms. For an interpretationIand a signatureΣ, theΣ-typest^I_Σ(x)andr^I_Σ(x, y)ofx, y∈∆^Iare defined by taking

t^I_Σ(x) ={Σ-concept nameA|x∈A^I}, r^I_Σ(x, y) ={Σ-roleR|(x, y)∈R^I}.

SupposeIi is a materialisation of Ki,i = 1,2. A functionh:∆^I2 → ∆^I1 is aΣ- homomorphismfromI2toI1if, for anya∈ind(K2)and anyx, y∈∆^I2,

– h(a^I2) =a^I1whenevert^I_Σ²(a)6=∅orr^I_Σ²(a, y)6=∅for somey∈∆^I2, and – t^I_Σ²(x)⊆t^I_Σ¹(h(x)), r^I_Σ²(x, y)⊆r^I_Σ¹(h(x), h(y)).

As answers toΣ-CQs are preserved underΣ-homomorphisms,K1Σ-query entailsK2

if there is aΣ-homomorphism fromI2toI1. However, the converse does not hold.

Example 4. Suppose I2 and I1 on the right are material- isations of KBs K2 and K1, whereais the only ABox indi- vidual. LetΣ={Q, R, S, T}.

Then there is no Σ-homo-

a u

P R S T S T

Q

Q Q Q

a x

T , Q R

S, Q R

T , Q R

S, Q

I2

I1

morphism from I₂ toI₁ (as r^I_Σ²(a, u) =∅, we can map uto, say, xbut then only the shaded part of I2 can be mapped Σ-homomorphically to I1). However, for any Σ-queryq(x),I2 |=q(c)impliesI1 |=q(c)as any finite subinterpretation ofI2can beΣ-homomorphically mapped toI1.

We say thatI₂isfinitelyΣ-homomorphically embeddable intoI₁if, for everyfinite subinterpretationI₂⁰ ofI₂, there exists aΣ-homomorphism fromI₂⁰ toI₁.

To prove the following theorem, one can regard any finite subinterpretation ofI₂as a CQ whose variables are elements of∆^I2, with the answer variables being inind(K2).

Theorem 5. SupposeKi is a consistent KB with a materialisationIi,i = 1,2. Then K1Σ-query entailsK2iffI2is finitelyΣ-homomorphically embeddable intoI1.

One problem with applying Theorem 5 is that materialisations are in general infinite for any of the DLs considered in this paper. We address this problem by introducing finite representations of materialisations. LetKbe a KB and letG= (∆^G,·^G, )be a finite structure such that∆^G=ind(K)∪Ω, forind(K)∩Ω=∅,·^Gis an interpretation function on∆^G withA^G_i ⊆∆^G,P_i^G ⊆ind(K)×ind(K), and(∆^G, )is a directed graph (containing loops) with nodes∆^Gand edges ⊆∆^G×Ω, in which every edge u vis labelled with a set(u, v)^G 6=∅of roles satisfying the condition: ifu1 v andu₂ v, then(u₁, v)^G = (u₂, v)^G. We callG agenerating structure forK if the interpretationMdefined below is a materialisation ofK. ApathinG is a sequence σ =u₀. . . u_n withu₀ ∈ ind(K)andu_i u_i+1fori < n. Lettail(σ) = u_n and let path(G)be the set of paths inG. The materialisationMis given by:∆^M=path(G), a^M=a, fora∈ind(K), A^M={σ|tail(σ)∈A^G},

P^M=P^G∪ {(σ, σu)|tail(σ) u, P ∈(tail(σ), u)^G}

∪ {(σu, σ)|tail(σ) u, P⁻∈(tail(σ), u)^G}.

DLLhasfinitely generated materialisationsif everyL-KB has a generating structure.

Theorem 6. Horn-ALCHIand all of its fragments defined above have finitely gener- ated materialisations. Moreover,

– for anyL ∈ {ALCHI,ALCI,ALCH,ALC}and anyHorn-LKB(T,A), a gen- erating structure can be constructed in time|A| ·2^{p(|T |)},pa polynomial;

– for anyLin theELandDL-Litefamilies introduced above and anyL-KB (T,A), a generating structure can be constructed in time|A| ·p(|T |),pa polynomial.

Finite generating structures have been defined forEL[16],DL-Lite[13] and more expressive Horn DLs [8]. With the exception of DL-Lite, however, the relation guiding the construction of materialisations was implicit. We show how the existing constructions can be converted to generating structures in the full paper.

Example 7. The materialisation I2 from Example 4 can be gen- erated by the structureG2shown

on the right. ^a

P R⁻

S⁻

T⁻ S⁻ Q

−

Q⁻

G₂

For a generating structure G for K and a signature Σ, the Σ-types t^G_Σ(u) and r^G_Σ(u, v)ofu, v∈∆^Gare defined by takingt^G_Σ(u) ={Σ-concept nameA|u∈A^G}, r^G_Σ(u, v)as{Σ-roleR|(u, v)∈R^G}ifu, v∈ind(K), as{Σ-roleR|R∈(u, v)^G} ifu v, and∅otherwise, where(P⁻)^Gis the converse ofP^G. We also definer¯^G_Σ(u, v) to contain the inverses of the roles inr^G_Σ(u, v); note thatr¯^G_Σ(u, v)is not the same as r^G_Σ(v, u). We writeu ^Σvifu vandr^G_Σ(u, v)6=∅.

In the next section, we show that, for a DLLhaving finitely generated material- isations, Σ-query entailment forL-KBs can be reduced to the problem of finding a winning strategy in a game played on the generating structures for these KBs.

Σ-Query Entailment by Games

Suppose a DLLhas finitely generated materialisations,Ki is a consistentL-KB, for i = 1,2, andΣa signature. LetGi = (∆^Gi,·^Gi, i)be a generating structure forKi

andMibe its materialisation;G_i^ΣandM^Σ_i denote the restrictions ofGiandMitoΣ.

We begin with a very simple game on the finite generating structureG₂^Σ and the possibly infinite materialisationM^Σ₁.

Infinite gameGΣ(G2,M1).This game is played by two players: player 2 and player 1.

Thestatesof the game are of the formsi= (ui 7→σi), fori≥0, whereui∈∆^G2and σi∈∆^M1 satisfy the following condition:

(s1) t^G_Σ²(ui)⊆t^M_Σ¹(σi).

The game starts in a states0= (u07→σ0)withσ0=u0in caseu0∈ind(K2). In each roundi >0, player 2 challenges player 1 with someu_i ∈∆^G2such thatu_i−1 ^Σ₂ u_i. Player 1 has to respond with aσ_i∈∆^M1 satisfying(s₁)and

(s2) r^G_Σ²(ui−1, ui)⊆r^M_Σ¹(σi−1, σi).

This gives the next statesi = (ui 7→ σi). Note that of all the ui onlyu0 may be an ABox individual; however, there is no such a restriction on theσi. Aplay of length n≥0starting froms0is any sequences0, . . . ,snof states obtained as described above.

For an ordinal λ ≤ ω, we say that player 1 has a λ-winning strategy in the game GΣ(G2,M1)starting from a states0if, for any play of lengthi < λ, which starts from s0and conforms with this strategy, and any challenge of player 2 in roundi+1, player 1 has a response. The following theorem gives a game-theoretic flavour to the criterion of Theorem 5 (see the full paper for a proof).

Theorem 8. M2is finitelyΣ-homomorphically embeddable intoM1iff (abox) r^M_Σ²(a, b)⊆r^M_Σ¹(a, b), for anya, b∈ind(K2), and,

(win) for anyu₀∈∆^G2andn < ω, there existsσ₀∈∆^M1 such that player 1 has an n-winning strategy in the gameGΣ(G2,M1)starting from(u07→σ0).

Example 9. Let Σ = {Q, R, S, T}. Consider G₂^Σ andM^Σ₁ shown in the picture on the right.

For any n < ω and u ∈ ∆^G2, player 1 has ann-winning strategy in

a

u

R⁻ S⁻

T⁻ S⁻ Q

−

Q⁻

a σ

T , Q R

S, Q R

T , Q R

S, Q

0 1 2 2 3

3

4 4

G₂^Σ

M^Σ₁

GΣ(G2,M1). A4-winning strategy starting from(u 7→ σ)is shown by dotted lines (in round 2, player 2 has two possible challenges). For a largern, a suitableσcan be chosen further away from the rootaofM1.

The criterion of Theorem 8 does not seem to be a big improvement on Theorem 5 as we still have to deal with an infinite materialisation. Our aim now is to show that condition(win)in the infinite gameG_Σ(G₂,M₁)can be checked by analysing a more complex game on the finitegenerating structuresG2 andG1. We consider four types of strategies inGΣ(G2,M1). For each strategy type,τ, we define a gameG^τ_Σ(G2,G1) such that, for anyu0∈∆^G2, the following conditions are equivalent:

(< ω) for everyn < ω, player 1 has ann-winningτ-strategy inGΣ(G2,M1)starting from some(u07→σ₀ⁿ);

(ω) player 1 has anω-winning strategy inG^τ_Σ(G₂,G₁)starting from some state de- pending onu0andτ.

We begin with ‘forward’ winning strategies sufficient for DLs without inverse roles.

Forward strategy and gameG^f_Σ(G₂,G₁).We say that aλ-strategy (λ≤ω) for player 1 in the gameGΣ(G2,M1)isforwardif, for any play of lengthi−1< λ, which conforms with this strategy, and any challenge u_i−1 ^Σ₂ ui by player 2, the responseσi of player 1 is such that eitherσ_i−1, σi∈ind(K1)orσi=σ_i−1v, for somev∈∆^G1.

For example, if theGi,i= 1,2, satisfy the condition (f) theΣ-labels on _i-edges contain no inverse roles,

theneverystrategy inG_Σ(G2,M1)is forward. This is clearly the case forHorn-ALCH, Horn-ALC,ELHandEL, which by definition do not have inverse roles.

The existence of a forwardλ-winning strategy for player 1 inG_Σ(G₂,M₁)is equiv- alent to the existence of anω-winning strategy in the gameG^f_Σ(G2,G1), which is de- fined similarly toGΣ(G2,M1)but with two modifications: (1) it is played onG2and G1; and (2) the responsexi∈∆^G1of player 1 to a challengeu_i−1 ^Σ₂ uimust be such that eitherx_i−1, xi ∈ ind(K1)orx_i−1 1 xi, and(s1)–(s2)hold (withG1 andxiin place ofM1andσi).

Example 10. LetG2andG1be as shown on the right. Then, for any u∈∆^G2, there isx∈∆^G1 such that player 1 has an ω-winning strategy in G^f_Σ(G2,G1) starting from(u7→x).

a

R R

Q

R R

a

R

R

Q

0

1

1 2

G₂^Σ

G₁^Σ

The next theorem follows from K¨onig’s Lemma:

Lemma 11. For any u0 ∈ ∆^G2, condition (< ω) holds for forward strategies in GΣ(G2,M1) iff (ω)holds inG^f_Σ(G2,G1)for some state(u07→x0).

G^f_Σ(G2,G1)is a standard simulation or reachability game on finite graphs, where the existence ofω-winning strategies for player 1 follows from the existence ofn-winning strategies forn=O(|G2| × |G1|), which can be checked in polynomial time [18, 5]. By Theorem 6 and(f), we obtain:

Theorem 12. For combined complexity, checkingΣ-query entailment is inP forEL andELHKBs, and inEXPTIMEforHorn-ALCandHorn-ALCHKBs. For data com- plexity, it is inPfor all these DLs.

In comparison to forward strategies, the winning strategies used in Example 9 can be described as ‘backward.’

Backward strategy and gameG^b_Σ(G2,G1).Aλ-strategy for player 1 inGΣ(G2,M1) isbackwardif, for any play of lengthi−1< λ, which conforms with this strategy, and any challengeu_i−1 ^Σ₂ uiby player 2, the responseσi of player 1 is theimmediate predecessorofσ_i−1inM1in the sense thatσ_i−1 =σiv, for somev ∈∆^G1 (player 1 loses in case σ_i−1 ∈ ind(K1)). Note that, since M1 is tree-shaped, the response of player 1 to any different challengeu_i−1 ^Σ₂ u⁰_imust be the sameσi; cf. Example 9.

That is why the states of the gameG^b_Σ(G2,G1)are of the form(Ξi 7→xi), where Ξi⊆∆^G2,Ξi6=∅, andxi∈∆^G1satisfy the following condition:

(s⁰₁) t^G_Σ²(u)⊆t^G_Σ¹(xi), for allu∈Ξi.

The game starts in a state(Ξ₀7→x₀)such that (s⁰₀) ifu∈Ξ0∩ind(K2), thenx0=u∈ind(K1).

For eachi >0, player 2 always challenges player 1 with the setΞ_i=Ξ_i−1, where Ξ ={v∈∆^G2 |u ^Σ₂ v, for someu∈Ξ},

provided that it is not empty (otherwise, player 2 loses). Player 1 responds withx_i∈∆^G1 such thatx_{i 1}x_i−1and(s⁰₁)and the following condition hold:

(s⁰₂) r^G_Σ²(u, v)⊆¯r^G_Σ¹(xi−1, xi), for allu∈Ξi−1,v∈Ξi.

Lemma 13. For any u0 ∈ ∆^G2, condition (< ω) holds for backward strategies in GΣ(G2,M1)iff(ω)holds inG^b_Σ(G2,G1)for some state({u0} 7→x0).

Although Lemmas 11 and 13 look similar, the gameG^b_Σ(G2,G1)turns out to be more complex thanG^f_Σ(G2,G1).

Example 14. To illustrate, con- sider G₂^Σ shown on the right (with concepts and roles omitted) and an arbitrary G1. A play in

G₂^{Σ a u}

w1

v1

w2

v2

v3

G^b_Σ(G2,G1) may proceed as: ({u} 7→x₀), ({v1, w₁} 7→x₁), ({v2, w₂} 7→x₂), ({v3, w₁} 7→x₃), etc. This gives at least 6 different setsΞ_i. But ifG2containedkcy- cles of lengthsp₁, . . . , p_k, wherep_iis theith prime number, then the number of states inG^b_Σ(G₂,G₁)could be exponential (p₁× · · · ×p_k). In fact, we have the following:

Lemma 15. Checking(ω)in Lemma 13 isCONP-hard.

Observe that in the case ofDL-LitecoreandDL-Litehorn, (which have inverse roles but no RIs), generating structures G = (∆^G,·^G, )can be defined so that, for any u∈∆^GandR, there isat most onevwithu vandR∈r^G(u, v)[13]. As a result, anyn-winning strategy starting from(u₀7→σ₀)inG_Σ(G₂,M₁)consists of a (possibly empty) backward part followed by a (possibly empty) forward part. Moreover, in the backward games for these DLs, the sets Ξi are alwayssingletons. Thus, the number of states in the combined backward/forward games on theGi is polynomial, and the existence of winning strategies can be checked in polynomial time.

Theorem 16. CheckingΣ-query entailment forDL-LitecoreandDL-LitehornKBs is in Pfor both combined and data complexity.

An arbitrary strategy for player 1 inGΣ(G2,M1)is a combination of a backward strategy and a number of start-bounded strategies to be defined next.

Start-bounded strategy and gameG^s_Σ(G2,G1).A strategy for player 1 in the game GΣ(G2,M1) starting from a state (u0 7→ σ0)is start-bounded if it never leads to (ui7→σi)such thatσ0=σiv, for somevandi >0. In other words, player 1 cannot use those elements ofM1that are located closer to the ABox thanσ0; the ABox individuals inM1can only be used ifσ0∈ind(K1).

Example 17. The strategy starting from (u₂ 7→ σ₁) and shown on the right is start-bounded.

u2

T W W₁⁻ T₁⁻

σ1 T , T₁ W, W₁

0

4

1

3

G₂^Σ 2

M^Σ₁

In the gameG^s_Σ(G2,G1), player 1 will have to guessallthe points ofG2that are mapped to the same point ofM1.

Thestatesof G^s_Σ(G2,G1)are of the form(Γ_i, Ξ_i 7→ x_i),i ≥ 0, whereΓ_i, Ξ_i ⊆

∆^G2,Ξ_i 6=∅,x_i ∈∆^G1 and(s⁰₁)holds. The initial state is of the form(∅, Ξ₀ 7→ x₀) such that (s⁰₀) holds. In each round i > 0, player 2 challenges player 1 with some u ^Σ₂ vsuch thatu∈Ξ_i−1and

(nbk) ifv∈Γ_i−1thenr^G_Σ²(u, v)6⊆¯r^G_Σ¹(x_i−2, x_i−1).

Player 1 responds with either a state(Ξ_i−1, Ξ_i 7→ x_i)such thatx_{i−1 1} x_i(and so x_i∈/ ind(K₁)) and(s⁰⁰₂)holds, or a state(∅, Ξ_i7→x_i)such thatx_i−1, x_i∈ind(K₁)and (s⁰⁰₂) r^G_Σ²(u, v)⊆r^G_Σ¹(x_i−1, x_i).

We make challengesu ^Σ₂ v, for whichu∈Ξ_i−1and(nbk)does not hold, ‘illegiti- mate’ becausex_i−2can always be used as a response. Because of this, player 1 always moves ‘forward’ inG₁, but has to guess appropriate setsΞ_iin advance. Note thatΓ_iis always uniquely determined byx_i−1,xiandΞ_i−1(and it is eitherΞ_i−1or empty).

Example 18. Let G₂^Σ and G₁^Σ be as shown on the right (cf. Example 17).

We show that player 1 has anω-winning strategy in

u2 T u6 W u7 W₁⁻ u8 T₁⁻ u9

x1 T , T₁ x3 W, W₁ x4

a ⁰

0 1

1 2

G₂^Σ

G₁^Σ

G^s_Σ(G2,G1)starting from(∅,{u2, u9} 7→ x1). Player 2 challenges withu2 Σ 2 u6, and player 1 responds with({u2, u9},{u6, u8} 7→x3). Then player 2 picksu6 Σ

2 u7

and player 1 responds with ({u6, u8},{u7} 7→ x4), where the game ends. Note the crucial guesses{u2, u9} 7→x1and{u6, u8} 7→x3made by player 1. If player 1 failed to guess thatu₈must also be mapped tox₃and responded with({u2, u₉},{u6} 7→x₃), then after the challengeu₆ ^Σ₂ u₇and response({u6},{u7} 7→x₄)), player 2 would picku₇ ^Σ₂ u₈, to which player 1 could not respond.

Lemma 19. For anyu0 ∈∆^G2, condition(< ω)holds for start-bounded strategies in GΣ(G2,M1)iff(ω)holds inG^s_Σ(G2,G1)for some state(∅, Ξ07→x0)withu0∈Ξ0.

Arbitrary strategies and game G^a_Σ(G₂,G₁). An arbitrary winning strategy in the gameG_Σ(G₂,M₁)can be composed of one backward and a number of start-bounded strategies.

Example 20.

Consider G₂^Σ and M^Σ₁ shown on the right. Starting from (u1 7→ σ2), player 1 can respond to the challenges u₁ ^Σ₂ u₂ ^Σ₂ u₃

u1 u2 R⁻

u3

u6 S⁻

T u7

W

u8 W₁⁻

u9 T₁⁻

u10 S⁻₁

u4

U u5

U⁻

σ2

σ1 R

σ3 T,T¹

σ4 W, W1

a

S, S1

U b

0

1 1 2 2

G₂^Σ

M^Σ₁

according to the backward strategy; the challengesu2 Σ

2 u6 Σ

2 u7 Σ

2 u8 Σ 2

u9according to the start-bounded strategy as in Example 17; the challenges u3 Σ 2

u4 Σ

2 u5also according to the obvious start-bounded strategy; finally, the challenge u9 Σ

2 u10needs a response according to the backward strategy. We will combine the two backward strategies into a single one, but keep the start-bounded ones separate.

The gameG^a_Σ(G2,G1)begins as the gameG^b_Σ(G2,G1), but with states of the form (Ξ_i7→x_i, Ψ_i),i≥0, whereΞ_i⊆∆^G2andx_i∈∆^G1satisfy(s⁰₁)andΨ_iis a (possibly empty) subset ofΞ_i , which indicates initial challenges in start-bounded games. The initial state satisfies(s⁰₀). In each roundi >0, ifx_i−1∈ind(K₁)then player 2 launches the start-bounded gameG^s_Σ(G₂,G₁)with the initial state(∅, Ξ_i−17→x_i−1). Otherwise, ifx_i−1 ∈/ ind(K1), player 2 has two options. First, he can challenge player 1 with the setΨ_i−1 (that is, similar to the backward game but with a possibly smallerΨ_i−1 in place ofΞ_i−1); player 1 responds to this challenge with a state(Ξi7→xi, Ψi)such that Ψ_i−1⊆Ξi,xi 1x_i−1and(s⁰₂)holds. Second, player 2 can launch the start-bounded gameG^s_Σ(G2,G1)with the initial state(∅, Ξ_i−17→x_i−1), where the first challenge of player 2 must be picked fromΦ_i−1=Ξ_i−1\Ψ_i−1.

Example 21. We illustrate theω-winning strategy for player 1 inG^a_Σ(G2,G1)starting from({u1} 7→x2,{u2}), whereG^Σ₂ is from Example 20 andG₁^Σlooks likeM^Σ₁ from Example 20 (but withxiin place ofσi):

{u1} 7→x2,{u₂}

{u2, u9} 7→x1,{u₃,u10}

{u3, u10} 7→a,∅ ∅,{u3, u10} 7→a

∅,{u4} 7→b

u₃ u₄

∅,{u5} 7→a

u₄ u₅

∅,{u2, u9} 7→x1 {u₂,u₉},{u6, u8} 7→x3

u2 u6

{u₆,u₈},{u7} 7→x4 u6 u7

Lemma 22. For anyu0 ∈∆^G2, condition(< ω)holds for arbitrary strategies in the gameGΣ(G2,M1)iff(ω)holds inG^a_Σ(G2,G1)for some(Ξ07→x0, Ψ0)withu0∈Ξ0. Condition(ω)in Lemma 22 is checked in timeO(|ind(K2)|×2^{|∆G2\ind(K2)|}×|∆^G1|), which can be readily seen by analysing the full game graph forG^a_Σ(G2,G1)(similar to that in Example 21). By Theorem 6, we then obtain:

Theorem 23. For combined complexity, the Σ-query entailment problem is in 2EXPTIMEforHorn-ALCHIandHorn-ALCI KBs and inEXPTIMEforDL-Lite^H_horn andDL-Lite^H_coreKBs. For data complexity, these problems are all inP.

Discussion

We have shown that, for all of our DLs,Σ-query entailment and inseparability are in P for data complexity. The next theorem establishes a matching lower bound:

Theorem 24. For data complexity,Σ-query entailment and inseparability are P-hard forDL-Lite_coreandELKBs.

For combined complexity, EXPTIME-hardness ofΣ-query inseparability forHorn-ALC can be proved by reduction of the subsumption problem: we have T |= A v B iff (T,{A(a)})and(T ∪ {AvB},{A(a)})are{B}-query inseparable. We now estab- lish matching lower bounds in the technically challenging cases.

Theorem 25. For combined complexity, Σ-query entailment and inseparability are (i)2EXPTIME-hard forHorn-ALCIKBs and(ii) EXPTIME-hard forDL-Lite^H_coreKBs.

The obtained tight complexity bounds apply to the membership problem for uni- versal UCQ-solutions and toΣ-query inseparability of TBoxes. As a consequence of Theorems 3, 23 and 25 we obtain:

Theorem 26. For combined complexity, the membership problem for universal UCQ- solutions is 2EXPTIME-complete forHorn-ALCHIandHorn-ALCI; EXPTIME-com- plete forHorn-ALCH,Horn-ALC,DL-Lite^H_horn andDL-Lite^H_core;and P-complete for ELandELH. For data complexity, all these problems areP-complete.

Σ-query inseparability ofDL-Lite^H_core TBoxes was known to sit between PSPACE

and EXPTIME[12]. Using the fact that witness ABoxes forDL-Lite^H_coreTBox separabil- ity can always be chosen among the singleton ABoxes [12, Theorem 8], we can modify the proof of Theorem 25 to improve the PSPACElower bound:

Theorem 27. Σ-query inseparability ofDL-Lite^H_coreTBoxes isEXPTIME-complete.

For more expressive DLs, TBoxΣ-query inseparability is often harder than KB inseparability: forDL-Lite_horn, the space of relevant witness ABoxes for TBox sepa- rability is of exponential size and, in fact, TBox inseparability is NP-hard, while KB inseparability is in P. Similarly,Σ-query inseparability ofELKBs is tractable, while Σ-query inseparability of TBoxes is EXPTIME-complete [17]. The complexity of TBox inseparability for Horn-DLs extendingHorn-ALCis not known.

As for future work, from a theoretical point of view, it would be of interest to inves- tigate the complexity ofΣ-query inseparability for KBs in more expressive Horn DLs (e.g.,Horn-SHIQ) and non-Horn DLs extendingALC. We conjecture that the game technique developed in this paper can be extended to those DLs as well. Our games can also be used to defineefficient approximationsofΣ-query entailment and insepa- rability for KBs. The existence of a forward strategy, for example, provides a sufficient condition forΣ-query entailment for all of our DLs. Thus, one can extract aΣ-query module of a given KBKby exhaustively removing fromKthose inclusions and asser- tionsαsuch that player 1 has a winning strategy in the gameG^f_Σ(G₂,G₁), whereG₁ andG₂are generating structures forK \ {α}andK, respectively. The resulting modules are minimal for our DLs without inverse roles, and we conjecture that in practice they are often minimal for DLs with inverse roles as well; see [12] for experiments testing similar ideas for module extraction from TBoxes.

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