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Botoeva, E. and Kontchakov, Roman and Ryzhikov, V. and Wolter, F.
and Zakharyaschev, Michael (2014) Query inseparability by games.
In:
Bienvenu, M. and Ortiz, M. and Rosati, R. and Simkus, M. (eds.) Informal
Proceedings of DL (Vienna, 17-20 July).
CEUR Workshop Proceedings
1193. CEUR, pp. 83-95.
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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.
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 becomanswer-ing 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
(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-LiteHcore [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-LiteHhorn
DL-Litehorn
DL-LiteHcore
DL-Litecore 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-LiteHcoreand 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-LiteHcore 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-LiteHcoreTBoxes 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
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 Pi−; we assume that(Pi−)− = 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⊥,>,Aior∃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-LiteHhornand
DL-LiteHcore TBox can also contain a finite number ofrole inclusions(RIs)R1 vR2,
where theRi are roles. InELHTBoxes, RIs have no inverse roles.DL-LiteTBoxes
may also containdisjointness constraintsB1uB2 v ⊥andR1uR2 v ⊥, for basic
conceptsBiand rolesRi. 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:aI
i 6=aIj 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 formAk(z1)orPk(z1, z2)withzi∈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
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-typestIΣ(x)andrIΣ(x, y)ofx, y∈∆Iare defined by taking
tIΣ(x) ={Σ-concept nameA|x∈AI}, rIΣ(x, y) ={Σ-roleR|(x, y)∈RI}. SupposeIi is a materialisation of Ki,i = 1,2. A functionh:∆I2 → ∆I1 is a
Σ-homomorphismfromI2toI1if, for anya∈ind(K2)and anyx, y∈∆I2,
– h(aI2) =aI1whenevertI2
Σ(a)6=∅orr
I2
Σ(a, y)6=∅for somey∈∆I2, and
– tI2
Σ(x)⊆t
I1
Σ(h(x)), r
I2
Σ(x, y)⊆r
I1
Σ(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 rΣI2(a, u) =∅, we can map uto, say, xbut then only
We say thatI2isfinitelyΣ-homomorphically embeddable intoI1if, for everyfinite
subinterpretationI0
2ofI2, 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 withAGi ⊆∆G,PiG ⊆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. . . un withu0 ∈ ind(K)andui ui+1fori < n. Lettail(σ) = un and let path(G)be the set of paths inG. The materialisationMis given by:∆M=path(G), aM=a, fora∈ind(K), AM={σ|tail(σ)∈AG},
PM=PG∪ {(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| ·2p(|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 tGΣ(u) and rGΣ(u, v)ofu, v∈∆Gare defined by takingtΣG(u) ={Σ-concept nameA|u∈AG}, rGΣ(u, v)as{Σ-roleR|(u, v)∈RG}ifu, v∈ind(K), as{Σ-roleR|R∈(u, v)G}
ifu v, and∅otherwise, where(P−)Gis the converse ofPG. We also definer¯GΣ(u, v)
to contain the inverses of the roles inrGΣ(u, v); note thatr¯GΣ(u, v)is not the same as rGΣ(v, u). We writeu Σvifu vandrG
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;GiΣandM Σ
i denote the restrictions ofGiandMitoΣ.
We begin with a very simple game on the finite generating structureGΣ
2 and the
possibly infinite materialisationMΣ
1.
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) tGΣ2(ui)⊆tMΣ1(σi).
The game starts in a states0= (u07→σ0)withσ0=u0in caseu0∈ind(K2). In each
roundi >0, player 2 challenges player 1 with someui ∈∆G2such thatui−1 Σ2 ui.
Player 1 has to respond with aσi∈∆M1 satisfying(s1)and (s2) rGΣ2(ui−1, ui)⊆rMΣ1(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) rM2
Σ (a, b)⊆r
M1
Σ (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
GΣ
2 andMΣ1 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 GΣ
2
MΣ
1
GΣ(G2,M1). A4-winning strategy starting from(u 7→ σ)is shown by dotted lines
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 gameGΣ(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 gameGτΣ(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 inGτ
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 gameGfΣ(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 ui−1 Σ2 ui by player 2, the responseσi of
player 1 is such that eitherσi−1, σi∈ind(K1)orσi=σi−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 inGΣ(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 inGΣ(G2,M1)is
equiv-alent to the existence of anω-winning strategy in the gameGfΣ(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 challengeui−1 Σ2 uimust be such
that eitherxi−1, xi ∈ ind(K1)orxi−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 GfΣ(G2,G1) starting from(u7→x).
a
R R Q
R R
a R
R
Q
0
1 1
2 GΣ
2
GΣ
1
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 inGΣf(G2,G1)for some state(u07→x0).
GfΣ(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
In comparison to forward strategies, the winning strategies used in Example 9 can be described as ‘backward.’
Backward strategy and gameGb
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 challengeui−1 Σ2 uiby player 2, the responseσi of player 1 is theimmediate
predecessorofσi−1inM1in the sense thatσi−1 =σiv, for somev ∈∆G1 (player 1
loses in case σi−1 ∈ ind(K1)). Note that, since M1 is tree-shaped, the response of
player 1 to any different challengeui−1 Σ2 u0imust be the sameσi; cf. Example 9.
That is why the states of the gameGbΣ(G2,G1)are of the form(Ξi 7→xi), where
Ξi⊆∆G2,Ξi=6 ∅, andxi∈∆G1satisfy the following condition:
(s01) tG2
Σ(u)⊆t
G1
Σ(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 setΞi=Ξi−1, where
Ξ ={v∈∆G2 |u Σ
2 v, for someu∈Ξ},
provided that it is not empty (otherwise, player 2 loses). Player 1 responds withxi∈∆G1
such thatxi 1xi−1and(s01)and the following condition hold: (s02) r
G2
Σ(u, v)⊆¯r
G1
Σ(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 inGΣb(G2,G1)for some state({u0} 7→x0).
Although Lemmas 11 and 13 look similar, the gameGbΣ(G2,G1)turns out to be more complex thanGfΣ(G2,G1).
Example 14. To illustrate, con-sider GΣ
2 shown on the right
(with concepts and roles omitted) and an arbitrary G1. A play in
GΣ
2
a u
w1 v1
w2 v2
v3
Gb
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 setsΞi. But ifG2containedk
cy-cles of lengthsp1, . . . , pk, wherepiis theith prime number, then the number of states
inGb
Σ(G2,G1)could be exponential (p1× · · · ×pk). 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∈rG(u, v)[13]. As a result, anyn-winning strategy starting from(u07→σ0)inGΣ(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
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 gameGs
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
2 GΣ
2
MΣ
1
In the gameGsΣ(G2,G1), player 1 will have to guessallthe points ofG2that are mapped to the same point ofM1.
Thestatesof Gs
Σ(G2,G1)are of the form(Γi, Ξi 7→ xi),i ≥ 0, whereΓi, Ξi ⊆
∆G2,Ξ
i 6=∅,xi ∈∆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 Σ
2 vsuch thatu∈Ξi−1and (nbk) ifv∈Γi−1thenrGΣ2(u, v)6⊆¯r
G1
Σ(xi−2, xi−1).
Player 1 responds with either a state(Ξi−1, Ξi 7→ xi)such thatxi−1 1 xi(and so
xi∈/ ind(K1)) and(s002)holds, or a state(∅, Ξi7→xi)such thatxi−1, xi∈ind(K1)and (s002) r
G2
Σ(u, v)⊆r
G1
Σ(xi−1, xi).
We make challengesu Σ
2 v, for whichu∈Ξi−1and(nbk)does not hold,
‘illegiti-mate’ becausexi−2can always be used as a response. Because of this, player 1 always
moves ‘forward’ inG1, but has to guess appropriate setsΞiin advance. Note thatΓiis
always uniquely determined byxi−1,xiandΞi−1(and it is eitherΞi−1or empty).
Example 18. Let GΣ
2 and GΣ
1 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 GΣ
2
GΣ
1
GsΣ(G2,G1)starting from(∅,{u2, u9} 7→ x1). Player 2 challenges withu2 Σ2 u6,
and player 1 responds with({u2, u9},{u6, u8} 7→x3). Then player 2 picksu6 Σ2 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
Arbitrary strategies and game Ga
Σ(G2,G1). An arbitrary winning strategy in the
gameGΣ(G2,M1)can be composed of one backward and a number of start-bounded
strategies.
Example 20.
Consider GΣ
2 and MΣ
1 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 U− u5
σ2
σ1
R
σ3
T,T1 σ4
W, W1
a S, S1
b U
0
1 1 2 2 GΣ
2
MΣ
1
according to the backward strategy; the challengesu2 Σ2 u6 2Σ u7 Σ2 u8 Σ2
u9according to the start-bounded strategy as in Example 17; the challenges u3 Σ2
u4 Σ2 u5also according to the obvious start-bounded strategy; finally, the challenge
u9 Σ2 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 gameGa
Σ(G2,G1)begins as the gameG b
Σ(G2,G1), but with states of the form
(Ξi7→xi, Ψi),i≥0, whereΞi⊆∆G2andxi∈∆G1satisfy(s01)andΨiis a (possibly
empty) subset ofΞi , which indicates initial challenges in start-bounded games. The initial state satisfies(s00). In each roundi >0, ifxi−1∈ind(K1)then player 2 launches
the start-bounded gameGs
Σ(G2,G1)with the initial state(∅, Ξi−17→xi−1). Otherwise,
ifxi−1 ∈/ ind(K1), player 2 has two options. First, he can challenge player 1 with the
setΨi−1 (that is, similar to the backward game but with a possibly smallerΨi−1 in
place ofΞi−1); player 1 responds to this challenge with a state(Ξi7→xi, Ψi)such that
Ψi−1⊆Ξi,xi 1xi−1and(s02)holds. Second, player 2 can launch the start-bounded
gameGs
Σ(G2,G1)with the initial state(∅, Ξi−17→xi−1), where the first challenge of
player 2 must be picked fromΦi−1=Ξi−1\Ψi−1.
Example 21. We illustrate theω-winning strategy for player 1 inGaΣ(G2,G1)starting from({u1} 7→x2,{u2}), whereGΣ2 is from Example 20 andG1Σlooks likeMΣ1 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 inGaΣ(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 forGaΣ(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-LiteHhorn
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-LitecoreandELKBs.
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-LiteHcoreKBs.
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-LitehornH andDL-LiteHcore;and P-complete for
ELandELH. For data complexity, all these problems areP-complete.
Σ-query inseparability ofDL-LiteHcore TBoxes was known to sit between PSPACE
and EXPTIME[12]. Using the fact that witness ABoxes forDL-LiteHcoreTBox 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-LiteHcoreTBoxes isEXPTIME-complete.
For more expressive DLs, TBoxΣ-query inseparability is often harder than KB inseparability: forDL-Litehorn, the space of relevant witness ABoxes for TBox
sepa-rability is of exponential size and, in fact, TBox insepasepa-rability 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 gameGfΣ(G2,G1), whereG1
andG2are generating structures forK \ {α}andK, respectively. The resulting modules
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