Conjunctive Query Answering for Description Logics with Transitive Roles

Transcription

1 Conjunctive Quey Anweing fo Deciption Logic with Tanitive Role Bite Glimm Ian Hoock Ulike Sattle The Univeity of Manchete, UK Abtact An impotant eaoning tak, in addition to the tandad DL eaoning evice, i conjunctive quey anweing. In thi pape, we peent algoithm fo conjunctive quey anweing in the expeive Deciption Logic SHQ and SHOQ. In paticula, we allow fo tanitive (o nonimple) ole in the quey body, which i a featue that i not uppoted by othe exiting conjunctive quey anweing algoithm. Fo SHQ, we achieve thi by extending the logic with a eticted fom of binde and tate vaiable a known fom Hybid Logic. We alo highlight, why the addition of invee ole make the tak of finding a deciion pocedue fo conjunctive quey anweing moe complicated. 1 Intoduction Exiting Deciption Logic (DL) eaone 1 povide automated eaoning uppot fo checking concept fo atifiability and ubumption, and alo fo anweing queie that etieve intance of concept and ole. The development of a deciion pocedue fo conjunctive quey anweing in expeive DL i, howeve, till a patially open quetion. Gounded conjunctive queie fo SHIQ ae uppoted by KAON2, Pellet, and Race quey language nrql. Howeve, the emantic of gounded queie i diffeent fom the uually aumed openwold emantic in DL ince exitentially quantified vaiable ae alway bound to named individual. Thi wok wa uppoted by an EPSRC tudenthip. 1 Fo example, FaCT++ KAON2 kaon2.emanticweb.og/, Pellet o Race Po

2 None of the exiting conjunctive quey anweing technique [14, 10, 4, 9, 13] i able to handle tanitive ole o nominal in the quey body. 2 In thi pape, we peent an extenion of SHQ with a eticted fom of the binde opeato and tate vaiable a known fom Hybid Logic [3]. Thi extended logic enable an extenion of the olling-up technique [14, 4] to tanitive ole; a featue which wa left open by Teai [14]. Unfotunately, the binde aleady make the DL ALC undecidable. Fo quey anweing, howeve, the binde occu in a vey eticted fom and, a we how in Section 4, thi extenion of SHQ i decidable. Futhe on, we adapt the tableaux algoithm fo SHOQ [7] in ode to decide conjunctive quey entailment in SHQ. Quey anweing fo a logic that include nominal, e.g., a logic uch a SHOQ, ha the additional difficulty that cycle ae no longe limited to ABox individual o hotcut via tanitive ole. Theefoe, the non-deteminitic aignment of individual name to vaiable in a cycle, a uggeted fo the DL DLR by Calvanee et al. [4], can no longe be applied. In Section 5, howeve, we how that, by alo intoducing new vaiable, the non-deteminitic aignment of individual name to vaiable can be ued in ode to povide a quey anweing algoithm fo SHOQ. Finally, we highlight why the extenion of SHQ o SHOQ with invee ole make the deign of a deciion pocedue fo conjunctive quey anweing moe complicated. 2 Peliminaie We aume eade to be familia with the yntax and emantic of the DL SHOIQ (fo detail ee [1]) and of SHOIQ knowledge bae. Theefoe, we intoduce only the yntax and emantic of the binde and of conjunctive queie hee. Let L be a Deciption Logic, C an L-concept, and N V a finite et of vaiable name with y N V. With L, we denote the language obtained by allowing, additionally, y and y.c a L-concept. Fo an intepetation I = ( I, I), an element d I, and a vaiable y N V, we denote with I [y/d] the intepetation that extend I uch that y I = {d}. The L -concept y.c i then intepeted a ( y.c) I = {d I d C I [y/d] }. Let y be a vecto of vaiable and c a vecto of individual name. A Boolean conjunctive quey q ha the fom conj 1 ( y; c)... conj n ( y; c). We call T(q) = y c the et of tem in q, 3 and we call each conj i ( y; c) fo 1 i n 2 Although the algoithm peented by Calvanee et al. [4] allow the ue of egula expeion in the quey (in paticula the tanitive eflexive cloue), it ha been hown that the algoithm i incomplete [8, 5]. 3 Fo eadability, we ometime abue ou notation and efe to y a a et. When efeing

3 an atom. Atom ae eithe concept o ole atom: a concept atom ha the fom t: C, and a ole atom the fom t, t :, fo {t, t } T(q), C a poibly complex L-concept, and an L-ole. Let K be an L-knowledge bae (L-KB), I = ( I, I) a model fo K, and q a Boolean conjunctive quey fo K. An aignment in I i a mapping A : T(q) I. We ay that q i tue in I and wite I = q if thee exit an aignment A in I.t. t A {t I } fo evey individual t c, t A C I fo evey concept atom t: C in q, and t A, t A I fo evey ole atom t, t : in q. We call uch an aignment a q-mapping w..t. I. If I = K implie I = q fo all model I of K, then we ay that q i tue in K, and wite K = q; othewie we ay that q i fale in K, and wite K = q. Since anweing non-boolean conjunctive queie can be educed to anweing (poibly eveal) Boolean queie, we conide only Boolean queie hee. 3 Reducing Quey Anweing to Concept Unatifiability The main idea ued in thi pape wee fit intoduced by Calvanee et al. [4] fo deciding conjunctive quey entailment fo the expeive DL DLR eg. Since quey anweing can be educed to quey entailment, the algoithm in [4] i capable of deciding conjunctive quey anweing a well. The autho how how a quey q can be expeed a a concept C q, uch that q i tue w..t. a given knowledge bae if adding C q make the KB unatifiable. In ode to obtain the concept C q, the quey q i epeented a a diected, labelled gaph. Thi gaph, called a tuple-gaph o a quey gaph, i taveed in a depth-fit manne and, duing the taveal, node and edge ae eplaced with appopiate concept expeion, leading to the concept C q afte completing the taveal. The node in a quey gaph coepond to the tem in the quey and ae labelled with the concept that occu in the coeponding concept atom. The edge coepond to the ole atom in q and ae labelled accodingly. Fo example, let q 1 be the quey x: C x, y : y : D and q 2 the quey x: C x, y : y, x : y : D. The quey gaph fo q 1 and q 2 ae depicted in Fig. 1 and Fig. 2 epectively. Figue 1: The (acyclic) quey gaph fo q 1. Figue 2: The (cyclic) quey gaph fo q 2. Since q 1 i acyclic, we can build the concept that epeent q 1 a follow: to a vecto y a a et, we mean the et {y i y i occu in y}.

4 tat at x and tavee the gaph to y. Since y i a leaf node, emove y and it incoming edge and conjoin.d to the label C of x, eulting in C.D fo C q1. A given KB K entail q 1 iff K { C q1 } i unatifiable. Since the gaph i collaped with each tep, the technique i often called olling-up. Full detail can be found in the oiginal pape [4], although the deciption thee i lightly diffeent ince eification and name fomulae ae ued in ode to educe DLR eg (which allow alo fo ole of aity n > 2) to convee-pdl. Teai [14] popoe a imila appoach fo the DL SHQ (without tanitive ole in the quey). Thi eduction i not diectly extendable to cyclic queie ince, due to the tee-model popety of mot DL, a concept cannot captue cyclic elationhip. Howeve, fo imple DL, thi alo mean that cycle can only be enfoced by ABox aetion and hence, Calvanee et al. ugget to eplace vaiable in a cycle, i.e., x and y in q 2, non-deteminitically with individual name fom the KB (o with tem fom the moe pecific quey in the cae of quey entailment). Although the technique peented by Calvanee et al. povide the baic foundation of eveal quey anweing algoithm, including the extenion peented hee, it wa found late that the algoithm in it oiginal fom i not a deciion pocedue. Hoock et al. [8] point out that, by identifying vaiable with each othe, ome cyclic queie become acyclic, and hence a eplacement of vaiable with named individual might not find all olution. Fo example, identifying y and y in the quey gaph depicted in Fig. 3, lead to the acyclic quey gaph in Fig. 4, which can be expeed a C..D. Clealy, the KB containing the ABox aetion a:.(c..d) would make the quey tue, although in the elevant aignment none of the vaiable can be bound to the individual name a. Figue 3: gaph. y y z:d The oiginal quey y=y z:d Figue 4: The quey gaph afte identifying y with y. Anothe poblem aie though the egula expeion in DLR eg, which ae, fo example, capable of expeing invee ole in the quey. Let q 3 be the quey x, y : y, x : (ee Fig. 5). Identifying x and y till leave u with a cyclic quey gaph. Let K be a knowledge bae containing the ole axiom and the ABox aetion a:... Clealy, we have that K = q 3 (ee Fig. 6), although again in the elevant aignment none of the vaiable can be bound to the individual name a. Simila example can be given fo egula expeion that ue the Kleene ta fo expeing the tanitive cloue. Thee dicoveie motivated the wok peented in the following ection, which in paticula taget the poblem aiing with cyclic queie.

5 x - Figue 5: The quey gaph fo q 3. y - -, - a, Figue 6: A epeentation of the canonical model of K. The pat that make q 3 tue ae highlighted. 4 The Rolling-Up Technique with Binde An extenion of the olling-up technique to cyclic queie i not diectly poible, a we have tied to explain in the peviou ection. Due to the tee-model popety of mot DL, a concept cannot captue cyclic elationhip, but it i alo not alway coect to eplace vaiable name in a cycle with individual name fom the knowledge bae. In thi ection, we how how cyclic queie fo an L-KB can be olled-up into L -concept. The binde can label element in a model with a vaiable, and thi vaiable can be ued to enfoce a co-efeence. Conide again the quey q 2 x: C x, y : y, x : y : D (ee Fig. 2). Uing the binde, we can contuct the following concept C q2 fo q 2 : x.(c.(d.x)). In ode to obtain thi concept, we till tavee the quey gaph (ee Fig. 7): we tat at x, tavee to y, then we each x again and eplace the -edge by conjoining.x to the label of y, now y i a leaf node and we eplace y and it incoming -edge by conjoining. y.(d.x) to the label of x, ince x i the lat node, we obtain C q2 a x.(. y.(d.x). Since y neve occu a a vaiable, we can omit the binde fo y. Now we have again that K = q 2 iff K { C q2 } i unatifiable. Futhe detail and a poof that C q2 indeed captue q 2 can be found in a technical epot [6]. u9.x u9.#y.(du9.x) Figue 7: The gaph taveal fo the quey gaph of q 2 tep by tep. Pat aleady een duing the taveal ae highlighted. Even fo cyclic queie, the olling-up i now taightfowad. If, howeve, the logic L unde conideation doe not povide fo invee ole, then thi technique i only applicable in cae each component in the quey gaph i alo tongly connected. In the peence of weakly connected component, we would need invee ole in ode to oll-up a quey. Fo example, let q 4 be the quey x: C x, y : x, y : y : D, i.e., both edge go now fom x to y and x i not eachable fom y. Theefoe, the quey gaph i compoed of two weakly connected component and a taveal of the quey gaph would get tuck at y. With invee ole, q 4 could eaily be expeed a x.(c.(d Inv().x)). Teai [14] how how abitay conjunctive queie fo SHQ can be olled-up

6 even without invee ole, and an extenion of thi technique might wok fo binde and vaiable a well. Figue 8: The quey gaph fo q 4. Unfotunately, thee ae no deciion pocedue fo expeive DL with the binde opeato. It i even known that ALC extended with binde and tate vaiable i undecidable [2]. Howeve, we can obeve ome inteeting popetie fo ou quey concept. (1) Afte the olling-up, all vaiable occu only poitively. (2) Only exitential etiction ae intoduced in the olling-up. Hence, negating the quey concept fo q 2 and tanfoming it into negation nomal fom yield x.( C.( D. x)) and we obeve that all vaiable occu negated and ae unde the cope of univeal quantifie. Fo ALC, thee etiction ae enough to egain decidability [11]. 4.1 A Tableaux Algoithm fo SHQ Quey Concept Ou algoithm i baed on the tableaux algoithm fo SHOQ [7], which obviouly decide KB atifiability fo SHQ. Uing thi algoithm ha the advantage that we can ue nominal fo the contant in the quey. Altenatively, epeentative concept and additional ABox aetion fo each contant in a quey could be ued [14]. A tableaux algoithm build a finite epeentation of a model, which i called a completion gaph. The node in the completion gaph epeent element in the domain and ae labelled with a et of concept. Labelled edge epeent the elational tuctue among the element. An initial completion gaph i expanded accoding to a et of expanion ule. The expanion top when an obviou contadiction, called a clah, occu, o when no moe ule ae applicable. In the latte cae, the completion gaph i called complete. Temination i guaanteed by a technique called blocking. A complete and clah-fee completion gaph can be unavelled into a model fo the given KB. In ode to handle quey concept that may contain binde and tate vaiable, ome adaptation of the exiting algoithm ae neceay. In ode to toe the binding of vaiable, we modify the node label.t. they contain tuple of the fom C, B, whee C i a concept and B i a (poibly empty) et of binding of the fom {y 1 /v 1,..., y n /v n }, whee y 1,..., y n ae the fee vaiable in C and v 1,..., v n ae node. The next obviou addition i a ule to handle concept of the fom y.c: if y.c, B i in the label of a node v, we add C, {y/v} B and y, {y/v} to the label of v. The latte tate that y i bound to v at v, and the fome that the fee vaiable y in C i bound to v. All othe exiting ule

7 have to popagate the binding a well, e.g., the -ule applied to.c, B in the label of a node v add C, B to the label of v -ucceo. The et B contain all and only the binding fo the fee vaiable in C. Anothe obviou conequence i the addition of a new clah condition: if both y, {y/v} and y, {y/v} ae in the label of the node v, then thi i a clah. A moe challenging tak i the extenion of the blocking condition. Fo SHQ, howeve, we ague that we can imply ignoe the binding, i.e., if C, B i in the label, we conide only C in the blocking condition. Thi clealy enue temination. But why doe thi guaantee that we can unavel a complete and clah-fee completion foet into a model? Obviouly, in SHQ, thee i no way fo a node to popagate infomation back to it anceto, and clahe accoding to the new clah condition can only occu though a cyclic tuctue. Thi i o becaue a node v i only labelled with y, {y/v} by an application of the new -ule to ome concept y.c, B in the label of v. Futhemoe, the only way y can occu with v a a binding i when C, {y/v} B i expanded to y, {y/v} via a cyclic path back to v. Thi i obviouly only poible among individual node in SHQ and, theefoe, no clah in the tableau can by caued by unavelling. Hence, tanitive ole alone ae not cauing majo poblem. An inteeting conequence i, howeve, that we looe the finite model popety. Fo example, let K contain the axiom. and C q with a tanitive ole and C q := x.(. x). The fit axiom enfoce an infinite -chain fo evey individual. Nomally, a finite model could contain an -cycle intead of an infinite chain, but thi would clealy violate C q. Hence, evey model of K mut be acyclic and theefoe contain an infinite -chain. 5 A Rolling-up Technique fo SHOQ In thi ection, we how how the olling-up technique can be extended to the DL SHOQ, i.e., SHQ plu nominal. We do not ue binde hee, but athe popoe an extenion of the gueing technique a ketched in Section 3. In the peence of nominal, a imple non-deteminitic aignment of individual name to vaiable that occu in a cycle i not ufficient. Fo example, Fig. 9 epeent a model fo the KB containing the axiom {a} C D.(C.(D.{a})) and tan(). The quey x: C y : D x, y : y, x : (ee Fig. 10) would clealy be tue, although in the elevant aignment a cannot be bound to eithe x o y. Since the quey gaph fo thi quey contain jut one tongly connected component, no invee ole ae equied in the olling-up and we hould be able to deal with thi quey. Fo SHOQ, ou main agument fo the coectne of an extenion of the gueing of vaiable aignment i that a cycle among new node can only occu due to a tanitive ole that povide a hotcut fo kipping the nominal.

8 a: Cu D n1:c n2:d Figue 9: The dahed line indicate the elationhip added due to being tanitive. Theefoe, thee i a cycle not diectly containing the nominal a. Hence, a nominal i alway at leat indiectly involved in a cycle. In thi cae, we have only one nominal a, and we may gue that it i eithe in the poition of x, in the poition of y, o it i plitting the (tanitive) ole (ee Fig. 11). In each cae, we can oll-up the quey gaph into the nominal, obtaining the thee quey concept C 1 q := {a} C.(D.{a}), C 2 q := {a} D.(C.{a}), C 3 q := {a}.(c.(d.{a})). Identifying x with y give the additional foth quey concept C 4 q := {a} C D.{a}.{a}. The quey i tue jut in cae one of thee quey concept i entailed by the KB. Clealy, in ou example, only the concept C 3 q that coepond to the new gue i entailed. If we had n nominal, then we would need to ty 4n guee. Figue 10: The oiginal quey gaph. a Figue 11: An altenative quey gaph in which the nominal a i aumed to be involved in the cycle. Since it wa, to the bet of ou knowledge, not even known if conjunctive quey anweing fo logic with nominal and tanitive ole i decidable, thi technique i clealy valuable, although it i, due to it highly non-deteminitic natue, not vey pactical. 6 The Challenge of Invee and Nominal Neithe the agument ued fo the extenion of SHQ with binde (i.e., blocking can ignoe diffeent binding), no the extended gueing tategy fo SHOQ, can be ued with invee ole. Fig. 12 how a epeentation of a model fo the concept {a}.(. ) fo a tanitive and ymmetic ole. The quey x, x : i obviouly tue in thi model. The nominal a i, howeve, not even indiectly involved in the cycle. a n1 n2 Figue 12: The dahed line epeent the additional elationhip added fo being tanitive and ymmetic.

9 Since completion gaph ae finite epeentation of model, the quetion i how fa do we have to expand a completion gaph befoe blocking i afe, i.e., unavelling into a tableau doe not lead to a clah. In ode to ee the ame poblem fom anothe pepective, we biefly ketch a diffeent quey anweing algoithm in the tyle of CARIN [10]. CARIN povide a conjunctive quey anweing algoithm fo the DL ALCN R. Recall fom Section 2 that a quey q i tue in a knowledge bae K, if thee i a q- mapping fo evey model I = ( I, I) of K. Similaly, we define a mapping φ fom tem in a quey to node in a completion gaph G and we call φ a q-mapping w..t. G if, fo each contant t in q, {t} i in the label of φ(t) (i.e., contant ae mapped to the coeponding nominal node), fo each concept atom t: C in q, C i in the label of φ(t) and, fo each ole atom t, t : in q, φ(t ) i an -decendant of φ(t), whee -decendant implicitly cloe the tanitive ole. Since φ i puely yntactic, we extend the KB with an axiom C C fo each concept C.t. t: C i a concept atom in q. Hence, we obtain a deciion pocedue fo conjunctive quey anweing if we can how the following: Claim 6.1 Fo each model I of K, thee i a q-mapping w..t. I iff fo each completed and clah-fee completion gaph G of K, thee i a q-mapping w..t. G. In ode to take the length of the path in the quey into account, blocking mut be delayed appopiately. In CARIN (i.e., fo a logic without tanitive ole) thi i achieved by uing two iomophic tee (intead of two iomophic pai a it i the cae fo nomal paiwie blocking a in SHIQ).t. the depth of the tee coepond to the longet path in the quey. The if diection of Claim 6.1 i elatively taight fowad but, fo the only if diection, we have to how that (in contapoitive fom), if thee i a completion gaph G fo K.t. thee i no q-mapping φ w..t. G, then thee i a model I of K.t. thee i no q-mapping σ w..t. I. We now give an example that how that, even if we find uch a completion gaph G, we cannot guaantee that thee i no mapping σ w..t. the canonical model I of G, if q contain a cycle with a tanitive ole occuing in it. Of coue thee may be anothe completion gaph, fo which thi poblem doe not occu, but it i had to pove that thee i alway a canonical counte-model fo the quey. Let K be a KB containing the axiom.. fo a tanitive and ymmetic ole and let q be the quey x, x : x, y : y, z : z, z : x: C z : C, ee Fig. 13 ight. The uppe pat of Fig. 13 how a poible (implified) completion gaph G fo K, whee C o C i added to the node label to allow fo a puely yntactic mapping φ. The gey undelying tiangle hape illutate the two iomophic tee ued fo blocking, and clealy thee i no q-mapping w..t. G. The patial epeentation of a model depicted in the lowe pat of Fig. 13, howeve, how that, by unavelling G, we would get

10 a model I of K.t. thee i a q-mapping w..t. I. Thi i the cae becaue thee i no longe only one element in the extenion of C, and all ole elationhip fo q ae atified ince i tanitive and ymmetic (infeed elationhip ae only pictued whee they ae elevant fo the mapping). Even chooing a lage depth fo the two tee would allow thi to happen, ince the deciion fo C o C and fo an - o -ucceo wa puely non-deteminitic. C C C C C C C C C C C C C C C C C C C C C C C C C C C C Figue 13: A completion gaph G with tee fo blocking (top), a patial unavelling of G (bottom), and a quey gaph (ight). Theefoe, the abence of a q-mapping fo the completion gaph doe not pove that thee i a model fo which thee i no q-mapping. Requiing eveal iomophic tee o pai befoe blocking occu obviouly doe not help eithe ince only one pat between the iomophic tee/pai could contain the twice equied C in the node label. The poblem i that, by epeating the unique pat between the blocking and the blocked node, we can poduce a tuctue that allow fo a q-mapping. A tempting olution fo afe blocking condition would be to equie a path of n iomophic equence diectly following each othe, whee n i the length of the longet path in the quey. Howeve, thi will not guaantee temination, a the following agument how: we could intepet each path fom a oot o nominal node a a wod ove the edge label, e.g., the above example would poduce a wod tating with. Fo the uggeted blocking condition, uch a path mut contain a ubting of the fom W n, i.e., n epetition of the wod W, whee W i any equence of lette ove ou alphabet of ole. The additional blocking condition fo the node label ae ielevant hee. In 1944 Moe and Hedlund [12] howed that the o called Thue-Moe equence, which fom a wod ove an alphabet of only two lette, i cube-fee, which would coepond to a path of unbounded length in which no blocking occu. Hence, the algoithm would not teminate fo n 3. Fo a thee lette alphabet, thi hold aleady fo n 2. We encounte imila difficultie when uing binde and vaiable fo a logic like SHIQ o SHOIQ. It i difficult to detemine how deep the completion gaph ha to be befoe blocking i afe. If the completion gaph i not deep enough, unavelling doe not yield a model. Although fo the CARIN technique, ome canonical model with a q-mapping have completion gaph without a q-mapping, it eem a if by teting all poi- y z:c

11 ble completed and clah-fee completion gaph, one would alway find at leat one completion gaph uch that alo it canonical model doe not povide fo a q-mapping. Howeve, thi i only an obevation, which could uppot the view that the poblem i, depite all thee poblem, decidable. A diffeent poof technique might be neceay to how thi, and ou futue wok i aimed at invetigating thi. 7 Concluion In the peviou ection, we have peented two idea that allow an extenion of the olling-up technique alo to cyclic conjunctive queie fo SHQ and SHOQ. Fo SHQ, we achieved thi by extending the logic with a eticted fom of binde and tate vaiable a known fom Hybid Logic. Thi allow the expeion of cyclic queie a concept ince, with binde and vaiable we can expe a co-efeence. Quey entailment can then be educed to deciding concept atifiability fo the extended DL. Howeve, adding the binde, even in the vey eticted fom needed fo quey anweing, ha a notable impact on the eulting logic; e.g., fo SHQ, thi extenion lead to the lo of the finite model popety. In Section 4.1, we illutate how a tableaux algoithm fo SHQ can be extended in ode to handle quey concept. Although each intoduce a feh nominal on the fly, we how that, fo SHQ, temination can be egained by ignoing them in the blocking condition. Thu we have hown how conjunctive queie with tanitive ole in the quey body can be anweed. In Section 5, we extend the wok by Calvanee et al. [4] to SHOQ, i.e., to a logic that allow fo nominal. We do thi by popoing a moe ophiticated gueing technique, which then again enable the olling-up of a quey into a SHOQ-concept. In Section 6, we highlight why none of the popoed technique extend eaily to a logic with invee ole. We alo how thi fo a quey entailment algoithm in the tyle of CARIN [10]. The main poblem i to decide when a completion gaph i expanded fa enough in ode to afely pedict that the quey i alo not entailed in it poibly infinite canonical model. The olling-up technique with binde and vaiable integate eamlely into the exiting tableaux algoithm, wheea, fo the CARIN-tyle algoithm, the each fo a q-mapping i an additional and completely epaated tep. Moeove, the added axiom C C fo evey concept C in the quey, can ignificantly inceae the amount of non-deteminim. Thi make binde and vaiable the moe attactive choice fom an implementation point of view. Ou futue wok will include effot to how that anweing abitay conjunctive queie fo SHIQ and SHOIQ i decidable. If that i the cae and we believe it i we aim at extending the olling-up technique with binde

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