Optimizing the Nominal Introduction Rule in (Hyper)Tableau Calculi

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1 Optimizing the Nominl Introdution ule in (Hyper)Tleu Cluli oris Motik, o herer, nd In Horroks Oxford University Computing Lortory 1 Introdution A tleu lulus for HIQ hs een known for some time, nd forms the sis for severl highly suessful implementtions [1, 8, 9]. Extending this lulus to HOIQ, however, ws notoriously diffiult due to n intertion etween nominls, inverse roles, nd numer restritions whih results in (prtil) loss of the forest-model property for models of HOIQ knowledge ses. In this pper, we nlyze the prolems tht rise from this omintion of fetures nd present n overview of two solutions. The first solution, desried in [3], extends the HIQ lulus with n NNrule whih extends the non-forest-like portion of model with new individuls. Applitions of this rule n e highly nondeterministi, however, nd n sustntilly inrese the size of the generted models, whih leds to ineffiieny. We present n lterntive solution sed on new NI-rule. Our pproh does not introdue new individuls; insted, existing individuls re inorported into the non-forest-like portion of the model. We implemented our solution in HermiT. 1 As we disuss in [7], this pproh seems to work well in prtie. 2 Preliminries To show tht desription logi knowledge se K = (, T, A) is stisfile, tleu lgorithm onstruts sequene of Aoxes A = A 0, A 1,..., A n lled derivtion, where eh A i is otined from A i 1 y n pplition of one inferene rule. 2 The inferene rules mke the informtion impliit in the xioms of nd T expliit, nd thus evolve the Aox A towrds (representtion of ) model of K. The lgorithm termintes either if no inferene rule is pplile to some A n, in whih se A n represents model of K, or if A n ontins n ovious ontrdition, in whih se the model onstrution hs filed. The following inferene rules re ommonly used in DL tleu luli. -rule: Given (C 1 C 2 )(s), derive either C 1 (s) or C 2 (s). -rule: Given (C 1 C 2 )(s), derive C 1 (s) nd C 2 (s) ome formliztions of tleu lgorithms work on ompletion grphs, whih hve nturl orrespondene to Aoxes.

2 -rule: Given (.C)(s), derive (s, t) nd C(t) for t fresh individul. -rule: Given (.C)(s) nd (s, t), derive C(t). -rule: Given GCI C D nd n individul s, derive ( C D)(s). -rule: Given n.c(s), (s, t 0 ),..., (s, t n ), nd C(t 0 ),..., C(t n ), hoose t i nd t j for some 0 i < j n, prune t i, nd merge it into t j. The individuls present in the originl Aox A re lled nmed individuls, nd they n e onneted y role ssertions in n ritrry wy. The individuls introdued y the - nd -rules re lled lokle individuls. For exmple, if.c() is expnded into (, s) nd C(s), then s is lled lokle individul nd it is the -suessor of. It is not diffiult to see tht no tleu inferene rule n onnet s with some element of A: the individul s n prtiipte only in inferenes tht derive n ssertion of the form D(s) or rete new suessor or s. Hene, eh Aox A otined from A n e seen s forest : eh nmed individuls n e ritrrily onneted to other nmed individuls nd to tree of lokle suessors. The onept lel L A (s) is defined s the set of ll onepts C suh tht C(s) A, nd the edge lel L A (s, s ) s the set of ll tomi roles suh tht (s, s ) A. A nïve pplition of the tleu rules does not terminte if the Tox ontins existentil quntifiers in yles: given Tox T = {. }, nïve pplitions of the -rule produe n infinite hin of lokle individuls. To ensure termintion is suh ses, tleu lgorithms employ loking [4]. DLs suh s HIQ nd HOIQ llow for inverse roles nd numer restritions, whih require nestor pirwise loking [4]: for s nd t lokle individuls nd s nd t ny individuls of n Aox A, t loks s if nd only t is n nestor of s, L A (s) = L A (t), L A (s ) = L A (t ), L A (s, s ) = L A (t, t ), nd L A (s, s) = L A (t, t). In tleu lgorithms, the - nd -rules re pplile only to nonloked individuls, whih ensures termintion: the numer of different onept nd edge lels is exponentil in K, so n exponentilly long rnh in forest-like Aox must ontin loked individul, thus limiting the length of eh rnh in n Aox. Let A e n Aox to whih no tleu inferene rule is pplile, nd in whih s is loked y t. We n onstrut model from A y unrveling tht is, y repliting the frgment etween s nd t infinitely often. Intuitively, loking ensures tht the prt of the Aox etween s nd s ehves just like the prt etween t nd t, so unrveling indeed genertes model. If our logi were le to onnet lokle individuls in non-tree-like wy, then unrveling would not generte model; in ft, the notion of nestors, desendnts, nd loking would itself e ill-defined. 3 Nominls, Inverse oles, nd Numer estritions We now disuss the diffiulties tht rise when extending tleu luli to hndle nominls, inverse roles, nd numer restritions. We summrize oth the originl solution presented in [3], s well s our novel pproh whih we inorported into our hypertleu lulus.

3 A. A..C s1 s3.c C.{} s2 s4 C.{} A. A..C s1 s3.c C.{} s2 () Non-Tree-Like trutures due to Merging A..{} 2.A d A..{} e.a () loking nd Unrveling Fig. 1: Prolems used y nominls. Nmed individuls re shown in lk, loked individuls in white, nd loking is indited with dshed line. 3.1 The Min Prolems There re two primry prolems tht must e ddressed when extending HIQ tleu lulus with nominls. Nonforest models. Nominls n mke Aoxes non-forest-like, s the following simple exmple demonstrtes. (1) A 1 = { A(), A() } T 1 = { A.,.C, C.{} } uessive rule pplitions on A 1 nd T 1 n produe the Aox A 1 1 shown on the left-hnd side of Figure 1. This Aox is lerly not forest-shped: the two -pths in A 1 1 strt t the nmed individuls nd, respetively, nd end in nmed individul. This y itself is not prolem: termintion of model onstrution on suh extended forest-like Aoxes n e ensured esily if the DL llows for nominls, ut not in the presene of oth inverse roles nd numer restritions [2]. Assume now tht we extend T 1 with the xiom 1. On A 1 1, this might merge s 2 into s 4. Note tht oth s 2 nd s 4 re lokle individuls; furthermore, neither individul is n nestor of the other, so we n merge, sy, s 4 into s 2. This produes the Aox A 2 1 shown on the right-hnd side of Figure 1. The ssertion (s 3, s 2 ) mkes A 2 1 not forest-shped. y extending the exmple, it is possile to use nominls, inverse roles, nd numer restritions to rrnge lokle individuls in yles. This loss of forest-shped models prevents the use of loking nd pruning, thus invlidting the stndrd termintion rguments. loking nd unrveling. Even if the lokle individuls in tleu mintin forest struture, nominls n prevent the suessful pplition of loking nd unrveling. Consider the following knowledge se. (2) A 2 = { A() } T 2 = { A., A.{},.A, 2 }

4 s1 s2 s1 s2 s1 s2 s3 s4 s3 s4 s3 Fig. 2: The Introdution of oot Individuls A tleu expnsion of A 2 nd T 2 would produe the Aox A 1 2 shown in Figure 1. Individul e is loked in A 1 2 y the individul, so the derivtion termintes. Note tht the lst xiom from T 2 is stisfied: is the only individul in A 1 2 tht hs -suessors nd it hs two suh suessors. To onstrut model from A 1 2, we unrvel the loked prts of the Aox tht is, we onstrut n infinite pth tht extends pst e y dupliting the frgment of the model etween nd e n infinite numer of times. This, however, retes dditionl -suessors of e, whih invlidtes the lst xiom from T 2 ; thus, the unrveled Aox does not define model of A 2 nd T A Nïve olution To solve the ove prolems, we need to extend the ritrrily interonneted prt of the Aox: in ddition to nmed nd lokle individuls, we introdue root individuls freshly introdued individuls whih n e ritrrily onneted to nmed individuls nd other root individuls. We pply the following test (*): if n Aox A ontins ssertions (s, ), A(s), nd n.a(), where is root or nmed individul nd s is lokle individul tht is not suessor of, then we hnge s into root individul. Applied to A 2 1, this ondition hnges the sttus of s 2 from lokle individul into root individul. After this hnge, only s 1 nd s 3 re lokle in A 2 1, so the Aox hs the extended forest-like shpe nd we n pply loking nd pruning s usul. This is shemtilly shown in Figure 2. Promotion of lokle individuls to roots lso elegntly solves the loking nd unrveling prolem. In A 2 nd T 2, euse must stisfy n t-most restrition of the form 2, s soon s (d, ) is derived, ondition (*) is fulfilled nd we turn d into root individul. This solution, however, introdues nother prolem: the numer of root individuls n now grow ritrrily, s shown in the following exmple. (3) A 3 = { A() } T 3 = { A.A, A.{} 2 } On A 3 nd T 3, rule pplitions n produe the derivtion sequene shown in Figure 3: fter nd d re introdued s desendents of, they stisfy ondition (*) so we hnge them into root individuls. usequently, the third xiom from T 3 is not stisfied, so we merge two neighors of ; we hoose to merge into. ine d is now not lokle individul, we nnot prune it. The first xiom of T 3 is not stisfied for d, so we n extend the Aox with new suessor e whih lso stisfies (*) nd eomes root individul. If we ontinue to merge suessors of into we n repet the sme inferenes forever.

5 A.A.{} d d d e 3.3 The Existing olution Fig. 3: A Modified Yo-Yo Exmple To gurntee termintion, the lulus from [3] uses n NN -rule whih refines ondition (*). Assume tht n Aox A ontins ssertions (s, ) nd A(s) where s is lokle individul tht is not suessor of root or nmed individul ; furthermore, ssume tht must stisfy n t-most restrition of the form n.a. If A lredy ontins root individuls z 1,..., z n suh tht 1 i n {A(z i), (z i, )} {z i z j 1 i < j m} A, then the -rule simply merges s into some z i ; no new root individul needs to e introdued. If A does not ontin suh z 1,..., z n, the NN -rule nondeterministilly guesses the ext numer m n of -neighors of tht re memers of A, genertes m fresh root individuls w 1,..., w m, nd extends A with the ssertions {A(w i ), (w i, ) 1 i m} {w i w j 1 i < j m} { m.a()}. This llows the NN -rule to e pplied t most one for eh onept of the form n.a nd eh root individul, whih ensures termintion: unlike in the nïve solution outlined in the previous setion, fresh root individuls re not reted whenever root individuls gins lokle neighor. For exmple, Figure 4 shows pplitions of the NN - nd -rules to the Aox A 4 = {..{}(), 3. () }. In A 1 4, shown t the left of the figure, the nmed individul hs lokle -neighor nd must stisfy the restrition 3.. The NN -rule nondeterministilly hooses whether to introdue one, two, or three fresh root individuls; the prllel rnhes of the derivtion re shown in the enter of the figure. The introdution of single individul, shown in the top rnh, results in deterministi pplition of the -rule s the lokle individul is merged into the fresh root individul z 1, shown in the upper right of the figure. For derivtion pths on whih more thn one fresh root individul is introdued, pplition of the -rule is nondeterministi: n e merged into ny of the new root individuls, nd the derivtion rnhes further. Although the N N-rule does ensure termintion of the tleu lgorithm, it is potentil soure of ineffiieny in knowledge ses in whih lrge numers pper within t-most onepts: n pplition of the N N-rule involving onept n.c guesses mong n different possile sizes for the neighor set, nd susequent pplitions of the -rule must hoose how to merge the new roots with lokle individuls. In the se of just single lokle neighor, this

6 ..{}.{} {}.{} {}.{} {}.{} 3...{}.{} {}.{} {}.{} 3...{}.{} 3. z3 z3..{}.{} 3. z3..{}.{} 3. z3 Fig. 4: An pplition of the NN-rule results in derivtion tree with n 2 rnhes. Furthermore, the introdution new root individuls n result in unneessry proessing nd lrge models. 3.4 The NI -rule As replement for the NN -rule desried ove, we introdue new NI - rule, whih refines the genertion of root individuls. Let us gin onsider the Aox A ontining ssertions (s, ) nd A(s), nd n.a(), where s is lokle individul tht is not suessor of root or nmed individul. In ny model of A, we n hve t most n different individuls i tht prtiipte in ssertions of the form ( i, ) nd A( i ). Hene, we ssoite with in dvne set of n fresh root individuls { 1,..., n }. Insted of hoosing some suset of these individuls to introdue nd relying upon the -rule to merge them with lokle neighors, however, we promote lokle individuls to root individuls diretly: to turn s into root individul, we nondeterministilly hoose j from this set nd merge s into j. In this wy, the numer of new root individuls tht n e introdued for numer restrition in the lel of n individul is limited to n. An pplition of our NI-rule to the Aox A 4 = {..{}(), 3. () } is given in Figure 5. y exploiting ound on the length of pths of lokle individuls in n Aox, we n estlish ound on the numer of root individuls introdued in derivtion, whih ensures termintion of the lgorithm. When formulting the NI -rule, we re fed with tehnil prolem: onepts of the form n.a re trnslted in our lulus into DL-luses, whih

7 ..{}.{} 3. z 1..{}.{} 3...{}.{} z {}.{} 3. z 3 Fig. 5: An pplition of the NI-rule mkes testing the ondition from the previous prgrph diffiult. For exmple, n pplition of the Hyp-rule to the third DL-luse in (3) (otined from the xiom 2. ) n produe n equlity suh s. This equlity lone does not reflet the ft tht must stisfy the t-most restrition 2.. To enle the pplition of the NI -rule, we introdue notion of nnotted equlities, in whih the nnottions estlish n ssoition with the t-most restrition. The detils of our proedure re presented in the next setion. 4 A Hypertleu Clulus for HOIQ We now summrize the hypertleu lgorithm tht n e used to hek stisfiility of HOIQ knowledge se K. This is extension of the proedure desried in [6] nd [5], nd full version of the lulus is presented in [7]. Our lgorithm first preproesses HOIQ knowledge se into n Aox nd set of DL-luses implitions of the form n i=1 U i m j=1 V j, where U i re of the form (x, y) or A(x), nd V j re of the form (x, y), A(x),.C(x), n.c(x), or x y. Due to lk of spe, we leve the detils of this trnsformtion to [7]. In ft, the preproessing produes DL-luses of ertin syntti struture, whih gurntees termintion of the model onstrution. In the rest of this pper, we often use the following funtion r. Given role nd vriles or onstnts s nd t, this funtion returns n tom tht is semntilly equivlent to (s, t) ut tht ontins n tomi role. r(, s, t) = { (s, t) if is n tomi role (t, s) if is n inverse role nd = Definition 1 (HT-Cluse). We ssume tht the set of tomi onepts N C ontins nominl gurd onept O for eh individul ; these onepts should not our in ny input knowledge se. An t-most equlity is n tom of the form s u n., where s, t, nd u re onstnts or vriles, n is nonnegtive integer, is role, nd is literl onept. emntilly, this tom is equivlent to s t; the u n. is only used to ensure termintion of the hypertleu lulus.

8 An HT-luse is DL-luse r of the form (4) U 1... U m V 1... V n, in whih it must e possile to seprte the vriles into enter vrile x nd set of rnh vriles y i suh tht the following properties hold, where A is n tomi onept, is literl ut not nominl gurd onept, O is nominl gurd onept, nd re tomi roles, nd T is role. Eh tom in the nteedent of r is of the form A(x), (x, y i ), (y i, x), or A(y i ). Eh tom in the onsequent of r is of the form (x), (y i ), (x, y i ), (y i, x), x y i, or y i y x n T.. For eh tom (x, y i ) or (y i, x) in the onsequent of r, the nteedent of r ontins n tom (x, y i ), (y i, x), O (x), or O (y i ). For eh equlity x y i in the onsequent of r, the nteedent of r ontins n tom O (x) or O (y i ). Eh equlity y i y x h T.A in the onsequent of r must our in suluse of r of the form (5) nd no y k with 1 k h + 1 should our elsewhere in r. (5) h+1... [r(t, x, y k ) A(y k )] k=1 1 k<l h+1 y k y x h T.A... Eh equlity y i y x n T. A in the onsequent of r must our in suluse of r of the form (6) nd no y k with 1 k h + 1 should our elsewhere in r. (6) h+1 h+1... r(t, x, y k ) A(y k ) k=1 k=1 1 k<l h+1 y k y x h T. A... We now present the hypertleu lulus for deiding stisfiility of n Aox A nd set of HT-luses C. In our lgorithm, we ll the individuls tht our in the input Aox nmed. Furthermore, for nmed individul, the NI -rule might need to introdue individuls tht re unique for, role, literl onept, nd some integer i; we represent suh individuls s.,, i. ine the NI -rule might e pplied to these individuls s well, we introdue the notion of root individuls finite strings of the form.γ γ n where is nmed individul nd eh γ l is of the form..i. In stndrd tleu lgorithms, the tree struture of the model is enoded in the edges etween individuls, whih lwys point from prents to hildren. In ontrst, our lgorithm enodes the prent-hild reltionships into individuls themselves: it represents individuls s finite strings of the form s.i 1, i 2,..., i n, where s is root individul nd i j re integers. For exmple,.2 is the seond hild of the nmed individul. Individuls with n 1 re lled lokle.

9 Definition 2 (Hypertleu Algorithm). Individuls. Given set of nmed individuls N I, the set of root individuls N O is the smllest set suh tht N I N O nd, if x N O, then x.,, i N O for eh role, literl onept, nd integer i. The set of ll individuls N A is the smllest set suh tht N O N A nd, if x N A, then x.i N A for eh integer i. The individuls in N A \ N O re lokle individuls. A lokle individul x.i is suessor of x, nd x is predeessor of x.i. Desendnt nd nestor re the trnsitive losures of suessor nd predeessor, respetively. Aoxes. The hypertleu lgorithm opertes on generlized Aoxes, whih re otined y extending the stndrd notion of Aoxes s follows. In ddition to stndrd ssertions, n Aox n ontin t-most equlities nd speil ssertion tht is flse in ll interprettions. Furthermore, ssertions n refer to the individuls from N A nd not only from N I. Eh (in)equlity s t (s t) lso stnds for the symmetri (in)equlity t s (t s). The sme is true for nnotted t-most equlities. An Aox A n ontin renmings of the form where nd re root individuls. The reltion in A must e yli, A n ontin t most one renming for n individul, nd, if A ontins, then should not our in ny ssertion or (in)equlity in A. An individul is the nonil nme of root individul in A, written = n A (), iff + A nd no individul exists suh tht + A, where + A is the trnsitive losure of in A. An Aox tht ontins only nmed individuls nd no t-most equlities is lled n input Aox. Pirwise Anywhere loking. For A n tomi onept nd role, onepts of the form A, n.a, or n. A, re lled loking-relevnt. The lels of n individul s nd of n individul pir s, t in n Aox A re defined s follows: L A (s) = {C C(s) A nd C is loking-relevnt onept} L A (s, t) = { (s, t) A} Let e strit ordering (i.e., trnsitive nd irreflexive reltion) on N A ontining the nestor reltion tht is, if s is n nestor of s, then s s. y indution on, we ssign to eh individul s in A sttus s follows: lokle individul s.i is diretly loked y lokle individul t.j iff t.j is not loked, t.j s.i, L A (s.i) = L A (t.j) nd L A (s) = L A (t), nd L A (s, s.i) = L A (t, t.j) nd L A (s.i, s) = L A (t.j, t); s is indiretly loked iff it hs predeessor tht is loked; nd s is loked iff it is either diretly or indiretly loked.

10 Pruning. The Aox prune A (s) is otined from A y removing ll ssertions ontining desendent of s. Merging. The Aox merge A (s t) is otined from prune A (s) y repling the individul s with the individul t in ll ssertions (ut not in renmings) nd, if oth s nd t re root individuls, dding the renming s t. Derivtion ules. Tle 1 speifies derivtion rules tht, given n Aox A nd set of HT-luses C, derive the Aoxes A 1,..., A n. In the Hyp-rule, σ is mpping from the set of vriles N V to the individuls ourring in the ssertions of A, nd σ(u) is the result of repling eh vrile x in the tom U with σ(x). ule Preedene. The -rule n e pplied to (possily nnotted) equlity s t in n Aox A only if A does not ontin n equlity s u n. to whih the NI-rule is pplile. Clsh. An Aox A ontins lsh iff A; otherwise, A is lsh-free. Derivtion. For set of HT-luses C nd n Aox A, derivtion is pir (T, λ) where T is finitely rnhing tree nd λ is funtion tht lels the nodes of T with Aoxes suh tht, for eh node t T, λ(t) = A if t the root of T, t is lef of T if λ(t) or no derivtion rule is pplile to λ(t) nd C, nd t hs hildren t 1,..., t n suh tht λ(t 1 ),..., λ(t n ) re extly the results of pplying one (ritrrily hosen, ut respeting the rule preedene) pplile rule to λ(t) nd C otherwise. We stress severl importnt spets of Definition 2. The NI -rule is never pplied to n t-most equlity of the form s u n. if u is not root individul, so suh n equlity n e egerly simplified into s t. The NI -rule, however, must e pplied to s u n. even if s = t; hene, suh n equlity must e derived even though it is logil tutology. Finlly, if C hs een otined y trnsltion of DL knowledge se tht does not use nominls, inverse roles, or numer restritions, then the preondition of the NI -rule will never e stisfied, so we need not keep trk of nnottions t ll. Theorem 1. Cheking whether HOIQ knowledge se K is stisfile n e performed y omputing K = (Ω(K)) nd then heking whether some derivtion for Ξ(K ) ontins lef node leled with lsh-free Aox. Furthermore, suh n lgorithm n e implemented in 2NExpTime in K. 5 Conlusion In this pper, we nlyzed the prolems rising in tleu luli due to n intertion etween nominls, inverse roles, nd numer restritions. We lso presented new hypertleu lulus for HOIQ, whih we implemented in our resoner HermiT. As we report in [7], our resoner seems to perform well on mny prtil prolems.

11 Tle 1: Derivtion ules of the Tleu Clulus If 1. U 1... U m V 1... V n C, nd 2. mpping σ of vriles to the individuls of A exists suh tht Hyp-rule 2.1 σ(x) is not indiretly loked for eh vrile x N V, 2.2 σ(u i) A for eh 1 i m, nd 2.3 σ(v j) A for eh 1 j n, then A 1 = A { } if n = 0; A j := A {σ(v j)} for 1 j n otherwise. If 1. n.c(s) A, 2. s is not loked in A, nd -rule 3. A does not ontin individuls u 1,..., u n suh tht 3.1 {r(, s, u i), C(u i) 1 i n} {u i u j 1 i < j n} A, nd 3.2 either s is lokle or no u i, 1 i n, is indiretly loked in A then A 1 := A {r(, s, t i), C(t i) 1 i n} {t i t j 1 i < j n} where t 1,..., t n re fresh distint suessors of s. If 1. s t A (the equlity n possily e nnotted), nd -rule 2. s t then A 1 := merge A (s t) if t is nmed individul, or t is root individul nd s is not nmed individul, or s is desendnt of t; A 1 := merge A (t s) otherwise. If s s A or {A(s), A(s)} A -rule then A 1 := A { }. If 1. s u n. A or t u n. A, 2. u is root individul, NI -rule 3. s is lokle nonsuessor of u, nd 4. t is lokle individul then A i := merge A (s n A(u.,, i )) for eh 1 i n. eferenes 1. V. Hrslev nd. Möller. ACE ystem Desription. In Pro. IJCA 2001, pges , ien, Itly, June I. Horroks nd U. ttler. Ontology esoning in the HOQ(D) Desription Logi. In Pro. IJCAI 2001, pges , I. Horroks nd U. ttler. A Tleux Deision Proedure for HOIQ. In Pro. IJCAI 2005, pges , Edinurgh, UK, July 30 August I. Horroks, U. ttler, nd. Toies. esoning with Individuls for the Desription Logi HIQ. In Pro. CADE-17, pges , Pittsurgh, UA, Motik,. herer, nd I. Horroks. A Hypertleu Clulus for HIQ. In Pro. of the 2007 Desription Logi Workshop (DL 2007), pges , Motik,. herer, nd I. Horroks. Optimized esoning in Desription Logis using Hypertleux. In Pro. CADE-21, volume 4603 of LNAI, pges 67 83, Motik,. herer, nd I. Horroks. Hypertleu esoning for Desription Logis. Tehnil report, University of Oxford, umitted to n interntionl journl Prsi nd E. irin. Pellet: An OWL-DL esoner. Poster, In Pro. IWC 2004, Hiroshim, Jpn, Novemer 7 11, D. Tsrkov nd I. Horroks. FCT++ Desription Logi esoner: ystem Desription. In Pro. IJCA 2006, pges , ettle, WA, UA, 2006.