Automata for Analyzing and Querying Compressed Documents

Transcription

1 Automt or Anlyzin nd Queryin Compressed Douments Brr Fil, Siv Annthrmn To ite this version: Brr Fil, Siv Annthrmn. Automt or Anlyzin nd Queryin Compressed Douments. Projet PRV (du LIFO), CATz. Rpport de reherhe LIFO <hl v3> HAL Id: hl Sumitted on 15 De 2006 HAL is multi-disiplinry open ess rhive or the deposit nd dissemintion o sientii reserh douments, whether they re pulished or not. The douments my ome rom tehin nd reserh institutions in Frne or rod, or rom puli or privte reserh enters. L rhive ouverte pluridisiplinire HAL, est destinée u dépôt et à l diusion de douments sientiiques de niveu reherhe, puliés ou non, émnnt des étlissements d enseinement et de reherhe rnçis ou étrners, des lortoires pulis ou privés.

2 Automt or Anlyzin nd Queryin Compressed Douments Brr FILA, LIFO, Orléns (Fr.) Siv ANANTHARAMAN, LIFO, Orléns (Fr.) Rpport N o

3 Automt or Anlyzin nd Queryin Compressed Douments Brr Fil, Siv Annthrmn LIFO - Université d Orléns (Frne), e-mil: {il, siv}@univ-orlens.r Astrt. In irst prt o this work, tree/d utomt re deined s extensions o (unrnked) tree utomt whih n run indierently on trees or ds; they n thus serve s tools or nlyzin or queryin ny semi-strutureddoument,whetherornotiveninompressedormt. In seond prt o the work, we present method or evlutin positive unry queries, expressed in terms o Core XPth xes, on ny d t representin n XML doument possily iven in ompressed orm; the evlution is done diretly on t, without unoldin it into tree. To eh Core XPth query o ertin si type, we ssoite word utomton; these utomt run on the rph o dependeny etween the non-terminls o the miniml strihtline reulr tree rmmr ssoited to the iven d t, or lon omplete silin hins in this rmmr. AnyivenpositiveCoreXPthquerynedeomposedintoquerieso the si type, nd the nswer to the query, on the d t, nthene expressed s su-d o t whose nodes re suitly leled under the runs o suh utomt. Keywords: Tree utomt, Tree rmmrs, Ds, XML, Core XPth. 1 Introdution Severl lorithms hve een optimized in the pst, y usin strutures over ds insted o over trees. Tree utomt re widely used or queryin XML douments (e.., [8, 9, 15, 16]); on the other hnd, the notion o ompressed XML doument hs een introdued in [2, 7, 12], nd possile dvnte o usin d strutures or the mnipultion o suh douments hs een rouht out in [12]. It is leitimte then to investite the possiility o usin utomt over ds insted o over trees, or queryin ompressed XML douments. D utomt (DA) were irst introdued nd studied in [5]; DA ws deined there s nturl extension o tree utomton, i.e. s ottom-up tree utomton runnin on ds; nd the lnue o DA ws deined s the set o ds tht et epted under (ottom-up) runs, deined in the usul sense; the emptiness prolem or DAs ws shown there to e NP-omplete, nd the memership prolem proved to e in NP; ut the prolem o stility under omplementtion o the lss o d utomt losely linked with tht o determiniztion ws let open. These two issues hve sine een settled netively in [1]: the reson is tht the set o ll terms (trees) represented y the set o ds epted y non-deterministi DA is not neessrily reulr tree lnue; onsequene is tht the lss o tree lnues reonized y DAs (s sets o epted ds) is strit superlss o the lss o reulr tree lnues. It is well-known however, tht nswers to MSO-deinle queries on (semi-)strutured trees orm reulr tree lnues ([18]); it is thus neessry to deine the lnues o DAs in mnner dierent rom tht o [5, 1], i they re to serve s tools or nlyzin nd queryin doument, independently o whether it is iven in (prtilly or ully) ompressed ormt, or s tree. Our irst im in this work is thereore to redeine the notion o the lnue o DA suitly, with suh n ojetive. 2

4 For hievin tht, we irst present (in Setion 2) the notion o ompressed doument s tree/d (trd, or short), desintin direted yli rph tht my e prtilly or ully ompressed. The terminoloy trd hs een hosen to distinuish it rom tht o td employed in [1]; this ltter term will e employed in this pper when reerrin to ully ompressed d. A Tree/D utomton (TDA, or short) is then deined s n utomton whih runs on trds. The essentil dierenes with the DAs o [1] re the ollowin: (i) our TDAs n e unrnked, nd (ii) lthouh the trnsition rules o TDA look quite like those o the DAs in [1], or those o TAs, run o TDA on ny iven trd t will rry with it not only ssinments o sttes to the nodes o t, ut lso to the edes o t; runs will e so deined tht TDA epts ny iven trd t i nd only i it epts the tree ˆt otined y unompressin t, stree utomton runnin on the tree ˆt, in the usul sense. In the seond prt o the pper, we present n pproh sed on word utomt or evlutin queries on trds tht represent XML douments in prtilly or ully ompressed ormt; the terms trd nd doument will thereore e onsidered synonymous in the sequel. Any iven trd t is irst seen s equivlent to miniml strihtline reulr tree rmmr L t, tht one n nturlly ssoite with t,. e.., [3, 4]. From the rmmr L t,weonstrut the rph o dependeny D t etween its non-terminls, nd lso the hilins (liner rphs ormed o omplete hins o silin non-terminls) o L t.the word utomt tht we uild elow will run on D t or the hilins o L t,rther thn on the doument t itsel. We shll only onsider positive unry queries expressed in terms o Core XPth xes. (The view we dopt llows us to deine the vrious xes o Core XPth on ompressed douments, in mnner whih does not modiy their semntis on trees.) For evlutin ny suh query on ny doument (trd) t, we proeed s ollows. We irst rek up the iven query into si su-queries o the orm Q= //*[xis::] where xis is Core XPth xis o ertin type. To eh suh si query Q, we ssoite word utomton A Q.The utomton A Q runs on the rph D t when xis is non-silin, nd on the hilins o L t when xis is silin xis. An essentil point in our method is tht the runs o A Q re uided y some well-deined semntis or the nodes trversed, inditin whether the urrent node nswers Q, or is on pth ledin to some other node nswerin Q. The utomton is not deterministi, ut its runs re mde eetively unmiuous y deinin suitle priority reltion etween the trnsitions, sed on the semntis. A si query Q n then e evluted in one sinle top-down pss o A Q, under suh n unmiuous run. An ritrry positive unry Core XPth query n e evluted on t y ominin the nswers to its vrious si su-queries, nd its nswer set is expressed s su-trd o t, whose nodes et leled in onormity with the semntis. It is importnt to note tht the evlution is perormed on the iven trd t; s suh, on two dierent trds orrespondin to two dierent ompressions o sme XML tree, the nswers otined my not e the sme, in enerl. The pper is strutured s ollows: Setion 2 presents the notions o trds, nd o Tree/D utomt. In Setion 3, we onstrut rom ny trd t its normlized strihtline reulr tree rmmr L t, s well s the dependeny rph D t nd the hilins o L t ; these will e seen s rooted leled yli rphs (rls, or short); the si notions o Core XPth re lso relled. Setion 4 is devoted to the onstrution o the word utomt or ny si Core XPth query, sed on the semntis, nd n illustrtive exmple. In Setion 5 we prove tht the runs o these utomt, uniquely nd eetively determined under mximl priority ondition, enerte the nswers to the queries. Setion 6 shows how non si (omposite, or imrited) Core XPth query n e evluted 3

5 in stepwise shion. In Setion 7, we show how to reine our pproh, so s to derive rom the nswer or ny iven Core XPth query Q on trd t, the nswer set or the sme query Q on the tree-equivlent ˆt o t (without resortin to ny unompressin opertion). In the ppendies, we show how to trnslte the usul Core XPth queries into one in stndrd orm on whih our pproh is pplile; this trnsltion is done in liner time on the size o the iven query; we lso present n lorithm or onstrutin the mximl priority run, or ny si query utomton over ny iven doument (trd), with omplexity ound o O(m), where m is the numer o edes o the rl D t ssoited to the trd. (Note: the numer m o edes on D t n e exponentilly smller thn the numer o edes on the trd t. Whent is tree, D t is isomorphi to t, sothe omplexity o our lorithm redues to O(n), where n is the numer o nodes on the tree t.) A omplete illustrtive exmple, on omposite imrited query, is iven in the lst ppendix. 2 Tree/D Automt Deinition 1 A tree/d (trd or short) over not neessrily rnked lphet Σ is rooted d (direted yli rph) t =(Nodes(t),Edes(t)), where, or ny node u Nodes(t): - u hs nme nme t (u) =nme(u) Σ; - the edes oin out o ny node re ordered; -ndinme(u) is rnked, then the numer o outoin edes t u isthernkonme(u). (Itisssumedthtnytrdtis onneted, nd hs unique root node.) Given ny node u on trd t, the notion o the su-trd o t rooted t u is deined s usul, nd denoted s t u.iv is ny node, γ(v) =u 1...u n will denote the strin o ll its not neessrily distint hildren nodes; or every 1 i n, the i-th outoin ede rom v to its i-th hild node u i γ(v) will e denoted s i e(v, i); we shll lso write then v u i ; the set o ll outoin (resp. inomin) edes t ny node v will e denoted s Out v (t), or Out v (resp. In v (t), or In v ); nd or ny node u, weset:prents(u) ={v Nodes(t) u is hild o v}. Atrdt is sid to e tree i In v (t) isemptyiv is root, nd In v (t) is sinleton otherwise. On ny trd t, wedeinethesetpos(t) stheseto ll the positions pos t (u) o ll its nodes u, these ein deined reursively, s ollows: i u is the root node on t, thenpos t (u) =ɛ, otherwise,pos t (u) ={α.i α pos t (v),v is prent o u, u is n i-th hild o v}. ThesetPos(t) onsists o (some o the) words over nturl inteers. To ny ede e : u i v on trd t, is nturlly ssoited the suset pos t (e) =pos t (u).i o Pos(t). The untion nme t is extended nturlly to the positions in Pos(t) sollows: or every u Nodes(t) ndα pos t (u), we set nme t (α) =nme t (u). Given trd t, we deine its tree-equivlent s tree ˆt suh tht: Pos(ˆt) = Pos(t), nd or every α Pos(t) wehvenme t (α) =nmeˆt (α). It is immedite tht ˆt is uniquely determined, up to tree isomorphism; it n tully e onstruted nonilly (. [7]), y tkin or nodes the set Pos(t), nd or direted edes the set {(α, α.i) α, α.i Pos(t)}, ehnodeα ein nmed with nme t (α). There is then nturl, nme preservin, surjetive mp rom Nodes(ˆt) ontonodes(t); it will e reerred to in the sequel s the ompression mp, nd denoted s. A trd is sid to e td,orully ompressed, i or ny two dierent nodes u, u on t, the two su-ds t u nd t u hve non-isomorphi tree-equivlents; otherwise, the trd is sid to e prtilly ompressed when it is not tree. For exmple, the tree to the let o Fiure 1 is the tree-equivlent o the prtilly 4

6 ompressed trd to the riht, nd lso to the ully ompressed td to the middle. Tree Fully Compressed Prtilly Compressed Fi. 1. tree, td, nd trd We deine now the notion o Tree/D utomton, irst over rnked lphet Σ, to ilitte understndin. The deinition is then esily extended to the unrnked se. Deinition 2 A Tree/D utomton (TDA, or short) over rnked lphet Σ is tuple (Σ,Q,F,Δ), whereq is inite non-empty set o sttes, F Q is the set o inl (or eptin) sttes, nd Δ is set o trnsition rules o the orm: (q 1,..., q k ) q, where Σ is o rnk k, ndq 1,...,q k,q Q. It will e onvenient to write the trnsition rules o TDA in dierent (ut equivlent) orm: trnsition o the orm (q 1,...,q k ) q is lso written s (,q 1...q k ) q, whereq 1...q k is seen s word in Q,olenth=rnk() in the rnked se. The notion o TDA is then extended esily to the unrnked se, i.e., where the sinture symols nmin the nodes re not ssumed to e o ixed rnk: it suies to deine the trnsitions to e o the orm (,ω) q, where ω is reulr expression on the lphet set Q. A TDA is sid to e ottom-up deterministi i whenever there re two trnsition rules o the orm (,ω) q, (,ω ) q,withq q,wehve neessrily ω ω = ; otherwise it is sid to e non-deterministi. We lso ree to denote the trnsitions o the orm (, ) q simply s q, ndreerto them s initil trnsitions. For deinin the notion o runs o TDAs on trd in ottom-up style, we need some preliminries. Let A e TDA with stte set Q nd trnsition set Δ. Suppose t is trd nd ssume iven mp M : Edes(t) Q. Iu is ny node on t with u 1...u n s the strin o ll its (not neessrily distint) hildren, the strin M(e(u, 1))...M(e(u, n)) Q, ormed o sttes ssined y M to the outoin edes t u, will e denoted s M(Out u ). We then deine, reursively in ottom-up style, inry reltion t u on the sttes o Q, with respet to (w.r.t. or wrt, or short) the iven mp M; this reltion, denoted s M u = u,is deined s ollows: Deinition 3 Let A,t,M e s ove, nd u ny iven node on the trd t. I u is le with nme(u) =,thenq u q i whenever q Δ we lso hve q Δ; otherwise q u q i: (i) (nme(u),m(out u )) q is n instne o trnsition rule in Δ; i.e., Δ hs rule (nme(u),ω) q suh tht M(Out u ) is in ω; (ii) there exists mp q : Q Q, suh tht: 5

7 - q (q) =q,ndtherule(nme(u), q (M(Out u ))) q is lso n instne o trnsition rule in Δ; - or ny ede e : u i u Out u, we hve: M(e) u q (M(e)). Deinition 4 Let A =(Σ,Q,F,Δ) e ny iven TDA, nd t ny iven trd. A run o A on t is pir (r, M ), wherer : Nodes(t) Q nd M : Edes(t) Q re mps suh tht the ollowin onditions hold, t ny node u on t: (1) i nme(u) =, then the rule (,M(Out u )) r(u) is n instne o trnsition rule in Δ; (2) thereisninominedee In u with M(e) =r(u); nd or every e In u suh tht M(e )=q q = r(u), we hve q M u q Arun(r, M ) is eptin on trd t i r(ɛ) F, i.e, r mps the root-node o t to n eptin stte. A trd t is epted y TDA i there is n eptin run on t. The lnue o TDA is the set o ll trds tht it epts. Remrk 1. i)notethtit is tree, then In u is sinleton t every non-root node u on t, sorun(r, M )onytdaont n e identiied with its irst omponent r; we et then the usul notion o runs o tree utomt on trees. Exmple 1. Over the unrnked sinture {,, } onsider TDA A, withthe ollowin trnsitions: p, q, p, q, (, p) q, (, q) p, (, q ) q, (, q Q ) q, (, p q) p, (, q pq) q in, (, pq ) q in, with Q = {p, q, q,q in },ndq in s the unique eptin stte. An eptin ottom-up run o A on td is depited on the let o Fiure 2, nd on its riht, the sme run s seen on the tree equivlent o the td. q in q in p q p p p q q p q p q q p q q q q p Fi. 2. A ottom-up eptin run o the TDA o Exmple 1 on trd, nd the sme seen on its tree equivlent. A ew omments on the ove run my e o help: we strt with ssinin stte q to the le node, under r; the ssinments o stte q under M to ll the inomin edes t this node poses no prolem; we n then ssin stte p to node, nd susequently lso p to the node, under r, vi the trnsition rule (, pq) p; we then ssin p under M to the irst inomin ede t ; tossin stte q under M to the seond inomin ede t, we just need to hek tht: - or mp : Q Q suh tht (p) =q, (q) =p,therule(, (p)(q)) q is n instne o trnsition rule o the TDA; 6

8 - or the outoin ede, leled with p y M, wehvep q = (p); - or the outoin ede, leled with q y M, wedohveq p = (q); rehin q in t the root-node is trivil vi the lst trnsition rule. (Note tht we ould hve s well ssined p under M to the seond inomin ede t, with no onditions to hek, then reh q in.) Remrk 1 (ontd.). ii) Unlike the DAs o [5] or [1], the ollowin ottom-up non-deterministi TDA: q 1, q 2,(q 1,q 2 ) q,withq 0,q 1,q s sttes where q is eptin, hs non-empty lnue: s TDA it epts (, ). For deterministi TDA, we hve the ollowin result (s expeted): Proposition 1 Let A e ottom-up deterministi TDA, nd t ny iven trd; then there is t most one run o A on t. Proo. Let Q e the set o sttes o A, ndm : Edes(t) Q ny iven mp ssinin sttes to the edes on t. We shll show y indution tht the hypothesis o determinism on A implies tht, t ny node u on t, the inry reltion M u = u deined ove (Deinition 3), w.r.t. the mp M, istheidentity reltion on the set Q. The proposition will then ollow rom onditions (1) nd (2) on runs,. Deinition 4; we will et, in prtiulr, tht or every inomin ede e t u, M(e) must e the sme s r(u); so the run n e identiied with its irst omponent r (s on tree). The indution will e on non-netive inteer d u, tht we deine t ny node u o t ndreertositsheiht on t s the mximl numer o rs on t rom u to the le nodes. I d u =0,thenu is le node; tht u is the identity reltion on Q in this se is immedite, rom the determinism o A, nd the deinition o u. So, ssume tht d u > 0, nd let v 1...v n e the strin o ll the hildren nodes o u on t. By the indutive hypothesis, or every i, 1 i n, the reltion vi is the identity reltion on Q; it ollows then, rom the onditions (i) nd(ii) onthereltion u (Deinition 3), tht this ltter must lso e the identity reltion on Q. We my now ormulte the prinipl result o the irst prt o this pper: Proposition 2 A TDA epts trd t i nd only i it epts the tree equivlent o t. Proo. Let ˆt e the tree equivlent o the trd t, nd the nturl surjetive ompression mp rom Nodes(ˆt) onto Nodes(t). For provin the only i prt o the ssertion, one uses the ollowin resonin, oupled with indution on the heiht untion t the nodes o t (deined in the proo o the previous proposition): Let (r, M ) e n eptin run o the iven TDA on the trd t; onsider node s on the tree equivlent ˆt, owhih the node u on t is the ime under the ompression mp ; letr(u) =q under the iven run o the TDA on t; then, or every stte q o the TDA suh tht q M u q, one n onstrut prtil run o the TDA seen s usul tree utomton on the tree ˆt, limin up rom le elow s on ˆt to the node s, nd ssinin the stte q to this node (or n illustrtive exmple, see the tree to the riht o Fiure 2). Provin the i prt o the ssertion is little more omplex. We strt with iven eptin run ˆρ o the iven TDA, s ottom-up tree utomton runnin in the usul sense on the tree ˆt; rom this run ˆρ, we shll onstrut run (r, M ) o the TDA on the trd t, y n indutive, top-down trversl o the td t; or this top-down trversl, we will e usin n inteer vlued untion deined t ny node u o t nd reerred to s its depth on t s the mximl numer o rs on t rom the root node on t to the node u. We shll lso use the t tht the 7

9 nodes o ˆt re in nturl ijetion with the set Pos(t) o positions on t. The topdown onstrution o the run (r, M ) is done y the ollowin pseudo-lorithm, where d stnds or the mximl depth on t t its le nodes. BEGIN /* deine irst r t the root node on t, nd M on its outoin edes */ r(ɛ t )=ˆρ(ɛˆt ); For every outoin ede e j, 1 j k, t ɛ t, set M(e j )=ˆρ(ɛ.j); i =1; /* Now o down */ while (i <d) do { For every node u t depth i do { hoose e In u (t), nd α pos t (e) suh tht M(e) =ˆρ(α); set r(u) =M(e); For every e j Out u (t), 1 j m, outoin rom u, set M(e j )=ˆρ(α.j); } i = i +1; } END. It is not diiult to hek then, tht y onstrution, the pir o mps (r, M ) ives n eptin run o the TDA on the trd t. We illustrte here the resonin employed in the proo o the i prt o the ove proposition, with the td t o Exmple 1. We strt with the run ˆρ on its tree-equivlent ˆt, s depited to the riht o Fiure 2. At strt, to the root node on t (t depth 0) is ssined the stte q in, nd to its three outoin edes, re ssined the three sttes p, q, q respetively; t, whih is the only node on t t depth 1, we hoose the irst inomin ede (o position 1, nd leled with p y M), nd set r(u) =ˆρ(1) = p; the two outoin edes t on t hve s positions the sets {11, 21}, {12, 22} respetively; to these two outoin edes t on t, we ssin the sttes tht ˆρ ssins to the two sons o the node t position 1 on ˆt, nmelyp, q respetively (this mens in essene tht we hve seleted the positions 11 nd 12 on the two outoin edes t on t); next, we o to depth 2 on t, where is the unique node, to whih we then hve to ssin the stte ˆρ(11) tht M hs lredy ssined to its inomin ede; the rest o the resonin is ovious, so let out. Remrk 2. i)lett t e two iven trds suh tht Pos(t )=Pos(t), nd there is nme preservin surjetive mp rom Nodes(t )ontonodes(t). We n then deine t to e ompression, or ompressed orm, o t ;ndreertot s n unompressed equivlent o t, nd to the surjetive mp on Nodes(t ) s ompression mp. It is esily heked tht t nd t hve then the sme tree-equivlent; nd it ollows rom Proposition 2 ove tht ny iven TDA A epts t i nd only i it epts t. It is leitimte then, to deine the lnue o TDA s the set o ll tds tht it epts (or trees tht it epts), or s the set o ll trds epted, up to tree-equivlene. ii) Unrnked trees re oten studied in the literture y trnsormin them into rnked inry trees, usin the well-known irst-hild, next-silin enodin or the trnsormtion (done in liner time wrt the numer o nodes o the iven tree). However, suh n enodin is meninless on trds, sine node n stnd or severl distint nodes o its tree-equivlent, nd the notions o irst-hild nd next-silin n e meninul on trds only when reerrin to the position sets o the nodes. Wht the ove proposition sys is tht tree utomt n run on trds without ny need or trnsormin the trd into (rnked) tree, 8

10 or trnsormin the utomton itsel in some wy. In prtiulr, the unrnked query utomt, e.., s deined in [8], n e used or queryin semi-strutured douments tht re iven in the orm o trds. However, we shll propose, in the setions to ome. n entirely dierent pproh or query evlution on trds. 3 Queryin Compressed Douments: Preliminries Given trd t, one n nturlly onstrut reulr tree rmmr ssoited with t, whihisstrihtline (. [4]), in the sense tht there re no yles on the dependeny reltions etween its non-terminls, nd eh non-terminl produes extly one su-trd o t. Suh rmmr will e denoted s L t,iitis normlized in the ollowin sense: (i) or every non-terminl A i o L t, there is extly one prodution o the orm A i (A j1,...,a jk ), where i<j r or every 1 r k; weshllthenset Sons(A i )={A j1,...,a jk },ndsym Lt (A i )=; (ii) the numer o non-terminls is the numer o nodes on t. Suh normlized rmmr L t is uniquely deined up to renmin o the nonterminls. For instne, or the trd t to the let o Fiure 3 we et the ollowin normlized rmmr: A 1 (A 2,A 3,A 4,A 5,A 2 ), A 2, A 3 (A 5 ), A 4, A 5. Suh rmmr is esily onstruted rom t, or instne y usin stndrd lorithm whih omputes the depth o ny node (s the mximl distne rom the root), to numer the non-terminls so s to stisy ondition (i) ove. t: D t : A _ 1 (, ) A 2 (, _ ) A (, _ 3 ) A, _ 4 ( ) A 5 (, F 1 : A 2 (, _ ) A (, _ 3 ) A, _ 4 ( ) A 5 (, A 2 (, _ ) F 0 : A 1 (, F3: A 5 (, Fi. 3. trd t, ssoited rl D t, nd hilins o L t The dependeny rph o the normlized rmmr L t ssoited with t, nd denoted s D t, onsists o nodes nmed with the non-terminls A i, 1 i n, nd one sinle direted r rom ny node A i tonodea j whenever A j is son o A i. The root o D t is y deinition the node nmed A 1. The notion o Sons o the nodes on D t is derived in the ovious wy rom tht deined ove on L t. Furthermore, to ny prodution A i (A j1,...,a jk )ol t, we ssoite rooted liner rph omposed o k nodes respetively nmed A j1,...,a jk,with root t A j1 nd suh tht or ll l {2,...,k} the node nmed A jl is the son o the node nmed A jl 1. This rph will e lled the hilin o L t ssoited with the (unique) A i -prodution; it is denoted s F i. We lso deine urther hilin denoted F 0, s the liner rph with sinle node nmed A 1,whereA 1 is the xiom o L t. In the sequel, we desinte y G either D t or ny o the hilins F o L t. We omplete ny o these yli rphs G into rooted leled yli rph (rl, or short), y tthin to eh node u on G, withnme(u) =A i,lel denoted lel(u),nddeinedslel(u)=(sym Lt (A i ), );. Fiure 3. 9

11 3.1 Positive Core XPth Queries on trds In this pper we restrit our study to positive Core XPth queries on trds. Rell tht Core XPth is the nvitionl sement o XPth, nd is sed on the ollowin xes o XPth (. [10, 19]): sel, hild, prent, nestor, desendnt, ollowin-silin, preedin-silin. A lotion expression is deined s predite o the orm [xis::], wherexis is one o the ove xes, nd is symol o Σ. Given ny trd t over Σ, ontext node u on t nd Σ, thesemntisorxis is deined y evlutin this predite t u. Thesemntisorthexessel, hild, desendnt re esily deined, extly s on trees (. [19]). For deinin the semntis o the reminin xes, we irst rell tht Prents(u) ={v Nodes(t) u is hild o v}. Deinition 5 Given ontext node u on trd t, nd Σ: i) [prent::] evlutes to true t u, i nd only i there exists -nmed node in Prents(u); ii) [nestor::] evlutes to true t u, ieither[prent::] evlutes to true t u, orthereexistsnodev Prents(u) suh tht [nestor::] evlutes to true t v; iii) [ollowin-silin::] evlutes to true t u, i there exists -nmed node u,ndnodev on t suh tht γ(v) is o the orm...u...u...; iv) [preedin-silin::] evlutes to true t u, i there exists -nmed node u,ndnodev on t suh tht γ(v) is o the orm...u...u... For the omposite xes desendnt-or-sel nd nestor-or-sel, the semntis re then dedued in n ovious mnner. We shll lso need position predites o the orm [position()= i]; their semntis is tht the expression [hild:: [position()= i]] evlutes to true t ontext node u, i: [hild::] evlutes to true t u, ndu is n i-th hild o some prent. Positive Core XPth query expressions re usully deined in the literture (. e.., [7]), s those enerted y the ollowin rmmr: A ::= sel hild desendnt prent nestor preedin-silin ollowin-silin S n ::= A:: position()= i S n nd S n S n or S n E n ::= A:: [S n ] E n [E n ] Q n ::= /S n /E n Q n /Q n We shll reer to the query expressions enerted y this rmmr s nonil; they n e shown to e o the type /C 1 /C 2 /.../C n,whereehc i is o the orm A::[X n ],orotheorma::[x n ] onn A :: [X n ],with onn {nd, or}, ndx n,x n {S n,e n,true}; we ree here to identiy A::[true] with A::. Any suh positive Core XPth query expression n e trnslted into one tht is in stndrd orm, i.e., where the ormt o the su-queries is o the type xis:: ; we ormlize this ide now. We shll reer to the xes sel, hild, desendnt, prent, nestor, preedin-silin, ollowin-silin s si. A si Core XPth query is query o the orm //*[xis::], where xis is si xis. More enerlly, the queries we propose to evlute on trds re deined ormlly s the expressions Q std enerted y the ollowin rmmr, where stnds or ny node nme on the douments, or or (menin ny ): A ::= sel hild desendnt prent nestor preedin-silin ollowin-silin S ::= A:: position()= i S nd S S or S Root E ::= A:: [S] E[E] Q std ::= //* //*[S] //*[E] Core XPth queries Q std o the ormt enerted y this rmmr re sid to e in stndrd orm; to e le to hndle ny positive Core XPth query with 10

12 suh rmmr, we hve introdued speil predite lled Root, deemed true only t the root node o the trd onsidered. By the evlution o iven query expression Q on ny trd t, wemen the ssinment: t the set o ll ontext nodes on t where the expression Q evlutes to true (ollowin the onventions o Deinition 2); this ltter set is lso lled the nswer or Q on t. TwoivenqueriesQ 1,Q 2 re sid to e equivlent i, on ny trd t, the nswer sets or Q 1 nd Q 2 re the sme. Any positive Core XPth query Q n n e trnslted into n equivlent one in stndrd orm; e.., /[ollowin-silin::]/d is equivlent to //*[sel::d nd prent::*[root nd sel:: [ollowin-silin::]]] in stndrd orm. An indutive proedure perormin suh trnsltion in the enerl se (o liner omplexity w.r.t. the numer o lotion steps in Q n ) is iven in Appendix I. The ollowin proposition results rom Deinition 5. Proposition 3 (1) For ny set o nodes X on trd t, ndnyxisa, we hve: A(X) = {/hild:: [position()= i 1 ]/.../hild:: [position()= i k ]/A:: } x X, α pos t (x) α = i 1...i k (2) For ny trd t, nd ny node with nme on t, we hve: (i) //*[preedin::] = {desendnt-or-sel(ollowin-silin( u //*[sel::u nd(desendnt:: or sel::)] ))} (ii) //*[ollowin::] = {desendnt-or-sel(preedin-silin( u //*[sel::u nd(desendnt:: or sel::)] ))} Finlly, ollowin [2], or ny set S o nodes on t, the sets o nodes ollowin(s) nd preedin(s) n now e deined ormlly, s ollows: ollowin(s) = desendnt-or-sel(ollowin-silin(nestor-or-sel(s))), preedin(s) = desendnt-or-sel(preedin-silin(nestor-or-sel(s))). Note: Unlike on tree, the nestor, desendnt, ollowin, sel nd preedin xes do not prtition the set o nodes on trd t, in enerl. 4 Automt or the Bsi Core XPth Queries 4.1 The Semntis o the Approh We irst onsider si Core XPth queries. Composite or imrited queries will susequently e evluted in stepwise shion; see Setion 6. To ny si query Q = //*[xis::], we shll ssoite word utomton (tully trnsduer), reerred to s A Q. It will run top-down, on the rl D t i xis is non-silin, nd on eh o the hilins F o L t otherwise. In either se, run will tth, to ny node trversed, pir o the orm ( l,x), where the omponent l o the pir hs the intended semntis o seletion or not, y Q, o the orrespondin node on t, nd the omponent x will e 1 or 0, with the intended semntis tht x = 1 i the orrespondin node on t hs desendnt nswerin Q. At the end o the run, lel(u), t ny node u o D t, will e repled y new lel derived rom the ll-pirs tthed to u y the run. To ormlize these ides, we introdue set o new symols L = {s, η,, } reerred to s llels (the term llel is used so s to void onusion with the term lel). We deine ll-pirs s elements o the set L {0, 1}, nd the sttes o 11

13 A Q s elements o the set {init} (L {0, 1}). For ny Q, the utomton A Q is over the lphet Σ {s, η}, hsinit s its initil stte, nd hs no inl stte. The set Δ Q o trnsitions o A Q will onsist o rules o the orm (q, τ) q where q {init} (L {0, 1}), q (L {0, 1}), nd τ Σ {s, η}. For ny rl G, we deine untion ll: Nodes(G) Σ {s, η}, y settin ll(u) =π 1 (lel(u)), the irst omponent o lel(u). The utomton A Q ssoited to si query Q =//*[xis::] will run top-down on the rl G, where G is D t i xis is si non-silin xis, nd G is ny hilin F o L t i xis is si silin xis. A run o A Q on G is mp r : Nodes(G) L {0, 1}, suh tht, or every u Nodes(G), the ollowin holds: -iu is root G, then the rule (init, ll(u)) r(u) isinδ Q ; - otherwise, or every v γ(u) therules(r(u), ll(v)) r(v) rellinδ Q. (Note: when xis is non-silin, this mounts to requirin tht, or ny node v, the stte r(v) must e in onormity with the sttes r(u) orevery prent node u o v, with respet to the rules in Δ Q.) From the run o the utomton A Q nd rom the sttes it tthes to the nodes o D t, we will dedue, t every node u o t, well-determined ll-pir s ( new) lel t u, vi the nturl ijetion etween Nodes(t) ndnodes(d t ). The ll-pirs thus tthed to the nodes o t will hve the ollowin semntis (where x stnds or the nme o the node u on t, orrespondin to the urrent node on D t ): -(, 1) : x =, urrent node on t is seleted y (i.e., is n nswer or) Q; -(, 1) : x =, urrent node is not seleted, ut hs seleted desendnt; -(, 0) : x =, urrent node is not seleted, nd hs no seleted desendnt; -(s, 1) : x, urrent node is seleted; -(η, 1) : x, urrent node is not seleted, ut hs seleted desendnt; -(η, 0) : x, urrent node is not seleted, nd hs no seleted desendnt. Only the nodes on D t, to whih the run o A Q ssoites the lels (s, 1) or (, 1), orrespond to the nodes o t tht will et seleted y the query Q. The ll-pirs with oolen omponent 1 will lel the nodes o D t orrespondin to the nodes o t whihreonpthtonnswerorthequeryq; thusthe utomt A Q will hve no trnsitions rom ny stte with oolen omponent 0 to stte with oolen omponent 1. Moreover, with view to deine runs o suh utomt whih re unique (or unmiuous in sense tht will e presently mde ler), we deine the ollowin priority reltions etween the llpirs: (η, 0) > (η, 1) > (s, 1), nd (, 0) > (, 1) > (, 1). A run o the utomton A Q will lel ny node u on G with n ll-pir either rom the roup {(, 0), (, 1), (, 1)} or rom the roup {(η, 0), (η, 1), (s, 1)}; nd this roup is determined y ll(u). For ese o presenttion, we ree to set η := s, nd oten denote either o the ove two roups o ll-pirs under the uniorm nottion {(l, 0), (l, 1), (l, 1)}, where l {η, }, with the orderin (l, 0) > (l, 1) > (l, 1). We shll onstrut run r o A Q on G tht will e uniquely determined y the ollowin mximl priority ondition: (MP): t ny node v on G, r(v) is the mximl ll-pir ( l,x)ortheorderin> in the roup {(l, 0), (l, 1), (l, 1)} determined y ll(v), suh tht A Q ontins trnsition rule o the orm (r(u), ll(v)) ( l,x), or every prent u o v. Suh run will ssin lel with oolen omponent 1 only to the nodes orrespondin to those o the miniml su-trd t ontinin the root o t nd ll the nswers to Q on t. 12

14 4.2 Re-lelin o D t y the Runs o A Q We irst onsider non-silin si query Q on iven doument t, ndiven run r o the utomton A Q on the D t ; t the end o the run, the nodes on D t will et re-leled with new ll-pirs, omputed s elow or every u Nodes(D t ): l r (u) =(s, 1) i r(u) {(s, 1), (, 1)}, l r (u) =(η, 1) i r(u) {(η, 1), (, 1)}, l r (u) =(η, 0) i r(u) {(η, 0), (, 0)}. The rl otined in this mnner rom D t, ollowin the run r nd the ssoited re-lelin untion l r, will e denoted s r(d t ). For si query Q over silin xis, the sitution is little more omplex, euse severl dierent nodes on one hilin o L t n hve the sme nme (non-terminl), or severl dierent hilins n hve nodes nmed y the sme non-terminl, or oth. Thus, to ny node o D t, nmed with non-terminl A, will orrespond in enerl set o ll-pirs, ssined y the vrious runs o A Q to the A-nmed nodes on the vrious hilins o L t. We thereore proeed s ollows: or every omplete set r o runs o A Q,ormedoonerunr F on eh hilin F, we will deine r(d t ) s the re-leled rl derived rom D t, under r. With tht purpose we ssoite to r nd ny u Nodes(D t ), set o ll-pirs: ll r (u) = {r F (v) v Nodes(F), nd nme(v) =nme(u)}. r F r We then derive, t eh node o D t unique ll-pir in onormity with the semntis o our pproh, y usin the ollowin untion: λ r (u) =s ll r (u) {(s, 1), (, 1)}, λ r (u) =η ll r (u) {(s, 1), (, 1)} =. From D t nd this untion λ r, we next derive n rl λ r (D t ) y re-lelin eh node u on D t with the pir (λ r (u), ). And inlly we deine r(d t )sthe rl otined rom λ r (D t ), y runnin on it the utomton or the si nonsilin query //*[sel::s], s indited t the einnin o this susetion. In prtil terms, suh run mounts in essene to settin, s the seond omponent o lel(u) t ny node u,theoolen1iuison pth to some node with ll s, nd 0 otherwise. All these detils re illustrted with n exmple in the ollowin susetion. 4.3 The Automt We irst present the utomt or the si queries //*[sel::] nd or //*[ollowin-silin::], nd ive n illustrtive exmple usin the ormer or = s, nd the ltter or =. The utomt or the other si queries re iven ter the exmple. Automt: or //*[sel::] nd or //*[ollowin-silin::] γ= init γ= η, 1 γ= γ= γ= γ= η, 0 T, 1 γ= init T, 1 η, 0 T, 0 s, 1 Fiure 4 elow illustrtes the evlution o Q = //*[ollowin-silin::], on the trd t o Fiure 3. We irst use the utomton or the si query 13

15 //*[ollowin-silin::] with =, nd then the utomton or //*[sel::] with = s. The su-trd o t, ormed o nodes orrespondintothoseo r(d t ) with lels hvin oolen omponent 1, ontins ll the nswers to Q on t. r 1 on F 1 : A 2 (, _ ) A (, _ 3 ) A, _ 4 ( ) A 5 (, A 2 (, _ ) ( s, 1 ) ( s, 1) (T,1) ( T, 0) ( η, 0) r0 on F0 : A _ 1 (, ) (η, 0) r 3 on F 3 : A 5 (, ( T, 0) r 0 r 1, r 3, on D t : (η, 0) A 1 (, A 2 (, _ ) A (, _ 3 ) A, _ 4 ( ) ( s, 1 ) ( s, 1) (T,1) ( η, 0) ( T, 0) A 5 (, D t ) λ r ( : A 1 ( η, _ ) run o the utomton or //*[sel : : s] on ( η, 1) A 1 ( η, _ ) inl re leled λ r ( D ): t rl:r(d t ) A 1 ( η, 1) A 2 (, _ ) s A 3 (, s _ ) A 4 (s, (, _ ) A 2 s (T,1) A 3 s (T,1) (, _ ) A 4 ( s, A2 (,) s 1 A 3 (,) s 1 A 4 ( s, 1) (T,1) A 5 (η _, ) (η, 0) A 5 (η _, ) A 5 (η, 0) Fi. 4. Automton or the query //*[prent::] init η, 1 η, 0 T, 0 T, 1 s, 1 T, 1 14

16 Automton or the query //*[nestor::] T, 1 γ= γ= γ= T, 1 η, 1 init T, 0 γ= s, 1 γ= γ= γ= η, 0 γ= γ= Automton or the query //*[hild::] T, 1 init T, 0 η, 0 η, 1 T, 1 s, 1 Automton or the query //*[preedin-silin::] s, 1 η, 1 init T, 1 T, 0 η, 0 T, 1 15

17 Automton or the query //*[desendnt::] init γ= γ= T, 1 T, 0 γ= η, 0 γ= γ= γ= s, 1 γ= γ= A ew words on some o the utomt y wy o explntion. First, the reson why the utomton or sel does not hve the sttes (, 0), (, 1), (s, 1): or (, 0), (, 1), y the semntis o susetion 4.1 wemusthvex =, where x is the nme o the urrent node on t, ut then the query //*[sel::] should selet the urrent node, so one nnot e t suh stte; s or (s, 1), the resonin is just the opposite. Next, the reson why the utomton or desendnt does not hve the sttes (η, 1), (, 1): i the semntis ttriute one o these pirs to ny node u, tht would men the node u hs seleted desendnt u ; whih mens tht u hs some -desendnt node, whih would then e -desendnt or u too, so Q should selet u. 5 Mximl Priority Runs o Bsi Query Automt Note tht the ollowin properties, required y our semntis o susetion 4.1, hold on the utomt A Q onstruted ove, or ny si Core XPth query Q = //*[xis::]: i) There re no trnsitions rom ny stte with oolen omponent 0 to stte with oolen omponent 1; ii) The -trnsitions hve ll their tret sttes in {(, 0), (, 1), (, 1)}; nd or ny γ, the tret sttes o γ-trnsitions re ll in {(η, 0), (η, 1), (s, 1)}. Theorem 1 Let Q e ny si Core XPth query, t ny iven trd, nd let G denote either the rl D t, or ny iven hilin F o L t. Assume iven lelin untion L romnodes(g) into the set o ll-pirs, whih is orret with respet to Q, i.e., in onormity with the semntis o susetion 4.1. Then there is run r o the utomton A Q on G, suh tht : i) r is omptile with L; i.e., r(u) = L(u) or every node u on G; ii) r stisies the mximl priority ondition (MP) osusetion4.1. Proo. We irst onstrut, y indution, omplete run (i.e., deined t ll the nodes o G) stisyin property i). For tht, we shll employ resonins tht will e speii to the xis o the si query Q. We ive here the detils only or the xis prent; they re similr or the other xes. Q = //*[prent::]: (The xis onsidered is non-silin so G = D t here.) At the root u node o D t,wesetr(u) = L(u); we hve to show tht there is trnsition rule in A Q o the orm (init, ll(u)) L(u). Oviously, or the xis prent, the root node u nnot orrespond to node on t seleted y Q, sothe only ll-pirs possile or L(u) re(l, 0), (l, 1), with l {η, }; or eh o these hoies, we do hve trnsition rule o the needed orm, on A Q. Consider then node v on D t suh tht, t eh o its nestor nodes u on D t, the prt o the run r o A Q hs een onstruted suh tht r(u) = L(u); 16

18 ssume tht the run nnot e extended t the node y settin r(v) = L(v). This mens tht there exists prent node w o v, suh tht ( L(w), ll(v)) L(v) is not trnsition rule o A Q ; we shll then derive ontrdition. We only hve to onsider the ses where the oolen omponent o L(w) is reter thn or equl to tht o L(v). The possile ouples L(w), L(v) re then respetively: L(w) :(, 0) (, 1) (, 1) (, 1) (, 1) L(v) : (η, 0) (, 1) (η, 1) (, 1) (η, 1) In ll ses, we hve ll(w) = euse o the semntis, so the node (on t orrespondin to the node) v hs -prent, so must e seleted; thus the ove hoies or L(v) re not in onormity with the semntis; ontrdition. We now prove tht the omplete run r thus onstruted, stisies property ii). For this prt o the proo, the resonin does not need to e speii or eh Q; so, write Q more enerlly, s //*[xis::] or some iven. Suppose the run r does not stisy the mximl priority ondition t some node v on G; ssume, or instne, tht the run r mde the hoie, sy o the ll-pir (l, 1), lthouh the mximl lelin o the node v, in mnner omptile with the ll-pirs o ll its prents, ws the ll-pir (l, 0). Sine L is ssumed orret, nd r is omptile with L, the mximl possile lelin (l, 0) would men tht the node (on t orrespondin to the node) v hs no desendnt seleted y Q; wheres, the hoie tht r is ssumed to hve mde t v, nmely the ll-pir (l, 1), hs the opposite semntis whether or not ll(v) =; in other words, the lelin L would not e orret with respet to Q; ontrdition. The other possiilities or the d lelins under r lso et eliminted in similr mnner. Theorem 2 Let Q, t, D t, F, G e s ove. Let r e (omplete) run o the utomton A Q on G, whih stisies the mximl priority ondition (MP) o susetion 4.1. Then the lelin untion L onnodes(g), deined s L(u) =r(u) or ny node u, is orret with respet to the semntis o susetion 4.1. Proo. Let us suppose tht the lelin L dedued rom r is not orret with respet to Q; we shll then derive ontrdition. The resonin will e y se nlysis, whih will e speii to the xis o the si query Q onsidered. We ive the detils here or Q = //*[desendnt::]. The xis is non-silin, so we hve G = D t here. The sets Nodes(t),Nodes(D t ) re in nturl ijetion, so or ny node u on D t we shll lso denote y u the orrespondin node on t, in our resonins elow. We sw tht the utomton A Q or the desendnt xis does not hve the sttes (η, 1), (, 1). Consider then node u on D t suh tht: or ll nestor nodes w o u, the llel r(w) is in onormity with the semntis, ut the ll-pir r(u) is not in onormity. Now, A Q hs only 5 sttes: (init), (, 1), (s, 1), (, 0), (η, 0), o whih only the lst our n llel the nodes. So the possile d hoies tht r is ssumed to hve mde t our node u, re s ollows: () r(u) =(, 1), ut the node u is not n nswer to the query Q. Here nme(u) muste, so the hoie o r ouht to hve een (, 0); () r(u) =(s, 1), ut the node u is not n nswer to the query Q. Here nme(u), so the hoie o r ouht to hve een (η, 0); () r(u) = (η, 0), ut the node u is n nswer to the query Q. Here nme(u), so the hoie o r ouht to hve een (s, 1); (d) r(u) = (, 0), ut the node u is n nswer to the query Q. Here nme(u) muste, so the hoie o r ouht to hve een (, 1). In ll the our ses, we hve to show: i) tht the ouht-to-hve-een hoie ll-pir is rehle rom ll the prent nodes o u; ii) nd tht, with suh new nd orret hoie mde t u, r n e ompleted rom u, intorunontheentiredd t. 17

19 The resonin will e similr or ses (), (), nd or the ses (), (d). Here re the detils or se (): Tht u is not n nswer to Q mens tht u hs no -desendnt node, so or ll nodes v elow u on D t,wehvell(v). Thereore, ssertions i) nd ii) ove ollow rom the ollowin oservtions on the utomton or Q= //*[desendnt::]: i) i r ould reh the stte (, 1) t node u (vi -trnsition) rom ny prent node o u, then(, 0) is lso rehle thus t u, romnyothem; ii) i, romthestte(, 1), r ould reh ll the nodes on D t elow u (with stte (η, 0)), vi trnsitions over γ, then it n do extly the sme now, with the orret hoie ll-pir (, 0) t u. As or se (): Node u is n nswer to Q here, so u hs -desendnt; let v e -node elow u on D t ; the ll-pir r(v) thtr ssins to v must then e either (, 1) or (, 0); this implies tht r pssed rom the stte (η, 0) supposedly ssined y r to u to(, 1) or (, 0) somewhere etween u nd v; whih is impossile, s is esily seen on the utomton A Q or the xis desendnt onsidered. The resonin or se (d) is even esier: rom stte (, 0), no stte with n outoin -trnsition is rehle. 6 Evlutin Composite Queries A omposite query is query in stndrd orm, ut is not si. We propose to evlute suh query inrementlly. For this, it suies to onsider queries tht re o the orm //*[A::x onn A ::x ],whereonn {nd, or}, oro the orm //*[A 1 ::*[A 2 ::]]. For those o the ormer type, we oserve irst tht the omponents in disjuntion (resp. onjuntion) under * ne evluted seprtely. Indeed, the nswer or Q = //*[A::x onn A ::x ] n e otined s union (resp. intersetion) o the nswers or the two omponent queries //*[A::x], nd//*[a ::x ],whenonn is n or (resp. n nd). We pply the method desried erlier, seprtely or Q 1 = //*[A::x] nd or Q 2 = //*[A ::x ], thus ettin two respetive evlutin runs r 1,r 2. Any node u o the d D t will then e re-leled, y the omposite query Q, with ll-pirs omputed y untion AND when onn = nd (resp. OR when onn = or), in onormity with the semntis presented in the Setion 4.1: AND(u) =(s, 1) i r 1 (u) =(l, 1) = r 2 (u); AND(u) =(η, 0) i r 1 (u) =(l, 0) or r 2 (u) =(l, 0); AND(u) =(η, 1) otherwise. OR(u) =(s, 1) i r 1 (u) =(l, 1) or r 2 (u) =(l, 1); OR(u) =(η, 0) i r 1 (u) =(l, 0) = r 2 (u); OR(u) =(η, 1) otherwise. Fiure 5 elow illustrtes the ove resonin, or the evlution o the omposite query Q = //*[sel:: ndprent::], on the trd t o Fiure 3: We next onsider the queries o the orm Q = //*[A 1 ::*[A 2 ::]], with imrited predites. For their evlution, we irst onsider mximl priority run evlutin r 2 (resp. set o runs r 2 ) o the utomton ssoited to the inner query //*[A 2 ::], ond t (resp. the set o ll hilins o L t ). This run (resp. set o runs) will output the rl r 2 (D t )(resp. r 2 (D t )), s desried in Setion 4.2. Evlutin the imrited query Q on the d t is then done y runnin the utomton or the si outer query //*[A 1 ::s] on r 2 (D t )(resp. r 2 (D t )). Finlly, the nswer or query o the type Q = //*[hild::x[position()= k]], is the suset o the nodes nswerin //*[hild::x], whih orrespond to k-thnodeonsomehilin. 18

20 ( η,1) //*[sel : : ] //*[prent : : ] nd(d t ) A 1 (, ( η,1) A 1 (, A 1 (η,1 ) A2(, _ ) A (, _ 3 ) A, _ 4 ( ) ( η,0) ( η,1) ( T, 1) A 2 (, _ ) A (, _ 3 ) A, _ 4 ( ) η,0 ( η,0) ( ) ( T, 1) A 2 (,) η 0 A 3 (,1 η ) A 4( η, 0) ( T, 1) A 5 (, ( s,1) A 5 (, A 5 ( s, 1 ) Fi Derivin the Answer on the Tree-Equivlent GivenCoreXPthqueryQ nd its nswer set on trd t, weshowherehow to derive the nswer or the sme query Q on the tree-equivlent ˆt o t; thisis o importne, sine the stndrd model or n XML doument (even when iven in ompressed orm) is enerlly onsidered s the tree representtion o the doument. We oserve, to strt with, tht the nswer set or Q on t is in enerl superset o the nswer set or Q on the tree-equivlent ˆt. This n e so or the ollowin two resons: (i) I ertin node u on t is seleted y Q, not ll o the nodes u on ˆt, tht re lits o u under the ompression mp on Nodes(ˆt), my nswer the query Q on the tree ˆt, evenwhenq is si query. For instne, onsider the si query //*[prent::]; on the ully ompressed td ((),()), the (unique) node nmed is n nswer; it hs two -nmed nodes s lits on the tree-equivlent ˆt, o whih only one is n nswer or the query. (ii) A node u on trd t my nswer omposite query Q, ut none mon the lits o u on ˆt my nswer the sme query Q on the tree ˆt. For instne, the unique -nmed node on the ompressed td ((),()) nswers the query //*[prent:: nd prent::], ut there is no node on the tree-equivlent nswerin this query. Atully, suh situtions rise only or queries involvin the upwrd xes prent, nestor, whih deine reltions tht re less trivil on trds thn on trees. We n ormulte this oservtion more preisely, s ollows: Lemm 1. Let A e one o the xes sel, hild, desendent, Q the si query //*[A::x], t ny iven trd, ˆt its tree-equivlent, u ny iven node on t, nd u 1 (u) ny node lit o u on ˆt. Then: wrt the mximl priority runs o the utomton or the xis A, respetively on D t nd Dˆt, the nodes u on t, ndu on ˆt, et leled y the sme ll-pir; in prtiulr, the node u nswers Q on t i nd only i the node u nswers thesmequeryq on the tree ˆt. Proo. Follows y oservin tht the semntis o Setion 4.1 hve een deined in mnner whih is top-down, nd tht the ompression mp : Nodes(ˆt) Nodes(t) mpsthesetnodes(ˆt u ), o nodes elow u on ˆt, ontothesetonodes o the su-trd t u. The ove lemm is irst step towrds the ojetive o this setion. As seond step, we propose to distinuish, on the utomt or the two si queries 19

21 //*[prent::], //*[nestor::], two speil types o trnsitions, esides the usul ones. (i) Type oret: The trnsitions tht will not e irle on tree; suh trnsitions will e represented y dotted rs on the utomton. The trnsitions onerned re the ones rom (η, 1) to (, 1) nd to (s, 1) on the utomt or prent nd desendnt, swellstheonerom(s, 1) to (, 1) on the prent utomton. A word o explntion miht help here: or instne, in order tht the trnsition rom stte (η, 1) to stte (, 1) e irle on the prent utomton, we hve to reh node orrespondin to -nmed node on the trd, whih must then lso hve s (unique) prent on the tree i.e., the node rom whih the trnsition is to e ired -nmed node; this prent node nnot orrespond then to node leled with (η, 1). The resonin is similr, lthouh not identil, or the nestor utomton. In reoverin the nswer or iven (si) query Q on the tree equivlent ˆt, the role plyed y the dotted r trnsitions is s ollows: i the urrent node on ˆt orresponds to position α on D t tht is rehed (y the utomton or Q) vi dotted trnsition, then the position α does not nswer Q on ˆt. Theide then is tht suh position α n then e orotten lon iven run, while lookin or nodes nswerin Q. Nevertheless, suh position nnot e orotten orever, in enerl: onsider or instne, the si query //*[nestor::] on trdt ontinin (mon others) two nodes u, v suh tht v is hild o u, u with -nestor, nd the nme t u is ; then the run o the utomton or this query must ttriute the lel (, 1) to u, nd(s, 1) to v; ndinsuh se, ll the positions o v tht extend the positions o u on t hve to e kept s nswers, even i some o the trnsitions o the utomton hve een dotted on the wy rom the root o D t to the prent node u. This remrk leds us to distinuish seond type o trnsitions: (ii) Type restore: Trnsitions tht restore ll the positions urrently visited t node, inludin the ones ontinin preixes orotten erlier under the iven run; suh trnsitions will e represented y thik rs on the utomton. The utomt thus revised or the two upwrd xes, re s ollows: Automton or the query //*[prent::] -revised init η, 1 η, 0 T, 0 T, 1 s, 1 T, 1 20

22 Automton or the query //*[nestor::] -revised T, 1 γ= γ= γ= η, 1 T, 1 init T, 0 γ= γ= s, 1 γ= γ= η, 0 γ= γ= Our next step towrds the ojetive o this setion is to omplete mximl priority run r o the utomton or Q, y ssoitin to ny node u on D t, prtition o Pos t (u) y two susets, respetively denoted s L r u nd L r u:ir(u) is (s, 1) or (, 1) then the set L r u will ontin ll the positions t the node u tht nswer Q on the tree equivlent ˆt o t. The rules or omputin these sets will e iven presently. Two oservtions re essentil in uildin these rules properly, nd optimlly: (i) The set o positions t ny node u, on the iven trd t, n e omputed symolilly nd dynmilly, under ny top-down trversl o D t ; suies to tth distint position symols X, X, to the vrious rs on D t ;ix is suh symol tthed to n r on D t oin rom non-terminl A o the rmmr L t tonon-terminlb o L t,thenx is ment to stnd or the set o inteers {j 1,,j m }, ivin the positions where B ppers on the rhs o the unique A-prodution o L t. Thus, or ny node u on t, ny o its positions is symolilly represented y word over the position symols. (ii) The si query onsidered Q n e the outer query in non si imrited query Q 0. Here, the prtition o Pos t (u) sy s L u L u tht results rom the inner query Q immeditely elow Q in Q 0, should e tken into ount. For tht purpose, note tht t the end o the run r or the inner query Q, every node u o D t ets re-leled s l r (u), this ltter ein either n (s, ), or n (η, ),. Setion 4.2; the si outer query Q is then o the orm //*[A::s] (. Setion 6). Aountin or the prtition L u L u o Pos t (u) risin rom the inner run r, is then done s ollows: l r (u) is(η, ): ssoite to u one llel position-set pir η, Pos t (u) ; l r (u)is(s, ): ssoite to u two llel position-set pirs: s, L u, η, L u. (Note: suh vision does not mount to unoldin the trd into its treeequivlent; indeed, the numer trnsitions onsider etween ny two nodes o D t, or the utomton o the outer query, is t most 4.) The rules or omputin the position sets L r u nd L r u,tnyivennodeu on t, under top-down trversl o t y run r o the utomton or the urrent query, re then ormulted s ollows, where w is ny prent node o u on t, α is word over the position symols stndin or position o w on t, X is the position symol on the r rom w to u on D t ; ll the position sets re seen here s unry predites: L r usul,oret w (α) L r w (α) restore L r w(α) usul,restore L r w (α) oret L r u (α.x) L r u (α.x) L r u(α.x) L r u (α.x) 21