On Testing Satisfiability of Tree Pattern Queries

Transcription

1 On Tsting Stisfiility of Tr Pttrn Quris Lks V.S. Lkshmnn, Gnsh Rmsh, Hui (wny) Wng, Zhng (Jssic) Zho Dprtmnt of Computr Scinc Univrsity of ritish Columi strct XPth n XQury (which inclus XPth s sulngug) r th mjor qury lngugs for XML. n importnt issu rising in fficint vlution of quris xprss in ths lngugs is stisfiility, i.., whthr thr xists ts, consistnt with th schm if on is vill, on which th qury hs non-mpty nswr. Our xprinc shows stisfiility chck cn ffct sustntil svings in qury vlution. W systmticlly stuy stisfiility of tr pttrn quris (which cptur usful frgmnt of XPth) togthr with itionl constrints, with or without schm. W intify css in which this prolm cn solv in polynomil tim n vlop novl fficint lgorithms for this purpos. W lso show tht in svrl css, th prolm is NP-complt. W rn comprhnsiv st of xprimnts to vrify th utility of stisfiility chck s prprocssing stp in qury procssing. Our rsults show tht this chck tks ngligil frction of th tim n for procssing th qury whil oftn yiling sustntil svings. Introuction With XML coming th stnr for t xchng, sustntil work hs n on on XML storg, n qury procssing n optimiztion [7, 6, 9, 7, 5,, 8, 2, 23, 3]. Howvr, rltivly littl work hs n on on tcting whthr givn qury is stisfil, i.., whthr thr is ny ts stisfying th qury. This is n importnt prolm for th following rsons. () Formulting quris ginst XML tss cn mor chllnging thn for rltionl tss. s prviw, w will show y xmpl, how vry similr quris cn grtly vry in trms Prmission to copy without f ll or prt of this mtril is grnt provi tht th copis r not m or istriut for irct commrcil vntg, th VLD copyright notic n th titl of th puliction n its t ppr, n notic is givn tht copying is y prmission of th Vry Lrg Dt s Enowmnt. To copy othrwis, or to rpulish, rquirs f n/or spcil prmission from th Enowmnt. Procings of th 30th VLD Confrnc, Toronto, Cn, 2004 of stisfiility. In, vn in th contxt of rltionl tss, Lvy t l. [8] stui stisfiility for vrious frgmnts of th tlog qury lngug n stlish complxity n ciility rsults. Hirs [5] is th only work on XPth stisfiility w r wr of. til comprison with our work pprs in Sction 6. (2) XML is intn to ctr for situtions whr no priori schm is vill for t. Qurying n XML ts in th snc of ny schm knowlg cn tricky. Th intrction twn vrious structurl constrints, tht rstrict structurl rltionships mong lmnts, n vlus constrints, tht constrin th contnts of lmnts or thir ttriut vlus, cn intrict. (3) Evn whn schm is known, gtting th qury right cn still non-trivil for th usr. For, th schm imposs structurl constrints on its own which tn to intrct with structurl n vlu-s constrints in th qury in sutl wys n my mk th qury unstisfil. Our xprinc shows tht chcking stisfiility of quris cn py sustntil ivins in sving consirl tim in qury vlution, whil ing ngligil ovrh to th ovrll qury vlution. sis, givn th consirl similrity twn stisfil qury n n unstisfil on, it woul usful to hv th systm ssist th usr in gtting thir quris right. Stisfiility tsting is ncssry first stp in uiling ny such tool. This ws th motivtion hin our work. Nxt, w shll illustrt ths points with xmpls. XQury [3] is th fcto stnr qury lngug for XML n inclus XPth [3] s sulngug. oth ths lngugs r s on sic prigm of fining inings of vrils y mtching tr pttrns ginst ts. nikt t l. [9] stuy th xprssiv powr of tr pttrn quris in rltion to XPth n xistntil first-orr logic. E.g., consir th XPth xprssion //[/// = /c//]. It corrspons to th tr pttrn qury (TPQ) Q 4 in Figur (ignor sh lins for now). Singl (oul) lins rprsnt prnt-chil (ncstor-scnnt) rltionship twn nos. s nothr xmpl, consir th XQury sttmnt: FOR $ IN ocumnt( oc.xml )//, TPQs r formlly fin in Sction 2. 20

2 Q D $6 $5 Q3 C D C C $5 $6 Q4 Q2 E $5 C D D C E F $6 D F $7 $5 $6 $7 Q5 C D D Figur : Exmpls in th snc of Schm $ IN $///, $f IN $///f, $c IN $//c, $ IN $c//, $f IN $c//f WHERE $ = $ ND $f = $f RETURN{$} This corrspons to th TPQ Q 2 in Figur. In, ch TPQ in th figur corrspons to n XPth xprssion, or n XQury qury. Ech qury is unstisfil, ut whn th sh lins r (i.., whn crtin prnt-chil rltionships r rlx to ncstor-scnnt), th quris com stisfil. W xplin this low. First, consir Q 3. It sks for nos tht hv no tht is oth chil n is scnnt vi n intrmit no C. This is clrly unstisfil in ny tr. Howvr, rlxing th chil to scnnt mks it stisfil. Nxt, consir Q 4, which sks for nos which hv scnnt D vi chil s wll s vi C chil. Sinc ch lmnt hs uniqu tg, this is unstisfil. gin, rlxing on of th chil constrints to scnnt rnrs th qury stisfil. Q 5 is unstisfil, ut for sutlr rson. If th two D nos r th sm, th C scnnt of must scnnt of th chil. Th scnnt D of th C no must thus t istnc 3 or mor whrs th D chil of is t istnc 2 from, which is impossil. Onc gin, rlxing ny of th chil constrints to scnnt rnrs th qury stisfil. Nxt, consir Q 2. Th constrint tht th two E lvs must inticl rquirs nos,, C n th two E nos to li on th sm (root-to-lf) pth. Similrly, th intity of th two F nos rquirs nos,, C n th two F nos to li on th sm pth. This is impossil, sinc C, hving iffrnt tg thn th two chilrn of, is forc to scnnt of oth, whrs n D cnnot li on th sm pth. Rlxing th chil constrint on, D,.g., to scnnt mks th qury stisfil. Finlly, consir Q. ll gs xcpt on, D r scnnt constrints. Th qury is unstisfil cus th two nos r th sm no, sy v, n v hs chil D, which is scnnt of scnnt C no of v. contriction riss cus of th inconsistncy in th rquir istnc twn v n th D no. gin, rlxing th only chil constrint in th qury to scnnt rnrs it stisfil. gnrl rmrk out ll quris is inst of rlxing chil constrint, ropping ny othr constrint in th qury (.g., intity of two nos) lso rnrs it stisfil. Th xmpls show tht rsoning out stisfiility is intrsting n non-trivil. W mk th following contriutions in this ppr. Rsoning out stisfiility cn ruc to mking infrncs out rltionships twn nos n/or thir contnts or ttriut vlus. W vlop infrnc ruls for ucing itionl structurl rltionships twn qury nos from thos stt in th qury (Sctions 3 n 4). W propos constrint grph for tr pttrn qury. It consists of structurl prt tht cpturs structurl constrints in th qury n vlu-s prt tht cpturs vlu-s constrints. Using our infrnc ruls, w vlop chs procur for closing th constrint grph w.r.t. ll constrints impli y givn ons. W show th chs is complt whn th qury contins no wilcrs: qury is stisfil iff its constrint grph, whn chs, os not rsult in ny violtions, in prcisly fin sns. Our infrnc ruls n chs r vlop for oth whn no schm is known n whn schm is givn (Sctions 3 n 4). W intify conitions unr which tsting stisfiility is NP-complt (Sctions 3.3 n 4.2.) n whn it cn on in polynomil tim. For th lttr css, w vlop fficint lgorithms for stisfiility tsting (Sctions 3.2 n 4.2). Finlly, w rn comprhnsiv st of xprimnts on synthticlly gnrt t st on svrl wll-known DTDs incluing uction.t n protin.t, tst vrious kins of stisfil n unstisfil quris, n msur oth th itionl ovrh incurr on stisfil quris n th mount of svings on unstisfil quris. Whil th svings r mor thn n orr of mgnitu, our rsults show tht th ovrh is smll frction of ovrll qury vlution tim (Sction 5). Som sic finitions n prolm sttmnt r givn in Sction 2. Rlt work pprs in Sction 6, whil Sction 7 summrizs th ppr n iscusss futur work. 2 ckgroun n Prolms Stui (n XML) ts is finit root orr tr D = (N, E, r, λ), whr N rprsnts lmnt nos, E rprsnts prnt-chil rltionship, λ, th lling 2

3 function, ssigns tg with ch no, n r is th root. ssocit with ch no is st of ttriutvlu pirs. In this ppr, w o not consir orr ny furthr. Fig. 2() shows n xmpl ts D. Tr pttrn quris, introuc in [], cptur usful frgmnt of XPth. tr pttrn qury (TPQ) is tripl Q = (V, E, F), whr (V, E) is root tr, with nos V ll y vrils, n with E = E c E consisting of two kins of gs, cll pc- (E c ) n -gs (E ), corrsponing to th chil n scnnt xs of XPth. istinguish no in V (shown ox in Figur ) 2 corrspons to th nswr lmnt. F is conjunction of tg constrints (TCs), vlu-s constrints (VCs), n no intity constrints (NICs). TCs r of th form $x.tg = t, whr t is tg nm. VCs inclu slction constrints $x.vl rlop c, $x.ttr rlop c, n join constrints $x.ttr rlop $y.ttr, n $x.vl rlop $y.vl, whr rlop {=,, >,,, <}, ttr, ttr r ttriuts, vl rprsnts contnt, n c is constnt. With fw clrly intifi xcptions, w ssum no isjunctions ppr in VCs, throughout th ppr. Whn w o llow isjunctions, thy r confin to slction conitions. NICs r $x iop $y, whr iop {, }. 3 W opt th trm structurl constrints to rfr to NICs n pricts of th form pc($x, $y), ($x, $y), rprsnting pc- n -gs. Figur 2() shows n xmpl TPQ Q. n xmpl us of isjunction in slction constrints is th constrint.typ = pprck.typ = spirloun on no in plc of th xisting constrint thr. nswrs for TPQs r formliz using mtchings. mtching of TPQ Q to ts D is function h : Q D tht mps nos of Q to nos of D such tht: (i) structurl rltionships r prsrv whnvr (x, y) E c h(y) is chil of h(x) in D n whnvr (x, y) E, thr is pth from h(x) to h(y) in D; n (ii) th formul F is stisfi. W sy tht ts D stisfis qury Q provi thr is mtching h : Q D. mtching of th qury Q in Figur 2() to th ts D of Figur 2() is schmticlly illustrt with numrs sis ts nos. qury Q is stisfil provi thr is ts D tht stisfis Q. For rility, whnvr th tg constrint $x.tg = t pprs in TPQ Q = (V, E, F), w rop tht constrint from th formul prt F, n writ t right nxt to no $x in Q. If no $x is not tgg, w ssocit wilcr nxt to no $x in Q. This is illustrt for qury Q in Figur 2()-(c). This corrspons to th XPth xprssion /i[///txt()= Rymon Smullyn ]/[@typ=pprck]/uthor[txt()=. Russl ]. If th constrint.typ pprck wr rplc y.typ = pprck in Q, ts D in Figur 2() wouln t stisfy it, s no mtching 2 Sinc istinguish nos o not ply ny rol in stisfiility, w o not consir thm furthr. 3 Th constrints $x $y n $x $y sy nos $x n $y r (not) th sm. ook typ: pprck i titl uthor yr titl uthor uthor "Wht is th nm of this ook?" "Rymon Smullyn" (2) () "986" ().tg=i &.vl="rymon Smullyn" &.typ!=pprck &.tg=uthor &.vl=".russl" () ook (3) typ: hroun "Principi ".N. Mthmtic" ". Russl" Whith" (4) i uthor (c).vl="rymon Smullyn" &.typ!=pprck &.vl=".russl" Figur 2: n xmpl: () ts D n: () TPQ Q, (c) Q m mor rl. is possil. In th squl, w writ lmnt tgs or wilcrs nxt to qury nos s pproprit. Thus, qury Q is si to hv wilcrs if on or mor nos o not hv thir tgs constrin y TC. Othrwis it wilcr-fr. Q is join-fr if it contins no join constrints n no NICs. W strct th schm of ts (in our ppr, w only consir DTDs) s grph with nos corrsponing to tgs n gs ll y on of th quntifirs?,,, + with thir stnr mning of optionl, on, zro or mor, n on or mor rspctivly. n xmpl of schm grph pprs in Figur 8. It sys,.g., tht ctgoris consists of ctgory lmnts, ch of which hs uniqu scription. Prolms Stui: W consir tsting stisfiility of vrious clsss of TPQs (with/without VCs, with/without isjunction in VCs, with/without join n no intity constrints, with/without wilcrs) oth in th snc of schm n in th prsnc of schm without isjunction (i.., choic) n cycls. 3 Stisfiility without Schm Givn TPQ Q, trmining whthr Q is stisfil in th snc of schm, solly pns on th structurl constrints n ny VCs prsnt in Q. In ition, it my ncssry to consir isjunctions n wilcrs in th qury, if prsnt. W systmticlly stuy th prolm for vrious TPQ clsss. 3. Join-fr TPQs with Wilcrs Rcll tht join-fr TPQs o not contin join or no intity constrints. Not tht Q my still involv 22

4 vlu-s slction constrints. In th spcil cs tht Q hs no VCs, it is lwys stisfil. In, stisfying instnc D for Q cn construct s follows. D is tr isomorphic to th qury tr Q xcpt ll gs r pc-gs. For vry qury no tht is tgg, th corrsponing no in D hs th sm tg; if th qury no is wilcr, th corrsponing no in D my hv n ritrry tg. It is sy to s tht D lwys stisfis Q. Suppos Q = (V, E, F) os contin VCs. Sinc it os not contin ny join constrints, vry VC constrins uniqu no in Q. Lt F x th mximl suformul of F tht constrins no x. To vrify tht Q is stisfil, it thn suffics to vrify if F x is stisfil for ch no x. Th following proposition summrizs th sitution for join-fr TPQs. Proposition 3. For join-fr tr pttrn qury Q, possily contining wilcrs, th following hols:. If Q contins no VCs ssocit with ny no, thn Q is stisfil. 2. If Q contins vlu-s slction constrints (ut is join-fr), thn Q is stisfil iff for vry no, th ssocit st of VCs is consistnt. Th complxity of vrifying stisfiility thus pns on th kins of formuls F x constrining ch no x. If no isjunction occurs, consistncy of F x n hnc of F cn vrifi in polynomil tim using th soun n complt xiom systm givn in [2]. If VCs F x ssocit with no x cn involv ritrry isjunctions, tsting consistncy of F x coms quivlnt to ST n hnc is NP-complt. If F x is isjunction of conjunctions, thn th mtho propos in [2] cn sily xtn to yil polynomil tim tst for stisfiility of Q. 3.2 Wilcr-fr TPQs with Joins Lt Q TPQ contining join n/or no intity constrints, ut no wilcrs n no isjunction. W rlx th lttr rstrictions in Sction 3.3. Th prsnc of join n no intity constrints intrcts in n intrict wy with th structurl constrints. E.g., th constrint x y implis ny ncstors of x n y in th qury Q must li on th sm pth in stisfying ts. low, w sprt th rsoning into structur n vlu-s prts n pin own xctly how thy hnshk Rsoning out Structur In this sction, w consir quris with just NICs. Th ffct of VCs of th form $x.vl rlop $y.vl tc. r rss in Sction Som issus involv in rsoning out stisfiility r illustrt y th following xmpl. Exmpl 3. [Structurl rsoning] Consir th qury in Figur 3, which is inticl to qury Q2 in Figur. s iscuss in th introuction, it is unstisfil. Th rsoning involvs infrring tht no pirs n must li on th sm root-to-lf pth s wll s tht thy must cousins of ch othr, ling to contriction. Th xmpl illustrts svrl points.. Tsting stisfiility involvs infrring rltionships twn pirs of nos s on structurl constrints stt in th qury. Thus, w n infrnc ruls. 2. Som of th intrmit rltionships infrr ov cnnot irctly rprsnt in th lngug of TPQs (.g., x n y must li on th sm pth ). Thus, th lngug is not clos w.r.t. stisfiility rsoning. W coul rprsnt th nw rltionships y prmitting isjunction in structur. E.g., x n y li on th sm pth iff (x y (x, y) (y, x)). Howvr, prmitting ritrry isjunctions cn l to high complxity. W show tht ll w n to o is th following pricts: s(x, y) mning x y or (x, y), OTSP(x, y) mning s(x, y) or (y, x), COUS(x, y) mning OTSP(x, y). Not tht th pricts OTSP, COUS,, r symmtric whil pc,, s r not. This xpn st of pricts is in clos w.r.t. stisfiility rsoning. Q: c r $5 $6 $7 $8 $5 $6 & $7 $8 & Mrg Nos $5 n $6 Mrg Nos $7 n $8 Q is ST if (r,) is inst of pc Infrncs:. (2,5), (3,6) > OTSP(2,3) 2. (4,8), (3,7) > OTSP(4,3) 3. pc(,2), pc(,4), (2!=4) > COUS(2,4) 4. pc(,2), (,3), OTSP(2,3) > s(2,3) 5. pc(,4), (,3), OTSP(4,3) > s(4,3) 6. s(2,3), s(4,3) > OTSP(2,4) >VIOLTION: OTSP(2,4), COUS(2,4)< Figur 3: Infrring Structurl Pricts Dtrmining stisfiility of qury works s follows. First, w us infrnc ruls to otin th closur of structurl pricts. Thn, w chck th rsulting st of pricts for violtions (fin low). Th qury is stisfil iff th st of pricts is violtionfr (consistnt). Structurl Constrint Grph: In orr to fficintly implmnt procur for stisfiility chcking, w construct (structurl) constrint grph G Q for th qury Q s follows. G Q contins on no for ch qury no. For ch prict φ(x, y) in Q, G Q contins irct g ll φ from x to y. For symmtric pricts, th g is iirct. Infrnc Ruls n Chs: Nw structurl pricts r infrr from xisting ons in th qury y using st of infrnc ruls. n infrnc rul is of th form P,... P k R n sys if pricts P,..., P k r tru, thn R is tru. Infrnc ruls r us for chiving closur of structurl pricts n thus for ctching inconsistncis cus y conflicting pirs of pricts. For th structurl pricts, w hv - 23

5 vlop totl of 22 infrnc ruls. 4 For rvity, w show only som intrsting ruls in Figur 4 n xplin som slct ons. Th complt tils cn foun in []. W xplin thr of th ruls. Rul 2 sys whnvr x lis on th sm pth s ch of pir of cousins y n z, thn x must thir ncstor 5. Rul 3 sys two unqul nos x, y t n qul istnc from no z must cousins. Th qul istnc implis th pths from z to x n y must involv only pc-gs. Rul 7 sys whnvr x n y r on th sm pth, x is chil of n ncstor of y, thn y must slf or scnnt of x, i.., s(x, y). Th chs procur is to simply pply th infrnc ruls until no nw infrncs r possil. If violtion, fin nxt, is tct t ny point, w cn xit from chs rly. W will iscuss mor fficint implmnttion of chs shortly.. s(x, z), s(y, z) OTSP(x, y) 2. OTSP(x, y), OTSP(x, z), COUS(y, z) (x, y) 3. x y COUS(x, y), whnvr x, y r t th sm istnc from thir lst common qury ncstor z. 4. pc(x, z), pc(y, z) x y. 5. pc(z, x), pc(z, y), OTSP(x, y) x y. 6. (x, z), pc(y, z) s(x, y). 7. pc(z, x), (z, y), OTSP(x, y) s(x, y). Figur 4: Slct Infrnc Ruls (no schm). Violtions: violtion is pir of conflicting pricts twn pir of nos. Exmpls of conflicting pirs of pricts r x y, x y; (x, y), s(y, x); n OTSP(x, y), COUS(x, y). In, ths thr pirs cptur ll possil violtions, sinc othr violtions r susum y thm. For instnc, pc(x, y) conflicts with COUS(x, y). ut sinc pc(x, y) implis OTSP(x, y) this conflict is covr y th pir OTSP(x, y), COUS(x, y). Violtions mk th qury unstisfil. Figur 3 monstrts th chs s logicl infrncs. t th n of stp 6, w fin violtion cus of th conflicting pricts OTSP() n COUS(). To implmnt th chs mor fficintly, w mploy th constrint grph. Spcificlly, givn TPQ Q, w initiliz its constrint grph CG Q. For vry pir of nos $i, $j, whnvr thir tgs r iffrnt, w iirct g ll twn $i n $j. W pply th infrnc ruls rptly. Whnvr prict p($i, $j) is riv, (irct) g from $i to $j ll p if p is on of, s n mk it iirct if p is on of, OTSP, COUS. Whn $i $j is riv, w mrg nos $j n $j. W rpt until no nw infrncs r m or violtion is tct. constrint grph, with chs ppli on it, is chs constrint grph. Hr is n xmpl. Th qury of Figur 3 coms stisfil if th pc-g from to is chng to n -g (shown ott in th figur). Figur 5() shows th constrint grph 4 Incluing trivil ons such s pc(x, y) (x, y). 5 Not tht th rul is symmtric for this qury n Figur 5(c) shows th chs constrint grph. Figur 5() shows stisfying instnc of th qury. c Constrint Grph $5 pc $6 r $7 () Constrint Grph Chs Constrint Grph c L $5,$6 r COUSIN forc L = {pc,,s,otsp, } $8 $7,$8 r c () Instnc = {,s,otsp, } (c) Chs Constrint Grph Figur 5: Dtrmining Stisfiility Th min rsult of this sction is th following: Thorm 3. (Compltnss of Chs) : Lt Q tr pttrn qury contining no intity constrints ut no wilcrs. It is stisfil iff th chs constrint grph of Q is violtion-fr. W rfr th rr to [] for th proof. Hr, w giv th ky intuition. Th If irction is sy to s sinc vry infrnc rul is soun n thrfor prsrvs stisfiility. For th Only If irction, suppos G is th chs constrint grph of qury Q n G is violtion-fr. W construct stisfying tr instnc s follows. Procur FstChs(CGrph CG) fin ll -clsss of nos; for ch quivlnc clss E, fin th mximl OTSP st s pr(x) x E in G, whr pr(x) is th st of prcssors of x in G; for ll x, y s.t. x y G, pply th istnc rul (#3) to riv COUS(x, y); propgt COUS() ownwr using infrnc ruls; if COUS(x, y) is riv, x y to G; if violtion is foun rturn unstisfil ; if x y is riv, mrg x n y; whil thr is no chng { pply ruls for infrring,, s; if nos r qut, mrg thm; if violtion is foun rturn unstisfil ; } rturn tru; Figur 6: pply Chs in CGrph 24

6 Cll st S of nos in Q n OTSP st provi x, y S: OTSP(x, y) G. OTSP sts r upwr clos, i.., whn x S, n s(y, x) G, thn y S. Hncforth, w consir mximl OTSP sts, i.., OTSP sts whos propr suprsts r not OTSP sts. Th i is to forc rltionships twn pirs of nos until G coms complt st, i.., nos x, y G n for ny prict p, ithr p(x, y) or p(x, y) hols in G. In prticulr, ll nos in mximl OTSP sts r totlly orr using topologicl sort. Diffrnt mximl OTSP sts r incorport in iffrnt rnchs of th tr. W nxt rifly commnt on n fficint implmnttion of th chs. niv implmnttion woul tk tim O(n 5 ), whr n is th numr of nos in th qury. This is cus ch rul involvs 3 nos n thr r O(n 2 ) itrtions possil in th worst cs for no nw infrncs r m. mor fficint implmnttion is suggst in Figur 6. Th i is to xploit th upwr closur (ownwr closur) of OTSP (COUS) prict. It cn shown tht mximl OTSP sts cn comput stticlly s on th constrints givn in th qury. Similrly, w cn infr COUS gs fficintly. Infrnc ruls for th rmining pricts n to ppli rptly until ithr violtion is foun or no nw infrncs r possil. Th worst-cs complxity of this lgorithm rmins th sm. Howvr, in prctic it is much ttr thn th niv lgorithm Intrction with VCs Up to this point, w hv not consir VCs. Evn whn qury is stisfil w.r.t. its structurl constrints, th VCs my rnr it unstisfil. s mntion rlir, consistncy of conjunction of VCs cn chck in polynomil tim using th soun n complt xiom systm provi in [2]. Th chcking lgorithm cn implmnt fficintly using sprt vlu-s constrint grph using is similr to th structurl constrint grph. Th tils r similr n r omitt. Wht out intrctions twn structurl constrints n VCs? It cn shown tht th intrction hppns vi two min links: (i) Th structurl constrints my imply x y for nos x, y. ll VCs pplicl to x r pplicl to y. This is utomticlly cptur y mrging x n y. (ii) VCs cn imply x y for nos x, y. This cn in turn triggr infrncs of structurl pricts. Th procur for tsting stisfiility of qury Q with structurl constrints n VCs is thn s follows: (i) Chs th VCs (using sprt vlus constrint grph); if ny violtion is foun rturn unstisfil. (ii) Construct th (structurl) constrint grph G of Q; propgt ll constrints x y riv from VC chs to G n chs it; (iii) Q is stisfil iff th chs trmints with no violtion. W cn show: Thorm 3.2 (TPQs with VCs) : Lt Q tr pttrn qury with structurl constrints n Disjunction NICs/join constrints Wilcrs Complxity X PTIME X PTIME X X NP-Complt X X NP-complt Figur 7: Complxity of chcking Stisfiility without Schm VCs n no wilcrs. Thn tsting stisfiility of Q cn on in polynomil tim using th procur ov. 3.3 TPQs with Wilcrs, Joins, n Disjunction W rlx th rstrictions on TPQs w.r.t. wilcrs n isjunctions in this sction. Th first osrvtion is tht whn wilcrs r llow, stisfiility tsting coms NP-complt, vn whn thr is no isjunction. This follows from th following rsult, prov y Hirs [5]. Thorm 3.3 ([5]) : Suppos Q is tr pttrn qury with wilcrs n only constrints, whr th qury uss only pc- n s-gs. Thn tsting whthr Q is stisfil is NP-complt. Whil Hirs rsult is couch in trms of syntcticlly iffrnt lngug, th frgmnt for which this rsult pplis corrspons to tr pttrn quris with wilcrs n constrints, whr th ntir qury rucs to singl mximl OTSP st. It is trivil to pt his proof for tr pttrn quris with rgulr pc- n -gs. Nxt, wht if w isllow wilcrs ut llow isjunction in VCs. Th prolm gin coms NPcomplt. Thorm 3.4 (TPQs with isjunction) : Lt Q tr pttrn qury contining VCs, with isjunction llow in slction constrints ssocit with nos. Thn tsting stisfiility of Q is NP-complt. Th proof is y ruction from 3ST, n only mks us of pc-gs, isjunctiv vlu-s slction constrints, n constrints. It continus to hol whn constrints r rplc y join constrints. Th complxity rsults for th schmlss cs r summriz in Figur 7. 4 Stisfiility in th Prsnc of cyclic Schm schm provis itionl knowlg for infrring structurl pricts in qury. E.g, consir th schm n qury Q4 in Figur 8. It is not stisfil. Th qury Q4 sks for txt which is oth chil of scription n scnnt of prlist, ut th schm os not prmit this. Howvr, if txt is chng to scnnt of prlist, Q4 coms stisfil. Suppos Q4 is ccoringly chng. In th snc of 25

7 schm, th st w cn conclu out scription n txt thn is tht thy must li on th sm pth ut using th schm, w cn conclu tht scription is th ncstor of txt. schm (uction.t) togthr with st of unstisfil quris s wll s minor vrints which r stisfil r givn in Figur 8. Th rr is ncourg to rson out thir stisfiility. In th rst of this sction, w consir cyclic (DG) schm. Extnsions to cyclic schms will ppr in th full ppr. 4. TPQs without VCs DTD sit rgions ctgoris + si fric ctgory itm scription Q3 prlist ctgory scription txt scription Q4 Q scription sit ctgory txt ctgoris prlist Q2 itm Q5 si sit txt txt txt txt sit ctogry fric Figur 8: Exmpls in th Prsnc of Schm Whn no VCs r prsnt, for qury to stisfil with rspct to schm, its structurl constrints n to consistnt with th schm. n ming of qury into schm, fin low, prcisly cpturs this consistncy. Dfinition 4. [Eming] n ming of qury Q into schm is function f : Q stisfying th following conitions: (i) f mps ch tgg no to no with th sm tg; (ii) whnvr (x, y) is pc-g (-g) in Q, thr is n g (pth) from f(x) to f(y) in. Consir qury Q4 in Figur 8, ut without th join conition. Th rr cn vrify th xistnc of n ming into th schm in Figur 8. In th snc of wilcrs, th tsting th xistnc of n ming rucs to tsting for ch g (x, y) in Q with tg(x) =, tg(y) = (sy), whthr n g or pth from to xists in, which cn sily tst. Th following rsult is strightforwr: Proposition 4. Lt schm n lt Q tr pttrn qury with no wilcrs or VCs. Thn Q is stisfil with rspct to iff thr is n ming f from Q into. 4.. With Wilcrs Consir th xmpls in Figur 9. Qury Q is not stisfil cus no vli instnc of th schm cn hv pth from to of lngth 2. On th othr hn, qury Q2 is stisfil cus thr xists vli instnc of th schm which hs pth of lngth t lst 2 from to. For ch of th quris, th possil schm nos it coul m to r illustrt s st, right nxt to th no in Figur 9. For th no in qury Q, th st of schm nos it cn m to is mpty. Not tht if th qury contins only wilcr nos, chcking stisfiility trivilly rucs to chcking if th schm is of givn pth. Schm + c Q {} {,c,,} {} {} Q2 {} {} {,c,,} {} Figur 9: Quris with Wilcrs Whn wilcrs r prsnt, smnticlly w cn ssign ny tgs to th wilcrs n chck for th xistnc of n ming. This pproch tks xponntil tim. Wht w n is mrly confirm th xistnc of n ming. This cn ccomplish y ssociting with ch qury no x ll st L(x). For ch tgg no x, w initiliz L(x) to th uniqu schm no with tht tg. For wilcr no x, w initiliz L(x) to th st of ll tgs in th schm. W nxt prun L(x) s follows. First, in ottom-up phs, w mrk ll lvs. Whnvr ll chilrn y,..., y k of no x r mrk, w lt tg t from L(x) provi for som y i, r(x, y i ) hols ccoring to th qury Q, whr r is pc or, ut thr is no tg u L(y i ) such tht contins n g or pth to vrify r(t, u). Thn mrk no x. If t ny stg ny ll st coms mpty, w know th qury is unstisfil. Onc th root is mrk, w o top-own swp s follows. First unmrk th root. For ny no x whos prnt y is unmrk, lt from L(x) ny tg t if thr is no tg u L(y) such tht r(y, x) ccoring to Q, n contins n g or pth vrifying this. Thn unmrk x. Th procur trmints whn n mpty ll st is tct or whn ll nos r unmrk. Th psuoco for this lgorithm is shown in Figur 0. W cn show: Thorm 4. (Lling) : Lt Q TPQ contining wilcrs ut no VCs n no NICs. Thn Q is stisfil with rspct to schm iff for ch x Q, L(x), whr L(x) is th st of schm lls comput y th procur in Figur 0. y prcomputing rchility on, givn t, t, w cn tst if vrifis r(t, t ), whr r {pc, }, in constnt tim. W visit ch qury no n ch qury g t most twic. During ch visit, w my 26

8 ChckLl(Q, ) For ch no x tgg t in Q, L(x) = {t} For ch wilcr lf l in Q, L(l) = {tgs of } Mrk ll lf nos. Lt r {pc, } Rpt { // ottom-up Phs nos x Q whos chilrn y,... y k r ll mrk For ch chil y i of x { Initiliz S i = {}; For ch u L(y i ) { S i = S i {t r(t, u) }; } k L(x) = S i; } i= Mrk x; If L(x) is mpty, rturn(q is not ST); } Until ll nos r mrk. Unmrk th root; Rpt { // Top-own Phs For ch x whos qury prnt y is unmrk { For ch u L(x) { If t L(y) s.t. r(t, u) rmov u from L(x); } Unmrk x; if L(x) is mpty, rturn(q is not ST); } Until ll nos r unmrk. rturn(q is ST) Figur 0: Chck Wilcr Eming n to compr ll pirs of tgs in th ll sts of th two nos in th g. Thus, th worst-cs tim complxity is O(m 3 +n m 2 ), whr m is th numr of nos in n n is th numr of nos in Q. 4.2 Rsoning in th prsnc of No Intity Constrints Lt us consir th clss of TPQs tht contin no wilcrs ut my contin NICs (, ), n VCs (without isjunctions). prt from th intrction twn structurl pricts n VCs, thr is intrction twn schm n th structurl constrints impos y th qury. Exmpl 4. [Impct of Schm] Consir th xmpls in Figur. Qury Q is stisfil with rspct to th schm. Th rsoning hin this is s follows. Sinc nos n $5 r inticl, nos n must li on th sm pth. From th schm, w cn thn conclu tht is n ncstor of. In, n instnc cn otin from th chs constrint grph tht stisfis th qury. Qury Q2 is not stisfil with rspct to th schm. Hr is why. From th schm, vry occurrnc of ncssrily hs grnchil f, which is uniqu. Th qury sks for two istinct scnnts of tgg f, on s grnchil n on s ny scnnt. Howvr, from th schm, w cn conclu tht nos n $5 r inticl i.., $5 which contricts th qury constrint $5. Th xmpl illustrts svrl points.. Th qury contins no wilcrs. Thus, for two nos if ithr s or OTSP prict hols, thn from th schm, it is possil to conclu strict pc or rltionship twn thm. Hnc, th pricts s n OTSP which w us in th snc of schm, now com runnt. 2. W cn us th schm to trmin whn two qury nos r inticl. In th schm of Figur, thr is uniqu pth from no to f, with ll g lls ithr or?. Hnc ny two scnnts f of in th instnc shoul inticl. 3. Following th sm rgumnt, using th schm it is lso possil to infr tht two nos must cousins, y trmining whn th nos li on istinct pths. Schm + + f c Q $5 $5 Q2 f Figur : Infrnc from Schm Dtrmining stisfiility of qury works s follows. W us th schm to infr structurl pricts twn ny pir of qury nos (which r tgg). W us infrnc ruls to comput th closur of structurl pricts n chck th rsulting st for violtions. Th qury is stisfil iff th rsulting st is violtion-fr. s for, w us constrint grph n st of infrnc ruls to comput th closur, with som iffrncs in th infrnc ruls us. Infrnc Ruls n Chs: Th st of infrnc ruls r pt from thos vlop for th schmlss cs. Ruls involving s or OTSP r ropp, sinc th schm llows us to riv n unmiguous rltionship whnvr s or OTSP hols. itionlly, w n to infr rltionships twn lmnt typs from th schm. Th schm cn tll us tht two tgs t, t r rlt y pc-/-rltionship, or tht two qury nos must inticl or tht thy must cousins. This sttic nlysis of th schm cn prform using th ruls shown in Figur 2, xplin nxt. Th complt st of infrnc ruls cn foun in []. Rul corrspons to isjoint nos. Lt x, y ny nos in qury Q n suppos z is thir lst common ncstor in Q. Lt givn schm. Suppos (z, u,..., u k, x) n (z, v,..., v m, y) r th pths in Q from z to x n y rspctivly. W cll ths pths th qury contxt of x n y. Not tht ll nos r tgg in Q. For simplicity, not th tg of ch no y its prim vrsion, i.., no x hs tg x. Suppos thr is no pth in tht psss through ll th nos z, u,..., u k, v,..., v m, x, y n in n orr comptil with th qury contxts ov, which rspcts ny pc-rltionships prsnt in th qury contxts. Thn w cn conclu tht x n y must cousins in vry vli instnc of, which stisfis Q. Whn this conition hols, w sy x n y r isjoint. s n xmpl, consir qury Q4 in Figur 8, without th sh lin. Thn qury nos n (with tg txt) r ncssrily cousins. This is cus thr is no pth in th schm tht psss through ctgoris, scription, prlist, n txt in ny comptil orr, such tht thr is f 27

9 . whnvr x, y r isjoint, infr COUS(x, y). 2. whnvr z is lc(x, y), x n y r uniqu w.r.t. z, th pth from z to x tht stisfis th qury contxt of x is inticl to th pth from z to y tht stisfis th qury contxt of y, tg(x) = tg(y), infr (x, y) 3. whnvr z is th lc(x, y), x n y r uniqu w.r.t. z, th uniqu pth from z to y tht stisfis th qury of x n y contins g (x, y ), tg(x) tg(y), infr pc(x, y) 4. whnvr (x, z), (y, z), : xctly on pth from x to y n tht pth is n g, infr pc(x, y) Figur 2: Slct Infrnc Ruls(with Schm) irct g from scription to txt, so n r isjoint. Howvr, if th g is rlx to n -g (i.., sh lin is ), such pth xists in th schm, so n r not isjoint, hnc n r not ncssrily cousins. Disjointnss cn chck fficintly using vrint of mrg sort n in tim linr in th sum of sizs of th two qury contxts. Ruls 2-3 corrspon to uniqu nos. Lt Q qury n x n y th nos in Q, such tht (y, x) G. Lt schm. Thn x is uniqu with rgr to y whnvr hs xctly on pth from y to x stisfying th qury contxt of x n y, n no g on this pth is ll or +. Th intuition hin rul 2 is tht th qury pths from z to x s wll s from z to y will oth mpp ncssrily to on pth in vry vli instnc of. So, if x n y hv th sm tg, thy must mp to th sm instnc no. Rul 3 hs similr intuition. Rul 4 sys whnvr (x, z), (y, z) hols, clrly on of x, y must prnt/ncstor of th othr (whn x n y hv iffrnt tgs). This is trmin y th schm. Finlly, th chs procur for TPQs (with NICs ut no wilcrs) in th prsnc of schm is s follows. First, construct th constrint grph G of Q s for th schmlss cs. Nxt, using sttic nlysis of th schm, infr ll COUS,, pc, rltionships n thm to G. Chs G using th infrnc ruls intifi ov until sturtion or violtion tction. W cn show: Thorm 4.2 (Chs Compltnss with Schm) : Lt n cylic schm without choic n Q tr pttrn qury with NICs ut no wilcrs. Thn Q is stisfil w.r.t. iff thr is n ming of Q into n no violtion is tct whn th constrint grph of Q is chs. To unrstn th implictions of Thorm 4.2 for th complxity of chcking stisfiility of tr pttrn qury w.r.t. givn cyclic schm without choic, w consir this prolm t two lvls. Firstly, lt ny schm. Thn w fin th lngug ST ={Q Q is qury & Q is stisfil w.r.t. }. W cll th complxity of chcking this mmrship qury complxity, y nlogy with th notion of t complxity in [22]. Sconly, w fin th lngug ST to ST={(, Q) is schm & Q is qury & Q is stisfil w.r.t }. W cll th complxity of chcking this mmrship th comin complxity of stisfiility chcking, y nlogy to th wllknown notion of comin complxity[22]. W hv th following rsults. Thorm 4.3 (Qury Complxity) : Th qury complxity of stisfiility chcking in th prsnc of cyclic schm without choic is PTIME. Th i is tht w cn pply th infrnc ruls to sturtion or until violtion is tct, which is procss tht tks polynomil tim in th siz of qury. W cn lso tst whthr thr is n ming from th qury to th schm in PTIME. This yils polynomil tim lgorithm for tsting stisfiility in th prsnc of schm. Efficint implmnttion, similr to tht iscuss in Sction 3.2., is possil. Th tils r omitt. finl not is tht VCs cn sily incorport in th sm wy thy wr for th schmlss cs. Thus, w cn tst stisfiility in polynomil tim in th prsnc of schm n VCs. Thorm 4.4 (Comin Complxity) : Th comin complxity of stisfiility in th prsnc of cylic schm without choic is co-np-complt. Th complxity coms not irctly from th chs, ut from violtion chcking. Figur 3 illustrts th violtion chcking procur. DTD C? D? E Q E() E() E(),, Figur 3: Violtion Dtction Exmpl Th qury Q in Figur 3 is not stisfil cus it ssrts thr must xist t lst thr iffrnt Es unr. Howvr, thr r only two pths from to E in th DTD, ll of whos gs r ll /?. Thus thr xist t most two Es unr in ny vli instnc. Th proof of Thorm 4.4 is y ruction from Mximl Cliqu. Th tils cn foun in []. Whil th comin complxity is high, in prctic, w will oftn wnt to chck th stisfiility of mny quris ginst fix schm, illustrting th significnc of qury complxity n of Thorm

10 4.2. No Intity Constrints n Wilcrs In th prsnc of schm, tsting stisfiility of tr pttrn qury with wilcrs n NICs is NPcomplt. Similrly, whn thr r no wilcrs ut th qury contins vlu-s isjunctiv slction constrints, gin th prolm is NP-complt. Thorm 4.5 (Hrnss rsults) : Lt schm n Q tr pttrn qury. Thn stisfiility of Q w.r.t. is NP-complt in th following css: () Q contins wilcrs n NICs. (2) Q contins isjunctiv VCs (n no NICs). Th first rsult is y ruction of 3-colorility n th proof only uss constrints. Th scon rsult is y ruction of 3ST. oth proofs only mk us of tr schms n only pc-gs in Q. Th complxity rsults for th schm cs r summriz in Figur 4. ll rsults shown corrspon to qury complxity. Disjunction NICs/join constrints Wilcrs Complxity X PTIME X PTIME X X NP-complt X NP-complt Figur 4: Complxity of chcking Stisfiility with schm. 5 Exprimntl Rsults To stuy th ffctivnss of tsting stisfiility, w systmticlly rn rng of xprimnts to msur th impct of vrious prmtrs. In ition to msuring svings n ovrh, w lso msur how stisfiility chcking tim vris s function of th numr n kins of constrints. W rn our xprimnts on th XMrk nchmrk tst [24] n iomicl tst [25] from th Ntionl iomicl Rsrch Fountion. For ch tst w construct th ocumnts of vrious siz using th IM XMLGnrtor [26]. W us Wutk DTDprsr [27] to prs th DTD, which is n for sttic nlysis of schm. For qury vlution, w us n XQury ngin XQEngin [28] for convninc n flxiility. oth tools r opn sourc, vlop in Jv. W implmnt our stisfiility tsts in Jv. Stup: W rn our xprimnts on sprc worksttion running SunOS vrsion 5.9 with 8 procssors ch hving sp of 900MHz n 32G of RM. ll vlus rport r th vrg of 5 trils ftr ropping th mximum n minimum, osrv uring iffrnt worklos. Qury St: ll quris chosn for xprimnttion corrspon to clsss of tr pttrn quris stui in this ppr. Pls not tht whn multipl no qulitis r prsnt in TPQ, w n to us XQury for its implmnttion. For stisfiility tsting without schm n with schm css, w us Q-Q3 in Figur 8. lthough Q: for $ in oc( uction.xml )//ctgory, 2 $ in $//scription, 3 $C in $//prlist, 4 $2 in $//scription 5 whr $//txt is $C//txt n $2//prlist is $C 6 rturn $ Q2: 7 for $ in oc( uction.xml )//scription, 8 $ in $/prlist, 9 $C in $//listitm, 0 $D in $//txt whr $//ol is $C//ol n $D//kywor is $C//kywor 2 rturn $ Q3: 3 for $ in oc( uction.xml )//ctgoris 4 whr $/scription/txt is $//prlist//txt 5 rturn $ Figur 5: Exmpls for Schmlss cs w us th sm st of quris, w us iffrnt nlysis for no schm mo n schm mo sprtly. W lso xprimnts with th iomicl tst ut w i not inclu th tils for spc limittions. Th tils cn foun in th full vrsion of this ppr. Sving&Ovrh Rtio: Lt c th tim tkn to trmin th stisfiility of qury Q n lt th tim it tks to vlut th qury ovr th ocumnt (without using stisfiility chck). Th svings rtio S Q otin y using stisfiility chck on unstisfil quris is fin s S Q = c n th ovrh rtio incurr y oing stisfiility chck on stisfil quris is fin s O Q = c+. Intuitivly, th closr to th two rtios r th ttr. Sving Rtio: Not surprisingly, on unstisfil quris, stisfiility chck ls to phnomnl svings. Our sving rtio is clos to (usully twn out 0.8 n 0.9) whthr th schm is prsnt or not. W omit ths rsults for rvity. Ovrh Rtio: Figurs 6 n 7 show th vrition of svings rtio with ocumnt siz for th thr stisfil quris Q Q3 in Figur 8 (with schm) n Q Q3 in Figur 5 (without schm). W xpct th ovrh rtio to crs s th ocumnt siz incrss. O v rh R tio S T - N o D T D k 0k 00k m D o c u m n t Siz ( y t ) Q ' Q 2 ' Q 3 ' Figur 6: Ovrh Rtio - Without Schm 29

11 O v rh R tio S T - u c tio n. t k 0k 00k m 0m D o c u m n t Siz ( y t ) Figur 7: Ovrh Rtio - With Schm In, this hvior cn osrv from th figurs. Ovrll, our rsults show tht th ovrh is ngligil frction of th vlution tim. In ition, w lso tst th impct of numr of constrints on stisfiility chck tim. For stisfil quris, s xpct th tim incrss, whil for unstisfil quris, it crss s violtions r foun fstr. W lso vri th structur of rsulting OTSP sts y ing constrints n stui thir ffct on stisfiility chck tim. W foun fw lrg OTSP sts incrs th tsting tim mor thn svrl smll OTSP sts. Th sm conclusions wr lso otin from th xprimnts on th iomicl tst. 6 Rlt Work Continmnt: Thr hs n much work on qury procsssing, continmnt n minimiztion of vrious XPth frgmnts [8, 7, 5, 2, 3,, 23]. Kupr t l. [9] stuy xprssiv powr n closur proprtis of vrious XPth frgmnts n tr pttrn quris. Lvy t l. [8] stui qury quivlnc n stisfiility for tlog xtnsions. Stisfiility cn ruc to continmnt: qury Q is unstisfil iff Q is contin in (fix) unstisfil qury Q. Howvr, our rsults on stisfiility in this ppr cnnot otin from known rsults on continmnt. Spcificlly, w show stisfiility cn tst in polynomil tim for th following clsss of tr pttrn quris quris: (i) TP /,//,[], n (ii) TP /,//,[],NIC, oth in th snc of schm n in th prsnc of n cyclic DTD without isjunction. In th snc of schm, continmnt for th formr clss is co-npcomplt [2] whil for th lttr it is Π p 2-complt [8]. Whil [8] consir continmnt in th prsnc of intgrity constrints, s point out y th uthors, thy o not cptur DTD compltly. Continmnt for TP /,//,[], w.r.t. DTD ws shown to EXP- TIME complt [3], ut it shoul not tht th DTD is llow to contin choic n cycls. Complxity of continmnt whn th DTD is cyclic n/or choic-fr is opn. Finlly, complxity of continmnt for TP /,//,[],NIC w.r.t. DTD is opn, lthough [3] Q ' Q 2 ' Q 3 ' show tht continmnt for TP /,[] n TP //,[] w.r.t. DTD is co-np-hr, whn th DTD is llow to contin choic n cycls. In sum, our PTIME rsults for stisfiility cnnot otin from known rsults on continmnt. Continmnt cn ruc to stisfiility: givn quris Q, Q, Q is contin in Q iff Q Q is unstisfil. ut this cnnot us to riv th hrnss rsults in this ppr sinc Q Q os not long to th clss of tr pttrn quris stui in this ppr. Stisfiility: Th closst work is Hirs [5], whr h consirs th complxity of stisfiility tsting for XPth frgmnts in th snc of schm. Howvr, thr r importnt iffrncs in th contriutions of th two pprs, s w xplin in til low. Th min contriution of [5] ws showing tht tsting stisfiility of XPth xprssions is NP-complt for vrious XPth frgmnts: (i) XPth with chil n slf-or-scnnt n intrsction, (ii) prnt, union, n rnching, (iii) root, rnching, chil, prnt, slf-or-ncstor. ll ths rsults pn on wilcr ing prsnt in th qury. Sconly, h show tht whn only rnching (n ll forwr n ckwr xs s wll s orr) r prsnt, stisfiility cn tst in polynomil tim. For this, h uss tr scription grph, which is similr to our constrint grph, xcpt VCs r not consir. Th procur h opts for stisfiility tsting hs flvor similr to our chs, ut th infrnc ruls r consirly simplr. Th min rson is whn st oprtions (union, intrsction) r snt, on cnnot xprss qulity. In this cs, th infrncs com much simplr. H lso show tht whn ll th xs n root r prsnt, ut non of th st oprtions or rnching r llow, gin stisfiilty cn tst in polynomil tim. similr commnt pplis to infrncs in this cs. y contrst, ll our PTIME rsults llow rnching. In prticulr, whn th qury contins no wilcrs ut contins VCs n NICs, w giv n fficint polynomil tim tst for stisfiility. This rsult os not follow from th rsults of [5]. sis, w hv xtn th tchniqus n rsults for tsting stisfiility in th prsnc of schm. To th st of our knowlg, this hs not n rss for. Finlly, our NP-compltnss rsults r orthogonl to thos in [5]. On xcption is Thorm 3.3, which s w mntion, is n sy corollry of rsult in [5]. Tsting stisfiility of tr scriptions, s on prtil tr scriptions is of consirl intrst in computtionl linguistics [0, 2, 6]. Constrint grphs r on kin of prtil tr scription. Kutz n roirsky [0] rcntly prsnt n fficint lgorithm tht chcks th stisfiility of pur ominnc constrints, which scri unll root trs using prtil orr. For ritrry pirs of nos thy spcify sts of missil rltiv positions in tr. Howvr, th (pur) ominnc constrints r sust of th structur constrints stui in this ppr. sis, rltionships such s OTSP n COUS r not consir thr, nor is rsoning in th prsnc of 30

12 schm. Thr r lso othr work rlt to stisfiility prolm. Ppkonstntinou t l.[4] stui th infrnc of DTDs for viws of XML t. This ppr propos two xtnsions tht nhnc DTD s scriptiv powr. It mntion tht stisifiility for th viws prouc y th slctiv quris in th contxt of th xtn DTD cn chck in PTIME. Howvr, th slctiv quris r only sust of th TPQs w iscuss in our ppr; thy in t llow ithr wilcrs or no qulity. Thus th prolm of chcking th stisfiility of slctiv quris is quivlnt of fining th ming of th qury in our ppr. 7 Summry Whil thr hs n consirl work on continmnt n minimiztion for vrious XPth n tr pttrn qury frgmnts, th rlt prolm of stisfiility hs n lrgly ignor. W vlop mtho for tsting stisfiility of vrious clsss of tr pttrn quris, which r known to closly rlt to XPth n XQury n to of funmntl importnc [9]. W stuy this prolm oth with n without schm (cyclic n choic-fr) n intify css in which it is NP-complt n whn it is PTIME. For th lttr cs, w vlop fficint lgorithms s on chs procur. W complmnt our nlyticl rsults with n xtnsiv st of xprimnts. Whil stisfiility chcking cn ffct sustntil svings in qury vlution, our rsults monstrt tht it incurs ngligil ovrh ovr stisfil quris. Stisfiility, for lrgr qury clsss, in th prsnc of cycls n/or choic r intrsting prolms. Stisfiility in th prsnc of XML schm is n importnt prolm. Rsults on som of ths prolms will ppr in th full vrsion of this ppr. Rfrncs [] Sihm mr-yhi t l. Minimiztion of tr pttrn quris. In CM SIGMOD Confrnc, 200. [2] T. Cornll. On trmining th consistncy of prtil scriptions of trs. In 32n CL Confrnc, 994. [3] D. Drpr t. l. Xqury.0 n xpth 2.0 forml smntics. Tchnicl rport, W3C, [4] M. F. Frnnz t. l. Xqury.0 n xpth 2.0 t mol. Tchnicl rport, W3C, [5] J. Hirs. Stisfiility of xpth xprssions. In DPL [6] H. V. Jgish t. l. Timr: ntiv xml ts. VLD Journl, [7] C. Koch n G. Gottlo. Xpth qury procssing. In 9th Intrntionl Workshop on Dts Progrmming Lngugs (DPL), Potsm, Grmny, Sptmr [8].Y. Lvy t. l. Equivlnc, qury-rchility, n stisfiility in tlog xtnsions. In CM PODS Confrnc, 993. [9] G. M. Kupr t l. Structurl proprtis of xpth frgmnts. In ICDT [0] M. Kutz n M. roirsky. Pur ominnc constrints. In STCS [] Lks V.S. Lkshmnn t l. On Tsting Stisfiility of Tr Pttrn Quris. Tch. Rport, Dpt. of Computr Scinc, UC, Mrch vill from lks/pprs.html. [2] G. Miklu n D. Suciu. Continmnt n quivlnt for n xpth frgmnt. In PODS [3] F. Nvn n T. Sch. Xpth continmnt in th prsnc of isjunction, ts n vrils. ICDT [4] Ynnis Ppkonstntinou t l. DTD Infrncfor Viws of XML Dt In CM PODS Confrnc [5] R. Pichlr t l. Th complxity of xpth qury vlution. In CM PODS Confrnc, [6] J. Rogrs n K. Vijy-Shnkr. Rsoning with scriptions of trs. In CL Confrnc, 992. [7] J. Shnmugsunrm t.l. Rltionl tss for qurying xml ocumnts: Limittions n opportunitis. In VLD Confrnc, 999. [8] V. Tnnn n. Dutsch. Continmnt n intgrity constrints for xpth frgmnts. In 8th KRD, 200. [9] I.Ttrinov t. l. Storing n qurying orr xml using rltionl ts systm. In CM SIGMOD Confrnc, [20] R. Trinn t l. Dominnc constrints: lgorithms n complxity. In 3r confrnc on Logicl spcts of Computtionl Linguistics, 200. [2] Jffry D. Ullmn. Principls of Dts n Knowlg-s Systms Volum II: Th Nw Tchnologis. Computr Scinc Prss, 989. [22] Mosh Vri. Th complxity of rltionl qury lngugs. In CM STOC, 982,pp [23] Ptr T. Woo. Continmnt for xpth frgmnts unr t constrints. ict [24] XMrk: [25] iomicl ts: rsrch/xmltsts/www/rpository.html. [26] IM XML gnrtor: [27] Wutk DTD prsr: [28] XQury: 3