The Complexity of XPath Query Evaluation

Transcription

1 Th Complxity of XPath Qury Evaluation Gorg Gottlob Databas and AI Group Tchnisch Univrsität Win A-1040 Vinna, Austria Christoph Koch LFCS Univrsity of Edinburgh Edinburgh EH9 3JZ, UK Rinhard Pichlr Inst. für Computrsprachn Tchnisch Univrsität Win A-1040 Vinna, Austria ABSTRACT In this papr, w study th prcis complxity of XPath 1.0 qury procssing. Evn though havily usd by its incorporation into a varity of XML-rlatd standards, th prcis cost of valuating an XPath qury is not yt wllundrstood. Th first polynomial-tim algorithm for XPath procssing (with rspct to combind complxity) was proposd only rcntly, and vn to this day all major XPath ngins tak tim xponntial in th siz of th input quris. From th standpoint of thory, th prcis complxity of XPath qury valuation is opn, and it is thus unknown whthr th qury valuation problm can b paralllizd. In this work, w show that both th data complxity and th qury complxity of XPath 1.0 fall into lowr (highly paralllizabl) complxity classs, but that th combind complxity is PTIME-hard. Subsquntly, w study th sourcs of this hardnss and idntify a larg and practically important fragmnt of XPath 1.0 for which th combind complxity is LOGCFL-complt and, thrfor, in th highly paralllizabl complxity class NC INTRODUCTION XPath 1.0 is th nod-slcting qury languag cntral to most cor XML-rlatd tchnologis that ar undr th auspics of th W3C, including XQury, XSLT, and XML Schma. Evaluating XPath quris fficintly is ssntial to th ffctivnss and ral-world impact of ths tchnologis. Th most natural qustion rlatd to XPath qury procssing, its complxity, howvr, has rcivd surprisingly littl attntion. Th first polynomial-tim algorithms for XPath procssing (w.r.t. both th siz of th data and th qury, i.., combind complxity, cf. [10]) wr proposd only rcntly [3]. Apparntly, th fact that quris can b valuatd in polynomial tim with rspct to combind complxity for th full XPath 1.0 languag was not folklor. W bliv that at th tim of writing this, all publicly availabl XPath ngins and systms procssing languags containing XPath (such as XQury or XSLT procssors) tak Prmission to mak digital or hard copis of all or part of this work for prsonal or classroom us is grantd without f providd that copis ar not mad or distributd for profit or commrcial advantag and that copis bar this notic and th full citation on th first pag. To copy othrwis, to rpublish, to post on srvrs or to rdistribut to lists, rquirs prior spcific prmission and/or a f. PODS 2003, Jun 9-12, 2003, San Digo, CA. Copyright 2003 ACM /03/06...$5.00. tim xponntial in th sizs of th XPath xprssions in th input. This thsis is supportd by xprimntal vidnc on a numbr of popular systms in [3]. Morovr, immdiat functional implmntations of th standards documnts, of th currnt standard, XPath 1.0 [13], as wll as th now proposd XPath 2.0 languag 1 (through th nw XML Qury 1.0 Algbra) [14], lad to xponntial-tim procssing of XPath 1.0 quris. Th polynomial-tim rsult of [3] was shown using a form of dynamic programming. Basd on this, w prsntd algorithms that run in tim O( D 5 Q 2 ) and spac O( D 4 Q 2 ), whr D dnots th siz of th data and Q is th siz of th qury. W also introducd th logical cor fragmnt of XPath, calld Cor XPath, which includs th logical and path procssing faturs of XPath but xcluds arithmtics and string manipulations. Cor XPath quris can b valuatd in tim O( D Q ), i.. linar in th siz of th qury and of th data. In a scond papr [4], w improvd th abov uppr bounds on th complxity to tim O( D 4 Q 2 ) and spac O( D 2 Q 2 ). Morovr, w dfind a larg fragmnt of XPath for which w providd a quadratic-tim, linar-spac valuation algorithm. W also pointd out th faturs of XPath causing th various incrass in th dgrs of polynomials stablishd whn moving from a smallr XPath fragmnt to a largr on. Now that th combind complxity of XPath is known to b polynomial, a natural qustion mrgs, namly whthr XPath is also P-hard (i.., hard for polynomial tim), or altrnativly, whthr it is in th complxity class NC, and thus ffctivly paralllizabl. In cas th problm is P-hard, it is intrsting to undrstand th sourcs of this hardnss, and to find larg, ffctivly paralllizabl fragmnts. This papr thus studis th prcis complxity of XPath 1.0 qury procssing. Th contributions ar as follows. W stablish th combind complxity of XPath to b P-hard. This rmains tru vn for th Cor XPath fragmnt. W show that positiv Cor XPath, i.. Cor XPath without ngation, is LOGCFL-complt, and thus highly paralllizabl. Morovr, if th languag is furthr rstrictd to th path xprssions fragmnt (PF) without conditions, 1 XPath 2.0 now includs most of XQury and thus is Turingcomplt; howvr, most ral-world path quris will rmain xprssibl in XPath 1.0, which is a strict fragmnt of XPath 2.0.

2 P-complt Cor XPath WF pwf pos. Cor XPath LOGCFL LOGCFL-complt PF NL-complt XPath P pxpath Figur 1: Combind complxity of XPath. th complxity of valuating quris is complt for nondtrministic logarithmic spac. W xtnd Cor XPath by th arithmtics faturs of XPath, to th so-calld Wadlr Fragmnt (WF), and show that a larg fragmnt of it, which w call pwf ( positiv / paralll WF), is still in LOGCFL and can b massivly paralllizd. Th main faturs xcludd from WF to obtain pwf ar ngation and squncs of condition prdicats. This lads us to an vn largr fragmnt of XPath, calld pxpath, which w bliv contains most practical XPath quris and for which qury valuation can b massivly paralllizd (th combind complxity is still LOGCFL-complt). Finally, w complmnt our rsults on th combind complxity of XPath with a study of data complxity and qury complxity. Both problms fall into low (highly paralllizabl) complxity classs vn in th prsnc of ngation in quris. Th inclusion rlationships 2 btwn fragmnts discussd and thir (combind) complxitis ar shown in Figur 1. An arrow L 1 L 2 mans that languag L 1 is a fragmnt of languag L PRELIMINARIES 2.1 Complxity Classs W brifly discuss th complxity classs and som of thir charactrizations usd throughout th papr. For mor thorough survys of th rlatd thory s [6, 7]. 2 In th drawing, w assum that NL LOGCFL P. By P, L, and NL w dnot th wll-known complxity classs of problms solvabl in dtrministic polynomial tim, dtrministic logarithmic spac, and nondtrministic logarithmic spac, rspctivly, on Turing machins. It is conjcturd that problms complt for P ar inhrntly squntial and cannot profit from paralll computation. A problm is instad calld highly paralllizabl if it can b solvd within th complxity class NC of all problms solvabl in polylogarithmic tim on a polynomial numbr of procssors working in paralll [5]. By NC i, w dnot th class of problms solvabl in tim O(log i n) using O(n i ) procssors (in trms of th siz n of th input). A simpl modl of paralll computation is that of boolan circuits. By a monoton circuit, w dnot a circuit in which only -gats and -gats (but no -gats) ar usd. A family of circuits is a squnc G 0, G 1, G 2,..., whr th n-th circuit G n has n inputs. Such a family is L-uniform if thr xists an L-boundd dtrministic Turing machin which, on th input of n bits 1, outputs th circuit G n. A circuit or family of circuits has boundd fan-in if all of its gats hav fan-in boundd by som constant. A smi-unboundd circuit is a monoton circuit in which all -gats ar of boundd fan-in (w.l.o.g., w may rstrict th fan-in to two) but th -gats may hav unboundd fan-in. Dfinition 2.1. SAC 1 is th class of problms solvabl by L-uniform familis of smi-unboundd circuits of dpth O(log n) (SAC 1 circuits). LOGCFL is usually dfind as th complxity class consisting of all problms L-rducibl to a contxt-fr languag. Thr ar two important altrnativ charactrizations that w ar going to us. Proposition 2.2 ([11]). LOGCFL = SAC 1. SAC 1 circuit valu is LOGCFL-complt. A nondtrministic auxiliary pushdown automaton (NAux- PDA) is a nondtrministic Turing machin with a distinguishd input tap, a worktap, an output tap, and a stack (of which strictly only th topmost lmnt can b accssd at any tim). Proposition 2.3 ([9]). LOGCFL is th class of all dcision problms solvabl by an NAuxPDA with a logarithmic spac-boundd worktap in polynomial tim. ([1]). LOGCFL is closd undr com- Proposition 2.4 plmnt. Rgarding th containmnt of ths classs, w know that NC 1 L NL LOGCFL NC 2 NC P. P, LOGCFL, and NL ar closd undr L-rductions. 2.2 A Brif Introduction to XPath XPath 1.0 is a languag with a larg numbr of faturs and thrfor somwhat unwildy for thortical tratmnt. In this papr, w rstrict ourslvs to introducing only som of ths faturs, and to giving an informal xplanation of thir smantics. For a dtaild dfinition of th full XPath languag, w rfr to [13], and for a concis yt complt formal dfinition of th XPath smantics s [3]. In this sction, w dfin two basic fragmnts of XPath. Cor XPath, first dfind in [3], supports th most commonly usd faturs of XPath, path navigation and conditions with logical connctivs, but xcluds arithmtics,

3 string manipulations, and som of th mor sotric aspcts of th languag. Th scond fragmnt, which was first discussd in [12] by Wadlr, contains XPath s logical and arithmtic faturs, but xcluds string manipulations. W rfr to it as th Wadlr Fragmnt, short WF. W start by discussing Cor XPath. W sktch th fragmnt in trms of its syntax and thn informally discuss th smantics. Dfinition 2.5. Th syntax of Cor XPath is dfind by th grammar locpath ::= / locpath locpath / locpath locpath locpath locstp. locstp ::= axis :: ntst [ bxpr ]... [ bxpr ]. bxpr ::= bxpr and bxpr bxpr or bxpr not( bxpr ) locpath. axis ::= slf child parnt dscndant dscndant-or-slf ancstor ancstor-or-slf following following-sibling prcding prcding-sibling. whr locpath is th start production, axis dnots axis rlations (s blow), and ntst dnots tags labling documnt nods or th star * that matchs all tags ( nod tsts ). Th main syntactical fatur of Cor XPath ar location paths. Exprssions nclosd in brackts ar calld conditions or prdicats. Th main application of XPath is th navigation in XML documnt trs. This is don using th axis rlations, natural binary rlations such as child and dscndant btwn nods, which w do not dfin hr (but s [13, 3]; thy also hav th intuitiv manings convyd by thir nams). Th probably most common us of XPath is to compos axis applications with slctions of documnt nods by thir tags ( nod tsts ). For instanc, th qury /dscndant::a/child::b slcts all thos nods labld b that ar childrn of nods labld a that ar in turn dscndants of th root nod (dnotd by th initial slash). Conditions nclosd in squar brackts allow to impos additional constraints on nod slctions. For xampl, /dscndant::a/child::b[dscndant::c and not(following-sibling::d)] slcts xactly thos nods v from th nods in th rsult of /dscndant::a/child::b that hav at last on 3 dscndant labld c and do not hav a right sibling in th tr that is labld d (i.., thr is no child v of th parnt of v which follows v in th flow of th documnt and is labld d ). Dfinition 2.6. Th syntax of th WF-Quris is dfind by th Cor XPath grammar with th following xtnsions. bxpr is now bxpr ::= bxpr and bxpr bxpr or bxpr not( bxpr ) locpath nxpr rlop nxpr. 3 Location paths occurring in conditions i.., within squar brackts hav an xists -smantics, maning that at last on nod must match th location path starting from th currnt nod. Morovr, xpr ::= locpath bxpr nxpr. nxpr ::= position() last() numbr nxpr arithop nxpr. arithop ::= + - * div mod. rlop ::= =!= < <= > >=. xpr (rathr than locpath ) is now th start production and numbr dnots constant ral-valud numbrs. Evn though XPath is mainly undrstood as a languag for slcting a subst of th nods of an XML documnt tr, qury rsults can also b of diffrnt typs, namly for th WF numbrs and boolans (as wll as charactr strings for full XPath). XPath xprssions ar valuatd rlativ to a contxt, which by dfinition is a tripl of a contxt nod and two intgrs, th so-calld contxt position and th contxt siz. For dtails, w rfr to [13, 3], but considr th xampl qury child::a[position() + 1 = last()]. Rlativ to a contxt-tripl (v, i, j), i and j ar ignord whn th location stp child::a slcts thos childrn of v that ar labld a. Lt {w 1,..., w m} b this st of nods, whr th indics corrspond to th rlativ ordr of th nods in th documnt, simply spaking 4. Th application of an axis causs a chang of contxt to which th condition [position() + 1 = last()] is applid. Th condition is trid on ach of th tripls (w 1, 1, m),..., (w m, m, m). It will slct all thos nods w k for which k + 1 = m, i.. th position k in th slction is by on smallr than th last indx m, th siz of th slction. Proposition 2.7 ([3]). XPath qury valuation is in P with rspct to combind complxity. Cor XPath quris can vn b valuatd in tim O( Q D ), whr Q dnots th siz of th qury and D dnots th siz of th data. 3. COMPLEXITY OF CORE XPATH In this sction, w show that XPath and vn Cor XPath ar P-hard with rspct to combind complxity. Rmark 3.1. In th proof of th following thorm, w oftn assign svral labls to on and th sam nod, vn though ach nod of an XML documnt can hav only on tag. W assum ths labls to b assignd, say, using attributs or by childrn additionally introducd for this purpos 5. W add condition xprssions of th form T (l) (whr l is a labl) to Cor XPath. For instanc, w can writ child::*[t (a)] in plac of child::a and now raliz and vrify multipl labls of on and th sam nod (imagin child::*[t (a) and T (b) and T (c)]). Thorm 3.2. Cor XPath is P-complt with rspct to combind complxity. Proof. Mmbrship of th combind complxity vn of full XPath was shown to b in P in [3], thus all w nd to show is P-hardnss. This is don by rduction from th monoton boolan circuit valu problm, which is P-complt [7]. 4 To b prcis, for som axs this ordr is rvrsd, s [13]. 5 T (l) could b a shortcut for child::l.

4 G 9 G 6 G 7 G 1 G 2 G 3 G 4 (a 1 ) (b 1 ) (a 0 ) (b 0 ) G 8 G 5 Figur 2: A 2-bit full addr carry-bit circuit. Givn an instanc of this problm (a monoton boolan circuit), lt M dnot th numbr of input gats and lt N dnot th numbr of all othr gats in th circuit. Lt th gats b namd G 1... G M+N. Without loss of gnrality 6, w may assum that th gats G 1... G M+N ar numbrd in som ordr such that no gat G i dpnds on th output of anothr gat G j with j > i. In particular, th input gats ar namd G 1... G M and th output gat is G M+N. An xampl of a circuit with appropriatly numbrd gats is shown in Figur 2. This circuit computs th carry-bit of a two-bit full-addr, i.. it tlls whthr adding th two-bit numbrs a 1a 0 and b 1b 0 lads to an ovrflow. Th carry-bit c 1 is computd as (a 1 b 1) (a 1 c 0) (b 1 c 0) whr c 0 = a 0 b 0 is th carry-bit of th lowr digit (a 0 and b 0). Th documnt tr is of vry simpl and rgular structur; it consists of a root nod v 0 with M + N childrn v 1... v M+N, of which ach v i again has xactly on child v i (thus, th tr has dpth thr). For our carry-bit xampl of Figur 2 with M = 4 and N = 5, th tr is v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 Nod labls ar takn from th alphabt {0, 1, G, R, I 1,..., I N, O 1,..., O N } and ach tr nod is assignd a st of such labls. This is don as follows. Th root nod v 0 has no labls. Th nods v 1... v M+N ar assignd th labl G ach. (In a way dscribd latr, nod v i rprsnts th valu of gat G i). Nod v M+N is also assignd labl R (for rsult ). Each nod out of v 1... v M is assignd th truth valu at th input gat of th sam indx (i.., out of G 1... G M), rspctivly. This is ithr th labl 0 or 1. Morovr, if th output of gat G i is an input of gat G M+k (thus, by our gat ordring rquirmnt, i < M + k), w add I k to th labls of v i and O k to th labls of v M+k. In our xampl, th nods v 1... v 9 ar labld 6 Th gats can b sortd to adhr to such an ordring in logarithmic spac. L 5 L 4 L 3 L 2 L 1 v 1 v 6 v 2 v 7 v 9 v 8 v 3 v 5 v 4 I 3 O 2 I 2 G 6 G 1 G 2 v(a 1 ) v(b 1 ) I I 4 4 G 8 I 2 G 7 G 9 I 5 I 5 I 5 O 3 O 5 O 4 I 1 G 5 I 3 O 1 I1 G 3 G 4 v(a 0 ) v(b 0 ) Figur 3: Circuit of Figur 2 with gats srializd. v 1: {G, v(a 1), I 2, I 3} v 2: {G, v(b 1), I 2, I 4} v 3: {G, v(a 0), I 1} v 4: {G, v(b 0), I 1} v 5: {G, O 1, I 3, I 4} v 6: {G, O 2, I 5} v 7: {G, O 3, I 5} v 8: {G, O 4, I 5} v 9: {G, R, O 5} whr v(a 1), v(b 1), v(a 0), v(b 0) {0, 1} ar th truth valus a 1, b 1, a 0, and b 0, rspctivly, at th input gats. Figur 3 shows how th I k and O k labls ar assignd to th nods v 1... v M+N. (Figur 3 is xplaind in mor dtail blow.) Finally, th nods v 1... v M ar labld {I 1,..., I N, O 1,..., O N } ach and th nods v M+i, for 1 i N, ar labld {I k, O k i k N}. Th qury valuating a circuit uss th intuition of procssing on gat out of G M+1... G M+N at a tim, in th ordr of ascnding indx. It is /dscndant-or-slf::*[t (R) and ϕ N ] with, for 1 k N, th condition xprssions and ϕ k := dscndant-or-slf::*[t (O k ) and parnt::*[ψ k ]] ψ k := not(child::*[t (I k ) and not(π k )]) if th typ of gat G M+k is and othrwis, and ψ k := child::*[t (I k ) and π k ] π k := ancstor-or-slf::*[t (G) and ϕ k 1 ]. Morovr, ϕ 0 := T (1). It is asy to s that th rduction can b ffctd in logarithmic spac. W nxt argu that it is also corrct. Discussion. W us th ordring of th circuit in that w, intuitivly, will valuat th circuit in Cor XPath on gat at a tim. W trat th circuit as if layrd, with all gats of a layr of th sam typ ( or ) and only

5 xactly on with fan-in gratr than on. (Our ncoding prmits unboundd fan-in, including on.) Figur 3 shows this altrnativ viw of th xampl circuit of Figur 2. Th N = 5 non-input gats hav bn alignd using fiv layrs L 1... L 5. Th smallr mpty circls dnot dummy gats of fan-in on, which ar ndd to propagat th valus of gats that ar alrady availabl to th layrs abov. In our ncoding, intuitivly, all gats of layr L k hav to hav th sam typ. Th typ of th dummy gats 7 in layr L k is thus dtrmind by th typ of th on gat of fan-in gratr than on (namly G M+k ). In th xampl, all gats of layrs L 1... L 4 ar of typ and th gats of layr L 5 ar all of typ. Th ϕ k, ψ k, and π k all ar condition xprssions, and thr is a natural maning to ϕ k matchs nod w or quivalntly nod w satisfis ϕ k, which w will dnot as w [ϕ k ] blow. Formally, w [ϕ k ] if and only if qury /dscndant-or-slf::*[ϕ k ] slcts nod w. W dfin w [ψ k ] and w [π k ] analogously. This notation hlps to imagin th qury (tr) bing procssd bottom-up. Claim. Lt 0 k N and 1 i M + k. Thn, v i [ϕ k ] gat G i valuats to tru. This can b shown by an asy induction. Induction start (k = 0). Th gats G 1... G M+k ar prcisly th input gats, which hav bn assignd thir initial valu (ithr 0 or 1) as labl. Th labl 1 is not usd lswhr in th tr. By dfinition, ϕ 0 is th xprssion T (1), so v i [ϕ 0 ] iff th valu of th input gat G i is 1. Thus our claim holds for k = 0. Induction stp. Now assum that our claim holds for k 1 0 (i.., v i [ϕ k 1 ] iff gat G i has bn stablishd to b tru by stp k 1). W show that it also holds for k. W procd by computing first [π k ], thn [ψ k ], and finally [ϕ k ]. On can vrify by inspction of π k that [π k ] = {v i, v i 1 i M + k, v i [ϕ k 1 ]} (Not that only th nods v 1... v M+k ar labld G.) ψ k is at th hart of our construction and prforms th actual computation of th M + k gats at layr k. Ths gats ar th M + k 1 dummy gats of fan-in on which just propagat th input and thus mak sur that th truth valu of gat G i (1 i < M + k), onc computd, rmains availabl to layrs abov, and th gat G M+k of fan-in gratr than on. In our documnt, nod v i is labld I k iff gat G M+k taks input from gat G i. By th dfinition of ψ k, v 0 [ψ k ] if and only if all childrn (for G M+k a -gat) or at last on child (for G M+k a -gat) that is labld I k matchs π k. Thus, v 0 [ψ k ] G M+k valuats to tru. For th dummy gats, th I k labls ar on lvl dpr down in th tr. (Th sol purpos of π k was to push th prviously computd valu of G i as a matching on nod v i to two diffrnt dpths to allow for diffrnt handling of dummy gats and gat G M+k by th sam XPath xprssion ψ k.) Nod v i has th labl I k for ach 1 i < M + k. Th parnt of v i howvr, v i, has only on child (namly, v i), so 7 In fact, th typs of gats of fan-in on do not mattr in th circuit: th conjunction as wll as th disjunction of a singl truth valu is th idntity. For this rason, and to sav spac, w do not show th typs of th dummy gats in Figur 3. G, I k v i π k v 0 ϕ k, ψ k π k I k, O k v ϕ k G v i π k, ψ k, ϕ k v i (a) π k, ϕ k G, I k v j π k π k G, O k v i v j v k+m (b) v k+m ϕ k Figur 4: Schmatic dsign of rlvant tr rgion and ϕ k /ψ k /π k -matchings mad for (a) dummy gats and (b) gats of fan-in gratr than on (hr, two). it dos not mattr whthr ψ k is of th - or th -typ. For 1 i < M + k, v i [ψ k ] gat G i valuats to tru. [ψ k ] computs (or prsrvs, in th cas of dummy gats) th truth valus of th gats, but th matchings that witnss ths truth valus nd up bing at diffrnt dpths in th tr, dpnding on th gat. ϕ k stors th matchings at th sam dpth (th nods v 1... v M+k ) for 1 i M + k: v i [ϕ k ] gat G i valuats to tru. This provs our claim. (ϕ k also matchs othr nods abov and blow v 1... v M+k, but this dos not mattr bcaus ths nods ar labld nithr G nor R.) Th schmatic dsigns of Figur 4 show th rgions of th documnt tr that w ar intrstd in, th rlvant labls, and th matchings of condition xprssions ϕ k, ψ k, and π k at stp k, for both cass of gats (dummy gats in Figur 4 (a) and gats G M+k in Figur 4 (b)). Th ovrall qury /dscndant-or-slf::*[t (R) and ϕ N ] has a nonmpty rsult xactly if th output gat G M+N of th circuit valuats to tru, bcaus v M+N is th only nod labld R and v M+N [ϕ N ] if and only if G M+N valuats to tru. Corollary 3.3. Cor XPath rmains P-hard vn if 1. th documnt tr is limitd to dpth thr and 2. only th axs child, parnt, and dscndant-or-slf ar allowd. Proof. Th prvious proof has th statd proprtis, xcpt that it uss th ancstor-or-slf axis in th dfinition of π k. All w nd to do is to rplac ancstor-or-slf::* in

6 π k by dscndant-or-slf::*/parnt::*. π k thn additionally matchs th root nod v 0, but this dos not mattr to th rmaindr of th construction bcaus v 0 nvr carris an I k labl and thus nvr has an impact on ψ k. W ovrstatd th rquird tr dpth in Corollary 3.3 to allow for multipl nod labls to b ncodd as additional childrn, as discussd in Rmark 3.1. Th documnt trs of th ncoding of th proof of Thorm 3.2 ar only of dpth two. Not also that th quris usd in th ncoding ssntially do not branch out in trms of axis applications. That is, in ach conjunction ( or is not usd) of xprssions, thr is at most on subxprssion that contains an axis application. 4. INSIDE CORE XPATH Th rsult of th prvious sction is ssntially ngativ: As Cor XPath is P-hard, it is considrd unlikly that a paralll algorithm xists for valuating all quris of this languag. It is thus natural to sarch for fragmnts of Cor XPath that w can show to b in NC and thrfor highly paralllizabl. In fact, such a fragmnt is obtaind by rmoving ngation ( not ) from Cor XPath. This fragmnt will b calld positiv Cor XPath. Thorm 4.1. Th combind complxity of positiv Cor XPath is in LOGCFL. W postpon th proof of this thorm to th nxt sction, whr w will nginr a strictly and considrably largr LOGCFL fragmnt of XPath (s Thorm 5.5). As w will s nxt, th gnral proof tchniqu usd to show Thorm 3.2 can b mployd to prov furthr hardnss rsults of XPath fragmnts (insid P) using circuits. (Rcall that th proof of Thorm 3.2 dos not rquir gats to hav boundd fan-in.) Thorm 4.2. Positiv Cor XPath is LOGCFL-hard w.r.t. combind complxity. Proof (Sktch). By rduction from SAC 1 circuit valu, which is LOGCFL-complt (s Proposition 2.2). Givn a SAC 1 circuit, w us th construction of th proof of Thorm 3.2 with th following changs: In th documnt, thr ar xactly two I k -labls now for ach -layr. W call thm I 1 k and I 2 k. For dummy - gats propagating th valu of gat G i, th singl input lin, nod v i, is assignd both I 1 k and I 2 k. W construct quris as usual, but instad of ngation (which allows to xprss an unboundd for all ), w us th XPath languag construct and with two inputs, on labld I 1 k and on I 2 k. That is, for gats of typ, ψ k is rplacd by ψ k := child::*[t (I 1 k) and π k ] and child::*[t (I 2 k) and π k ] Thus at vry -stp of th qury, th subxprssion of th qury nds to b insrtd twic. Although th qury grows xponntially in th dpth of th circuit, it can b computd in L bcaus th dpth of th circuit (and thus th siz of subxprssions to b copid) is only logarithmic. v 1 v 4 v 2 v 3 (a) (b) (c) c Figur 5: Graph (a), its (transposd) adjacncy matrix (b), and tr of th ncoding (c). All ls of th rduction rmains th sam. Lt PF b th fragmnt of Cor XPath containing only th location paths, without conditions (i.., no xprssions nclosd in brackts ar prmittd). Thorm 4.3. With rspct to combind complxity, PF is NL-complt undr L-rductions. Proof (Sktch). Mmbrship in NL is obvious: w can just guss th path whil w vrify it in L. NL-hardnss follows from a L-rduction from th dirctd graph rachability problm, which is NL-complt (cf. [7]). Th rduction is quit simpl, so w just provid an xampl (s Figur 5). Lt G = (V, E) b th dirctd input graph. Assum w look for a path from nod v i to nod v j. W abbrviat th n-tims rpatd application of an axis χ as χ n ::*. By χ n ::c, w dnot (χ::*/) n 1 χ::c. As is asy to vrify, givn a positiv intgr m, th qury /dscndant::v i/ϕ m with ϕ k := child::c/dscndant::/parnt 2 V ::*/child V ::c/ parnt::*/ϕ k 1 and ϕ 0 := slf::v j computs th nod labld v j if and only if nod v j is rachabl from nod v i in m stps. To xtnd this to rachability, w add a loop for ach nod of th graph (or quivalntly, st th main diagonal of th adjacncy matrix to ons only) and hav m := E. 5. PARALLELIZING WF W ar now going to sarch for rstrictions on WF that push down th complxity of th qury valuation problm to th highly paralllizabl complxity class LOGCFL. To achiv this, w rquir that scalar valus (i.., valus diffrnt from nod sts) can b stord in logarithmic spac. Morovr, w also hav to xclud two important constructs from WF, namly itratd prdicats of th form χ :: t[ 1]... [ k ] with k 2 and th not-function. Th rsulting XPath fragmnt will b rfrrd to as th positiv (or paralll ) WF (short pwf). It is formally dfind as follows: Dfinition 5.1. pwf is obtaind by rstricting WF in th following way: c c c v 1 v 2 v 3 v 4

7 1. Exprssions of th form χ :: t[ 1]... [ k ] with k 2 ar not allowd, whr χ dnots an axis, t is a nod tst and th i s ar XPath xprssions. 2. Th not-function may not b usd. 3. Th nsting dpth of arithmtic oprators is boundd by som constant k. Th first two rstrictions abov man that th grammar from Dfinitions 2.5 and 2.6 has to b modifid as follows: locstp ::= axis :: ntst [ bxpr ] bxpr ::= bxpr and bxpr bxpr or bxpr locpath nxpr rlop nxpr. Rmark 5.2. Not that positiv Cor XPath is strictly a fragmnt of pwf. This is du to th fact that th first rstriction abov plays no rol in Cor XPath. Mor gnrally, an XPath xprssion of th form χ :: t[ 1]... [ k ] is quivalnt to χ :: t[ 1 and... and k ] as long as position() and last() ar not usd. Th classical dcision problm rgarding qury valuation (also known as th Succss problm) is, givn a databas, a qury, and a qury rsult, to dcid whthr th givn qury rsult is corrct for th givn qury on th input databas. In our contxt, th XML documnt taks th plac of th databas, th XPath qury in conjunction with a contxttripl th plac of th qury, and finally, a valu that is ithr of typ boolan, numbr, string, or nod st assums th plac of th qury rsult to b chckd. For our purposs, it is convnint to work with th following slightly diffrnt dcision problm. Dfinition 5.3 (Singlton-Succss). Input: A tupl (D, Q, c, v), whr D is an XML documnt, Q an XPath qury, c a contxt-tripl, and v a valu. If Q is of typ numbr or string, thn v is a valu of this typ. If Q is of typ boolan, thn v is th valu tru. Finally, if Q is of typ nod st, thn v is a singl nod. Qustion: Dos qury Q on documnt D and contxt c valuat to v (in cas of rsult typ numbr, string or boolan) or dos it valuat to som nod st X with v X? Lmma 5.4. Th Singlton-Succss problm for pwf can b dcidd in LOGCFL. Proof (Sktch). W dscrib a NAuxPDA that dcids th Singlton-Succss problm for quris from pwf. Th LOGCFL-mmbrship of this problm will follow immdiatly from th corrctnss and th fact that th NAux- PDA runs simultanously with a logarithmic spac-boundd worktap and in polynomial tim. Notation. Lt an instanc of th Singlton-Succss problm b givn through som XML documnt D, XPath qury Q, contxt-tripl c and valu v. By T Q, w dnot th pars tr of Q. Th root nod of T Q will b dnotd by R. Rcall that vry nod N in T Q corrsponds to a subxprssion of Q, which w shall dnot by xpr(n). Finally, w writ K to dnot th maximum numbr of child nods of th nods in T Q. Actually, for pwf, K = 2 holds. Basics of th NAuxPDA. Th principal ida of th NAux- PDA is to travrs T Q along its dgs in dpth-first, lft-toright ordr. Along this travrsal, w basically pass through vry nod N in th qury tr T Q at most onc in downward dirction and at most K tims in upward dirction. If w visit a nod N in downward dirction, thn w mak som gusss (namly, a contxt c and th corrsponding rsult of valuating th subxprssion xpr(n) of Q on th documnt D for this contxt c ). If w procss a nod N in upward dirction and no furthr child nod of N has to b procssd (i.., ithr, thy hav all bn procssd or th rsult valu of xpr(n) is fully dtrmind by thos which hav alrady bn procssd), thn w carry out crtain consistncy chcks btwn th gusss at th currnt nod and at its child nods. Similarly, if w rach a laf nod N, thn w hav to chck whthr th contxt and th rsult valu gussd at N ar consistnt with th xprssion xpr(n). Our travrsal of th qury tr T Q starts at th root R of T Q in downward dirction. Evntually, w shall hav visitd all nods of T Q that ar rquird to dtrmin th ovrall rsult and w shall com back to th root R of th qury tr. If all th consistncy chcks thus carrid out wr succssful, thn th ovrall rsult of our computation is succss. As soon as on such chck fails, w halt with th ovrall rsult failur. Worktap and stack of th NAuxPDA. On our worktap, w maintain two intgrs CurrN and AuxN as wll as a variabl Dir which can hav on of th valus down or up. Morovr, our worktap contains th K + 2 main data structurs CurrVal, AuxVal, and ChildVal[i] with i {1,..., K}, ach consisting of four componnts cnod, cpos, csiz, and rs, which stand for contxt-nod, contxt-position, contxt-siz, and rsult, rspctivly. Th variabls CurrN and AuxN hold nod-ids of nods in th qury tr. CurrN dnots th currnt nod in th qury tr and AuxN is an auxiliary variabl. Dir dnots th dirction of th last mov (i.., ithr downward or upward) of our travrsal of T Q. CurrVal and ChildVal[i] with i {1,..., K} contain th valus of cnod, cpos, csiz, and rs that wr gussd whn procssing th currnt nod in th qury tr and th i-th child of this nod, rspctivly. AuxVal is usd as an auxiliary variabl for copying purposs. All of ths data structurs ar initially st to th valu undf. Th valus of CurrVal, ChildVal[1],..., ChildVal[K], and CurrN ar pushd onto th stack whn a nod is lft in downward dirction. Convrsly, ths valus ar poppd from th stack bfor a nod is ntrd in upward dirction. Consquntly, whnvr w start to procss a nod N, thn th stack contains th valus CurrVal, ChildVal[1],..., ChildVal[K], and CurrN for all nods along th path from R to th parnt of N. Initialization and main procdur. In th bginning, w slct th root nod R of T Q as th currnt nod CurrN and fill in th contxt and rsult valu from th input into th variabl CurrVal (rathr than gussing ths valus, as w shall do for all furthr nods). All th othr data structurs ChildVal[i] ar initializd to undf. If R is a laf nod (i.., T Q consists of th root R only), thn w chck th consistncy of th contxt and rsult valu in CurrVal with th XPath xprssion xpr(r). If this chck is succssful, thn th ovrall rsult of th NAuxPDA is succss, othrwis th NAuxPDA halts with failur. On th othr hand, if R is not a laf nod, thn w hav to mov downward in th qury tr. If th XPath xprssion xpr (R) is of th form 1 or 2 or of th form π 1 π 2, thn w choos nondtr-

8 xprssions xpr(currn) at laf nods of th qury tr T Q xpr(currn) local consistncy condition χ :: t r can b rachd from n via χ :: t position() r = p last() r = s c (= constant numbr) r = c xprssions xpr(currn) at intrnal nods of th qury tr T Q xpr(currn) local consistncy condition /π n = root r = r 1 π 1 π 2 (n = n 1 r = r 1) (n = n 2 r = r 2) π 1/π 2 (n = n 1 n 2 = r 1 r = r 2) χ :: t[] lt Y = {y dom y can b rachd from n via χ :: t} (first child of CurrN corrsponds r Y (lt p nw = position of r in Y, lt s nw = Y to, th scond on to χ :: t) n 1 = r p 1 = p nw s 1 = s nw r = tru) boolan(π) r = tru (n 1 = n p 1 = p s 1 = s r 1 dom) 1 and 2 r = tru [(n 1 = n p 1 = p s 1 = s r 1 = tru) (n 2 = n p 2 = p s 2 = s r 2 = tru)] 1 or 2 r = tru [(n 1 = n p 1 = p s 1 = s r 1 = tru) (n 2 = n p 2 = p s 2 = s r 2 = tru)] 1 RlOp 2 r = tru r 1 RlOp r 2 [(n 1 = n p 1 = p s 1 = s) (both 1 and 2 ar numbrs) (n 2 = n p 2 = p s 2 = s)] 1 ArithOp 2 r = r 1 ArithOp r 2 [(n 1 = n p 1 = p s 1 = s) (both 1 and 2 ar numbrs) (n 2 = n p 2 = p s 2 = s)] lgnd: n, p, s, r: CurrVal.cnod CurrVal.cpos CurrVal.csiz CurrVal.rs n 1, p 1, s 1, r 1 : ChildVal[1].cnod ChildVal[1].cpos ChildVal[1].csiz ChildVal[1].rs n 2, p 2, s 2, r 2 : ChildVal[2].cnod ChildVal[2].cpos ChildVal[2].csiz ChildVal[2].rs Tabl 1: Local consistncy chcks for pwf. ministically a singl child of R (and ignor th whol subtr of T Q rootd at th othr child nod of R). Othrwis w mov on to th first child of R. In ithr cas, th currnt valus of CurrN, CurrVal, ChildVal[1],..., ChildVal[K] ar pushd onto th stack and CurrN is assignd th nod-id of th lmnt nod to b visitd nxt. Procssing a nod in downward dirction. If a nod is ntrd in downward dirction, thn w guss th componnts of CurrVal. Aftr that w basically procd lik in th main procdur xplaind abov. In particular, th data structurs ChildVal[i] ar initializd to undf. Morovr, if th currnt nod is not a laf nod, thn w mak th sam downward mov as in th main procdur. Othrwis, if th currnt nod is a laf nod, thn w carry out th sam consistncy chck as bfor. In cas of a ngativ rsult of this chck, w again halt with failur. Howvr, in cas of a succssful chck, w ar of cours not yt allowd to halt with succss. Instad, w mov upward in th qury tr. For this upward mov, w sav th currnt valu of CurrN and CurrVal to th auxiliary variabls AuxN and AuxVal, rspctivly. Thn w pop CurrN, CurrVal, ChildVal[1],..., ChildVal[K] from th stack and finally assign AuxVal to th appropriat variabl ChildVal[1],..., ChildVal[K], i.., if th upward mov startd at th i-th child of its parnt, thn AuxVal is assignd to ChildVal[i]. Procssing a nod in upward dirction. Now suppos that w hav ntrd th currnt nod by an upward mov from its i-th child. If th currnt nod contains a child that has to b procssd yt, thn w mov on to this child by a downward mov. Of cours, prior to this mov, th variabls CurrN, CurrVal, ChildVal[1],..., ChildVal[K] hav to b pushd onto th stack. Othrwis, if thr ar no mor child nods lft to b procssd, thn th consistncy of th valus CurrVal and ChildVal[1],..., ChildVal[K] with th xprssion xpr(currn) has to b chckd. If this chck is succssful, thn w mak th sam kind of upward mov as in cas of a laf nod that is procssd in downward dirction. Othrwis, w halt with failur. Consistncy chcks for th currnt nod. If a laf nod in th qury tr has bn rachd, thn w hav to chck whthr th chosn combination of th componnts of CurrVal is indd allowd for th subxprssion xpr(currn). Similarly, if th contxt and rsult valu hav alrady bn dtrmind for a non-laf nod plus th rquird child nods, thn w hav to chck whthr th nondtrministic choics wr consistnt with th subxprssion xpr(currn). All possibl kinds of chcks thus rquird ar givn in Tabl 1, whr w us th following notation: W writ χ :: t for a location stp consisting of an axis χ and a nod tst t. stands for any XPath xprssion whil π stands for a location path. By RlOp and ArithOp w dnot rlational oprators (=,,,... ) and arithmtic oprators (+,,,... ), rspctivly. Th st of all lmnt nods in D is dnotd by dom and root dnots th concptual root nod in th XPath data modl (cf. [13]). Morovr, w assum w.l.o.g., that typ convrsions from nod sts to boolan valus ar mad xplicit via th XPath-function boolan. Finally, w do not trat th cas of undfind componnts sparatly. In gnral, w assum that conditions on undfind valus yild th rsult undf. But of cours, w assum that tru undf tru holds. Discussion. As for th corrctnss of this NAuxPDA, it

9 has to b shown that an instanc (D, Q, cn, cp, cs, v) of th Singlton-Succss problm of pwf yilds th answr ys iff thr xists a run of th NAuxPDA that halts with succss. A dtaild proof of this is providd in th full papr. It rmains to b shown that th NAuxPDA works simultanously in L and P. As for th tim complxity, rcall that th NAuxPDA travrss (parts of) th qury tr T Q in dpth-first, lft-to-right ordr. Along this travrsal, vry nod N is procssd at most onc in downward dirction and at most K tims (with K = 2 in cas of pwf) in upward dirction. Morovr, th actions rquird to procss a nod onc can b clarly don in polynomial tim. As for th spac complxity, not that th variabls of th worktap plus a fixd numbr of countrs and auxiliary variabls clarly fit into logarithmic spac. Th crucial obsrvation for th logarithmic spac complxity is that w nvr hav to xplicitly comput nod sts,.g., chcking r Y and dtrmining th position of r in Y and th siz of Y can b don without xplicitly computing th nod st Y itslf (cf. th consistncy chck for χ :: t[] in Tabl 1). Thorm 5.5. pwf is in LOGCFL with rspct to combind complxity. Proof (Sktch). Th NAuxPDA of th prvious proof nondtrministically gusss and vrifis a rsult, or in th cas of quris rturning a nod st, a singl nod of th rsult. Chcking whthr a givn XPath qury valuats to som nod st X (or quivalntly, computing that nod st) can b don by dciding th Singlton-Succss problm in a loop ovr all lmnts v X without a significant incras of th ovrall complxity. In our dfinition of th Singlton- Succss problm, w assumd th tchnical rstriction that rsults of boolan XPath quris can b only chckd to b tru. Chcking whthr a givn XPath qury with boolan rsult valu valuats to fals is th co-problm of chcking whthr a qury valuats to tru. Howvr, by Proposition 2.4, LOGCFL is closd undr complmntation. Rmark 5.6. It is wll-known that th complxity class LOGCFL is insid th class NC 2 of problms solvabl in tim O(log 2 n) with quadratically many procssors working in paralll. In fact, givn this intuition and th insight obtaind from th rduction to NAuxPDA, it is not hard to find a highly paralll algorithm for valuating pwf quris. Th intuition for matching straight-lin path quris (cf. our PF fragmnt from Sction 4) is similar to paralll algorithms for graph rachability (cf. [7]); howvr, rathr than conncting nods in a graph, th goal is to connct contxts with nods in th qury rsult. Additional synchronization is rquird for branchs in th qury tr (.g. and ), which is not surprising as graph rachability is in NL and thus prsumably simplr than a LOGCFL-complt problm. Howvr, at th branchs, th subxprssions blow can b valuatd in paralll bfor finalizing th branch (i.., procding bottom-up). Th nxt rsult shows that pwf is in a sns a maximal LOGCFL fragmnt of WF. Of cours, w ar not rally intrstd in daling with arbitrarily big numbr xprssions. As far as th othr two rstrictions in Dfinition 5.1 ar concrnd, non of thm can b simply omittd. This is clar for ngation, as Cor XPath is strictly a fragmnt of pwf xtndd by ngation. As shown nxt, th final rstriction is ssntial as wll. By itratd prdicats, w again rfr to location stps of th form χ :: t[ 1]... [ k ] with k 2. Thorm 5.7. Th combind complxity of pwf quris xtndd by itratd prdicats is P-complt. Proof (Sktch). Th proof gos by an appropriat modification of th XML documnt D and th location paths ϕ, ψ, and π from th proof of Thorm 3.2. XML documnt. W xtnd D by adding on additional child w i to vry nod v i with i {0,..., M + N} (as th right-most child, say). Each nod w i is labld W. Hnc, th condition T (W ) is fulfilld xactly by ths nw nods. Morovr, for th nod v 0, w introduc an additional labl A ( auxiliary ). Thus, th condition T (A) is only fulfilld by th root nod v 0. Th nw documnt tr corrsponding to th xampl in th proof of Thorm 3.2 looks as follows: v 0 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 1 w v 2 v 3 1 w 2 w 3 w 6 w 8 w 9 v 4 w v 5 w v w 0 v 7 w v 8 7 XPath qury. Th dsird qury Q (ncoding th valu of th gat G M+N ) is /dscndant-or-slf::*[t (R) and ϕ N ] whr th auxiliary location paths ϕ k, ψ k, and π k with 1 k N ar dfind as follows: and ϕ k := dscndant-or-slf::*[t (O k ) and parnt::*[ψ k]] ψ k := child::*[(t (I k ) and π k[last()=1]) or T (W )][last()=1], if th typ of gat G M+k is and othrwis, and ψ k := child::*[t (I k ) and π k[last() > 1]] π k := ancstor-or-slf::*[(t (G) and ϕ k 1) or T (A)]. Morovr, w st ϕ 0 := T (1) as in Thorm 3.2. In ordr to prov th corrctnss of this problm rduction, w introduc th following notion: Lt ρ and σ b XPath quris that ar valuatd on th documnt D or on D, rspctivly. Thn w call ρ and σ quivalnt on a nod x (that occurs both in D and D ), iff th valuation of boolan(ρ) on D and th valuation of boolan(σ) on D for th contxtnod x yild th sam rsult. Thn th following quivalncs hold for any k 1 (in cas of ψ k and π k ) and for any k 0 (in cas of ϕ k ): (1) ϕ k and ϕ k ar quivalnt on v 1,..., v M+N. (2) ψ k and ψ k ar quivalnt on v 0,..., v M+N. (3) π k and π k[last() > 1] as wll as not(π k ) and π k[last() = 1] ar quivalnt on v 1,..., v M+N, v 1,..., v M+N. Ths quivalncs can b shown by an asy induction argumnt. W only discuss (3) hr, in ordr to point out v 9

10 th cntral ida of ncoding th not-function by itratd prdicats: By th induction hypothsis, ϕ k 1 and ϕ k 1 ar quivalnt on v 1,..., v M+N, that is, on th nods for which th condition T (G) is tru. By th nw disjunct T (A) in th prdicat of π k, th location path π k valuats to th sam st as π k plus th nod v 0. Morovr, by th condition T (G), th nod st rsulting from π k nvr contains th nod v 0. Th quivalnc of π k with π k[last() > 1] and th quivalnc of not(π k ) with π k[last() = 1] (on v 1,..., v M+N, v 1,..., v M+N ) ar thus obvious. Th corrctnss of th whol problm rduction follows immdiatly from th quivalnc (1) for k = N. Not that in th abov proof of Thorm 5.7, w only mad us of prdicat squncs [ 1]... [ k ] whos lngth k was boundd by 2. W thus hav: Corollary 5.8. Th combind complxity of pwf xtndd by itratd prdicats of th form χ :: t[ 1][ 2] is P- complt. Dspit th ngativ rsults of Thorms 3.2 and 5.7, on possibl dirction into which pwf can b xtndd without laving th complxity class LOGCFL is to bound th dpth of ngation, i.., th maximum dpth of nstd occurrncs of th not-function in quris. Thorm 5.9. Th combind complxity of pwf quris augmntd by ngation with boundd dpth is in LOGCFL. Proof (Sktch). W modify th NAuxPDA ncoding in th proof of Lmma 5.4 as follows: First, w transform th input qury Q by mans of d Morgan s laws in such a way that all occurrncs of th not-function ar ithr shiftd immdiatly in front of rlational oprators RlOp or location paths π. Exprssions of th form 1 RlOp 2 whr both oprands ar numbrs can b rplacd by 1 not(rlop) 2. In othr words, = is rplacd by, < is rplacd by, tc. W thus gt an quivalnt qury Q whr th not only occurs in th form not(π). Thn w hav to modify our NAuxPDA in such a way, that it trats subxprssions of th form not(π) by chcking in a loop ovr all lmnt nods x in D whthr x is in th nod st rsulting from th valuation of π for th contxt-nod CurrVal.cnod. This chck is don by calling th NAuxPDA rcursivly. Th corrctnss of this algorithm follows immdiatly from th corrctnss of th NAuxPDA in th proof of Lmma 5.4. Th spac complxity clarly dos not significantly incras compard to th NAuxPDA in Lmma 5.4. Morovr, by th xistnc of a constant bound K on th nsting of ngation (and, hnc, by th sam bound K on th nsting of th loops in th abov dscribd modifid vrsion of our NAuxPDA), th polynomial tim uppr bound on this computation is also prsrvd. 6. PARALLELIZING XPATH In Sction 5, w provd th LOGCFL-mmbrship for a fragmnt of XPath that was drivd from WF by imposing som rstrictions. In fact, w can gt a much largr LOGCFL fragmnt of XPath by starting from full XPath and dfining th analogous rstrictions. Th important fact is again that th valuation can b don without th nd to vr comput nod sts xplicitly and without having to dal with scalars (i.., valus diffrnt from nod sts) that do not fit into L. Analogously to pwf, w thus dfin: Dfinition 6.1. Positiv (or paralll) XPath (pxpath) is obtaind by imposing th following rstrictions on XPath: 1. Exprssions of th form χ :: t[ 1]... [ k ] with k 2 ar not allowd. 2. Th following functions may not b usd: not, count, sum, string, and numbr as wll as th string functions local-nam, namspac-uri, nam, string-lngth, and normaliz-spac. 3. Constructs of th form 1 RlOp 2 whr at last on of th xprssions i is of typ boolan, ar forbiddn. 4. Th dpth of nsting of arithmtic oprators and of th concat-function is boundd by som givn constant K. Likwis, th arity of th concat-function is boundd by K. Th abov rstrictions xtnd th ons in Dfinition 5.1 in th following way: Th valuation of xprssions of th form count() and sum() rquirs th xplicit computation of th nod st valu of unlss w again introduc loops ovr dom into th NAuxPDA as in Thorm 5.9. With th functions string and numbr as wll as th string functions listd abov, w would hav to manipulat itms of information in th documnt D whos siz is not ncssarily logarithmically boundd. Morovr, th functions string and numbr could also b usd to ncod ngation,.g., numbr() = 0 for a boolan xprssion valuats to tru, iff valuats to fals. Similarly, constructs of th form 1 RlOp 2 whr at last on of th xprssions i is of typ boolan ar forbiddn sinc thy can also b usd to ncod ngation,.g., by an xprssion of th form tru(). Analogously to Thorm 5.5, w hav Thorm 6.2. Th combind complxity of pxpath is in LOGCFL. Proof (Sktch). Th LOGCFL-mmbrship of th problm Singlton-Succss and (as a consqunc, s th proof of Thorm 5.5) th combind complxity of pxpath can b stablishd by almost th sam NAuxPDA as in Sction 5. Th only adaptation rquird is an xtnsion of Tabl 1. In principl, for ach of th additionally allowd XPath constructs, a nw lin with th corrsponding local consistncy chck has to b addd. Altrnativly, w can covr ths nw lins via th ffctiv smantics function F of XPath oprators Op that was introducd in [3] for all XPath constructs xcpt for location paths and th functions position() and last() (whos smantics was dfind sparatly in [3]). Thn th consistncy chck for an xprssion of th form Op( 1,..., l ) coms down to th condition ( l i=1 (n i = n p i = p s i = s)) r = F [Op](r 1,..., r l ) Morovr, in cas of a boolan xprssion, w add th conjunct r = tru. Not that th last four lins in Tabl 1 ar (quivalnt formulations of) spcial cass of this principl with F [and] =, F [or ] =, F [RlOp] = RlOp, and F [ArithOp] = ArithOp, rspctivly. Of cours, th polynomial tim and logarithmic spac uppr bound on th complxity also holds for th thus xtndd NAuxPDA. Finally, w mntion that also pxpath can b xtndd by ngation with boundd dpth without dstroying th

11 LOGCFL-mmbrship. Th following rsult is statd without proof. Thorm 6.3. Th combind complxity of pxpath augmntd by ngation with boundd dpth is in LOGCFL. Concptually, this can b shown xactly lik Thorm 5.9. Howvr, thr ar now quit a fw nw constructs which hav to b considrd sparatly sinc ngation cannot b shiftd insid thm,.g., not( 1 RlOp 2) whr at last on of th oprands i is a nod st has to b tratd in a loop ovr all nods x dom just lik xprssions of th form not(π) in Thorm QUERY AND DATA COMPLEXITY In this papr, w hav addrssd th combind complxity of various fragmnts of XPath. Whil th gnral problm is P-hard, w hav nginrd larg fragmnts that can b massivly paralllizd. W conclud this tratmnt with an outlook towards th two main othr complxity masurs, th complxity of quris whn ithr th siz of th qury or of th data is fixd. Thorm 7.1. PF is L-hard undr NC 1-rductions (with rspct to data complxity). Proof. Givn a tr in which all nods hav a uniqu labl, th qury /dscndant-or-slf::v 1/dscndant::v 2 from our path xprssions fragmnt PF (s Sction 4) slcts a nod if and only if v 2 is rachabl from v 1 in th tr. This is dirctd tr rachability, which is L-complt undr NC 1- rductions [2]. Th qury is constant and can b assumd to work on th sam tr as th dirctd tr rachability problm. Th rsult follows. Thorm 7.2. XPath is in L w.r.t. data complxity. Proof (Sktch). Th basic ida for an XPath valuation algorithm that runs in L is motivatd by th bottom-up dynamic programming algorithm for full XPath of [3], which was basd on th notion of so-calld contxt-valu tabls, rlations consisting of tupls containing a contxt and a corrsponding valu for (a subxprssion of) th givn qury, on tupl for ach maningful contxt. W comput on such contxt-valu tabl for ach nod of th qury tr. Givn th contxt-valu tabls for th dirct subxprssions 1,..., n, computing th contxt-valu tabl of xprssion Op( 1,..., n), whr Op is an atomic XPath opration (a nod in th qury tr), only rquirs a vry simpl computational task which can b carrid out in L. Sinc w considr data complxity, th qury and th numbr of oprations in its qury tr is assumd fixd. W can compos a fixd numbr of stps that individually run in L into an algorithm that runs in L ovrall. Thorm 7.3. XPath without multiplication or th concat opration is in L w.r.t. qury complxity. Proof (Sktch). Lt Q b th input qury and D th (fixd) documnt. All oprations in Q hav a fixd arity not gratr than K = 3. Lt us first assum that Q dos not contain oprations such as + that mak strings or numbrs grow (logarithmically) with th siz of th qury. Thn, it is known from [3] that th siz of ach contxt-valu tabl is boundd by th constant D 4. To comput th contxt-valu tabl holding th rsult of Q on D, w simply hav to mak a bottom-up travrsal of th qury tr of Q, which can b prformd in L. Rgarding storag rquirmnts, only a stack boundd by K log Q contxt-valu tabls is ndd, which holds contxt-valu tabls computd bottom-up but not usd yt and waiting to b mployd for th computation of contxtvalu tabls highr up in th qury tr. (Not that this is not th dpth of th qury tr, which is not ncssarily boundd by O(log Q).) Computing contxt-valu tabls bottom-up stp by stp is important for handling path xprssions and ngation wll. For string- or numbr-typd xprssions, ths rlations do not hav to b matrializd, but rsults can b gnratd and chckd top-down whn computing a nod st-typd contxt-valu tabl for an xprssion that contains as a dirct subxprssion with only an additional D log Q sizd mmory window. W did not provid a lowr bound for th qury complxity of XPath, but conjctur a considrabl fragmnt of XPath to b ALOGTIME-complt with rspct to qury complxity. Acknowldgmnts This work was supportd by th EU Rsarch Training Ntwork GAMES. Th first author was partially fundd by grant Z29-N04 of th Austrian Rsarch Fund (FWF). Th scond author s visit to th Univrsity of Edinburgh s Laboratory for Foundations of Computr Scinc was sponsord by Erwin Schrödingr grant J2169 of th FWF. Th papr [8], also to b found in this procdings volum, contains a numbr of rlatd and ovrlapping rsults on th complxity of XPath, and is work indpndnt from ours. 8. REFERENCES [1] A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo, and M. Tompa. Two Applications of Inductiv Counting for Complmntation Problms. SIAM Journal of Computing, 18: , [2] S. A. Cook and P. McKnzi. Problms Complt for Dtrministic Logarithmic Spac. J. Algorithms, 8: , [3] G. Gottlob, C. Koch, and R. Pichlr. 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