Pattern Queries for XML and Semistructured Data

Transcription

1 INSTITUT FÜR INFORMATIK Lhr- un Forschungsinhit für Programmir- un Mollirungssprachn Ottingnstraß 67, D Münchn Pattrn Quris for XML an Smistructur Data François Bry an Sbastian Schaffrt Tchnical Rport, Computr Scinc Institut, Munich, Grmany Forschungsbricht/Rsarch Rport PMS-FB , March 2002

2 Pattrn Quris for XML an Smistructur Data François Bry an Sbastian Schaffrt Institut for Computr Scinc, Univrsity of Munich 1 Introuction Essntial to smistructur ata is th slction of ata from incompltly spcifi ata itms. For such a ata slction, a path languag such as XPath [1] is convnint bcaus it provis with rgular xprssions such as, +,?, an wilcars that giv ris to a flxibl no rtrival. Qury an transformation languags vlop sinc th mi 90s for XML [1] an smistructur ata.g. XQury [1], th prcursors of XQury [], an XSLT [1] rly upon such a path-orint slction. Thy us pattrns (also call tmplats) for xprssing how th slct ata, xprss by paths, ar r-arrang (or r-construct) into nw ata itms. Thus, such languags intrtwin construct parts, i.. th construction pattrns, an qury parts, i.. path slctors. This intrtwining has som avantags: For simpl qury-construct rqusts, th approach is rathr natural an rsults in an asily unrstanabl co. Howvr, intrtwining construct an qury parts also has rawbacks: 1. Qury-construct rqusts involving a complx ata rtrival might b confusing, 2. unncssarily complx path slctions,.g. XPath xprssions involving both forwar an rvrs axs, ar possibl [2], 3. in cas of svral path slctions, th ovrall structur of th rtriv ata itms might b ifficult to grasp. This papr arsss using pattrns insta of paths for qurying XML an smistructur ata. A mtaphor for this approach is to s quris as forms, answrs as form fillings yiling atabas itms. With this approach, pattrns ar us not only in construct xprssions, but also for ata slction. In th following, a basic qury languag is introuc. An answr to a qury in this languag is formaliz as a simulation [3] of a groun instanc of th qury in a atabas itm. This formalization yils a compositional smantics. 2 A Basic Qury Languag Th following principls hav prvail to th finition of th qury languag: Pattrn-bas or positional insta of navigational. A qury shoul corrspon to a form, an answr to a filling yiling a atabas itm. Th rlativ positions of variabls in a qury shoul b asily rcognizabl. Rfrntial transparncy. Th maning of an xprssion, spcially of a variabl, shoul b th sam whrvr it appars. Thrfor, structiv assignmnts ar prohibit an variabls must b functional or logic programming variabls. Compositional smantics. A (structurally) rcursiv finition of th smantics of a qury in trms of th smantics of its parts, i.. a Tarski-styl mol thory, is sought for.

3 Multipl variabl binings. Lik with SQL an othr qury languags, quris might hav svral answrs, ach answr bining th qury variabls iffrntly. Symmtry. Quris shoul allow similar forms of incomplt spcifications in brath, i.. concrning siblings, an in pth, i.. concrning chilrn. Not that th rquirmnts of [4] ar fulfill by or compatibl with th basic qury languag fin blow. Blow, th following pairwis isjoint sts of symbols ar rfrr to: A st I of intifirs, a st L of labls (or tags or strings), a st V l of labl variabls, a st V t of trm (or ata itm) variabls. Intifirs ar not by i, labls (variabls, rsp.) by lowr (uppr, rsp.) cas lttrs with or without inics. Th following mta-variabls (with or without inics an/or suprscripts) ar us: i nots an intifir, l nots a labl, L a labl variabl, X a trm variabl, t a trm (as fin blow), v a labl or a trm, an V a labl or trm variabl. A atabas is a st (or multist) of atabas trms. Th chilrn of a ocumnt no may b ithr orr (as in stanar XML), or unorr. In th following, a trm whos root is labll l an has orr (unorr, rsp.) chilrn t 1,..., t n is not l[t 1,..., t n ] (l{t 1,..., t n }, rsp.). Dfinition 1 (Databas Trms). Databas trms ar xprssions inuctivly fin as follows an satisfying Conitions 1 an 2 givn blow: 1. If l is a labl, thn l is a (atomic) atabas trm. 2. If i is an intifir an t is a atabas trm nithr of th form i 0 : t 0 nor of th form i 0, thn i: t is a atabas trm. 3. If i is an intifir, thn i is a atabas trm. 4. If l is a labl an t 1,..., t n ar n 1 atabas trms, thn l[t 1,..., t n ] an l{t 1,..., t n } ar atabas trms. Conition 1: For a givn intifir i an intifir finition i: t 0 occurs at most onc in a trm. Conition 2: For vry intifir rfrnc i occurring in a trm t an intifir finition i: t 0 occurs in t. A qury trm is a pattrn that spcifis a slction of atabas trms vry much lik logical atoms an SQL slctions. Th valuation of qury trms (cf. blow Dfinition 9) iffrs from th valuation of logical atoms an SQL slctions as follows: 1. Answrs might hav aitional subtrms to thos mntion in th qury trm. 2. Answrs might hav anothr subtrm orring than th qury. 3. A qury trm might spcify subtrms at an unspcifi pth. In qury trms, th oubl squar an curly brackts, [[ ]] an {{ }}, not xact subtrm pattrns, i.. oubl (squar or curly) brackts ar us in a qury trm to b answr by atabas trms with no mor subtrms than thos givn in th qury trm. [[ ]] is us if th subtrm orr in th answrs is to b that of th qury trm, {{ }} is us othrwis. Thus, possibl answrs to th qury trm t 1 = a[[b, c{, }, f]] ar th atabas trms a[b, c{,, g}, f] an a[b, c{,, g}, f{g, h}] an a[b, c{, {g, h}, g}, f{g, h}] an a[b, c[, ], f]. In contrast, a{b, c{, }, f, g} an a[b, c{, }, f, g] an a{b, c{, }, f} ar no answrs to t 1. 2

4 In a qury trm, a trm variabl X can b constrain to som qury trms using th construct, ra as. Thus, th qury trm t 2 = a[x 1 b[c, ], X 2, ] constrains th trm variabl X 1 to such atabas trms that ar possibl answrs to th qury trm b[c, ]. Not that th trm variabl X 2 is unconstrain in t 2. Possibl answrs to t 2 ar a[b[c, ], f, ] which bins X 1 to b[c, ] an X 2 to f, a[b[c, ], f[g, h], ] which bins X 1 to b[c, ] an X 2 to f[g, h], a[b[c,, ], f, ] which bins X 1 to b[c,, ] an X 2 to f, an a[b[c,, ], f, ] which bins X 1 to b[c,, ] an X 2 to f. In qury trms, th construct sc, ra scnant, spcifis a subtrm at an unspcifi pth. Thus, possibl answrs to th qury trm t 3 = a[x sc f[c, ], b] ar a[f[c, ], b] an a[g[f[c, ]], b] an a[g[f[c, ], h], b] an a[g[g[f[c, ]]], b] an a[g[g[f[c, ], h], i], b]. Dfinition 2 (Qury Trms). Qury trms ar xprssions inuctivly fin as follows an satisfying Conitions 1 an 2 of Dfinition 1: 1. If l is a labl an L is a labl variabl, thn l, L, l{{}}, an L{{}} ar (atomic) qury trms. 2. A trm variabl is a qury trm. 3. If i is an intifir an t is a qury trm nithr of th form i 0 : t 0 nor of th form i 0, thn i: t is a qury trm. 4. If i is an intifir, thn i is a qury trm. 5. If X is a variabl an t a qury trm, thn X t is a qury trm. 6. If X is a variabl an t is a qury trm, thn X sc t is a qury trm. 7. If l is a labl, L a labl variabl an t 1,..., t n ar n 1 qury trms, thn l[t 1,..., t n ], L[t 1,..., t n ], l{t 1,..., t n }, L{t 1,..., t n }, l[[t 1,..., t n ]], L[[t 1,..., t n ]], l{{t 1,..., t n }}, an L{{t 1,..., t n }} ar qury trms. A qury trm in which no variabls occur is groun. Qury trms that ar not of th form i, ar strict. Lftmost labls of strict groun qury trms ar fin as follows: For l, l{{}}, l[t 1,..., t n ], an l{t 1,..., t n } it is l; for i: t it is that of t; an for sc t it is sc l if l is th lftmost labl of t. Databas trms ar (simpl kins of) qury trms. Howvr, th st of answrs to a atabas trm t (consir as a qury trm) in a atabas D in gnral contains not only t (cf. blow Dfinition 9). E.g. th atabas trms f an f{a} an f{b} ar possibl answrs to th qury f. Howvr, f is th only possibl answr to th qury trm f{{}}. In a qury trm, multipl occurrncs of a sam trm variabl ar not prclu. E.g. a possibl answr to th qury trm a{x b{c}, X b{}} is a{b{c, }}. Th qury trm a[x b{c}, X f{}], howvr, has no answrs, for labls b an f ar istinct. Chil subtrms an subtrms of qury trms ar fin such that if t = f[a, g{y sc b{x}, h{a, X k{c}}], thn a an g{y sc b{x}, h{a, X k{c}} ar th only chil subtrms of t an.g. a an X an Y sc b{x} an h{a, X k{c}} an X k{c} an t itslf ar subtrms of t. Not that f is not a subtrm of t. Dfinition 3 (Variabl Wll-Form Qury Trms). A trm variabl X pns on a trm variabl Y in a qury trm t if X t 1 is a subtrm of t an Y is a subtrm of t 1. A qury trm t is variabl wll-form if t contains 3

5 no trm variabls X 0,..., X n (n 1) such that 1. X 0 = X n an 2. for all i = 1,..., n, X i pns on X i 1 in t. E.g. f{x g{x}} an f{x g{y }, Y h{x}} ar not variabl wllform. Variabl wll-formnss prclus quris spcifying infinit answrs. In th following, qury trms ar assum to b variabl wll-form. 3 Qury Smantics Th smantics is bas on graph simulation. Th graphs consir ar irct, orr an root an thir nos ar labll. Dfinition 4 (Simulation). Lt G 1 = (V 1, E 1 ) an G 2 = (V 2, E 2 ) b two graphs. Lt b an quivalnc rlation on V 1 V 2. A rlation S V 1 V 2 is a simulation with rspct to of G 1 in G 2 if: 1. If (v 1, v 2 ) S, thn v 1 v If (v 1, v 2 ) S an (v 1, v 1 ) E 1, thn thr xists v 2 V 2 such that (v 1, v 2 ) S an (v 2, v 2 ) E 2. Lt S b simulation S of G 1 = (V 1, E 1 ) in G 2 = (V 2, E 2 ). S is total, if for ach v 1 V 1 thr xists at last on v 2 V 2 such that (v 1, v 2 ) S. If G 1 has a root r 1, G 2 has a root r 2 an (r 1, r 2 ) S, thn S is a root simulation. S is minimal, if thr ar no simulations S S of G 1 in G 2 such that S S. Not that vry root simulation is total. Dfinition 5 (Graph Inuc by a Groun Qury Trm). Lt t b a groun qury trm. Th graph G t = (N t, V t ) inuc by t is fin by: 1. N t is th st of strict subtrms (cf. Dfinition 2) of t an ach t N t is labll with th lftmost labl (cf. Dfinition 2) of t. 2. V t is th st of pairs (t 1, t 2 ) such that ithr t 2 is a chil subtrm of t 1, or i is a chil subtrm of t 1 an th intifir finition i: t 2 occurs in t. 3. Th chilrn of a no ar orr in G t lik in t. Not that t is th root of G t. Figur 1 in Appnix illustrats Dfinition 5. Blow, a atabas trm is oftn intifi with th graph it inucs. Dfinition 6 (Groun Qury Trm Simulation). is th rlation on groun qury trms fin by t 1 t 2 if thr xists a (minimal) root simulation with rspct to labl intity S of t 1 in t 2 such that: 1. if v 1 = l{{}} occurs in t 1 an (v 1, v 2 ) S, thn v 2 has no chilrn in t if v 1 = l[t 1 1,..., t1 n] (n 1) occurs in t 1, (v 1, v 2 ) S an if (t 1 i, t2 j ) S (1 j m n), thn t 2 1,..., t2 m occur in this orr as chilrn of v 2 in th graph inuc by t if v 1 = l[[t 1 1,..., t1 n]] (n 1) occurs in t 1, (v 1, v 2 ) S an if (t 1 i, t2 j ) S (1 j m n), thn th t 2 j ar pairwis istinct (i.. m = n) an occur in this orr as chilrn of v 2 in th graph inuc by t 2. 4

6 4. if v 1 = {{t 1 1,..., t1 n}} occurs in t 1, (v 1, v 2 ) S an (t 1 i, t2 j ) S (1 j m n), thn v 2 has no othr chilrn than th t 2 i in t 2. Figur 2 in Appnix illustrats Dfinition 6. By Dfinition 4, is rflxiv an transitiv, i.. it is a prorr on th st of atabas trms. is not a partial orr, for although t 1 = f{a} t 2 = f{a, a} an t 2 = f{a, a} t 1 = f{a} (both a of t 2 can b simulat by th sam a of t 1 ), t 1 = f{a} t 2 = f{a, a}. Root simulation with rspct to labl quality is a first notion towars a formalisation of answrs to qury trms: If thr xists a root simulation of a atabas trm t 1, consir as a qury trm, in a atabas trm t 2, thn t 2 is an answr to t 1. An answr in a atabas D to a qury trm t q is charactris by binings for th variabls in t q such that th atabas trm t rsulting from applying ths binings to t q is an lmnt of D. Consir.g. th qury t q = f{x g{b}, X g{c}} against th atabas D = {f{g{a, b, c}, g{a, b, c}, h}, f{g{b}, g{c}}}. Th constructs in t q yil th constraint g{b} X g{c} X. Matching t q with th first atabas trm in D yils th constraint X g{a, b, c}. Matching t q with th scon atabas trm in D yils th constraint X g{b} X g{c}. g{b} X g{c} X is not compatibl with X g{b} X g{c}. Thus, th only possibl valu for X is g{a, b, c}, i.. th only possibl answr to t q in D is f{g{a, b, c}, g{a, b, c}, h}. Dfinition 7 (Groun Instancs of Qury Trms). A grouning substitution is a function which assigns a labl to ach labl variabl an a atabas trm to ach trm variabl of a finit st of (labl or trm) variabls. Lt t q b a qury trm, V 1,..., V n b th (labl or trm) variabls occurring in t q an σ b a grouning substitution assigning v i to V i. Th groun instanc t q σ of t q with rspct to σ is th groun qury trm that can b construct from t q as follows: 1. Rplac ach subtrm X t by X. 2. Rplac ach occurrnc of V i by v i (1 i n). Rquiring in Dfinition 2 sc to occur to th right of maks it possibl to charactris a groun instanc of a qury trm by a grouning substitution. This is hlpful for formalising answrs but not ncssary for languag implmntions. Not all groun instancs of a qury trm ar accptabl answrs, for som instancs might violat th conitions xprss by th an sc constructs. Dfinition 8 (Allow Instancs). Th constraint inuc by a qury trm t q an a substitution σ is th conjunction of all inqualitis tσ Xσ such that X t is a subtrm of t q not of th form sc t 0, an of all xprssions Xσ tσ (ra tσ subtrm of Xσ ) such that X sc t is a subtrm of t q, if t q has such subtrms. If t q has no such subtrms, th constraint inuc t q an σ is th formula tru. Lt σ b a grouning substitution an t q σ a groun instanc of t q. t q σ is allow if: 1. Each inquality t 1 t 2 in th constraint inuc by t q an σ is satisfi. 2. For ach t 1 t 2 in th constraint inuc by t q an σ, t 2 is a subtrm of t 1. 5

7 Dfinition 9 (Answrs). Lt t q b a qury trm an D a atabas. An answr to t q in D is a atabas trm t b D such that thr xists an allow groun instanc t of t q satisfying t t b. 3.1 Concluing Rmarks In prvious works, simulation has bn us for vrifying th conformity of smistructur ata to a schma cf..g. [5,6]. Hr, it is us for qury answring. Th authors ar not awar of formr uss of simulation for qury answring. [7] scribs a languag call fxt that has variabls for trms, corrsponing to trs, an forsts. In fxt, no slction is on with rgular xprssions. In contrast to xcrpt an th basic languag scrib abov uss trm variabls for this purpos. In [8], a qury an transformation languag is scrib that is rlat to logic an Prolog. This languag has (in th trminology us abov) only labl variabls. In contrast, th basic qury languag introuc abov also has trm variabls. In [9], th notion of matching subjacnt to Dfinition 9 is shown to b ciabl. A languag call xcrpt is unr vlopmnt which buils upon th basic languag introuc abov. xcrpt has construct trms in which variabls but no sc an may occur. A construct trm with variabls V 1,..., V n is associat with a conjunction or isjunction of (possibly ngat) qury trms in which all of V 1,..., V n occur. xcrpt has svral aitional faturs that for spac rasons cannot b mntion hr. In [9] som of ths faturs an part of an oprational smantics ar introuc. A prototyp has bn raliz that implmnt a st-orint backwar rasoning oprational smantics. First xprimnts suggst that th xcrpt approach to qurying is convnint in practic. Rfrncs 1. Worl Wi Wb Consortium (W3C) (2002) 2. Oltanu, D., Muss, H., Furch, T., Bry, F.: Xpath: Looking forwar. In: Procings of Workshop on XML Data Managmnt (XMLDM), Springr LNCS (2002) 3. Hnzingr, M.R., Hnzingr, T.A., Kopk, P.W.: Computing Simulations on Finit an Infinit Graphs (1996) 4. Mair, D.: Databas Dsirata for an XML Qury Languag. In: Procings of QL 98 - Th Qury Languags Workshop. (1998) 5. Frnanz, M., Suciu, D.: Optimizing Rgular Path Exprssions Using Graph Schmas. In: Procings of th Int. Conf. on Data Enginring. (1988) Abitboul, S., Bunman, P., Suciu, D.: Data on th Wb. From Rlations to Smistructur Data an XML. Morgan Kaufmann (2000) 7. Brla, A., Sil, H.: fxt A Transformation Languag for XML Documnts. J. of Computing an Information Tchnology (CIT), Spcial Issu on Domain-Spcific Languags (2001) 8. Grahn, G., Lakshmanan, L.V.S.: On th Diffrnc btwn Navigating Smi-structur Data an Qurying It. In: Procings of Workshop on Databas Programming Languags. (1999) Bry, F., Schaffrt, S.: Towars a Dclarativ Qury an Transformation Languag for XML an Smistructur Data: Simulation Unification. Tchnical Rport PMS-FB , Inst. for Computr Scincs, Univrsity of Munich, (2002) 6

8 Appnix f f[i :a,b{c{,,^i },^i },i :g[h,i]] f a b g i :a a b{c{,,^i },^i } b i :g[h,i] g c h i c{,,^i } 1 c h h i i (a) Abstract no rprsntation (b) Full no rprsntation (th labls ar print in gray) Fig. 1. Graph inuc by t = f[i 1 : a, b{c{,, i 1 }, i 2 }, i 2 : g[h, i]]. Not that th graph inuc by a groun qury trm os not fully convy th trm structur: Missing ar graphical rprsntations of th various nstings [ ], { }, [[ ]] an {{ }}. f f a b sc b c a Fig. 2. A minimal simulation of th (graph inuc by th) groun qury trm t q = f{i 1 : a, b[{{}}, i 1 ], sc} in th (graph inuc by th) atabas trm t b = f[b[, i 2 : a], i 2, c, {}]. 7