Efficient Static Analysis of XML Paths and Types

Transcription

1 Efficiet Static Aalysis f XML Paths ad Types Pierre Geevès, Nabil Layaïda, Ala Schmitt T cite this versi: Pierre Geevès, Nabil Layaïda, Ala Schmitt Efficiet Static Aalysis f XML Paths ad Types Prceedigs f the 2007 ACM SIGPLAN cferece Prgrammig laguage desig ad implemetati, Ju 2007, Sa Dieg, Uited States pp , 2007, <101145/ > <hal > HAL Id: hal Submitted 20 Nv 2007 HAL is a multi-discipliary pe access archive fr the depsit ad dissemiati f scietific research dcumets, whether they are published r t The dcumets may cme frm teachig ad research istitutis i Frace r abrad, r frm public r private research ceters L archive uverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusi de dcumets scietifiques de iveau recherche, publiés u, émaat des établissemets d eseigemet et de recherche fraçais u étragers, des labratires publics u privés

2 Efficiet Static Aalysis f XML Paths ad Types Pierre Geevès Ecle Plytechique Fédérale de Lausae pierregeeves@epflch Nabil Layaïda Ala Schmitt INRIA Rhôe-Alpes {abillayaida, alaschmitt}@iriafr Abstract We preset a algrithm t slve XPath decisi prblems uder regular tree type cstraits ad shw its use t statically typecheck XPath queries T this ed, we prve the decidability f a lgic with cverse fr fiite rdered trees whse time cmplexity is a simple expetial f the size f a frmula The lgic crrespds t the alterati free mdal µ-calculus withut greatest fixpit, restricted t fiite trees, ad where frmulas are cycle-free Our prf methd is based tw auxiliary results First, XML regular tree types ad XPath expressis have a liear traslati t cycle-free frmulas Secd, the least ad greatest fixpits are equivalet fr fiite trees, hece the lgic is clsed uder egati Buildig these results, we describe a practical, effective system fr slvig the satisfiability f a frmula The system has bee experimeted with sme decisi prblems such as XPath emptiess, ctaimet, verlap, ad cverage, with r withut type cstraits The beefit f the apprach is that ur system ca be effectively used i static aalyzers fr prgrammig laguages maipulatig bth XPath expressis ad XML type atatis (as iput ad utput types) Categries ad Subject Descriptrs E1 [Data Structures]: Trees; F41 [Mathematical Lgic ad Frmal Laguages]: Mathematical Lgic mdal lgic; F43 [Mathematical Lgic ad Frmal Laguages]: Frmal Laguages decisi prblems; H21 [Database Maagemet]: Lgical Desig; H23 [Database Maagemet]: Laguages Query Laguages Geeral Terms Keywrds 1 Itrducti Algrithms, laguages, thery, verificati Mdal lgic, satisfiability, type checkig, XPath This wrk is mtivated by the eed f efficiet type checkers fr XML-based prgrammig laguages where XML types ad XPath queries are used as first class laguage cstructs I such settigs, XPath decisi prblems i the presece f XML types such as DTDs r XML Schemas arise aturally Examples f such decisi prblems iclude emptiess test (whether a expressi ever selects des), ctaimet (whether the results f a expressi are always icluded i the results f ather e), verlap (whether tw Majr part f this wrk de whe the authr was at INRIA Rhôe-Alpes Permissi t make digital r hard cpies f all r part f this wrk fr persal r classrm use is grated withut fee prvided that cpies are t made r distributed fr prfit r cmmercial advatage ad that cpies bear this tice ad the full citati the first page T cpy therwise, t republish, t pst servers r t redistribute t lists, requires prir specific permissi ad/r a fee PLDI 07 Jue 11 13, 2007, Sa Dieg, Califria, USA Cpyright c 2007 ACM /07/0006 $500 expressis select cmm des), ad cverage (whether des selected by a expressi are always ctaied i the ui f the results selected by several ther expressis) XPath decisi prblems are t trivial i that they eed t be checked a pssibly ifiite quatificati ver a set f trees Ather difficulty arises frm the cmbiati f upward ad dwward avigati trees with recursi [31] The mst basic decisi prblem fr XPath is the emptiess test f a expressi [3] This test is imprtat fr ptimizati f hst laguages implemetatis: fr istace, if e ca decide at cmpile time that a query result is empty the subsequet bud cmputatis ca be igred Ather basic decisi prblem is the XPath equivalece prblem: whether r t tw queries always retur the same result It is imprtat fr refrmulati ad ptimizati f a expressi [17], which aim at efrcig peratial prperties while preservig sematic equivalece [23] The mst essetial prblem fr type-checkig is XPath ctaimet It is required fr the ctrl-flw aalysis f XSLT [25], fr checkig itegrity cstraits, ad fr XML security [12] The cmplexity f XPath decisi prblems heavily depeds the laguage features Previus wrks [28, 3] shwed that icludig geeral cmpariss f data values frm a ifiite dmai may lead t udecidability Therefre, we fcus a XPath fragmet which cvers all features except cutig [8] ad data values I ur apprach t slve XPath decisi prblems, tw issues eed t be addressed First, we idetify the mst apprpriate lgic with sufficiet expressiveess t capture bth regular tree types ad ur XPath fragmet Secd, we slve efficietly the satisfiability prblem which allws t test if a give frmula f the lgic admits a satisfyig fiite tree The essece f ur results lives i a sub-lgic f the alterati free mdal µ-calculus (AFMC) with cverse, sme sytactic restrictis frmulas, withut greatest fixpit, ad whse mdels are fiite trees We prve that XPath expressis ad regular tree type frmulas cfrm t these sytactic restrictis Blea clsure is the key prperty fr slvig the ctaimet (a lgical implicati) I rder t btai clsure uder egati, we prve that the least ad greatest fixpit peratrs cllapse i a sigle fixpit peratr Surprisigly, the traslatis f XML regular tree types ad a large XPath fragmet des t icrease cmplexity sice they are liear i the size f the crrespdig frmulas i the lgic The cmbiati f these igrediets lead t ur mai result: a satisfiability algrithm fr a lgic fr fiite trees whse time cmplexity is a simple expetial f the size f a frmula The decisi prcedure has bee implemeted i a system fr slvig XML decisi prblems such as XPath emptiess, ctaimet, verlap, ad cverage, with r withut XML type cstraits The system ca be used as a cmpet f static aalyzers fr prgrammig laguages maipulatig XPath expressis ad XML type atatis fr bth iput ad utput

3 2 Outlie The paper is rgaized as fllws We first preset ur data mdel, trees with fcus, ad ur lgic i 3 ad 4 We ext preset XPath ad its traslati i ur lgic i 5 Our satisfiability algrithm is itrduced ad prve crrect i 6, ad a few details f the implemetati are discussed i 7 Applicatis fr type checkig ad sme experimetal results are described i 8 We study related wrk i 9 ad cclude i 10 Detailed prfs ad implemetati techiques ca be fud i a lg versi f this paper [16] 3 Trees with Fcus I rder t represet XML trees that are easy t avigate we use fcused trees, ispired by Huet s Zipper data structure [20] Fcused trees t ly describe a tree but als its ctext: its previus sibligs ad its paret, icludig its paret ctext recursively Explrig such a structure has the advatage t preserve all ifrmati, which is quite useful whe csiderig laguages such as XPath that allw frward ad backward axes f avigati Frmally, we assume a alphabet Σ f labels, raged ver by σ t ::= σ[tl] tree tl ::= list f trees ɛ empty list t :: tl cs cell c ::= ctext (tl, Tp, tl) rt f the tree (tl, c[σ], tl) ctext de f ::= (t, c) fcused tree I rder t deal with decisi prblems such as ctaimet, we eed t represet i a fcused tree the place where the evaluati was started usig a start mark, fte simply called mark i the fllwig T d s, we csider fcused trees where a sigle tree r a sigle ctext de is marked, as i σ S [tl] r (tl, c[σ S ], tl) Whe the presece f the mark is ukw, we write it as σ [tl] We write F fr the set f fiite fcused trees with a sigle mark The ame f a fcused tree is ied as m(σ [tl], c) = σ We w describe hw t avigate fcused trees, i biary style There are fur directis that ca be fllwed: fr a fcused tree f, f 1 chages the fcus t the childre f the curret tree, f 2 chages the fcus t the ext siblig f the curret tree, f 1 chages the fcus t the paret f the tree if the curret tree is a leftmst siblig, ad f 2 chages the fcus t the previus siblig Frmally, we have: (σ [t :: tl], c) 1 = (t, (ɛ, c[σ ], tl)) (t, (tl l, c[σ ], t :: tl r)) 2 = (t, (t :: tl l, c[σ ], tl r)) (t, (ɛ, c[σ ], tl)) 1 = (σ [t :: tl], c) (t, (t :: tl l, c[σ ], tl 2 r)) = (t, (tl l, c[σ ], t :: tl r)) Whe the fcused tree des t have the required shape, these peratis are t ied 4 The Lgic We itrduce i this secti the lgic t which XPath expressis ad XML regular tree types are gig t be traslated, a sub-lgic f the alterati free mdal µ-calculus with cverse We als itrduce a restricti the frmulas we csider ad give a iterpretati f frmulas as sets f fiite fcused trees We fially shw that the lgic has a sigle fixpit fr these mdels ad that it is clsed uder egati L µ ϕ, ψ ::= frmula true σ σ atmic prp (egated) S S start prp (egated) X variable ϕ ψ disjucti ϕ ψ cjucti a ϕ a existetial (egated) µx iϕ i i ψ least -ary fixpit νx iϕ i i ψ greatest -ary fixpit V X V ϕ ψ V ϕ ψ V Figure 1 Lgic frmulas = F σ V = {f m(f) = σ} = V (X) σ V = ϕ V ψ V S V = = {f m(f) σ} f f = (σ S [tl], c) = ϕ V ψ V S V = {f f = (σ[tl], c)} a ϕ V = {f a f ϕ V f a ied} a V = {f f a uied} \ µx iϕ i i ψ V = let T i = T i F ϕ i V [Ti/Xi] Ti νx iϕ i i ψ V i ψ V [Ti /X i ] = let T i = [ T i F T i ϕ i V [Ti /X i ] i ψ V [Ti /X i ] Figure 2 Iterpretati f frmulas I the fllwig iitis, a {1, 2, 1, 2} are prgrams ad atmic prpsitis σ crrespd t labels frm Σ We als assume that a = a Frmulas ied i Fig 1 iclude the truth predicate, atmic prpsitis (detig the ame f the tree i fcus), start prpsitis (detig the presece f the start mark), disjucti ad cjucti f frmulas, frmulas uder a existetial (detig the existece a subtree satisfyig the sub-frmula), ad least ad greatest -ary fixpits We chse t iclude a -ary versi f fixpits because regular types are fte ied as a set f mutually recursive iitis, makig their traslati i ur lgic mre succict I the fllwig we write µxϕ fr µxϕ i ϕ We ie i Fig 2 a iterpretati f ur frmulas as sets f fiite fcused trees with a sigle start mark The iterpretati f the -ary fixpits first cmpute the smallest r largest iterpretati fr each ϕ i the returs the iterpretati f ψ We w restrict the set f valid frmulas t cycle-free frmulas, ie frmulas that have a bud the umber f mdality cycles idepedetly f the umber f ufldig f their fixpits A mdality cycle is a subfrmula f the frm a ϕ where ϕ ctais a tp-level existetial f the frm a ψ (By tp-level we mea uder a arbitrary umber f cjuctis r disjuctis, but t uder ay ther cstruct) Fr istace, the frmula µx 1 (ϕ 1 X) i X is t cycle free: fr ay iteger, there is a ufldig f the frmula with mdality cycles O the ther had, the frmula µx 1 (X Y ), Y 1 (Y ) i X is cycle free: there is at mst e mdality cycle Cycle-free frmulas have a very iterestig prperty, which we w describe T test whether a tree satisfies a frmula, e may i i

4 ie a straightfrward iductive relati betwee trees ad frmulas that ly hlds whe the rt f the tree satisfies the frmula, ufldig fixpits if ecessary Give a tree, if a frmula ϕ is cycle free, the every de f the tree will be tested a fiite umber f time agaist ay give subfrmula f ϕ The ituiti behid this prperty, which hlds a cetral rle i the prf f lemma 42, is the fllwig If a tree de is tested a ifiite umber f times agaist a subfrmula, the there must be a cycle i the avigati i the tree, crrespdig t sme mdalities ccurrig i the subfrmula, betwee e ccurrece f the test ad the ext e As we csider trees, the cycle implies there is a mdality cycle i the frmula (as cycles f the frm cat ccur) Hece the umber f mdality cycles i ay expasi f ϕ is ubuded, thus the frmula is t cycle free We are w ready t shw a first result: i the fiite fcused-tree iterpretati, the least ad greatest fixpits cicide fr cyclefree frmulas T this ed, we prve a strger result that states that a give fcused tree is i the iterpretati f a frmula if it is i a fiite ufldig f the frmula I the base case, we use the frmula σ σ as false DEFINITION 41 (Fiite ufldig) The fiite ufldig f a frmula ϕ is the set uf (ϕ) iductively ied as uf (ϕ) = {ϕ} fr ϕ =, σ, σ, S, S, X, a uf (ϕ ψ) = ϕ ψ ϕ uf (ϕ), ψ uf (ψ) uf (ϕ ψ) = ϕ ψ ϕ uf (ϕ), ψ uf (ψ) uf ( a ϕ) = a ϕ ϕ uf (ϕ) uf (µx iϕ i i ψ) = uf (ψ{ µx iϕ i i X i / Xi }) {σ σ} uf (νx iϕ i i ψ) = uf (ψ{ νx iϕ i i X i / Xi }) {σ σ} LEMMA 42 Let ϕ a cycle-free frmula, the ϕ V = uf (ϕ) V The reas why this lemma hlds is the fllwig Give a tree satisfyig ϕ, we deduce frm the hypthesis that ϕ is cycle free the fact that every de f the tree will be tested a fiite umber f times agaist every subfrmula f ϕ As the tree ad the umber f subfrmulas are fiite, the satisfacti derivati is fiite hece ly a fiite umber f ufldig is ecessary t prve that the tree satisfies the frmula, which is what the lemma states As least ad greatest fixpits cicide whe ly a fiite umber f ufldig is required, this is sufficiet t shw that they cllapse Nte that this wuld t hld if ifiite trees were allwed: the frmula µx 1 X is cycle free, but its iterpretati is empty, whereas the iterpretati f νx 1 X icludes every tree with a ifiite brach f 1 childre We w illustrate why frmulas eed t be cycle free fr the fixpits t cllapse Csider the frmula µx 1 1 X Its iterpretati is empty The iterpretati f νx 1 1 X hwever ctais every fcused tree that has e 1 child I the rest f the paper, we ly csider least fixpits A imprtat csequece f Lemma 42 is that the lgic restricted i this way is clsed uder egati usig De Mrga s dualities, exteded t evetualities ad fixpits as fllws: a ϕ = a a ϕ µx iϕ i i ψ = µx i ϕ i{ X i / Xi } i ψ{ X i / Xi } 5 XPath ad Regular Tree Laguages XPath [6] is a pwerful laguage fr avigatig i XML dcumets ad selectig sets f des matchig a predicate I their simplest frm, XPath expressis lk like directry avigati paths Fr L XPath e ::= XPath expressi /p abslute path p relative path e 1 e 2 ui e 1 e 2 itersecti Path p ::= path p 1/p 2 path cmpsiti p[q] qualified path a::σ step with de test a:: step Qualif q ::= qualifier q 1 ad q 2 cjucti q 1 r q 2 disjucti t q egati p path Axis a ::= tree avigati axis child self paret descedat desc-r-self acestr ac-r-self fll-siblig prec-siblig fllwig precedig example, the XPath expressi Figure 3 XPath Abstract Sytax /child::bk/child::chapter/child::secti avigates frm the rt f a dcumet (desigated by the leadig / ) thrugh the tp-level bk de t its chapter child des ad t its child des amed secti The result f the evaluati f the etire expressi is the set f all the secti des that ca be reached i this maer The situati becmes mre iterestig whe cmbied with XPath s capability f searchig alg axes ther tha child Fr istace, e may use the precedig-siblig axis fr avigatig backward thrugh des f the same paret, r the acestr axis fr avigatig upward recursively Furthermre, at each step i the avigati the selected des ca be filtered usig qualifiers: blea expressi betwee brackets that ca test the existece r absece f paths We csider a large XPath fragmet cverig all majr features f the XPath recmmedati [6] except cutig ad cmpariss betwee data values Fig 3 gives the sytax f XPath expressis Fig 4 gives a iterpretati f XPath expressis as fuctis betwee sets f fcused trees 51 XPath Embeddig We w explai hw a XPath expressi ca be traslated it a equivalet L µ frmula that perfrms avigati i fcused trees i biary style Lgical Iterpretati f Axes The traslati f avigatial primitives (amely XPath axes) is frmally specified i Fig 5 The traslati fucti, ted A a χ, takes a XPath axis a as iput, ad returs its L µ traslati, parameterized by the L µ frmula χ give as parameter This parameter represets the ctext i which the axis ccurs ad is eeded fr frmula cmpsiti i rder t traslate path cmpsiti Mre precisely, the frmula A a χ hlds fr all des that ca be accessed thrugh the axis a frm sme de verifyig χ Let us csider a example The frmula A child χ, traslated as µz 1 χ 2 Z, is satisfied by childre f the ctext χ These des csist f the first child ad the remaiig childre Frm the first child, the ctext must be reached immediately

5 S e : L XPath 2 F 2 F S e /p F = S p p rt(f ) S e p F = S p p {(σ S [tl],c) F } S e e 1 e 2 F = S e e 1 F S e e 2 F S e e 1 e 2 F = S e e 1 F S e e 2 F S p : Path 2 F 2 F S p p 1/p 2 F = f f S p p 2 (Sp p 1 F ) S p p[q] F = {f f S p p F S q q f } S p a::σ F = {f f S a a F m(f) = σ} S p a:: F = {f f S a a F } S q : Qualif F {true, false} S q q 1 ad q 2 f = S q q 1 f S q q 2 f S q q 1 r q 2 f = S q q 1 f S q q 2 f S q t q f = S q q f S q p f = S p p {f} S a : Axis 2 F 2 F S a self F = F S a child F = fchild(f ) S a fll-siblig fchild(f ) S a fll-siblig F = siblig(f ) S a fll-siblig siblig(f ) S a prec-siblig F = psiblig(f ) S a prec-siblig psiblig(f ) S a paret F = paret(f ) S a descedat F = S a child F S a descedat (Sa child F ) S a desc-r-self F = F S a descedat F S a acestr F = S a paret F S a acestr (Sa paret F ) S a ac-r-self F = F S a acestr F S a fllwig F = S a desc-r-self (Sa fll-siblig (Sa ac-r-self F )) S a precedig F = S a desc-r-self (Sa prec-siblig (Sa ac-r-self F )) fchild(f ) = {f 1 f F f 1 ied} siblig(f ) = {f 2 f F f 2 ied} psiblig(f ) = f 2 f F f 2 ied paret(f ) = {(σ [rev a(tl l, t :: tl r)], c) rev a(ɛ, tl r) = tl r (t, (tl l, c[σ ], tl r)) F } rev a(t :: tl l, tl r) = rev a(tl l, t :: tl r) rt(f ) = {(σ S [tl], (tl, Tp, tl)) F } rt(paret(f )) Figure 4 Iterpretati f XPath i terms f Fcused Trees A : Axis L µ L µ A self χ = χ A child χ = µz 1 χ 2 Z A fll-siblig χ = µz 2 χ 2 Z A prec-siblig χ = µz 2 χ 2 Z A paret χ = 1 µzχ 2 Z A descedat χ = µz 1 (χ Z) 2 Z A desc-r-self χ = µzχ µy 1 (Y Z) 2 Y A acestr χ = 1 µzχ 1 Z 2 Z A ac-r-self χ = µzχ 1 µyz 2 Y A fllwig χ = A desc-r-self η1 A precedig χ = A desc-r-self η2 η 1 = A fll-siblig A ac-r-self χ η 2 = A prec-siblig A ac-r-self χ Figure 5 Traslati f XPath Axes E : L XPath L µ L µ E /p χ = P p ((µz 1 2 Z) (µyχ S 1 Y 2 Y )) E p χ = P p (χ S) E e 1 e 2 χ = E e 1 χ E e 2 χ E e 1 e 2 χ = E e 1 χ E e 2 χ P : Path L µ L µ P p 1/p 2 χ = P p 2 (P p 1 χ) P p[q] χ = P p χ Q q P a::σ χ = σ A a χ P a:: χ = A a χ Figure 6 Traslati f Expressis ad Paths by gig ce upward via 1 Frm the remaiig childre, the ctext is reached by gig upward (ay umber f times) via 2 ad the fially ce via 1 Lgical Iterpretati f Expressis Fig 6 gives the traslati f XPath expressis it L µ The traslati fucti E e χ takes a XPath expressi e ad a L µ frmula χ as iput, ad returs the crrespdig L µ traslati The traslati f a relative XPath expressi marks the iitial ctext with S The traslati f a abslute XPath expressi avigates t the rt which is take as the iitial ctext Figure 7 illustrates the traslati f the XPath expressi child::a[child::b] This expressi selects all a child des f a give ctext which have at least e b child The traslated L µ frmula hlds fr a des which are selected by the expressi The first part f the traslated frmula, ϕ, crrespds t the step

6 d Traslated Query: a (µx 1 (χ S) 2 X) {z } ϕ a b χ ϕ ψ c a ϕ child::a [child::b] 1 µyb 2 Y {z } ψ Figure 7 XPath Traslati Example child::a which selects cadidates a des The secd part, ψ, avigates dwward i the subtrees f these cadidate des t verify that they have at least e immediate b child Nte that withut cverse prgrams we wuld have bee uable t differetiate selected des frm des whse existece is tested: we must state prperties bth the acestrs ad the descedats f the selected de Equippig the L µ lgic with bth frward ad cverse prgrams is therefre crucial fr supprtig XPath Lgics withut cverse prgrams may ly be used fr slvig XPath emptiess but cat be used fr slvig ther decisi prblems such as ctaimet efficietly XPath cmpsiti cstruct p 1/p 2 traslates it frmula cmpsiti i L µ, such that the resultig frmula hlds fr all des accessed thrugh p 2 frm thse des accessed thrugh p 1 frm χ The traslati f the brachig cstruct p[q] sigificatly differs The resultig frmula must hld fr all des that ca be accessed thrugh p ad frm which q hlds T preserve sematics, the traslati f p[q] stps the selectig avigati t thse des reached by p, the filters them depedig whether q hlds r t We express this by itrducig a dual frmal traslati fucti fr XPath qualifiers, ted Q q ad ied i Fig 8, that perfrms filterig istead f avigati Specifically, P ca be see as the avigatial traslatig fucti: the traslated frmula hlds fr target des f the give path O the ppsite, Q ca be see as the filterig traslatig fucti: it states the existece f a path withut mvig t its result The traslated frmula Q q χ (respectively P p χ) hlds fr des frm which there exists a qualifier q (respectively a path p) leadig t a de verifyig χ XPath traslati is based these tw traslatig mdes, the first e beig used fr paths ad the secd e fr qualifiers Wheever the filterig mde is etered, it will ever be left The traslati f paths iside qualifiers is als give i Fig 8 It uses the traslati fr axes ad is based XPath symmetry: symmetric(a) detes the symmetric XPath axis crrespdig t the axis a (fr istace symmetric(child) = paret) We may w state that ur traslati is crrect, by relatig the iterpretati f a XPath frmula applied t sme set f trees t the iterpretati f its traslati, by statig that the traslati f a frmula is cycle-free, ad by givig a bud i the size f this traslati We restrict the sets f trees t which a XPath frmula may be applied t thse that may be deted by a L µ frmula This restricti will be justified i Secti 52 where we shw that every regular tree laguage may be traslated t a L µ frmula PROPOSITION 51 (Traslati Crrectess) The fllwig hld fr a XPath expressi e ad a L µ frmula ϕ detig a set f fcused trees, with ψ = E e ϕ: Q : Qualif L µ L µ Q q 1 ad q 2 χ = Q q 1 χ Q q 2 χ Q q 1 r q 2 χ = Q q 1 χ Q q 2 χ Q t q χ = Q q χ Q p χ = P p χ P : Path L µ L µ P p 1/p 2 χ = P p 1 (P p 2 χ) P p[q] χ = P p (χ Q q ) P a::σ χ = A a (χ σ) P a:: χ = A a χ A : Axis L µ L µ A a χ = A symmetric(a) χ Figure 8 Traslati f Qualifiers 1 ψ = S e e ϕ 2 ψ is cycle-free 3 the size f ψ is liear i the size f e ad ϕ 52 Embeddig Regular Tree Laguages Several frmalisms exist fr describig types f XML dcumets (eg DTD, XML Schema, Relax NG) I this paper we embed regular tree laguages, which gather all f them [26] it L µ We rely a straightfrward ismrphism betwee uraked regular tree types ad biary regular tree types [19] Assumig a cutably ifiite set f type variables raged ver by X, biary regular tree type expressis are ied as fllws: L BT T ::= tree type expressi empty set ɛ leaf T 1 T 2 ui σ(x 1, X 2) label let X it i i T bider We refer the reader t [19] fr the detatial sematics f regular tree laguages, ad directly itrduce their traslati it L µ: : L BT L µ T = σ σ fr T =, ɛ T 1 T 2 = T 1 T 2 σ(x 1, X 2) = σ succ 1(X 1) succ 2(X 2) let X it i i T = µx i T i i T where we use the frmula σ σ as false, ad the fucti succ ( ) takes care f settig the type frtier: 8 < α if X is bud t ɛ succ α(x) = α α X if ullable(x) : α X if t ullable(x) accrdig t the predicate ullable(x) which idicates whether the type T ɛ bud t X ctais the empty tree

7 Nte that the traslati f a regular tree type uses ly dwward mdalities sice it describes the allwed subtrees at a give ctext N additial restricti is impsed the ctext frm which the type iiti starts I particular, avigati is allwed i the upward directi s that we ca supprt type cstraits fr which we have ly partial kwledge i a give directi Hwever, whe we kw the psiti f the rt, cditis similar t thse f abslute paths are added i the frm f additial frmulas describig the psiti that eed t be satisfied This is particularly useful whe a regular type is used by a XPath expressi that starts its avigati at the rt (/p) sice the path will t g abve the rt f the type (by addig the restricti µz 1 2 Z) O the ther had, if the type is cmpared with ather type (typically t check iclusi f the result f a XPath expressi i this type), the there is restricti as t where the rt f the type is (ur traslati des t impse the chse de t be at the rt) This is particularly useful sice a XPath expressi usually returs a set f des deep i the tree which we may cmpare t this partially ied type 6 Satisfiability-Testig Algrithm I this secti we preset ur algrithm, shw that it is sud ad cmplete, ad prve a time cmplexity budary T check a frmula ϕ, ur algrithm builds satisfiable frmulas ut f sme subfrmulas (ad their egati) f ϕ, the checks whether ϕ was prduced We first describe hw t extract the subfrmulas frm ϕ 61 Prelimiary Defiitis Fr ϕ = (µx iϕ i i ψ) we ie exp(ϕ) = ψ{ µx iϕ i i X i/xi } which detes the frmula ψ i which every ccurrece f a X i is replaced by (µx iϕ i i X i) We ie the Fisher-Lader clsure cl(ψ) f a frmula ψ as the set f all subfrmulas f ψ where fixpit frmulas are additially uwud ce Specifically, we ie the relati e L µ L µ as the least relati that satisfies the fllwig: ϕ 1 ϕ 2 e ϕ 1, ϕ 1 ϕ 2 e ϕ 2 ϕ 1 ϕ 2 e ϕ 1, ϕ 1 ϕ 2 e ϕ 2 a ϕ e ϕ µx iϕ i i ψ e exp(µx iϕ i i ψ) The clsure cl(ψ) is the smallest set S that ctais ψ ad clsed uder the relati e, ie if ϕ 1 S ad ϕ 1 e ϕ 2 the ϕ 2 S We call Σ(ψ) the set f atmic prpsitis σ used i ψ alg with ather ame, σ x, that des t ccur i ψ t represet atmic prpsitis t ccurrig i ψ We ie cl (ψ) = cl(ψ) { ϕ ϕ cl(ψ)} Every frmula ϕ cl (ψ) ca be see as a blea cmbiati f frmulas f a set called the Lea f ψ, ispired frm [27] We te this set Lea(ψ) ad ie it as fllws: Lea(ψ) = a a {1, 2, 1, 2} Σ(ψ) {S} { a ϕ a ϕ cl(ψ)} A ψ-type (r simply a type ) (Hitikka set i the tempral lgic literature) is a set t Lea(ψ) such that: a ϕ Lea(ψ), a ϕ t a t (mdal csistecy); 1 / t 2 / t (a tree de cat be bth a first child ad a secd child); exactly e atmic prpsiti σ t (XML labelig); we use the fucti σ(t) t retur the atmic prpsiti f a type t; S may belg t t t = (, ) ϕ Lea(ψ) ϕ t ϕ t = ({ϕ}, ) ϕ 1 t = (T1, F 1) ϕ 2 t = (T2, F 2) ϕ 1 ϕ 2 t = (T1 T 2, F 1 F 2) ϕ 1 t = (T1, F 1) ϕ 1 ϕ 2 t = (T1, F 1) ϕ / t = (T, F ) ϕ t = (T, F ) ϕ Lea(ψ) ϕ 2 t = (T2, F 2) ϕ 1 ϕ 2 t = (T2, F 2) exp(µx iϕ i i ψ) t = (T, F ) µx iϕ i i ψ t = (T, F ) ϕ t ϕ / t = (, {ϕ}) ϕ 1 / t = (T1, F 1) ϕ 2 / t = (T2, F 2) ϕ 1 ϕ 2 / t = (T1 T 2, F 1 F 2) ϕ 1 / t = (T1, F 1) ϕ 1 ϕ 2 / t = (T1, F 1) ϕ t = (T, F ) ϕ / t = (T, F ) ϕ 2 / t = (T2, F 2) ϕ 1 ϕ 2 / t = (T2, F 2) exp(µx iϕ i i ψ) / t = (T, F ) µx iϕ i i ψ / t = (T, F ) Figure 9 Truth assigmet f a frmula We call Types(ψ) the set f ψ-types Fr a ψ-type t, the cmplemet f t is the set Lea(ψ) \ t A type determies a truth assigmet f every frmula i cl (ψ) with the relati ied i Fig 9 Nte that such derivatis are fiite because the umber f aked µx iϕ i i ψ (that d t ccur uder mdalities) strictly decreases after each expasi We fte write ϕ t if there are sme T, F such that ϕ t = (T, F ) We say that a frmula ϕ is true at a type t iff ϕ t We w relate a frmula t the truth assigmet f its ψ-types PROPOSITION 61 If ϕ t = (T, F ), the we have T t, F Lea(ϕ) \ t, ad V ψ T ψ V ψ F ψ implies ϕ (every tree i the iterpretati f the first frmula is i the iterpretati f the secd) If ϕ / t = (T, F ), the we have T t, F Lea(ϕ) \ t, ad V ψ T ψ V ψ F ψ implies ϕ We ext ie a cmpatibility relati betwee types t state that tw types are related accrdig t a mdality DEFINITION 62 (Cmpatibility relati) Tw types t ad t are cmpatible uder a {1, 2}, writte a(t, t ), iff 62 The Algrithm a ϕ Lea(ψ), a ϕ t ϕ t a ϕ Lea(ψ), a ϕ t ϕ t The algrithm wrks sets f triples f the frm (t, w 1, w 2) where t is a type, ad w 1 ad w 2 are sets f types which represet every witess fr t accrdig t relatis 1(t, ) ad 2(t, ) The algrithm prceeds i a bttm-up apprach, repeatedly addig ew triples util a satisfyig mdel is fud (ie a triple whse first cmpet is a type implyig the frmula), r util

8 Upd(X) = X {(t, w 1(t, X ), w 2(t, X )) S / t Types(ψ) 1 t w 1(t, X ) 2 t w 2(t, X ) } (t, w 1(t, X ), w 2(t, X )) S S t Types(ψ) 1 t w 1(t, X ) 2 t w 2(t, X ) (t, w 1(t, X S ), w 2(t, X )) S S / t Types(ψ) 1 t w 1(t, X S ) 2 t w 2(t, X ) (t, w 1(t, X ), w 2(t, X S )) S S / t Types(ψ) 1 t w 1(t, X ) 2 t w 2(t, X S ) w a(t, X) = {type(x) x X a type(x) a(t, type(x))} X S = x X x = (,, ) S FialCheck(ψ, X) = x X S, dsat(x, ψ) a {1, 2}, a / type(x) X = {x X x = (,, )} dsat((t, w 1, w 2), ψ) = ψ t x, dsat(x, ψ) (x w 1 x w 2) type((t, w 1, w 2)) = t Figure 10 Operatis used by the Algrithm mre triple ca be added Each iterati f the algrithm builds types represetig deeper trees (i the 1 ad 2 directi) with pedig backward mdalities that will be fulfilled at later iteratis Types with backward mdalities are satisfiable, ad if such a type implies the frmula beig tested, the it is satisfiable The mai iterati is as fllws: X repeat X X X Upd(X ) if FialCheck(ψ, X) the retur ψ is satisfiable util X = X retur ψ is usatisfiable where X Types(ψ) 2 Types(ψ) 2 Types(ψ) ad the peratis Upd( ) ad FialCheck( ) are ied Fig 10 We te X i the set f triples ad T i the set f types after i iteratis: T i = type(x) x X i Nte that T i+1 is the set f types fr which at least e witess belgs t T i 63 Crrectess ad Cmplexity I this secti we ie the ecessary tis t prve the crrectess f the satisfiability testig algrithm, ad shw that its time cmplexity is 2 O( Lea(ψ) ) THEOREM 63 (Crrectess) The algrithm decides satisfiability f L µ frmulas ver fiite fcused trees Termiati Fr ψ L µ, sice cl(ψ) is a fiite set, Lea(ψ) ad 2 Lea(ψ) are als fiite Furthermre, Upd( ) is mtic ad each X i is icluded i the fiite set Types(ψ) 2 Types(ψ) 2 Types(ψ), therefre the algrithm termiates T fiish the prf, it thus suffices t prve sudess ad cmpleteess Prelimiary Defiitis fr Sudess First, we itrduce a ti f partial satisfiability fr a frmula, where backward mdalities are ly checked up t a give level A frmula ϕ is partially satisfied iff ϕ 0 V as ied i Fig 11 Fr a type t, we te ϕ c(t) its mst cstraied frmula, where atms are take frm Lea(ψ) I the fllwig, stads fr S if S t, ad fr S therwise ϕ c(t) = σ(t) ^ σ ^ a ϕ ^ a ϕ σ Σ,σ / t a ϕ t a ϕ / t We w itrduce a ti f paths, writte ρ which are ccateatis f mdalities: the empty path is writte ɛ, ad path ccateati is writte ρa V ϕ ψ V ϕ ψ V 1 ϕ 0 V 2 ϕ 0 V 1 ϕ >0 V 2 ϕ >0 V = F X V = ϕ V ψ V p V = ϕ V ψ V p V = F S V = F S V 1 ϕ V 2 ϕ V = V (X) = {f m(f) = p} = {f m(f) p} = f f = (σ S [tl], c) = {f f = (σ[tl], c)} = f 1 f ϕ 1 V f 1 ied = f 2 f ϕ 1 V f 2 ied = f 1 f ϕ +1 V f 1 ied = f 2 f ϕ +1 V f 2 ied a V = {f f a uied} \ µx iϕ i i ψ V = let T i = T i F ϕ i V [Ti/Xi] Ti i ψ V [T i /X i ] Figure 11 Partial satisfiability Every path may be give a depth: depth(ɛ) = 0 depth(ρa) = depth(ρ) + 1 if a {1, 2} depth(ρa) = depth(ρ) 1 if a {1, 2} A frward path is a path that ly metis frward mdalities We ie a tree f types T as a tree whse des are types, T ( ) = t, with at mst tw childre, T 1 ad T 2 The avigati i trees f types is trivially exteded t frward paths A tree f types is csistet iff fr every frward path ρ ad fr every child a f T ρ, we have T ρ ( ) = t, T ρa ( ) = t implies a t, a t, ad a(t, t ) Give a csistet tree f types T, we w ie a depedecy graph whse des are pairs f a frward path ρ ad a frmula i t = T ρ ( ) r the egati f a frmula i the cmplemet f t The directed edges f the graph are labeled with mdalities csistet with the tree This graph crrespds t what the algrithm ultimately builds, as every iterati discvers lger frward paths Fr every (ρ, ϕ) i the des we build the fllwig edges: ϕ Σ(ψ) Σ(ψ) {S, S, a, a }: edge i

9 ρ = ɛ ad ϕ = a ϕ with a {1, 2}: edge ρ = ρ a ad ϕ = a ϕ : let t = T ρ ( ) We first csider the case where a {1, 2} ad let t = T ρa ( ) As T is csistet, we have ϕ t hece there are T, F such that ϕ t = (T, F ) with T a subset f t, ad F a subset f the cmplemet f t Fr every ϕ T T we add a edge a t (ρa, ϕ T ), ad fr every ϕ F F we add a edge a t (ρa, ϕ F ) We w csider the case where a {1, 2} ad first shw that we have a = a As T is csistet, we have a i t Mrever, as t is a tree type, it must ctai a As a is a backward mdality, it must be equal t a as at mst e may be preset Hece we have ρ aa = ρ ad we let t = T ρ ( ) By csistecy, we have ϕ t, hece ϕ t = (T, F ) ad we add edges as i the previus case: t (ρ, ϕ T ) ad t (ρ, ϕ F ) ρ = ρ a ad ϕ = a ϕ : let t = T ρ ( ) If a is t i t the edge is added Otherwise, we prceed as i the previus case Fr dwward mdalities, we let t = T ρa ( ) ad we cmpute ϕ / t = (T, F ), which we kw t hld by csistecy We the add edges t (ρa, ϕ T ) ad t (ρa, ϕ F ) as befre Fr upward mdalities, as we have a i t, we must have a = a ad we let t = T ρ ( ) We cmpute ϕ / t = (T, F ) ad we add the edges t (ρ, ϕ T ) ad t (ρ, ϕ F ) as befre LEMMA 64 The depedecy graph f a csistet tree f types f a cycle-free frmula is cycle free LEMMA 65 (Sudess) Let T be the result set f the algrithm Fr ay type t T ad ay ϕ such that ϕ t, the ϕ 0 Prf utlie: The prf (detailed i [16] ) prceeds by iducti the umber f steps f the algrithm Fr every t i T ad every witess tree T rted at t built frm X, we shw that T is a csistet tree type ad we build a fcused tree f that is rted (ie f the shape (σ [tl], (ɛ, Tp, tl ))) We the prceed t shw that f satisfies ϕ c(t) at level 0 T d s, we use a further iducti the depedecy tree LEMMA 66 (Cmpleteess) Fr a cycle-free clsed frmula ϕ L µ, if ϕ the the algrithm termiates with a set f triples X such that FialCheck(ϕ, X) Prf utlie: As the frmula is satisfiable, we csider a smallest fcused tree f satisfyig it We the use Lemma 42 t derive a fiite satisfacti relati f ϕ that ctais f We the rely this relati t build a ru f the algrithm that prduces a type with backward mdality implyig the frmula LEMMA 67 (Cmplexity) Fr ψ L µ the satisfiability prblem ψ is decidable i time 2 O() where = Lea(ψ) 7 Implemetati Techiques Our implemetati relies a symblic represetati f sets f ψ-types usig Biary Decisi Diagrams (BDDs) [5] First, we bserve that the implemetati ca avid keepig track f every pssible witesses f each ψ-type I fact, fr a frmula ϕ, we ca test ϕ by testig the satisfiability f the (liear-size) plugig frmula ψ = µxϕ 1 X 2 X at the rt f fcused trees That is, checkig ψ 0 while esurig there is ufulfilled upward evetuality at tp level 0 Oe advatage f prceedig this way is that the implemetati ly eed t deal with a curret set f ψ-types at each step We w itrduce a bit-vectr represetati f ψ-types Types are cmplete i the sese that either a subfrmula r its egati must belg t a type It is thus pssible fr a frmula ϕ Lea(ψ) t be represeted usig a sigle BDD variable Fr Lea(ψ) = {ϕ 1,, ϕ m}, we represet a subset t Lea(ψ) by a vectr t = t 1,, t m {0, 1} m such that ϕ i t iff t i = 1 A BDD with m variables is the used t represet a set f such bit vectrs We ie auxiliary predicates fr prgrams a {1, 2}: isparet a ( t) is read t is a paret fr prgram a ad is true iff the bit fr a is true i t ischild a( t) is read t is a child fr prgram a ad is true iff the bit fr a is true i t Fr a set T 2 Lea(ψ), we te χ T its crrespdig characteristic fucti Ecdig χ Types(ψ) is straightfrward The predicate status ϕ( t) is the equivalet f the bit vectr represetati We w cstruct the BDD f the relati a fr a {1, 2} This BDD relates all pairs ( x, y) that are csistet wrt the prgram a, ie, such that y supprts all f x s a ϕ frmulas, ad vice-versa x supprts all f y s a ϕ frmulas: a( x, y) = ^ 1 i m 8 < : x i status ϕ( y) y i status ϕ( x) therwise if ϕ i = a ϕ if ϕ i = a ϕ Fr a {1, 2}, we ie the set f witessed vectrs: χ Wita(T )( x) = isparet a ( x) y [ h( y) a( x, y) ] where h( y) = χ T ( y) ischild a( y) The, the BDD f the fixpit cmputati is iitially set t the false cstat, ad the mai fucti Upd( ) is implemeted as: 0 χ Upd(T ) ( x) = χ T ( Types(ψ) ( x) ^ 1 χ Wita(T )( x) A a {1,2} Fially, the slver is implemeted as iteratis ver the sets χ Upd(T ) util a fixpit is reached The fial satisfiability cditi csists i checkig whether ψ is preset i a ψ-type f this fixpit with ufulfilled upward evetuality We use tw majr techiques fr further ptimizati First, BDD relatial prducts ( y [ h( y) a( x, y) ]) are cmputed usig cjuctive partitiig ad early quatificati [10] Secd, we bserved that chsig a gd iitial rder f Lea(ψ) frmulas des sigificatly imprve perfrmace Experiece has shw that the variable rder determied by the breadth-first traversal f the frmula ψ t slve, which keeps sister subfrmulas i clse prximity, yields better results i practice 8 Typig Applicatis ad Experimetal Results Fr XPath expressis e 1,, e, we ca frmulate several decisi prblems i the presece f XML type expressis T 1,, T : XPath ctaimet: E e 1 T1 E e 2 T2 (if the frmula is usatisfiable the all des selected by e 1 uder type cstrait T 1 are selected by e 2 uder type cstrait T 2) XPath emptiess: E e 1 T1 XPath verlap: E e 1 T1 E e 2 T2 XPath cverage: E e 1 T1 V 2 i E e i Ti Tw prblems are f special iterest fr XML type checkig: Static type checkig f a atated XPath query: E e 1 T1 T 2 (if the frmula is usatisfiable the all

10 e 1 e 2 e 3 e 4 e 5 e 6 e 7 /a[//b[c/*//d]/b[c//d]/b[c/d]] /a[//b[c/*//d]/b[c/d]] a/b//c/fll-siblig::d/e a/b//d[prec-siblig::c]/e a/c/fllwig::d/e a/b[//c]/fllwig::d/e a/d[precedig::c]/e *//switch[acestr::head]//seq//audi[prec-siblig::vide] XPath Decisi Prblem XML Type Time (ms) e 1 e 2 ad e 2 e 1 e 353 e 4 e 3 ad e 4 e 3 e 45 e 6 e 5 ad e 5 e 6 e 41 e 7 is satisfiable SMIL e 8 is satisfiable XHTML e 9 (e 10 e 11 e 12) XHTML Table 2 Sme Decisi Prblems ad Crrespdig Results e 8 e 9 e 10 e 11 e 12 descedat::a[acestr::a] /descedat::* html/(head bdy) html/head/descedat::* html/bdy/descedat::* Figure 12 XPath Expressis Used i Experimets DTD Symbls Biary Type Variables SMIL XHTML 10 Strict Table 1 Types Used i Experimets des selected by e 1 uder type cstrait T 1 are icluded i the type T 2) XPath equivalece uder type cstraits: E e 1 T1 E e 2 T2 ad E e 1 T1 E e 2 T2 (This test ca be used t check that the des selected after a mdificati f a type T 1 by T 2 ad a XPath expressi e 1 by e 2 are the same, typically whe a iput type chages ad the crrespdig XPath query has t chage as well) As third-party implemetati we kw f addresses reverse axes ad recursi, we simply prvide evidece that ur apprach is efficiet We carried ut extesive tests 1 [16], ad preset here ly a represetative sample that icludes the mst cmplex laguage features such as recursive frward ad backward axes, itersecti, large ad very recursive types with a reasable alphabet size The tests use XPath expressis shw Fig 12 (where // is used as a shrthad fr /desc-r-self::*/ ) ad XML types shw Table 1 Table 2 presets sme decisi prblems ad crrespdig perfrmace results Times reprted i millisecds crrespd t the ruig time f the satisfiability slver withut the (egligible) time spet fr parsig ad traslatig it L µ The first XPath ctaimet istace was first frmulated i [24] as a example fr which the prpsed tree patter hmmrphism techique is icmplete The e 8 example shws that the fficial XHTML DTD des t sytactically prhibit the estig f achrs Fr the XHTML case, we bserve that the time eeded is mre imprtat, but it remais practically relevat, especially fr static aalysis peratis perfrmed ly at cmpile-time 9 Related Wrk The XPath ctaimet prblem has attracted a lt f research atteti i the database cmmuity, where the fcus was give t the study f the impact f differet XPath features the ctaimet cmplexity (see [28] fr a verview) The cmplexity f XPath satisfiability i the presece f DTDs als is extesively studied i [3] Frm these results, we kw that XPath ctaimet with r withut type cstraits rages frm EXPTIME t udecidable 1 Experimets have bee cducted with a JAVA implemetati ruig a Petium 4, 3 Ghz, with 512Mb f RAM with Widws XP Mst frmalisms used i the ctext f XML are related t e f the tw lgics used fr uraked trees: first-rder lgic (FO), ad Madic Secd Order Lgic (MSO) FO ad relatives are frequetly used fr query laguages sice they icely capture their avigatial features [2] I a attempt t reach mre expressive pwer, the wrk fud i [1] prpses a variat f Prpsitial Dyamic Lgic (PDL) with a EXPTIME cmplexity MSO, specifically the weak madic secd-rder lgic f tw successrs (WS2S) [9], is e f the mst expressive decidable lgic used whe bth regular types ad queries [2] are uder csiderati WS2S satisfiability is kw t be -elemetary A drawback f the WS2S decisi prcedure is that it requires the full cstructi ad cmplemetati f tree autmata Sme tempral ad fixpit lgics clsely related t MSO have bee itrduced ad allw t avid explicit autmata cstructi The prpsitial mdal µ-calculus itrduced i [22] has bee shw t be as expressive as determiistic tree autmata [11] Sice it is trivially clsed uder egati, it cstitutes a gd alterative fr studyig MSO-related prblems Mrever, it has bee exteded with cverse prgrams i [31] The best kw cmplexity fr the resultig lgic is btaied thrugh reducti t the emptiess prblem f alteratig tree autmat which is i 2 O(4 lg ), where crrespds t the legth f a frmula [18] Ufrtuately the lgic lacks the fiite mdel prperty Frm [31], we kw that WS2S is exactly as expressive as the alterati-free fragmet (AFMC) f the prpsitial mdal µ-calculus Furthermre, the AFMC subsumes all early lgics such as CTL ad PDL (see [2] fr a cmplete survey tree lgics) The gal f the research preseted s far is limited t establishig ew theretical prperties ad cmplexity buds Our research differs i that we seek precise cmplexity buds, efficiet implemetati techiques, ad ccrete desig that may be directly applied t the type checkig f XPath queries uder regular tree types I this lie f research, sme experimetal results based WS2S, thrugh the Ma tl [21], have recetly bee reprted fr XPath ctaimet [15] Hwever, fr static aalysis purpses, the explsiveess f the apprach is very difficult t ctrl due t the -elemetary cmplexity Clser t ur ctributi, the recet wrk fud i [29] prvides a decisi prcedure fr the AFMC with cverse whse time cmplexity is 2 O( lg ) Hwever, mdels f the lgic are Kripke structures (ifiite graphs) Efrcig the fiite tree mdel prperty ca be de at the sytactic level [29], ad this has bee further develped i the XML settig i [14] Nevertheless, the drawback f this apprach is that the AFMC decisi prcedure requires expesive cycle-detecti fr rejectig ifiite derivati paths fr least fixpit frmulas The preset wrk shws hw this ca be avided fr fiite trees As a csequece, the resultig perfrmace is much mre attractive I a earlier wrk XML type checkig, a lgic fr fiite trees was preseted [30], but the lgic is t clsed uder egati I [7], a techique is preseted fr statically esurig crrectess f paths The apprach ly deals with emptiess f XPath

11 expressis withut reverse axes, whereas ur apprach slves the mre geeral prblem f ctaimet, icludig reverse axes The wrk [25] presets a apprximated techique that is able t statically detect errrs i XSLT stylesheets Their apprach culd certaily beefit frm usig ur exact algrithm istead f their cservative apprximati The XDuce [19], CDuce [4], ad XStatic [13] laguages supprt patter-matchig thrugh regular expressi types but t XPath A survey existig research statically type checkig XML trasfrmatis ca be fud i [25] 10 Cclusi The mai result f ur paper is a sud ad cmplete algrithm fr the satisfiability f decisi prblems ivlvig regular tree types ad XPath queries with a tighter 2 O() cmplexity i the legth f a frmula Our apprach is based a sub-lgic f the alteratifree mdal µ-calculus with cverse fr fiite trees Our prf methd reveals deep cectis betwee this lgic ad XPath decisi prblems First, the traslatis f XML regular tree types ad a large XPath fragmet are cycle-free ad liear i the size f the crrespdig frmulas i the lgic Secd, fiite trees, sice bth peratrs are equivalet, the lgic with a sigle fixpit peratr is clsed uder egati This allws t address key XPath decisi prblems such as ctaimet Fially, there are a umber f iterestig directis fr further research that build ideas develped here: extedig XPath t restricted data values cmpariss that preserves this cmplexity, fr istace data values a fiite dmai, ad itegratig related wrk cutig [8] t ur lgic We als pla ctiuig t imprve the perfrmace f ur implemetati Ackwledgmets We wuld like t thak Girgi Ghelli fr his helpful cmmets earlier drafts ad Bejami C Pierce fr his useful suggestis Refereces [1] L Afaasiev, P Blackbur, I Dimitriu, B Gaiffe, E Gris, M Marx, ad M de Rijke PDL fr rdered trees Jural f Applied N- Classical Lgics, 15(2): , 2005 [2] P Barceló ad L Libki Tempral lgics ver uraked trees I LICS 05: Prceedigs f the 20th Aual IEEE Sympsium Lgic i Cmputer Sciece, pages 31 40, New Yrk, NY, USA, 2005 [3] M Beedikt, W Fa, ad F Geerts XPath satisfiability i the presece f DTDs I PODS 05: Prceedigs f the twety-furth ACM Sympsium Priciples f Database Systems, pages 25 36, New Yrk, NY, USA, 2005 ACM Press [4] V Bezake, G Castaga, ad A Frisch CDuce: A XMLcetric geeral-purpse laguage I ICFP 03: Prceedigs f the Eighth ACM SIGPLAN Iteratial Cferece Fuctial Prgrammig, pages 51 63, New Yrk, NY, USA, 2003 ACM Press [5] R E Bryat Graph-based algrithms fr blea fucti maipulati IEEE Tras Cmputers, 35(8): , 1986 [6] J Clark ad S DeRse XML path laguage (XPath) versi 10, W3C recmmedati, Nvember /REC-xpath [7] D Clazz, G Ghelli, P Maghi, ad C Sartiai Static aalysis fr path crrectess f XML queries J Fuct Prgram, 16(4-5): , 2006 [8] S Dal Zili, D Lugiez, ad C Meyssier A lgic yu ca cut I POPL 04: Prceedigs f the 31st ACM SIGPLAN-SIGACT Sympsium Priciples f Prgrammig Laguages, pages , New Yrk, NY, USA, 2004 ACM Press [9] J Der Tree acceptrs ad sme f their applicatis Jural f Cmputer ad System Scieces, 4: , 1970 [10] J Edmud M Clarke, O Grumberg, ad D A Peled Mdel checkig MIT Press, Cambridge, MA, USA, 1999 [11] E A Emers ad C S Jutla Tree autmata, µ-calculus ad determiacy I Prceedigs f the 32d aual Sympsium Fudatis f Cmputer Sciece, pages , Ls Alamits, CA, USA, 1991 IEEE Cmputer Sciety Press [12] W Fa, C-Y Cha, ad M Garfalakis Secure XML queryig with security views I SIGMOD 04: Prceedigs f the 2004 ACM SIGMOD Iteratial Cferece Maagemet f Data, pages , New Yrk, NY, USA, 2004 ACM Press [13] V Gapeyev ad B C Pierce Regular bject types I Eurpea Cferece Object-Orieted Prgrammig (ECOOP), Darmstadt, Germay, 2003 A prelimiary versi was preseted at FOOL 03 [14] P Geevès ad N Layaïda A system fr the static aalysis f XPath ACM Tras If Syst, 24(4): , 2006 [15] P Geevès ad N Layaïda Decidig XPath ctaiemet with MSO Data & Kwledge Egieerig, t appear, 2007 [16] P Geevès, N Layaïda, ad A Schmitt A satisfiability slver fr XML ad XPath, Jue [17] P Geevès ad J-Y Vi-Dury Lgic-based XPath ptimizati I DcEg 04: Prceedigs f the 2004 ACM Sympsium Dcumet Egieerig, pages , NY, USA, 2004 ACM Press [18] E Grädel, W Thmas, ad T Wilke, editrs Autmata lgics, ad ifiite games: a guide t curret research Spriger-Verlag, New Yrk, NY, USA, 2002 [19] H Hsya, J Vuill, ad B C Pierce Regular expressi types fr XML ACM Tras Prgram Lag Syst, 27(1):46 90, 2005 [20] G P Huet The zipper J Fuct Prgram, 7(5): , 1997 [21] N Klarlud, A Møller, ad M I Schwartzbach MONA 14, Jauary [22] D Kze Results the prpsitial µ-calculus Theretical Cmputer Sciece, 27: , 1983 [23] M Y Levi ad B C Pierce Type-based ptimizati fr regular patters I DBPL 05: Prceedigs f the 10th Iteratial Sympsium Database Prgrammig Laguages, vlume 3774 f LNCS, Ld, UK, August 2005 Spriger-Verlag [24] G Miklau ad D Suciu Ctaimet ad equivalece fr a fragmet f XPath Jural f the ACM, 51(1):2 45, 2004 [25] A Møller ad M I Schwartzbach The desig space f type checkers fr XML trasfrmati laguages I Prc Teth Iteratial Cferece Database Thery, ICDT 05, vlume 3363 f LNCS, pages Spriger-Verlag, Jauary 2005 [26] M Murata, D Lee, M Mai, ad K Kawaguchi Taxmy f XML schema laguages usig frmal laguage thery ACM Trasactis Iteret Techlgy, 5(4): , 2005 [27] G Pa, U Sattler, ad M Y Vardi BDD-based decisi prcedures fr the mdal lgic K Jural f Applied N-classical Lgics, 16(1-2): , 2006 [28] T Schwetick XPath query ctaimet SIGMOD Recrd, 33(1): , 2004 [29] Y Taabe, K Takahashi, M Yamamt, A Tzawa, ad M Hagiya A decisi prcedure fr the alterati-free tw-way mdal µ- calculus I I TABLEAUX 2005, vlume 3702 f LNCS, pages , Ld, UK, September 2005 Spriger-Verlag [30] A Tzawa O biary tree lgic fr XML ad its satisfiability test I PPL 04: Ifrmal Prceedigs f the Sixth JSSST Wrkshp Prgrammig ad Prgrammig Laguages, 2004 [31] M Y Vardi Reasig abut the past with tw-way autmata I ICALP 98: Prceedigs f the 25th Iteratial Cllquium Autmata, Laguages ad Prgrammig, pages , Ld, UK, 1998 Spriger-Verlag