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1 Einugh Reseh Exploe Chteizing XML Twig Queies with Exmples Cittion fo pulishe vesion: Stwoko, S & Wiezoek, P 2015, Chteizing XML Twig Queies with Exmples. in 18th Intentionl Confeene on Dtse Theoy, ICDT 2015, Mh 23-27, 2015, Bussels, Belgium. pp DOI: /LIPIs.ICDT Digitl Ojet Ientifie (DOI): /LIPIs.ICDT Link: Link to pulition eo in Einugh Reseh Exploe Doument Vesion: Pulishe's PDF, lso known s Vesion of eo Pulishe In: 18th Intentionl Confeene on Dtse Theoy, ICDT 2015, Mh 23-27, 2015, Bussels, Belgium Genel ights Copyight fo the pulitions me essile vi the Einugh Reseh Exploe is etine y the utho(s) n / o othe opyight ownes n it is onition of essing these pulitions tht uses eognise n ie y the legl equiements ssoite with these ights. Tke own poliy The Univesity of Einugh hs me evey esonle effot to ensue tht Einugh Reseh Exploe ontent omplies with UK legisltion. If you elieve tht the puli isply of this file ehes opyight plese ontt openess@e..uk poviing etils, n we will emove ess to the wok immeitely n investigte you lim. Downlo te: 24. Sep. 2018

2 Chteizing XML Twig Queies with Exmples Słwek Stwoko 1 n Piot Wiezoek 2 1 Univesity of Lille 3, INRIA LINKS, CNRS Lille, Fne slwomi.stwoko@ini.f 2 Univesity of Wołw, Institute of Compute Siene Wołw, Poln piotek@s.uni.wo.pl Astt Typilly, (Boolen) quey is finite fomul tht efines possily infinite set of tse instnes tht stisfy it (positive exmples), n impliitly, the set of instnes tht o not stisfy the quey (negtive exmples). We investigte the following ntul question: fo given lss of queies, is it possile to hteize evey quey with finite set of positive n negtive exmples tht no othe quey is onsistent with. We stuy this question fo twig queies n XML tses. We show tht while twig queies e hteizle, they genelly equie exponentil sets of exmples. Consequently, we fous on ptil sulss of nhoe twig queies n show tht not only e they hteizle ut lso with polynomilly-size sets of exmples. This esult is otine with the use of geneliztion opetions on twig queies, whose pplition to n nhoe twig quey yiels popely ontine n minimlly iffeent quey. Ou esults illustte futhe inteesting n stong onnetions etween the stutue n the semntis of nhoe twig queies tht the lss of ity twig queies oes not enjoy. Finlly, we show tht the lss of unions of twig queies is not hteizle ACM Sujet Clssifition H.2.3 Quey lnguges, H.2.1 Noml foms Keywos n phses Quey hteiztion, Quey exmples, Quey fitting, Twig queies Digitl Ojet Ientifie /LIPIs.ICDT Intoution One of the entl, if not efining, instuments in ompute siene is using fomul, finite syntti ojet, to efine (possily infinite) set of its moels. A typil exmple e egul expessions tht efine lnguges of wos. Dtse queies lso fll into this tegoy, whih is est illustte with Boolen queies: quey q efines the set of instnes stisfying q, positive exmples, n impliitly the set of instnes tht o not stisfy q, negtive exmples. In this ppe, we stuy the question of (finite) hteizility: Cn evey quey e hteize with finite set of exmples? Moe peisely, given lss of queies Q is it possile fo evey q Q to fin set of exmples suh tht q is the only quey (moulo equivlene) in Q onsistent with it i.e., quey stisfying ll positive exmples n none of negtive exmples. An if it is possile, n we sy nything out the nume n the sizes of the neessy exmples? The question of hteizility ises ntully in the ontext of lening/tehing [9, 6] whih els with the polem of onstuting fomul (quey) onsistent with given set of exmples. Howeve, eseh on hteizility hs nume of potentil pplitions of inepenent inteest euse it yiels wy to genete set of exmples onsistent with Słwek Stwoko n Piot Wiezoek; liense une Cetive Commons Liense CC-BY 18th Intentionl Confeene on Dtse Theoy (ICDT 15). Eitos: Melo Aens n Mtín Ugte; pp Leiniz Intentionl Poeeings in Infomtis Shloss Dgstuhl Leiniz-Zentum fü Infomtik, Dgstuhl Pulishing, Gemny

3 S. Stwoko n P. Wiezoek 145 given quey. Suh exmples n e use, fo instne, fo elementy quey engine eugging, n quey visuliztion n explntion. The exhustive ntue of the exmples povie y hteizility i.e., thee is extly one quey stisfying the exmples, my lso e useful in finl, veifition, stges of evese engineeing of tse queies. In this ppe, we investigte the polem of hteizility fo XML tses n twig queies [2, 14]. XML ouments n e seen s lele unnke tees n twig queies e tee-shpe pttens tht itionlly use wil lel (mthing ny lel) n esennt eges (mthing pth of positive length). Twig queies e the oe of vitully ny XML quey lnguge n hve een extensively stuie in the litetue [4]. In ptiul, lening twig queies fom exmples hs een peviously investigte [19, 5] n the uent ppe n e seen s ontinution of this line of wok n n ttempt t eepening ou unestning of the eltionship etween quey n its exmples. In essene, ou esults show tht hteizility is mesue of ihness of the quey lss, whih is losely elte to its expessive powe. We show tht unions of twig queies e not hteizle euse vitully ny set of exmples hs infinitely mny onsistent queies i.e., the lss of unions of twig queies is vey ih. On the othe hn, the lss of twig queies is less ih, given tee the nume of twig queies stisfie in it is finite, whih llows to show tht twig queies e hteizle. While twig queies e hteizle with finite sets of exmples, we show tht the nume of neessy exmples my e exponentil. The min ontiution of this ppe is showing tht (polynomilly) smll sets of exmples e suffiient to hteize nhoe twig queies [19]. In essene, this sulss of twig queies fois esennt eges inient to noe, whih meely pevents fom imposing onitions on nesty with lowe oun on epth: noe x is nesto of noe y n the pth fom x to y is of length k, whee k > 1. While the expessive powe oes not seem to e signifintly estite fom the ptil point of view, this lss of twig queies exhiits vey lose eltionship etween the stutue n the semntis: ontinment of two twig queies is equivlent to the existene of n emeing etween the queies, n equivlene tht oes not hol fo ity twig queies [13]. This eltionship etween the stutue of the quey n its semntis goes even futhe [19]: the use of two positive exmples of n nhoe quey q llows to ientify ll queies tht ontin q. We ontinue to exploe this eltionship n eepen its unestning y hteizing the stutue of the semi-lttie of nhoe twig queies: fo given nhoe twig quey q we e inteeste in the nume n the sizes of the most speifi nhoe twig queies popely ontining q, n inteestingly, we show tht thei nume n thei sizes e polynomilly smll (w..t. the size of q). This esult is essentil in showing tht nhoe twig queies hve polynomilly-size hteizing sets of exmples. To unestn the existene of n emeing, n hene the ontinment, etween two nhoe twig queies, we stuy thee geneliztion opetions pplie to twig queies (f. Fig. 5): 1) hnging lel to, 2) hnging the type of n ege fom hil to esennt, n 3) emoving noe. While ntul n elementy, these opetions ptue peisely the sulss of injetive (nesto-peseving) emeings etween queies: quey p n e otine fom quey q y pplying sequene of geneliztion opetions if n only if thee is n injetive emeing of q into p. Consequently, when pplie, in iligent mnne, to nhoe twig queies they llow to hteize the semi-lttie of nhoe twig queies une injetive semntis. We lso show tht nhoe twig queies hve unique nonil foms whih n e effiiently otine y itetively pplying the opetions n the unique nonil fom is in ft the (size-)miniml equivlent quey. This futhe I C D T

4 146 Chteizing XML Twig Queies with Exmples illusttes the esile popeties of nhoe twig queies s minimiztion fo ity twig queies is known to e inttle [2, 13]. We point out, howeve, tht lsses of twig queies popely ontining nhoe twig queies e known to hve ttle minimiztion se on opetions euing the quey [11]. To exten the hteiztion esults fom injetive to stn semntis, we ientify mpping ovelp s the essentil iffeene etween the oesponing two type of emeings: unlike the injetive emeing, the stn emeing of quey q into quey p my mp iffeent fgments of q into the sme fgment of p eting n ovelp of the imges of the iffeent fgments of q. To ess this iffeene we exploe using uplition opetion tht etes septe opies of the fgment of p thus eliminting the ovelp. Howeve, this opetion intoues eunnies n my inese itily the size of the quey. Consequently, it is pplie togethe with iffeent geneliztion opetions to voi intouing eunnies n to voi unesile gowth of the quey we evise eusive ptten of pplying uplition opetions tht llows to polynomilly oun the nume of intoue opies. The nume of the geneliztions is line in the size of the oiginl quey, n thnks to pplying the uplition opetion in ontolle mnne, the size of eh geneliztion is t most quti. The geneliztions e then use to onstut set of hteizing exmples onsisting of polynomil nume of exmples eh of polynomilly oune size. The min ontiutions of the ppe e: We fomulte the polem of hteizility of Boolen lss of XML queies with exmples n stuy it fo lsses of twig queies. We show tht unions of twig queies e not hteizle while twig queies lone e ut the nume of exmples neessy to hteize twig quey my e exponentil in the size of the quey. We investigte hteizility of ih sulss of nhoe twig queies, n popose set of ntul geneliztion opetions tht llows to hteize with polynomillysize set of exmples ny nhoe twig quey une the injetive (nesto-peseving) semntis. We popose uplition opetion whose iligent use llows to exten the hteizility to nhoe twig queies to the stn semntis. Relte wok. Ou wok is losely elte to the senio of tehing [10, 9, 16] whee the set of exmples to e pesente is selete y helpful tehe. Golmn n Kens [9] efine sequene of positive n negtive exmples to e tehing sequene fo given onept if it uniquely speifies in the onept lss C. Hene, this is essentilly the sme ie s in ou se. They stuy the popeties of tehing imension of onept lss tht is minimum nume of exmples tehe hs to evel in oe to uniquely ientify ny onept in the lss. They onsie, howeve, iffeent onept lsses then ous, nmely othogonl etngles n oolen fomuls. Also, tehing sequenes fo othe lsses of oolen fomuls hs een stuie in [18, 3]. Reently, lening n veifition of qhon queies ws stuie in [1]. qhon is speil lss of Boolen quntifie queies whose unelying fom is onjuntions of quntifie Hon expessions. In oe to veify tht given quey is equivlent to the one intene y use, unique veifition set of polynomil size is onstute. The veifition set onsists of exmples uniquely etemining the given quey. The veifition lgoithm lssifies some questions in the set s nswes (positive exmples) n othes s non-nswes (negtive exmples). The quey is inoet if the use isgees with ny of the quey s lssifition of questions in the veifition set.

5 S. Stwoko n P. Wiezoek 147 A nume of notions of hteizility hs een stuie in the ontext of gmmtil infeene [6]. Thei wok is elte to Gol s lssil moel [7, 8]. In the fist ppoh hteisti smples e onstute fo given lgoithm. The lgoithm must etun onept onsistent with the given smple n fo ny smple extening the hteisti smple fo the onept, it hs to etun. In nothe vint hteizility is pmetize with set I of lgoithms, i.e., fo eh onept, its hteisti set must llow ny lgoithm fom I to ientify. Finlly, they intoue the following notion: onept lss C is polynomilly hteizle iff fo eh onept thee exists hteisti smple S of polynomil size suh tht if nothe non-equivlent onept is omptile with S then is not omptile with the hteisti smple fo. In ou se q is the only quey tht is onsistent with the hteisti smple fo q. In the lening senio the min ojetive is to fin lening lgoithm tht poues fomul (quey) onsistent with given set of exmples [7, 8, 6, 19, 20]. One typilly uses weke notion of hteisti smple w..t. given lening lgoithm. This llows lene is, ollusion, e.g., using fixe oe on the lphet in lnguge infeene. Chteizility is stonge euse it implies the existene of hteizing smple inepenent of the lening lgoithm (no is). Ogniztion. In Setion 2 we ell si notions on XML n twig queies. In Setion 3 we fomlize the polem of hteizing queies with exmples n show tht unions of twig queies e not hteizle. In Setion 4 we show tht twig queies e hteizle ut my equie exponentilly mny exmples. In Setion 5 we ell nhoe twig queies n thei funmentl popeties. In Setion 6 we pesent si geneliztion opetions, show thei onnetion with injetive emeings, n show how to use them to hteize nhoe twig queies une the injetive semntis with polynomilly smll exmples. In Setion 7 we show tht the geneliztion opetions n e use to minimize nhoe twig quey n then show how to exten the ppoh use in Setion 6 to onstut polynomilly smll sets of exmples hteizing nhoe twig queies in the stn semntis. In Setion 8 we summize ou esults n outline futue ietions of stuy. Aknowlegments. This ppe is ptilly suppote y the Polish Ntionl Siene Cente gnt DEC-2013/09/B/ST6/ Bsi notions In this setion we ell si notions use to moel XML ouments n twig queies. Thoughout this ppe we ssume n infinite set of noe lels Σ whih llows us to moel ouments with textul vlues. Also, we fix one speil lel Σ tht we use on the oot noes of ll tees n queies. Tees. We moel XML ouments with unnke tees whose noes e lele with elements of Σ. Fomlly, tee t is tuple (N t, oot t, pent t, l t ), whee N t is nonempty finite set of noes, oot t N t is the oot noe, pent t : N t \{oot t } N t is hil-to-pent funtion, n l t : N t Σ is leling funtion. By Tee we enote the set of ll tees. An exmple of tee is pesente in Fig. 1. We efine nume of itionl notions. A lef is ny noe tht hs no hilen. A pth in t fom n to m (of length k) is sequene of noes n = n 1,..., n k = m suh tht pent t (n i ) = n i 1 fo 1 < i k. Then we lso sy tht n is n nesto of m n m is esennt of n. Note tht those two tems e eflexive: evey I C D T

6 148 Chteizing XML Twig Queies with Exmples () Tee t 0. () Twig quey q 0. () Emeings of q 0 in t 0. Figue 1 Tee, twig quey, n emeings. noe is its own nesto n esennt. We the jetive pope to inite tht n n m e iffeent noes. The epth of noe is the length of the pth fom the oot to the noe. The height of tee t, enote height(t), is the epth of its eepest lef. The size of t, enote size(t), is the nume of its noes. Queies. In genel, lss of Boolen queies is set Q with n impliitly given funtion L : Q 2 Tee tht mps evey quey q Q to the set L(q) Tee of tees tht stisfy q. The se lss of queies, tht we stuy in this ppe, e (Boolen) twig queies, known lso s tee pttens [2]. Bsilly, twig quey is n unnke tee tht my itionlly use istinguishe wil symol s lel n hs two types of eges, hil n pope esennt, oesponing to the stn XPth xes. Fig. 1 ontins exmple of twig queies: hil eges e wn with single line n esennt eges with oule line. Fomlly, twig quey q is tuple (N q, oot q, pent q, l q, ege q ), whee N q is nonempty finite set of noes, oot q is the oot noe, pent q : N q \ {oot q } N q is hil-to-pent funtion, l p : N q Σ {} is leling funtion, n ege q : N q \ {oot q } {hil, es} is the funtion tht inites the type of the inoming ege of non-oot noe. By Twig we enote the set of ll twig queies. We pt the stn notions efine fo tees (lef, pth, et.) to twig queies y ignoing the ege q omponent of the quey. Emeings. We efine the semntis of twig queies using the notion of n emeing whih essentilly mps noes of the twig quey to the noes of the tee in mnne onsistent with the semntis of the eges n the noe lels. In the sequel, fo two x, y Σ {} we sy tht x mthes y if y implies x = y. Note tht this eltion is not symmeti: the lel mthes ut oes not mth. Fomlly, n emeing of twig quey q in tee t is funtion λ : N q N t suh tht: 1. λ(oot q ) = oot t, 2. l t (λ(n)) mthes l q (n) fo evey noe n of q, 3. λ(n) is pope esennt of λ(pent q (n)) fo evey n N q \ {oot q }, 4. λ(n) is hil of λ(pent q (n)) fo evey n N t \ {oot q } suh tht ege q (n) = hil. Then, we sy tht t is stisfies q. Fig. 6 pesents ll emeings of the quey q 0 in tee t 0. The lnguge of quey q Twig is the set of ll tees stisfying q L(q) = {t Tee t stisfies q}. The notion of n emeing extens in ntul fshion to pi of queies q, p Twig: n emeing of q in p is funtion λ : N q N p tht stisfies the onitions 1, 2, n 3 ove (with t eing eple y p) n the following onition (whih ensues tht hil eges e mppe to hil eges only):

7 S. Stwoko n P. Wiezoek λ(n) is hil of λ(pent q (n)) n ege p (λ(n)) = hil fo evey n N q \ {oot q } suh tht ege q (n) = hil. Then, we wite q p. Beuse tee n e seen s twig quey, we often use the nottion n wite t q to inite tht thee is n emeing of q in t. Quey ontinment n equivlene. Given two queies q n p, q is ontine in p, in symols q p, iff L(q) L(p). We sy tht q n p e equivlent, enote q p, if L(q) = L(p). It is well known tht fo twig queies, the existene of n emeing implies ontinment ut the onvese oes not hol in genel [13]. Thee e lso signifint omputtionl iffeenes: the ontinment of twig queies is onp-omplete [17, 15] whees testing the existene of n emeing is in PTIME. 3 Chteizing queies with exmples In this setion we fomlly efine hteizility of queies n show tht unions of twig queies e not hteizle. An exmple is n element of Tee {+, }, pi onsisting of tee n n inito of whethe the exmple is positive (+) o negtive ( ). Evey quey q efines the set of its exmples L ± (q): L ± (q) = L(q) {+} (Tee \ L(q)) { }. Given set of exmples S, we enote y S + = {t Tee (t, +) S} n y S = {t Tee (t, ) S} the sets of espetively positive n negtive exmples in S. A quey q is onsistent with exmples S iff S + L(q) n S L(q) = (o simply S L ± (q)). Definition 3.1. A lss of queies Q is hteizle iff fo evey quey q Q, thee exists finite set of exmples Ch(q) hteizing q i.e., suh tht q is the only quey in Q onsistent with Ch(q) (moulo quey equivlene). Chteizility lone oes not ensue ny oun on the inlity of the set of hteizing exmples no ny oun on thei size. As we show lte on, twig queies e hteizle ut the nume of neessy exmples my e exponentil, whih my e unesile fo ptil puposes. Theefoe, we fomlize vint of hteizility tht ensues moe mngele size of the hteizing set of exmples. Definition 3.2. A lss of queies Q is suintly hteizle iff thee exists polynomil poly(x) suh tht fo evey quey q Q thee exists set of exmples Ch(q) hteizing q n suh tht its inlity is oune y poly(size(q)) n so is the size of evey of its elements. 3.1 Non-hteizility of unions of twig queies We now onsie the lss UTwig onsists of finite susets of twig queies Q = {q 1,..., q k } Twig intepete in the ntul fshion: L(Q) = {L(q) q Q}. The height of nonempty union of twig queies Q is the mximum height of quey in Q. Note tht if the height of tee t is supeio to the height of twig quey q, then q is not stisfie in t. Ntully, the sme neessy onition hol fo unions of twig queies. We sy tht quey Q UTwig is unstute if the set of its negtive exmples Tee \ L(Q) is infinite n ontins tees of ity height. The univesl quey {} is not unstute euse it hs no negtive exmples. One n, howeve, esily see tht UTwig I C D T

8 150 Chteizing XML Twig Queies with Exmples ontins unstute queies e.g., the singleton quey Q 0 = {q 0 } with q 0 fom Fig. 1 is unstute euse ny tee tht whose oot noe oes not hve hil is negtive exmple of Q 0. Now, tke ny unstute quey Q UTwig n set of exmples S onsistent with Q. Let U = (Tee \ L(Q)) \ S e the set of ll negtive exmples of Q tht e not use in S. Fom U we pik ny element t whose height is gete thn the height of ny negtive exmple in S. We tet t s twig quey n onstut Q = Q {t}. Clely, Q is onsistent with S euse ll positive exmples S + e stisfie y Q Q n none of the negtive exmples stisfy the newly e quey omponent t euse the height of t is gete thn the height of ny of the negtive exmples. Also, Q n Q e not equivlent euse t stisfies Q ut not Q. Theoem 3.3. Unions of twig queies e not hteizle. Finlly, we point out tht when pplie with iligene the ove poeue n e itete infinitely thus geneting n infinite sequene of queies onsistent with the given set of exmples. 4 Chteizility of Twig queies In this setion we show tht Twig queies e hteizle ut my equie nume of exmples exponentil in the size of the quey. Poposition 4.1. Twig queies e hteizle. Poof. Fo tee t Tee we efine the Twig-theoy it genetes Th(t) = {q Twig t L(q)}, the set of ll twig queies tht e stisfie y t. Fist, we show tht fo ny tee its Twig-theoy is finite moulo quey equivlene. We oseve tht the height of ny quey q Th(t) is oun y the height of t n the nume of iffeent lels in q is lso oune y the nume of iffeent lels in t. Beuse we onsie only non-equivlent memes of Th(t), no noe of q hs two ientil sutees oote t ny two hilen of the noe. The two osevtions ove llow us to inutively pove tht the nume of non-equivlent iffeent queies in Th(t) is inee finite. Next, fo given twig quey q we outline the onstution of hteizing set of exmples. Fo this, we onstut positive exmple t q 0 otine fom q y epling evey esennt ege y hil ege n using fesh lel 0 (not use in q) to eple evey. Note tht q Th(t q 0 ) n tht fo ny p Th(tq 0 ) tht is not equivlent to q thee exists witness t of the non-equivlene of p n q, whih we n use s positive exmple, if t stisfies q ut not p, o s negtive exmple, if t stisfies p ut not q. Sine Th(t q 0 ) ontins finite nume of queies moulo equivlene, only finite nume of exmples is neessy to onstut set of exmples hteizing q. Now, we show tht twig queies my equie exponentil nume of exmples to hteize them. Poposition 4.2. Fo ny ntul nume n thee exists twig quey q suh tht ny set hteizing q ontins t lest 2 n exmples. Poof. Fo given ntul nume n we onstut quey q n set of twig queies U suh tht: 1. fo eh p U we hve p q;

9 S. Stwoko n P. Wiezoek 151 A 0 i : i q: B 0 i : i p v : A 1 i : i i i A 1 1 A 0 2 A 0 3 A 0 1 A 1 2 A A 0 1 A 0 2 A 0 3 B 1 i : i i B k1 1 B k2 2 i i A 0 n A 0 n A 1 n B kn n () Quey q () Quey p v fo v = (k 1,..., k n). Figue 2 Twig quey q equiing exponentilly mny exmples w..t. p v. 2. fo eh p U we hve q p; Moeove, no single positive exmple t n witness the ft tht ny two istint p 1, p 2 U e not equivlent to q; 3. U ontins 2 n queies. Hene ny set of exmples S hteizing q will hve t lest 2 n negtive exmples use to istinguish q fom queies in U. We onstut the quey q ove the set of lels { 1,... n, } s illustte in Fig. 2. We lso onstut n uxiliy set of queies W onsists of pttens p v with v nging ove ll {0, 1}-vetos of length n, onstute s pesente in Fig. 2. We show tht the set W stisfies the onitions 2 n 3 n lte we use it to onstut the set U tht stisfies ll onitions. Clely, fo evey veto v we hve q p v essentilly euse A 1 i Bi k fo ny i {1,..., n} n k {0, 1}. Now, tke two {0, 1}-vetos v = (k 1,..., k n ) n w = (k 1,..., k n) tht iffe t position j {1,..., n} n w.l.o.g. ssume tht k j = 0 n k j = 1. Suppose now tht thee exists tee t tht witnesses oth the fts q p v n q p w i.e., t stisfies q ut neithe p v n p w. Let s j e the j-th nh of q i.e., the nh using A 1 j. Sine thee is n emeing of q into t, tke the pth in whih the nh s j is emee into π = j 1 j 1 j 1 j τ j j+1 j+1 j+1... n n n, whee τ is possily empty pth fgment. Note tht if τ is empty, then π p v, n thus, t p v. Howeve, if τ is not empty then π p w, n thus, t p w. This ontits the ssumption tht t oes not stisfy oth p v n p w. Consequently, evey quey p W equies unique positive exmple to istinguish it fom q. Now, fo two twig queies p n q y p q we enote the twig quey otine y joining p n q t the oot noe (whih hs the sme lel). The set of queies U is efine s U = {q p p W }. Beuse L(p q) = L(p) L(q), evey element of U is popely ontine in q n evey element of U still equies septe exmple to istinguish it fom q. 5 Anhoe twig queies In this setion, we pesent the lss of nhoe twig queies n gue tht they onstitute sulss tht is funtionlly vey lose to twig queies. We lso pesent the onstution I C D T

10 152 Chteizing XML Twig Queies with Exmples q 0 q 0 q 1 p 0 p 0 p 0 not nhoe nhoe nhoe not nhoe Figue 3 Anhoe n non-nhoe twig queies. of epesenttive ouments fo given nhoe twig quey n ell the eltionships etween thei stutue n semntis with thei impotnt omputtionl implitions. The lss of nhoe twig queies imposes estitions on the mutul use of n the esennt eges. Definition 5.1. A twig quey is nhoe if the following two onitions e stisfie: 1. A //-ege n e inient to -noe only if the noe is lef. 2. A -noe my e lef only if it is inient to //-ege. By AnhTwig we enote the set of ll nhoe twig queies. A nume of nhoe n non-nhoe queies is pesente in Fig. 3. Note tht the seon onition equiing lef to e inient to esennt ege is meely tehnil: if the ule is violte y some lef, we n hnge its ege to esennt ege n otin n equivlent quey (f. q 0 q 0 in Fig. 3). In essene, nhoe twig queies o not llow the esennt eges touh exept fo leves n thus nnot expess onitions on nesty of noes with miniml istne etween them. Fo instne the quey p 0 (Fig. 3) heks fo the existene of noe lele t epth 2. We o not elieve this estition to e signifint one fom the ptil point of view. The eson we use nhoe queies is the lose eltionship etween the stutue of the quey n its semntis [19]: ontinment is equivlent to the existene of emeing. Moe peisely, fo ny p, q AnhTwig we hve p q iff p q. The sme eltionship oes not hol fo queies tht e not nhoe e.g., the non-nhoe quey q 0 n the nhoe quey q 0 in Fig. 3 e equivlent n in ptiul q 0 q 0 ut thee is no emeing of q 0 into q 0. Consequently, testing the ontinment is eue to testing the existene of n emeing, n theefoe, is in PTIME. This stns in ontst with onp-ompleteness of the ontinment of ity twigs [17, 15]. The min tool use in poving the equivlene of ontinment n emeing fo nhoe queies e ontinment hteizing tees [13, 19]. Fo n nhoe quey q of height k we onstut two tees: t q 0 is otine fom q y epling evey esennt ege y hil ege n evey with fesh lel 0 tht is not use y q. t q 1 is otine fom q y epling evey y 1 n evey esennt ege y 2 -pth of length k, 1 n 2 e two iffeent fesh lels not use in q n iffeent fom 0. An exmple of the onstution is pesente in Fig. 4. The instumentl esult follows. Lemm 5.2 ([19]). Tke ny nhoe quey q n onstut t 1 q s esie ove. Fo ny quey p whose height is oune y the height of q n tht oes not use the lels 1 n 2, t q 1 p implies q p.

11 S. Stwoko n P. Wiezoek 153 q 0 t q 0 0 t q { Figue 4 Constution of ontinment hteizing tees. x δ 1 y δ 2 x y x y z z x z z Figue 5 One-step geneliztion opetions ( ). The poof onsists of n nhoing tehnique tht nomlizes the emeing of p into t q 1 n then tnsltes it to n emeing of p into q. Note tht if t q 0 p, then the height of p is oune y the height of q n p oes not use 1 n 2. Theefoe if oth t q 0 n tq 1 stisfy p, then q p. The most impotnt implition of Lemm 5.2 is tht y using only two positive exmples t q 0 n tq 1 we only nee to e out the queies tht popely ontin q n povie negtive exmples to istinguish q fom queies popely ontining q. Although thei nume might e quite lge, we fous only on the most speifi ones: Φ(q) = {q AnhTwig q q q. q q q }. Fom the view of the semi-lttie of nhoe twig queies, we wish to gin n unestning of its topology to hteize the (outoun) neighohoo of quey. 6 Geneliztion opetions In this setion, we intoue set of geneliztion opetions n show thei onnetion with n injetive emeings n how the opetions llow us to nvigte the semi-lttie of nhoe twig queies une injetive semntis. We use those finings to suintly hteize nhoe twig queies une the injetive semntis. We employ simple geneliztion opetions tht we fist efine on ity twig queies n lte on tilo them to nhoe twig queies. Thee e 3 opetions, pesente in Fig. 5, whee she lines inite optionl ity eges, x, y n z my hve ity lels in Σ {} while nges ove Σ: δ 1 hnges the lel of non- noe to ; δ 2 hnges hil ege to esennt eges; emoves noe n onnets ll its hilen to the pent noe with esennt ege; We sy tht q is one-step geneliztion of p, in symols p q, iff q is otine y pefoming one of the 3 geneliztion opetions. I C D T

12 154 Chteizing XML Twig Queies with Exmples t 0 q 0 t 1 q 1 o o o o o q 2 o o o o Figue 6 Injetive emeings t 0 q 0 n q 2 q Injetive emeings Thee is lose onnetion etween pplying sequenes of geneliztion opetions n the existene of speil kin of injetive emeings. Fomlly, n emeing of λ of q into p is injetive if it itionlly stisfies the onition: 5. λ(n 1 ) is n nesto of λ(n 2 ) in p if n only if n 1 is n nesto of n 2 in q, fo ny two noes n 1 n n 2 of q. We wite p q if thee is n injetive emeing of q into p. We efine nlogously the injetive emeings of twig quey into tee. Fig. 6 pesents n exmple of n injetive emeing of q 0 into t 0. We point out tht ny injetive emeing is n injetive funtion ut the onvese oes not neessily hols (f. q 0 n t 1 in Fig. 6). This howeve will not le to onfusion s we o not onsie emeings tht e injetive funtions while violting onition 5. We efine the injetive semntis of twig queies s L (q) = {t Tee t q} n y q p enote L (q) L (p). The nhoing tehnique of Lemm 5.2 n e esily pte to injetive emeings euse injetive emeings e une lose omposition. Coolly 6.1. Fo ny p, q AnhTwig, p q iff p q iff p q. The onnetion etween the geneliztion opetions n injetive emeings is quite ntul. While it is quite ovious tht p q implies p q, the onvese is lso tue euse the existene of n injetive emeing of q into p ensues tht ll fgments of q mth es of p in onfigution tht llows to esily ientify the geneliztion opetions equie to tnsfom p into q, f. the injetive emeing of q 1 into q 2 in Fig. 6. Hene, Lemm 6.2. Fo ity twig queies p, q Twig we hve p q iff p q. Finlly, we point out, howeve, ltentive types of injetive emeings hve een ientifie n use in the litetue (f. [12]), fo ou puposes ny type of injetive emeings n e use. Howeve, the set of neessy si geneliztion opetions (Fig. 5) epens on the hosen type of injetive emeing, n we hve hosen the type of injetive emeings known s nesto-peseving emeings euse the geneliztion opetions seems to e the most ntul. 6.2 Geneliztions of nhoe twig queies We wish to use the onnetion etween geneliztion opetions n the ontinment in oe to mp out the semi-lttie AnhTwig, of nhoe twig queies une the injetive

13 S. Stwoko n P. Wiezoek 155 q 1 q 1 q 1 q 2 q 2 q 2 q 3 q 3 q 3 δ 1 δ 1 δ 1 δ 1 δ 2 δ 2 not nhoe not nhoe not nhoe q 4 q 4 q 4 q 4 q 4 q 5 q 5 q 5 δ 2 δ 1 δ 2 q 6 q 6 q 6 q 6 q 6 e f e f e f e f e f Figue 7 Applying geneliztion opetions to nhoe twig queies. semntis. Moe peisely, fo given nhoe quey p we wish to know the set of nhoe queies Φ (p) in the immeite (outgoing) neighohoo of p. Φ (q) = {q AnhTwig q q q AnhTwig. q q q }. Lemm 6.2 enouges us to ppoh this hllenge with the use of geneliztion opetions tiloe to nhoe queies. Given two nhoe queies p, q AnhTwig, we sy tht q is n immeite nhoe geneliztion of p, in symols p q, iff p + q n thee is no z AnhTwig suh tht p + z n z + q. Essentilly, the immeite nhoe geneliztion eltion efines the Hsse igm of the semi-lttie of the nhoe twig queies une injetive semntis, n theefoe, Φ (p) = {p AnhTwig p p }. We next show how n e efine in tems of smll nume of mos onsisting of sequenes of geneliztion opetions. To otin n immeite nhoe geneliztion of n nhoe twig quey we pply iligently the geneliztions opetions, mking sue tht: 1) the en esult is n nhoe twig quey n 2) thee is no intemeite nhoe twig quey tht n e otine on n ltentive pth. In ptiul, the fist two geneliztion opetions δ 1 n δ 2 n e use s long s they o not yiel quey tht is not nhoe (f. q 1, q 2, n q 3 in Fig. 7). The is moe involve. Fist of ll, une etin onitions pplying is not uthoize euse n intemeite quey n e ehe with opetions δ 1 o δ 2 (f. q 4 n q 5 in Fig. 7). Futhemoe, while possily violting the stutul onstints of nhoe queies, n e pplie to non-lef noe povie tht is pplie to ll neighoing -noes (f. q 6 in Fig. 7). We stte fomlly the mnne in whih the geneliztions shoul e use on nhoe twig queies. Lemm 6.3 ( -opetions). Fo ny nhoe twig queies p, q AnhTwig we hve tht p q iff q is otine fom p y pplying: I C D T

14 156 Chteizing XML Twig Queies with Exmples p 0 p0 Φ p 0 p 0 Figue 8 Immeite nhoe geneliztions. p 0 t p 0 0, + t p 0 1, + t p 0 1, t p 0 1, t p 0 1, Figue 9 Chteizing quey une the injetive semntis. 1. δ 1 to n inne noe inient to hil eges only; 2. δ 1 to lef noe inient to esennt ege; 3. δ 2 to n ege tht is not inient to noe; 4. to lef noe only if it is noe o if its pent is noe with othe hilen; 5. to non- noe inient to esennt eges only; 6. in exhustive mnne to onnete e of noes. In the sequel we efe to the mos fom Lemm 6.3 s -opetions. We point out tht fom the point of view of -opetions, ny onnete e of noes is seen s one ptiul noe n we ientify it with its top most noe. Note tht the nume of possile iffeent pplitions of -opetions to quey p is O(size(p)), n onsequently, the Φ (p) ontins polynomil nume of queies eh of size oune y the size of p. An exmple of onstution of Φ is pesente in Fig Chteizility of AnhTwig une injetive semntis Beuse the nume of possile immeite nhoe geneliztions of quey p is O(size(p)), we n hteize ny nhoe twig quey p une injetive semntis with the following set of exmples (with ll t q 1 s tees using 2-hins of the sme length s t p 1 ): Ch (p) = {(t p 0, +), (tp 1, +)} {(t q 1, ) q Φ (p)}. An exmple of onstution of the hteizing set of exmples is pesente in Fig. 9 fo the quey p 0 fom Fig. 8. To show tht Ch (p) oes inee hteizes p, tke ny quey q onsistent with Ch (p). Sine q is stisfie y oth t p 0 n tp 1, p q n if p woul e popely ontine y q, then q woul stisfy one of the negtive exmples in Ch (p). Theoem 6.4. Anhoe twig queies e suintly hteizle une injetive semntis.

15 S. Stwoko n P. Wiezoek 157 δ 1 Figue 10 Reuing quey. 7 Chteizility of Anhoe Twig queies In this setion, we exten the ppoh pesente in the pevious setion to the stn semntis of twig queies. The min hllenge is to hnle possile ovelps of non-injetive emeings n we intoue uplition opetion whose ontolle use ensues suint hteizility. Thee is, howeve, lesse hllenge tht we nee to hnle fist, mking sue tht pplying -opetion yiels moe genel quey. Solving this hllenge lso shows tht minimiztion of nhoe twig queies is ttle n n e implemente using geneliztion opetions whih futhe illusttes the goo ehvio of the lss of nhoe twig queies. 7.1 Reuing nhoe twig queies If we e to se the solution of hteizility on -opetions, we must e we of the iffeene etween the two semntis, how it n ffet the use of -opetions, n how to oveome the iffiulties tht ise. The iffeene etween the semntis n e illustte on the exmple of quey q 0 in Fig. 1: une the injetive semntis it ensues the existene of noe with t lest two iffeent hilen, while suh onstint nnot e expesse with the stn semntis (note tht neithe of the emeings in Fig. 6 n is nesto-peseving). In ft, the lef noe in the quey q 0 is eunnt, n e emove, n smlle equivlent quey (une the stn semntis) is otine. The implitions on use of -opetions e s follows: if we tke n nhoe quey p n ny p otine y pplying -opetion., then p nees not popely inlue in p euse p n p my e equivlent une the stn semntis. To ess this ostle we eue the quey p: itetively pply geneliztion opetions (f. Fig. 10), following t eh step, s long s the quey emins equivlent to the oiginl one. Note tht -opetion my emove some noes, enme noes to, n hnge hil eges to esennt eges. Essentilly, it is monotone poess tht finishes fte O(size(p)) steps. An nhoe twig quey p is eue if it is the en esult of this poeue, o moe peisely, if fo no p Φ (p) we hve p p. Inteestingly, not only e eue queies suitle fo use in Φ n n e otine effiiently, ut lso they e the unique miniml nonil epesenttives of ll equivlent queies. Inee, we show tht n nhoe twig quey p is not miniml iff thee exists n emeing of p into p othe thn the ientity mp, n futhemoe it mps t lest two noes to the sme tget (n ovelp). We n then efully onstut the imge p of this emeing, n nhoe quey whose set of noes is the nge of the emeing (see exmple in Fig. 10). I C D T

16 158 Chteizing XML Twig Queies with Exmples p 0 p 0 p 0 p 0 q 0 p 0 Figue 11 Immeite nhoe geneliztions vesus n emeing with ovelp p 0 q 0. This quey is smlle thn p n the ientity mp is n injetive emeing of p into p, thus p p. We point out tht ttility of minimiztion of nhoe twig queies is not novel esult. [11] pesents lss of twig queies popely ontining AnhTwig fo whih minimiztion is ttle. While the tehnique is simil to the pesente ove (ut not ientil), it n possily e use s moe effiient ltentive only euse we shown tht ou euing metho lso poues the miniml quey. Theoem 7.1. Fo evey nhoe twig quey q thee exists unique eue equivlent nhoe twig quey. Futhemoe, this quey is the size-miniml quey equivlent to q n n e otine in time polynomil in the size of q. 7.2 Duplition opetion Fom now on, we use the stn semntis only. While using -opetions on eue quey p oes onstut set of most speifi queies popely ontining p, it oes not ontin ll suh queies. Tke fo instne the quey p 0 in Fig. 11 togethe with its immeite nhoe geneliztions Φ (p 0 ) = {p 0, p 0, p 0 }. Now tke the quey q 0 (Fig. 11) tht hs n emeing into p 0, is not equivlent to p 0 (thee is no emeing of p 0 into q 0 ), n note tht it nnot e emee into ny of the immeite nhoe geneliztions of p 0. This is euse the emeing of q 0 into p 0 ovelps t two noes of q 0 whose suqueies hve een otine with pplying iffeent geneliztion opetions. We ess the polem of the ovelp with wht oul e seen s ing uplition opetion to ou system: this opetion eples suquey oote t given noe y nume of ientil opies (inluing the sme type of the ege to the pent). Suh n opetion woul not, howeve, yiel quey moe genel thn the oiginl quey, ut meely n equivlent quey with eunnies, n theefoe, not even eue. Consequently, we investigte n ugmente vint tht pplies the uplition opetion n then genelizes evey opy. In Fig. 11 the quey p 0 is quey otine s esult of pplying this opetion to p 0 t noe. We point out, howeve, tht the efinition of this opetion is not suffiiently peise: it is mutully epenent on the efinition of geneliztion, whih ielly shoul use the new opetion. Also, unlike peviously use opetions whih oul only eese the size of the input quey, this opetion hs the potentil to inese the size the quey n when use eusively multiple time, the ouns on the size of the output quey e not le. We next settle these onens with n ppopite eusive efinition of quey geneliztion. We efine geneliztions of quey eusively on its suqueies. A suquey p of q t noe n N q \ {oot q } is essentilly the quey oote t n ut hving itionlly the inoming ege (fom the pent) whih n e ltee y pplying -opetions to the oot noe of q. Ntully, we pply only those opetions tht e llowe in the ontext of the omplete quey q (thus s if knowing the lel of the pent of the oot noe of p). We fix quey q n let p e its suquey. A miniml geneliztion of p is ny quey otine y eithe:

17 S. Stwoko n P. Wiezoek 159 q q e f e f e f e f e f Figue 12 Constuting the miniml geneliztion q of quey q (q q ). 1. pplying -opetion t the oot noe of p (ppopitely to the ontext of the oot noe in q) 2. epling the suquey p oote t hil of the oot noe of p with the set of ll miniml geneliztions of p. The miniml geneliztions of q e otine y using only the seon pt of the efinition (euse we o not pply geneliztion opetion to the oot noe). We wite q q to inite tht q is miniml geneliztion of q. An exmple of onstuting the miniml geneliztion of quey is pesente in Fig. 12. A eue nhoe twig quey q hs t most line nume of miniml geneliztions (equl to the nume of hilen of the oot noe). It is not le how ig they n e given tht uplition tkes ple. We pove polynomil oun on the size of q suh tht q q. Let n e noe of q, with the pth n = n 0, n 1,..., n k = oot q fom n to the oot of q, n let l i e the nume of hilen of n i fo i {1,..., k}. We show with n inutive poof tht q ontins O(l l k ) opies of the noe n. Sine l l k is oune y the nume of noes of q, eh noe is uplite t most size(q) times, n theefoe, q is of size O(size(q) 2 ). With n inutive poof on the stutue of n ity emeing etween two queies tht e not equivlent, we show tht the miniml geneliztions of quey q e the only outoun neighos of q in the semi-lttie of nhoe twig queies une the stn semntis. Lemm 7.2. Fo ny nhoe twig quey q, Φ(q) = {q AnhTwig q q }. Consequently, ny nhoe twig quey q n e hteize with polynomilly-size set of exmples Ch(q) = {(t q 0, +), (tq 1, +)} {(t p 1, ) p Φ(q)} n thei sizes e polynomilly oune y the size of q. Theoem 7.3. Anhoe twig queies e suintly hteizle. 8 Conlusions n futue wok In the pesent ppe, we hve ientifie n stuie novel polem of hteizing queies with exmples. Ou esults hve emonstte tht hteizility is mesue of ihness of the quey lss, whih is losely elte to its expessive powe. We hve show tht while union of twig queies e not hteizle, twigs lone e ut my equie exponentil numes of exmples. Though the stuy of emeings n geneliztion opetions we hve shown tht the lss of nhoe twig queies is hteizle with polynomilly size sets of exmples. Futue wok. We envision nume of possile futue ietions. We woul like to exten ou stuy of emeings to othe types of queies tht employ this mehnism of efining I C D T

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