A Rewrite Approach for Pattern Containment

Transcription

1 A Rewrite Approh or Pttern Continment Brr Kory LIFO - Université Orléns, Frne Astrt. In this pper we introue n pproh tht llows to hnle the ontinment prolem or the rgment XP(/,//,[ ], ) o XPth. Using rewriting tehniques we eine neessry n suiient onition or pttern ontinment. This rewrite view is then pte to query evlution on XML ouments, n remins vli even i the ouments re given in ompresse orm, s gs. 1 Introution The ous in this pper is on the ontinment prolem ([1,2]) or the rgment XP(/,//,[ ], ) o XPth. XPth ([3]) is the min lnguge or nvigting n seleting noes in XML ouments. The segment XP(/,//,[ ], ) eines lss o Core XPth queries expressing esennt reltionships etween noes, possily ontining ilters, n llowing to use the on t re (or wilr) symol. The queries o this rgment n e moele y ptterns: tree like grphs hving two types o eges hil n esennt. Every XML oument t is n unrnke tree t = (Noes t, Eges t ), n n lso e seen s pttern. For ny two ptterns P n Q, we sy tht P is ontine in Q (P Q), i the query represente y Q is more generl thn the one represente y P. For exmple, / is ontine in //, sine hil (/) is prtiulr se o esennt (//). The ig interest in the query ontinment prolem ([1,2,4,5]) is motivte y its pplitions. Using the notion o pttern ontinment we n eine queries whih re equivlent, i.e., tht on ny XML oument, selet the sme set o noes. The query equivlene prolem is losely linke to the query minimiztion prolem, whih is essentil or t se reserhers. Sine the time require or the evlution o given query Q is liner with respet to the size o Q ([6]), the minimiztion possiility o repling Q y n equivlent query o smller size is o interest rom the point o view o omplexity ([7,8,9,10]). We propose to hnle the ontinment prolem using rewrite pproh. We eine set o rewrite rules se on the semntis o XP(/,//,[ ], ) query ontinment, n show tht or ny two ptterns P n Q, P is ontine in Q i n only i we n rewrite P to Q using these rules. This provies hrteriztion o the ontinment prolem using lgeri tehniques, whih ws missing in the literture. Suh rewrite view gives us uniorm rmework to tret lso other prolems, or instne query evlution. We exten our pproh on ompresse ouments enoe s strightline regulr grmmrs, n pply The originl pulition is ville t:

2 our rewrite tehnique in orer to evlute XP(/,//,[ ], ) queries on ompresse or unole (roresent) XML ouments. This pper is orgnize s ollows: In Setion 2 we introue terminology n nottion, n rell some results on the pttern ontinment prolem. Our rewrite metho is presente in Setion 3. Finlly, in Setion 4 we show how to pt this rewrite pproh to the query evlution prolem. 2 The Pttern Continment Prolem Let Σ e n lphet ontining the element nmes o ll XML ouments onsiere. In this work we onsier the rgment XP(/,//,[ ], ) o XPth, whih onsists o: noe tests (symols rom Σ { }), hil xis (/), esennt xis (//), n quliiers lso lle ilters ([...]). Any element o XP(/,//,[ ], ) is query tht n e represente s roote tree struture grph over Σ { }, lle unry pttern, hving: eges o two types: simple or hil, n oule or esennt, noes lele y the symols rom Σ { }, one istinguishe noe mrke with speil seletion symol s representing the output inormtion (lote t the en o the min pth in the query onsiere). For instne, the unry pttern in Figure 1 represents the XP(/,//,[ ], ) query ///[.///]/[./ //]. The notion o unry ptterns is esily extene to s Fig. 1. Unry pttern representing query ///[.///]/[./ //] tht o n ry ptterns, where we hve n istinguishe noes, tht moel n ry queries seleting n tuples o noes. Miklu n Suiu show in [1] tht, or the purpose o the ontinment prolem, it is suiient to onsier only the ptterns o rity zero, lle oolen, where there re no istinguishe noes. Thus, ll ptterns onsiere in the sequel will e oolen, n they will e simply lle ptterns. For given pttern P, we enote y Noes P the set o ll its noes. For ny u Noes P, nme P (u) stns or the element o Σ { } leling the noe u. By Eges (P ) n Eges (P ) we men respetively the set o hil n esennt 2

3 eges o P. We eine the size o P (enote y P ) to e the numer o ll eges in P. Deinition 1. An XML tree t is moel o pttern P i there exists n emeing rom P to t; i.e., untion e: Noes P Noes t, stisying the ollowing onitions: 1. e preserves the root: e(root P ) = root t ; 2. e preserves the nmes: u Noes P, nme P (u) =, or nme P (u) = nme t (e(u)); 3. e preserves the reltion hil: (u, v) Eges (P ), (e(u), e(v)) Eges t ; 4. e preserves the reltion esennt: (u, v) Eges (P ), (e(u), e(v)) (Eges t ) +, where (Eges t ) + is the trnsitive losure o the reltion Eges t. The notion o moel is illustrte in Figure 2. P e t Fig. 2. Pttern P, its moel t, n emeing e rom P to t Deinition 2. Given two ptterns P n Q, we sy tht P is ontine in Q (P Q) i every moel o P is lso moel o Q. The ptterns P n Q re equivlent (P Q) i P Q n Q P. Figure 3 represents two ptterns whih re esily seen to e equivlent. P Q Fig. 3. Equivlent ptterns P n Q Miklu n Suiu prove in [1] tht the ontinment prolem or XP(/,//,[ ], ) is CoNP omplete. They lso give suiient ut not neessry onition or pttern ontinment. For tht purpose, they exten the notion o emeing to pttern homomorphism: 3

4 Deinition 3. Given two ptterns P n Q, homomorphism rom Q to P is untion ϕ: Noes Q Noes P, whih is: root n nme preserving; hil preserving: (u, v) Eges (Q), (ϕ(u), ϕ(v)) Eges (P ); esennt preserving: (u, v) Eges (Q), (ϕ(u), ϕ(v)) (Eges (P ) Eges (P )) +. The uthors o [1] prove tht i there exists homomorphism rom Q to P, then P is ontine in Q. They give n lgorithm whih or two given ptterns P n Q veriies, in time O( P Q ), whether there exists homomorphism rom Q to P. Figure 4 shows the ptterns P n Q, n the homomorphism ϕ rom Q to P proving tht P Q. Nevertheless, the existene o homomorphism rom Q P Q P ϕ ϕ Q Fig. 4. Homomorphism ϕ rom Q to P proving tht P Q to P is not neessry onition or P Q (s is esily heke or the ptterns P n Q given in Figure 3, whih re equivlent, ut there is no homomorphism neither rom Q to P, nor rom P to Q). In the ollowing exmple we show wy to prove the ontinment P Q, i there is no homomorphism rom Q to P. Exmple 1. In Figure 5 we hve presente two ptterns P n Q (orrowe rom [1]) stisying P Q, suh tht there is no homomorphism rom Q to P. Here, P Q Fig. 5. Ptterns s.t. P Q, ut no homomorphism rom Q to P to show the ontinment P Q, we hve to reson y ses. Let t e moel o P, n onsier the mile ege // o pttern P. This ege n e relize on t: either y the hil ege / (s in Figure 6), 4

5 P t e e Q e e Fig. 6. Moel o P (n Q), where // is relize y / P e t e Q e e Fig. 7. Moel o P (n Q), where // is relize y // or y pth / /... /, hving length 2 (s in Figure 7). Suh n nlysis shows tht ny moel o P is lso moel o Q, thus P Q. However, it is impossile to eine one generl homomorphism rom Q to P, s the right rnh //// // o Q orrespons in eh se to ierent rnh o P. 3 Pttern Continment vi rewriting We propose to hnle the pttern ontinment prolem using n pproh se on rewriting tehniques. A key ie is tht heking ontinment requires se nlysis in generl, n this n e enoe s rewriting (s we illustrte in Exmple 2 elow). We onstrut rewrite system R tht permits to eine neessry n suiient onition (see Theorem 1) or pttern ontinment on the rgment XP(/,//,[ ], ). We strt y giving orml einition o pttern, lterntive to tht use in the previous setions. Deinition 4. We eine ptterns over n lphet Σ s the expressions P erive rom the grmmr o Tle 1, where n stn respetively or hil n esennt, ω Σ { }, n is the on t re symol o XPth tht n reple ny σ o Σ. This grmmr proues preisely the ptterns s eine in [7,1]. For instne, the grph P in Figure 8 orrespons to the expression P = { {, }, } 5

6 M : ε ω ω MM // pth S : {MS} S S // set o siling unroote terms P : ωms // ptterns Tle 1. Grmmr or ptterns erive rom the grmmr o Tle 1. P Fig. 8. Pttern By term we men ny expression o the type M, S or P erive rom the grmmr o Tle 1, s well s ny inite isjuntion P 1 P 2 P n o ptterns. The terms o the type M orrespon to the liner pths without rnhing, they strt y mol symol ξ {, }; those o the type S represent set o terms hving ommon prent noe; n those o the type P re ptterns. The terms in P re roote (they strt y symol rom Σ { }), those in M n S re unroote. To simpliy, we will oten ientiy the singleton {M S} with the term MS. Given ptterns P n P i, or 1 i n, the terms o the orm ε, P, or P 1 P n, will lso e lle ptterns. A tree t is moel o pttern P 1 P n i t is moel o t lest one pttern P i, or 1 i n. Deinition 2 o pttern ontinment is extene in nturl wy to pttern ontinment. A isjuntive pttern will e use in se nlysis to represent ierent moels o given pttern with unique term, s in the ollowing exmple. Exmple 2. Consier the ptterns P = n Q = given in Figure 3. We know tht P Q, thus in prtiulr P Q, ut there is no homomorphism whih proves it. Using the rules o our system R eine elow we will e le to rewrite P to Q, n prove the ontinment P Q. The ie is tht every esennt is either hil or hs epth 2; thus, the ege o P n e relize either y the hil ege, or y pth hving t lest one itionl noe etween n, tht we n enote y. 6

7 We will then rewrite the pttern P to the pttern epiting the two ses mentione. The two pttern omponents o this pttern will then e rewritten in prllel. A hil, s well s esennt o epth 2 re prtiulr ses o esennt. As onsequene, the ege will e rewritten to, iem or the pth. This will give us the ollowing term:, whih will e inlly rewritten to Q, sine eh pttern omposing this pttern is extly the pttern Q. To ormlize the ie employe in the exmples ove, we introue set R o rules tht serve to rewrite roote n unroote terms. Let M, S (possily with primes, susripts) e s in the grmmr o Tle 1; ξ, ξ {, }, σ Σ, n ω, ω Σ { }: 1. S, M ε //ut; 2. MσS M S //reple ny symol o Σ y the o XPth; 3. ωs ωs //every hil is lso esennt; 4. ξωξ ω S ω S //ignore n intermeite noe; 5. M{S 1, S 2 } {MS 1, MS 2 } //let istriutivity; 6. S S S, where S S // new silings; 7. S S 1 S S 1, i S S //rewrite some o the silings; 8. ωs ( ωs) ( ωs) //se nlysis: esennt is either hil or hs epth 2; 9. ωs ( ωs) ( ωs) //iem. By ontext pttern we men ny pttern hving speil itionl hole symol tht reples one o its unroote su terms. Let us onsier ontext pttern C n n unroote term X. We eine the ill in o C with X (enote s C X) to e the pttern otine rom C y repling its hole symol with the term X; e.g. or the ontext pttern C = {, {, }, }, n the unroote term X = x{ y, z}, we get the ill in: C X = {, { x{ y, z}, }, }. We lso suppose tht or ny ontext pttern C n unroote terms X n X, the nottion C (X X ) stns or the isjuntive pttern C X C X. To rewrite ptterns with the rules o R given ove, we use suix rewriting: Deinition 5. Given pttern P n pttern or pttern Q, we sy tht P n e rewritten to Q in one step using suix rewriting, i there exist ontext pttern C n two unroote terms X n X, suh tht: P = C X, Q = C X, n X X is n instne o rule in R. Moreover, isjuntive terms n e rewritten using the ollowing itionl two rules, where P is pttern, n D, D 1, D 2 stn or ptterns: 7

8 10. D 1 D D 2 D, i D 1 n e rewritten to D 2 //se rewriting; 11. P P D P D //onsier ny given se only one. Rules 10 n 11 re use s ollows: i pttern L is n instne (moulo ommuttivity) o the LHS o rule 10 or 11, n pttern R is n instne (moulo ommuttivity) o the RHS o the sme rule, then L n e rewritten to R. We will enote y A R B the t tht pttern or pttern A is rewritten in one step to pttern or pttern B, y using the rules o R. The min result o our work is the ollowing: Theorem 1. For ny two ptterns P n Q, P is ontine in Q i n only i P R Q, i.e., P n e rewritten to Q using the rules o R in zero or initely mny steps. Proo. The semntis o the rules in R gurntee tht P R Q implies P Q. Inee, i X X is n instne o one o the rules 1 9, then or every ontext pttern C, we hve C X C X ; i L R is n instne o rule 10 or 11, we oviously hve L R. To show the onverse, we strt with the ollowing lemm: Lemm 1. For ny ptterns P n Q, i there exists homomorphism rom Q to P, then P R Q. Proo. Given homomorphism ϕ rom Q to P, we onstrut pttern P, suh tht P R P R Q, s ollows: () or every noe u o Q, we onstrut orresponing noe u o P, n we set nme P (u ) = nme P (ϕ(u)); () we onstrut hil ege (u, v ) Eges (P ), i n only i (u, v) Eges (Q); () we onstrut esennt ege (u, v ) Eges (P ), i n only i (u, v) Eges (Q). The ost o suh onstrution is liner with respet to the size o Q. The pttern P n e rewritten to the pttern Q using rule 2 o R. Inee, the strutures (noes, simple n oule eges) o P n Q re the sme, ut the nmes o some u Noes Q n the orresponing noe u Noes P my e ierent. Conition () implies tht: either nme Q (u) = nme P (u ) = nme P (ϕ(u)), or nme Q (u) nme P (u ) = nme P (ϕ(u)). In the seon se we hve (see Deinition 3): nme Q (u) =, n nme P (u ) Σ, thus to rewrite P to Q we hve to use rule 2. It remins to e shown tht P n e rewritten to P : using rules 1 n 7 (with S = ), we n ignore ll su rnhes o P whih o not ontin the noes imges uner ϕ; i some noe w o P is n imge o m istint noes u 1,..., u m o Q, then we rewrite the unique noe w o P to m noes u 1,..., u m o P, y using rule 6 (with S = S) n/or rule 5; 8

9 se when ege (u, v ) is in Eges (P ): rom onition () we know tht (u, v) Eges (Q), thus y Deinition 3 we hve (ϕ(u), ϕ(v)) Eges (P ) (we hve nothing to o with the ege (ϕ(u), ϕ(v)) when rewriting P to P ); se when ege (u, v ) is in Eges (P ): rom onition () n Deinition 3 we n eue tht there exist k 1 n w 0,... w k Noes P, suh tht: w 0 = ϕ(u), w k = ϕ(v), n i {0,..., k 1} we hve (w i, w i+1 ) Eges (P ) Eges (P ). I k = 1 n (ϕ(u), ϕ(v)) Eges (P ), then we n rewrite P to P using rule 3. I k 2, then we use (k 1 times) rule 4 to ignore the noes w 1,... w k 1 while rewriting P to P. Finlly, we otin P R P R Q. Note tht i P is tree, we lso hve the onverse o Lemm 1. Inee, it is suiient to remrk tht i P R Q n P is tree, then one n rewrite P to Q y using only rules 1 7; i X X is n instne o one o those rules, then or every ontext pttern C, there exists homomorphism rom C X to C X. O ourse, in the se when P is tree, homomorphism rom Q to P is n emeing rom the pttern Q to the tree P. The ove onsiertions give us the ollowing hrteriztion: Remrk 1. A tree t is moel o pttern Q i t R Q. By homomorphism rom pttern Q to pttern D = P 1 P n, we men untion whih is homomorphism rom Q to P i, or every 1 i n. Thus, using Lemm 1, we otin the ollowing orollry: Corollry 1. For ny given pttern Q n pttern D, i there exists homomorphism rom Q to D, then D R Q. Proo. It suies to remrk tht rules 10 n 11 imply tht pttern P 1 P n n e rewritten to pttern Q i n only i, or every 1 i n, we hve P i R Q. To inish the proo o Theorem 1, we use the ollowing proposition: Proposition 1. For two ptterns P n Q, i P Q, then one n onstrut pttern D veriying P R D, suh tht there exists homomorphism rom Q to D. Proo. From the result o Miklu n Suiu ([1]) we know tht it is possile to hek i there exists homomorphism rom Q to P. I it is the se, the pttern D stisying the proposition is equl to P (see Lemm 1). I not, isjuntive pttern D stisying the proposition n e onstrute y using rules 8 n 9 initely mny times. We know tht every moel o P is lso moel o Q. The ie is to represent ll moels o P y n equivlent pttern D = P 1 P n representing se nlysis, suh tht or every 1 i n, there exists homomorphism rom Q to P i. This termintes the proo o Theorem 1. 9

10 The rewrite system R is non eterministi; nevertheless i P n Q re given, there exists well eine, gol irete strtegy or rewriting P to Q. The ie is to use only those rules mong 1 11 tht permit to onverge to Q. We illustrte this strtegy in the ollowing exmple: Exmple 3. Let P n Q e the ptterns represente in Figure 5. We show how to rewrite P to Q, n thus prove the ontinment P Q. The pttern P = { {, }, } n e seen s the ill in { {, }, }. Using rule 8 or the unerline term, we enoe the ses epite in Exmple 1: { {, }, } { {, }, }. We otin the pttern { {, }, } { {, }, }, whih n e seen uner the orm { {, }, } { {, }, }. We rewrite it using rule 10. We ut (rule 1) the unerline prts, n get { { }, } { {, }}. The pttern tht we hve otine is then ientiie with {, } {, }. Its irst omponent is equl to the pttern Q. To the seon one, seen s the ill in {, }, we pply rule 4, n get the term {, } = {, }. Thus we otin the pttern {, } {, } = Q Q, tht is inlly rewritten to Q using rule 11. Remrk 2. Our pproh is no longer vli, i it is not se on suix rewriting; e.g. or P = n Q =, we hve P Q (P R Q using rules 9, 1), ut P = is not ontine in Q = : or instne, the tree t = g is moel o, ut not o. 10

11 4 Applitions The ojetive o this setion is to show tht our rewrite pproh remins vli even i the moels o ptterns re given in ompresse orm (s gs), n tht it n e pte or query evlution on XML ouments. 4.1 Cse o Compresse Douments To moel ompresse ouments we use roote gs inste o trees (s in [11,12,13,14]). Figure 9 represents three ormts o the sme oument: tree, ully n prtilly ompresse ormt (see [11] or orml einitions). In the Tree Fully ompresse Prtilly ompresse Fig. 9. Tree, ully ompresse ormt, prtilly ompresse ormt sequel, y t we will enote ny given representtion (tree or g) o the oument onsiere. To istinguish etween ierent ormts o the sme oument we use regulr tree grmmrs. Given oument t, we ll normlize grmmr or t regulr tree grmmr G t : whih reognizes only t, where every noe o t is represente y extly one non terminl, the inexes o non terminls or hilren noes re greter then the inexes o non terminls or prent noes. Suh normlize grmmrs re strightline in the sense eine in [15], i.e., there is no yle on their epeneny grph. For this reson we will reer to them s SLR grmmrs. Exmple 4. The SLR grmmrs or the three gs rom Figure 9 re respetively: X 0 (X 1, X 2, X 3, X 4 ) Y 0 (Y 1, Y 1, Y 2, Y 1 ) Z 0 (Z 1, Z 1, Z 2, Z 3 ) X 1 Y 1 Z 1 X 2 Y 2 Z 2 X 3 Z 3. X 4 We exten the notion o SLR grmmr to ptterns. To eine normlize grmmr G P or pttern P, it is suiient tht every non terminl X i ppering 11

12 on the right hn sie o ny proution o G P, is preee y mol symol or, orresponing to the type o ege pointing to the noe represente y X i on P. In orer to hve uniorm nottion tht overs ptterns s well s ouments, we will o the sme on the normlize grmmr G t, or ny oument t: every non terminl X i ppering on the right hn sie o some proution in G t, will e preee y. For instne, the grmmrs G P n G t respetively or the pttern P n the tree t o Figure 11, re given in Figure 10. P 0 ( P 1, P 2) X 0 ( X 2, X 1) P 1 X 1 ( X 2) P 2 X 2. Fig. 10. SLR grmmrs G P n G t or P n t rom Figure 11 To eine n emeing e rom pttern P to g t, we reple the onitions 3 n 4 o Deinition 1 respetively y: 3. u, v Noes P, suh tht (u, v) Eges (P ), there exists n ege going orm e(u) to e(v) on t; 4. u, v Noes P, suh tht (u, v) Eges (P ), there exists pth going rom e(u) to e(v) in (Eges t ) +. The notion o (g) moel o pttern n the pttern ontinment prolem re eine in the sme wy s in the se o tree moels. Figure 11 shows pttern P, its ompresse moel t, n n emeing rom P to t. P t Fig. 11. Pttern P, its ompresse moel t, n emeing rom P to t SLR grmmrs n e use in our rewrite pproh. To prove tht given g t is moel o pttern P, it is suiient (oring to Remrk 1) to rewrite the grmmr G t representing t to the grmmr G P representing P. We illustrte this ie in the ollowing exmple. 12

13 Exmple 5. Consier the grmmrs G P n G t given in Figure 10. We show how to rewrite G t to G P using rules o R: X 0 ( X 2, X 1 ) 2 X 0 ( X 2, X 1 ) X 1 ( X 2 ) X 1 ( X 2 ) 1 1 X 0 ( X 2 ) X 2 X 2 X 2 6 The irst proution o G t is irst rewritten using rule 2; then we ut rnh represente y X 1 (rule 1). At the sme time, we n eliminte rom G t the proution X 1 ( X 2 ), sine it hs eome unproutive (there is no more proution hving X 1 on their right hn sies). X 0 ( X 2, X 2) 3 X 0 ( X 2, X 2) P 0 ( P 1, P 2 ) X 2 X 2 P 1 X 2 X 2 P 2. Then, using rule 6 we oule the numer o hilren o X 0 ; we introue new non terminl X 2, whih proues the sme su pttern s X 2. Finlly, y Rule 3, we get grmmr whih is equl, up to non terminl renming, to G P. 4.2 Query Evlution SLR grmmrs help us to pt the rewrite pproh o Setion 3 to XP(/,//,[ ], ) query evlution on (ompresse) ouments. To represent unry queries, we use unry ptterns (see Setion 2). Let us onsier the unry pttern P rep- P: P 0 t: X 0 P 1 X 1 X 2 P 2 P 3 s X 3 X 4 X 5 P 4 X 6 Fig. 12. Unry pttern P n its ompresse moel t resenting the query P = / //[./]//[./], n the ompresse oument t, given in Figure 12. The orresponing SLR grmmrs G P n G t re respe- 13

14 tively: P 0 ( P 1 ) X 0 ( X 1, X 2 ) P 1 ( P 2, P 3 ) X 2 ( X 6 ) P 2 X 1 ( X 6, X 3 ) P 3 (s) ( P 4 ) X 3 ( X 4, X 5 ) P 4 X 4 ( X 6 ) X 5 ( X 6 ) X 6. The non terminl P 3 o G P is mrke s, sine it represents the output noe o P. To in n nswer or P on t, we rewrite the grmmr G t to the grmmr G P, using the rules o R. The non terminl o G t whih will e rewritten to the seleting non terminl P 3 o G P, will represent n nswer or P on t. We illustrte this resoning elow: X 0 ( X 1, X 2 ) X 2 ( X 6 ) X 1 ( X 6, X 3 ) 1 X 0 ( X 1 ) 2 X 0 ( X 1 ) 1 X 1 ( X 3 ) X 1 ( X 3 ) X 3 ( X 4, X 5 ) X 3 ( X 4, X 5 ) X 4 ( X 6 ) 4 3 X 3 ( X 4, X 5 ) 1 X 4 X 4 X 5 ( X 6 ) X 5 ( X 6 ) X 5 ( X 6 ) X 6 X 6 X 6 X 0 ( X 3 ) P 0 ( P 1 ) X 3 ( X 4, X 5 ) P 1 ( P 2, P 3 ) X 4 P 2 X 5 ( X 6 ) P 3 (s) ( P 4 ) X 6 P 4, We hve otine n SLR grmmr, whih is (up to non terminl renming) the SLR grmmr G P or P. The non terminl X 5 o G t hs een rewritten to the non terminl P 3, thus the noe represente y X 5 is n nswer or P on t. Note tht, s ny query P o the rgment XP(/,//,[ ], ) is purely esennt, the nswer or P on oument t oes not epen on the orm uner whih t is given (tree or g); this is no longer vli or queries ontining sennt xes (.[11]). Remrk lso tht our rewrite pproh n e extene to ny n ry query o XP(/,//,[ ], ); n n ry query selets set o n tuples o noes ([16]), n is esily represente s n n ry pttern. 14

15 5 Conlusion We hve presente n pproh se on rewrite tehniques, tht llows to hnle the prolem o query ontinment or the segment XP(/,//,[ ], ) o XPth. Suh rewrite view is lso pproprite or ompresse ouments moele s gs, n n e pte to (unry s well s n ry) query evlution on (ompresse) ouments. Strightline regulr tree grmmrs n provie n exponentil spe ompression. Nevertheless there exist more eiient ompression tehniques, like those se on stightline ontext ree grmmrs (SLCF, [15]), giving etter (up to ouly exponentil) ompression rtes. Currently we re stuying the possiility o extening our rewrite pproh to suh more eiient ompressions. We lso hope to pt our results to lrger rgments o XPth, ontining queries moele y more generl ptterns, hving oth esennt n sennt eges. Reerenes 1. Miklu, G., Suiu, D.: Continment n Equivlene or Frgment o XPth. J. ACM 51(1) (2004) Neven, F., Shwentik, T.: XPth Continment in the Presene o Disjuntion, DTDs, n Vriles. In Clvnese, D., Lenzerini, M., Motwni, R., es.: ICDT 03. Volume 2572 o LNCS., Springer (2003) W3C: XML Pth Lnguge. (1999) 4. Woo, P.T.: On the Equivlene o XML Ptterns. In Lloy, J.W., Dhl, V., Furh, U., Kerer, M., Lu, K.K., Plmiessi, C., Pereir, L.M., Sgiv, Y., Stukey, P.J., es.: CL 00. Volume 1861 o LNCS., Lonon, UK, Springer (2000) Shwentik, T.: XPth Query Continment. SIGMOD Re. 33(1) (2004) Gottlo, G., Koh, C., Pihler, R.: Eiient Algorithms or Proessing XPth Queries. ACM Trns. Dtse Syst. 30(2) (2005) Fles, S., Furro, F., Msiri, E.: On the Minimiztion o XPth Queries. J. ACM 55(1) (2008) 8. Amer-Yhi, S., Cho, S., Lkshmnn, L.V.S., Srivstv, D.: Tree Pttern Query Minimiztion. VLDB J. 11(4) (2002) Kimelel, B., Sgiv, Y.: Revisiting Reunny n Minimiztion in n XPth Frgment. In: EDBT 08: Proeeings o the 11th Interntionl Conerene on Extening Dtse Tehnology, New York, USA, ACM (2008) Rmnn, P.: Eiient Algorithms or Minimizing Tree Pttern Queries. In: SIG- MOD 02: Proeeings o the 2002 ACM SIGMOD Interntionl Conerene on Mngement o Dt, New York, USA, ACM (2002) Fil, B., Annthrmn, S.: Automt or Positive Core XPth Queries on Compresse Douments. In Hermn, M., Voronkov, A., es.: LPAR 06. Volume 4246 o LNAI., Springer (2006) Bunemn, P., Grohe, M., Koh, C.: Pth Queries on Compresse XML. In: VLDB 03, Morgn Kumnn (2003) Mrx, M.: XPth n Mol Logis o Finite DAG s. In Cile-Myer, M., Pirri, F., es.: TABLEAUX 03. Volume 2796 o LNAI., Springer (2003)

16 14. Frik, M., Grohe, M., Koh, C.: Query Evlution on Compresse Trees (Extene Astrt). In: LICS 03: Proeeings o the 18th Annul IEEE Symposium on Logi in Computer Siene, Wshington, DC, USA, IEEE Computer Soiety (2003) Bustto, G., Lohrey, M., Mneth, S.: Eiient Memory Representtion o XML Doument Trees. In. Syst. 33(4-5) (2008) Niehren, J., Plnque, L., Tlot, J.M., Tison, S.: N ry Queries y Tree Automt. In Biermn, G.M., Koh, C., es.: DPL 05. Volume 3774 o LNCS., Springer (2005)