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XML Schema, Tree Logic and Sheaves Automata

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(1)XML Schema, Tree Logic and Sheaves Automata Silvano Dal Zilio, Denis Lugiez. To cite this version: Silvano Dal Zilio, Denis Lugiez. XML Schema, Tree Logic and Sheaves Automata. RR-4631, INRIA. 2002. �inria-00071954�. HAL Id: inria-00071954 https://hal.inria.fr/inria-00071954 Submitted on 23 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. XML Schema, Tree Logic and Sheaves Automata Silvano Dal Zilio — Denis Lugiez. N° 4631 Novembre 2002. ISSN 0249-6399. ISRN INRIA/RR--4631--FR+ENG. THÈME 1. apport de recherche.

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(65) \ËK­oÌyÍy­¼Ê ¡«\¯Ã©y¬Ìy¯ÃÌ. GIHpN•NMLk\]_^¡]_^žkphiqK€ NMqžxžhib`dZqKL–NVqA] h”U+m\NMqKdf]WN‘m Xfq2m mpNMqpdZ]WNVU ]WHpNµLXZkpkph”qp€ N[ž]WNVqKm\NVm ‹ hi]WHXZqXZU`Uld\nMh”Xf]Wh”dZq z2NM]_‹•NVNMq X Xfq2m S ™GIHpNm\∅djLXZhiq8Γ,dfXŽ Γ: ShU:mpNMqpdZ]WNVm dom(Γ) Xfq2m ΓΓ(X ) = S Ž›LdjbWN‘LXfqKN‘U*mÉ]`HžHK^žXyk4]}df]`]WHpHKNšNVUWxyh”XfUV)b`hm\XfNVzpqpŒ”N df]`XNVm h”U}Γ L`XZkp k2N‘UlYKmnoH]Wd ]`SHKXyh”]q ]`HpΓNM™b`NÎÃqhU]WHKdZqKNŒi^Ɏ›dZŒ”dZŒidyqp‹Nšh”U`qpno€2HpÛNM‹;LN–X8djXZqpU`Œ”UW^ÉdžnMnh”djXfqK]WN‘UWmh”m\NV]`db‹;NVXjNMnoŒ”Œ§H œ xyXZbWhXfzpŒ”NQhiq dom(Γ) ™ ×Û× Ø   ×   Ó  „‘ƒ ∅`. Γ `  X 6∈ dom(Γ) Γ, X : S ` . GIHpN¦bWNVŒ”Xf]Wh”dZq h”UšzKXZUWNVm¤djq¤XZqÑXfY\[\h”ŒihXfb`^“Ž›Ypq2n¼]Wh”dZqې (d) ‹HphnoHÑndjL–kKY\]WN‘U ]`HpN“¶·®K¯Ã­ 5i­oÌy͑¶·®AÐ>dfÎQ`]WHKNd NM:Œ”NMSLNMqA]oUhiq d ;]WHKXf]hUšXfŒ”Œ]WHKN¦m\d\nYpLNMqA]ointer UdZz\]oXfh”qKXfzKŒiN8Ž›b`dZL d XyŽ·]`NMb k]`4HpNMNMb`qLYp]`Xy]`hidjq*dfŽžhU h§]o]`U HpNMN{Œ”NMUWLN]¡NVqjnM]odZUMqA™)]`XZgŠhiqKdZhib`qpN €šŽ›dZ]Wb`HpLN–Xfm\Œ”Œid\^jn Ypinter f X K q À m ‘  i h Ž h”UÈ]`HpNµm\d\nMYpLNMqA]  () = {} d Ž›dZb¡XZŒiŒ+k4NMb`LY\]oXye]Wh”dZ·.q . .·edZŽ  L M N A q o ] U e · ... · e σ ]`HpN}h”qjinter ]`NMb`xyXf(d)Œ 1..n ™ ×Û× Ø  × ‡

(66)  Ó  „‘ƒ nMUl] ∈ Datatype Γ`d:T Γ`d:T Γ` Γ` nMUl] : Datatype Γ ` a[d] : a[T ] Γ ` d : a[T ]? Γ `  : a[T ]? Γ` 1. σ(1). Γ` Γ`: Γ`d:S Γ ` d : S | S0. Γ ` d 1 : S1 Γ ` d 2 : S2 Γ ` d 1 · d 2 : S1 , S2 Γ ` d : S0 Γ ` d : S | S0. n. σ(n). Γ(X ) = S Γ ` d : S Γ`d:X Γ ` d1 : S, . . . , Γ ` dn : S Γ ` d1 · . . . · dn : S∗. Γ, X1 : S1 , . . . , Xn : Sn `  Γ, X1 : S1 , . . . , Xn : Sn ` d : S Γ ` d : S where X1 = S1 , . . . , Xn = Sn Γ` Γ ` d : AnyT. d ∈ inter (e1 · . . . · en ) Γ ` e1 : E1 , . . . , Γ ` en : En Γ ` d : E 1 & . . . & En. ]`HpNÎÃm\qÉNM½K]WHpqpNhi]Wh”qKdZNq8[ž]dfŽ+UlN‘z2n¼dZ]W]Wh”H8dZqÛdZ Yp‹•bIN–]`bWh”NVqjN*]`bWŒid\djm\€ZhY2nQnNXZqKUlm!djLdZYpN{bzKqpXjNVUl‹ hn–nMLŒ”XjXyUW]WU;HKdZNMŽ)L]`XybW]`NVh”N*nVXfXZŒ+Y\]W]WdždjdZLŒUQXy]`]WXKH2™ Xy]}‹hiŒ”Œ?z4N–Y2UlNMŽ›YpŒch”q ü)ü 32ý54‘þ.

(67) O‘~. ¨–Ì Û¶  ¶´© ? «ZÐf¶´­.  $Û $)' ¼()$&  $ 8'!+ I _$ %Å o()8

(68)  ;()  } $ Æ\dZLNQnMdZLkpY\]oXy]`hidjqKXfŒÀXZUWk4NVn¼]oUIdfŽÛdjYpbI]Wb`NMN*XfYp]WdZLXf]WdZq!bWNVŒi^djq XZbWhi]WHKL–NM]WhnMXZŒ4kKbWdjk2NVbl]`hiN‘U•dyxjNMb•]WHpN €jbWdjYpk (N, +) dfŽ•k2dAUlhi]Wh”xZN{qKXy]`Ypb`XZŒÛqžYpL{z2NVb`U¡‹h§]`HÉXjmpm\hi]Wh”dZq ™*GIHpN½2b`Ul]lœdZbom\NMb:]WHpNVdZb`^!dZŽµNV¢AYKXZŒihi]_^ djqŠ]WHKh”UQU_]`bWY2n¼]WYKbWN{h”UQXfŒUld¿AqKdy‹qXZU¡acbWN‘UlzKYpbW€jNMb  U:Xfb`h§]`HpLN]Whnf™QÎÃqŠ]`Hph”U*UWNVn]Wh”dZqŠ‹;Nb`NMxžhiNV‹¹UldjLN z2XZUWh”nQb`NVUWYpŒ§]oUIdZq acb`NVUWzpYpb`€ZNMb  U;XZbWhi]WHKL–NM]Whn*XfqKm!]WHKNQb`NMŒXy]`hidjq!‹hi]WHŠUWNMLh”Œih”qpNVXZb UlNM]`UV™ djU`Ulh”zphUIŒ”N{Xfqškpb`h”NVqAUW]WNMNVqA€Z]oNMXyb;]Wh”NdZ[\qkpb`Ž›NVdZU`b*UWhiacdjb`qšNVUWXfzpqKYpm b`€ZNMb hU• U:X{Xfacb`hib`]WHpNVUWLzpNYp]`b`h”€Zn{NVb hU* U €ZŽ›h”dZxZb`NMLq¦YKzAŒ”XK^8ž]`XZHpŒ”UWN–dŽ›dZnVXfŒ”ŒiŒ”dyŒi‹N‘m h”qp €! W€Z­Êbo Xf«"L /ÐAL­Xf b‘ Ê  ‹HKNMÁ b`N k4Exp ªo©f®pʼ¯ `Ìy¶·®K¯<™ ‰ NYKUWN N, M, . . . ]`d“b`XZqp€ZN8ψdyxZNVbhiqA]`NM€ZNVbšxyXfb`h”XZzpŒ”NVUXfq2m n, m, . . . ]Wd7boXfqp€jN8dyxZNMb q2Xy]WYKb`XZŒÀqAYKLz4NMboUM™  Ó ƒV ‚

(69) ) Ó

(70) iƒ ×  ƒV„M y† " „ ÎÃqA]`NM€Zk4NVdjbUWh§N]`[\hixjkpN*b`NVhiU`qAUl]Wh”NVdZ€Zq NVb ndZq2U_]oXfqA] Exp ::= n k4djUWh§]`hixjN*hiqA]WNV€ZNVbxŸXZbWhXfzKŒiN N Xjmpm\hi]Wh”dZq Exp + Exp c a W b ‘ N l U zKYpbW€jNMb  UInMdZqKUl]WboXfh”qA] φ, ψ, . . . ::= ` ] NVUl]Ž›dZbNV¢AYKXZŒihi]_^ (Exp = Exp ) p q NV€jXf]Wh”dZq ¬φ \ m hU›e_YKqKn¼]`hidjq φ∨ψ M N ž [ hUl]WNMqA]`h”XZŒ ¢jY2XfqA]Whi½2nMXf]Wh”dZq ∃N.φ acb`NVUWzpYpb`€ZNV b  UnMdZqKUl]WboXfh”qA]`U–XfŒ”Œidy‹ ]`HpN8m\N½KqKh§]`hidjqÂdfŽKN[\h”zpŒiNjc^ZNM]mpNVnhmpXZzpŒiNjckpb`dZk4NMbW]Wh”NVU{dyxZNVb k]`4HpdjN*UWhiŽ›]WdZh”xZb`LNh”YKqjŒ”]`X NM€jNMb?Œ”hi¿jNIŽ›dZbcNM[\XZLkpŒiN ]`HpdZN b xyXfŒ”Yph”U NXZdfq!Ž Xd\mph”m U•qžU_]`YpbWL{hn¼]`z2ŒiNV^{bV€Zb`NVXf]WNMb+]`HKXfq]`HpN xyXfŒ”YpN™ dfŽ Y jYKUWhiqp€ ‰ N{m\NVqpdf]`∃Z.(X N φ(X =, . Y. . ,+XZ)+X1)aµbWN‘UlzpXYKbW€jNMb  U;Ž›djbWL{YpŒ”X‹hi]WHŽ›∃Z.(X bWNVNhiqA]`NM=€ZNVZb xy+Xfb`Zh”XZ+zpŒ”NV1)‰U X , . . . , X Œ”^‹b`Ž›h§dZ]`bN ]o|=XfY\φ(n Y2XZqKUlN mÄ]W‹•HKNN8UWnMHKdZXfqKŒ”UlŒ ]`XZUWqAhiL] kpTrue ]`dZŒ”dZ€j,hiN‘. .UM.c,Œ”nh”¿ZN!) Ž›‹dZbHpNVNq [pXZφ(n L–kKŒiN , .]W.HK.N!, nŽ›dj)bWL{h”UYpŒU`XZXyU ]`h”(NUl½KN‘mÀ=™ N )NŠdZ‹b hiŒ”(NŒ:XfŒUW=d ™ M ) ∨ ¬(N = M )     

(71) *  , (  , ( aµbWN‘UlzpYKbW€jNMb  U¡Xfb`h§]`HpLN]WhnH2XZUQXm\N‘nhmpXfzpŒ”N{]WHpNVdZb`^Z™*GIHph”UQb`NVUWYpŒi]QnMXZqz4N{kpb`dyxZN‘m8YKUWhiqp€8XnMdZqpqKNVn¼œ ]`XZhibWdjhiq“]WHKL–‹h§NM]`]WH¤hnÊMŽ›­dj¬–bWL{¶  ¶·Yp®4Œ”­oXjÌ U šh”q7ÊM­¯/Ž›Êb`NMdfN ŽxyqKXfXfb`]Wh”XZYpzpboŒiXfN‘ŒcU qžYpL{z2NVb`UV™8gŠdZXZb`bWNNškp]WHpb`NVN8nhUlUWNVNMLŒ”^Zhi)Œ”hiqK]WHpNVNšXfbLUWd\N]`m\U{NVŒ”dfUŽ dfŽac?b`NV]WUWHpzpN YpUlb`NM€Z]–NMb dZ UŽ h”nMqAdž]WdZNVbo€Zm\NMhib;q2XyxZ]WN‘N‘n¼UM]`™dZboUcdfŽ+Ulh MN n ™ ÁmKm\h§]`hidjq!djq NN ,h.UI.m\. N, ½KNqKNVmXjUc]`HpN*k2djhiqA]_‹hUlNQXZmpmph§]`hidjqNdZqš]WHpN*xZN‘n¼]`dZboU Á  ¶·®4­oÌ ÊM­¯:dfŽ  P ) Èh”UQXšUWN]¡dZŽcxZN‘n¼]Wdjb`U:€ZNVqpNMboXy]`NVm zž^8XzKXZUWhUM b ∈ N 4XZqKmXUWN] dZXZŽ+ŒiŒ)k4]WNMHKb`Nh”džk4mKdjUMU` UlPh”zpŒ”=NNM{pŒ”NMLN,NV.qj. ]o.L(b, U , pdjz\}]`XZphi‹qpN‘hi]WmŠH zžp^8Œi∈h”qpNNVXZb¡Ž›nMdZdZb:LXZzKŒiŒ hiqKiXf∈]Wh”dZ1..kqŠdZ™;Ž?Á¹]WHpN–Œih”qpk2NVNVXZbWb h”d\UlmpNMU ] ‹L(b, hi]WH¦P]WHp)N{nzKdZXZq2UWUlNZhU_È]oXfUq2dfm Ž X!ÊM­M¬{¶  ¶·®È­oÌ }ÊM­M¯•hUX–½Kqphi]WN}YpqKhidjq!dZŽ+Œ”hiqKNVXfb UWN]oUM™ 1. 2. 1. 2. . 1. p. 1. 1. 8

(72) . ?   ?. ?. p.   . A?.  @. 1.  . 1 n.  . ?.  . n. n. n. 1. p. p. n. k. i. L(b, P ) = {b +. n. X. λi p i. λ1 , . . . , λk ∈ N}. i∈1..k. +ü.  (.

(73) ȪoËp­M¬Ì*° W­o­ )©WÐZ¶´ªÌy®ÈÏ

(74) \ËK­oÌyÍy­¼Ê ¡«\¯Ã©y¬Ìy¯ÃÌ. OjO. Á:qh”L–k4dZbW]`XZqA]ÛbWN‘UlYKŒ§]+h”UÛ]WHKXf]+UWNMLh”Œih”qpNVXZb)UWN]oU)XZbWNµnMŒidAUlN‘m*YpqKm\NVb•«\®2¶´©y®2jʼ«\¬ Xfq2m¶·¯­ WÌf¯/¶´©f®+”OV~ < ‹ HKNMb`N L+M = {x+y x ∈ L, y ∈ M } jXfqKm L = L+. . .+L ¸ n ]Wh”LNVUo» XfqKm L∗ = S L ™ ÎÃUWq N]o]`U}Hp¸›Nh”q nMXj]WHpUlN*N}dZqžŽ YphiL{]WNVz2b`NVXfb]Wh”dfdZŽ q Œižh”qp]`HpNVNXZb UWNMUlLNM]`h”UIŒih”h”qpq NVXZb ™ » UlNM] L∗ LXŸ^šz4NXYpqph”dZq8dfŽ?NM[žk4dZqKNMqA]WhXfŒ”Œi^LXZqž^Œ”hiqpN‘Xfb L Á:qpdf]`HpNMbcY2UlNMŽ›YpŒpLXy]`HpNMLXy]`h”nVXfŒž]`dAdjŒpqpNVNVm\N‘m{h”q–]WHpNkKbWN‘UlNVqj]oXy]`hidjq{dZŽ4djYpb?qpNM‹>nŒXZU`U dfŽÀXfY\]`dZLXyœ ]`dZq h”U]WHpN}qpdZ]Wh”dZq dfŽ Ì ¼¶ ‘˦¬Ìo²Z²4¶·®žÐf™  ,  * -,   ]`df]`XZŒiŒ”^ŠdZbom\NVbWN‘mÀK]`HpN–a XZbWh”¿žH LÇQh”XZxZkpNVkpq¦h”qpUW€ŠdZLdfNŽX8½KqK‹•h§]`djNb`m XZŒiwkKHKXfdfz4Ž NΣ] ΣhU=X {anœYp, kp. .Œ”N. ,dfaŽIqK}Xy4]`]`YpHKboXyXf]*Œc‹•qžYpNL{ndjz2qKNVUWb`h”UVm\ NVb (w) +‹HKNMb`N = (m , . . . , m ) ‰ ” h ; U ` ] p H } N ž q p Y  L 4 z M N I b Z d ) Ž \ d M n M n p Y W b ` b M N 2 q  n V N ; U Z d Û Ž W ] K H Q N ” Œ  N W ] W ] V N b ” h q ™  N W U K H Z X i Œ À Œ f X  Œ W U { d K Y W U Q N W ] p H * N p q Z d ` ] f X W ] ” h Z d q m ]`XZdLm\zpNMh”€ZqKYKdfh§]W]_N^]`djHpq!Nš]`qžHpYpN}L{‹;z2dZNVbobm dfŽI™ d\nMnYKbWb`NMqKnMNVU*dZŽ•]`HpNšŒ”aN]l]`NMb wa hiq w  djbUlh”LkpŒi^ a ‹HpNVq’]WHpNVbWNhU}(w)qpd w I G p H : N. a Z X W b ” h ž ¿  H  L Z X p k p k ” h p q  € LXZkpkph”qp€¦nMXZq’z2N!NVXZUWh”Œi^ndZdZŽÀLX}kpUWYpN]W]•NVm7dfŽÀ‹‹•HpdjNMb`q’mKU ]`h”HpUcN]WHpUWNN:]{UWNdZ]•Ždf‹•ŽÀdja?b`Xfmpb`U*hi¿žhUHXLbWXfNVkp€ZkpYph”qpŒXf€AbU?ŒdfXfŽÀqKhi€Z]`YKUcXZNV€ZŒiNVNZL™!NMÎÃqAqK]`m\UVNV™+NVGImÀ HKh”hiUŽ hU{XbWNV€ZYpŒXfbNM[\kpbWN‘UWUWh”dZq )]WHpNVq (reg)  ]WHpN!a?Xfb`hi¿žH“LXfkpkKhiqp€ÉdfŽ reg  h”U]WHKN!UWNMLhiŒ”h”qpNVXZb{UWN] reg djz\]`XZhiqKNVm XZU;Ž›dZŒ”Œ”dy‹ UM™ ͎ hU?]WHKN Œ”N]W]WNMb;h”q]WHpN¡XfŒ”kpHKXfz4N]‘Z]`HpNMq h”Uµ]WHpN Ypqphi]•xjNVn]WdZbµdfŽ A]WHpN xjNVn]WdZb ‹h§a]`HŠXfŒ”Œ NMŒ”NMiLNMqA]oUINV¢AYKXfŒÈ]`d 0 N[pnNVk\]I]WHpN i ]`(aHKXy)]hUNV¢AYKiXZŒÈ]Wd 1 (0, . . . , 0, 1,N 0, . . . , 0) ™ GIHpN}a?Xfb`hi¿žH LXfkpkph”qp€–Ž›dZb noHKdZhnN*h”U;]`HpN}Ypqph”dZq8dZŽÛ]WHKN}a XZbWh”¿žH!LXZkpkph”qp€jU ™ (reg + reg ) = (reg ) ∪ (reg ) GIHpN}a?Xfb`hi¿žH LXfkpkph”qp€–Ž›dZb UWNV¢AYpNVqA]WhXfŒ ndjLk2dAUlhi]Wh”dZq h”U;]`HpN}k4dZh”qj]_‹hUWN*Xjmpm\hi]Wh”dZq ™ (reg .reg ) = (reg ) + (reg ) GIHpN}a?Xfb`hi¿žH LXfkpkph”qp€–Ž›dZbhi]WNVb`Xf]Wh”dZq  (reg∗) h”U]WHpN}hi]WNVb`Xf]WN*dfŽ (reg) ™ (reg∗) = (reg)∗ Áb`NM€jYpŒ”XZbN[\kpb`NVU`Ulh”dZqÂXjUWUWd\nhXy]`NVm“]`d7XbWNV€ZYpŒXfb–‹•djb`mÄXfY\]`dZLXy]`dZq A nVXfq>z2NnMdZLkpY\]`NVm>hiq ™cGIHpNVbWNMŽ›dZb`NZjYKUlh”qp€b`NM€jYpŒXfbcNM[žkKbWN‘UWUWhidjqKU?dZŽÀUWNMLh”Œih”qpN‘Xfb•UWN]oUMA‹;N¡nMXZqšndjL–kKY\]WN:]WHpN¡a XZbWh”¿žH O(|A| ) LXZkpkph”qp€dfŽ?X{b`NM€jYpŒXfb;‹•djb`m!Œ”XZqp€ZY2Xf€ZNQh”q!]`hiLN O(|A| ) ™  ,       *  , #%$'&)( +* -, ‰ NInŒXfb`h§Ž›^*]`HpNIbWNVŒ”Xf]Wh”dZqz4N]_‹;NMNVqacb`NVUWzpYpb`€ZNV b  UÛnMdZqKUl]WboXfh”qA]Vya?Xfb`hi¿žH  U)LXZkpkph”qp€*XfqKm]WHpNUWNMLXfqA]WhnMU dZŽ•]`HpNh”qj]`NMb`ŒiN‘XŸxžhiqp€ dZk4NMboXy]`dZb}Xfq2m¦]`bW^¦]Wd8€Zh”xZNXZqh”qA]WYphi]Wh”dZq’djqHKdy‹¹]WHKN œdZk4NMboXy]`dZb*LXŸ^¦XZmKm WndjYpqA]Wh”qp€nMXfk2Xfzph”Œihi]Wh”NVU ¡]`dšÆ\noHpNMLXK™ GIHpN¡a?Xfb`hi¿žHšLXfkKkphiqK€h”qj]`bWd\m\Y2nNVmh”qš]`HpN:kKbWNVxAh”dZY2UµUWNVn¼]`hidjqškpb`dyxžh”m\N‘UµXUl]WboXfh”€ZHA]WŽ›dZb`‹;XZb`m{LXfk\œ kKhiqp€z4N]_‹;NMNVqŠm\d\nYpLNMqA]oU:Xfq2m]`YpkpŒ”NVU:dfŽ qKXy]`Ypb`XZŒÀqžYpLz4NMboUM™;˜ NM] a , . . . , a z4Nm\hU_]`hiqKn]:NMŒ”NMLNVqj] . n. n≥0. . 8

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