XML and Databases. Outline. Previous Lecture. XML Type Definition Languages. Sebastian Maneth NICTA and UNSW

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1 Outline XML nd Dtses Lecture 5 XML Vlidtion using Automt Sestin Mneth NICA nd UNSW. Recp: deterministic Reg Expr s / Glushkov Automton. Complexity of DD vlidtion. Beyond DDs: XML Schem nd RELAX NG 4. Sttic Methods, sed on ree Automt CSE@UNSW -- Semester, 009 Previous Lecture XML ype Definition Lnguges 4 XML type definition lnguges wnt to specify certin suset of XML doc s = type of XML documents Rememer he specifiction/type definition should e simple, so tht vlidtor cn e uilt utomticlly (nd efficiently) the vlidtor runs efficient on ny XML input (similr demnds s for prser) DD (Document ype Definition, WC) Originted from SGML. Now prt of XML DD my pper t the eginning of n XML document XML Schem (WC) Now t version. HUGE lnguge, mny uilt-in simple types Schems themselves: written in XML Reg Exprs must e deterministic (=-unmiguous) sme!! Unique Prticle Attriution ype def. lnguge must e SIMPLE! (similrly: prser genertors use EBN or smller suclsses: LL / LR) See the Schem Primer t RELAX NG (Osis) or tree structure definition, more powerful thn Schems&DDs O(n^) prsing XML ype Definition Lnguges 5 XML ype Definition Lnguges 6 DD (Document ype Definition) DD (Document ype Definition) <!DOCYPE root-element [ doctype declrtion ]> <!DOCYPE root-element [ doctype declrtion ]> <!ELEMEN element-nme content-model> content-models EMPY ANY (#PCDAA elem-nme_ elem-nme_n)* deterministic Reg Expr over element nmes <!ELEMEN element-nme content-model> content-models EMPY ANY (#PCDAA elem-nme_ elem-nme_n)* deterministic Reg Expr Most interesting / chllenging spect of DDs <!ALIS element-nme ttr-nme ttr-type ttr-defult..> ypes: CDAA, (v..), ID, IDREs Defults: #REQUIRED, #IMPLIED, vlue, #IXED <!ALIS element-nme ttr-nme ttr-type ttr-defult..> ypes: CDAA, (v..), ID, IDREs Defults: #REQUIRED, #IMPLIED, vlue, #IXED

2 7 8 Summry In order to check whether (lrge) document is vlid wrt to given DD ( it vlidtes ) you need to check if children lists mtch the given Reg Expr s Summry In order to check whether (lrge) document is vlid wrt to given DD ( it vlidtes ) you need to check if children lists mtch the given Reg Expr s his cn e done efficiently, using finite-utomt (As)! his cn e done efficiently, using finite-utomt (As)! o check if Reg Expr e is llowed in DD we hve to construct prticulr finite utomton: the Glushkov utomton. o check if Reg Expr e is llowed in DD we hve to construct prticulr finite utomton: the Glushkov utomton. Glu(e) must e deterministic. Glu(e) Glu(e) must e deterministic. Glu(e) Note If Glu(e) is deterministic, then its size (# trnsitions) is liner in size(e)! Note If Glu(e) is deterministic, then its size (# trnsitions) is liner in size(e)! Cn you explin why this is the cse? 9 0 Summry More Notes In order to check whether (lrge) document is vlid wrt to given DD ( it vlidtes ) you need to check if children lists mtch the given Reg Expr s () rom deterministic A you cnnot necessrily otin deterministic (= -unmiguous) regulr expression!! Exmple: e = ( )* ( ) NO -unmigous reg exp exists for e his cn e done efficiently, using finite-utomt (As)! o check if Reg Expr e is llowed in DD we hve to construct prticulr finite utomton: the Glushkov utomton. o check if Reg Expr e is llowed in DD we hve to construct prticulr finite utomton: the Glushkov utomton. Glu(e) must e deterministic. Glu(e) Glu(e) must e deterministic. Glu(e) Note If Glu(e) is deterministic, then its size (# trnsitions) is liner in size(e)! Note If Glu(e) is deterministic, then its size (# trnsitions) is liner in size(e)! Cn you explin why this is the cse? not correct: liner in size(e) * #letters(e) Cn you explin why this is the cse? not correct: liner in size(e) * #letters(e) More Notes () rom deterministic A you cnnot necessrily otin deterministic (= -unmiguous) regulr expression!! Exmple: e = ( )* ( ) or more detils: See pper y Brüggemnn-Klein, Linked from the course we-pge. NO -unmigous reg exp exists for e () Glu(e) is closely relted to homson(e) [remove ε-trnsitions] nd to Berry/Sethi(e) [sme] nd Brzozowski(e) Glushkov utomton Glu(e) Ech letter-position in the Reg Expr e ecomes one stte of Glu; plus, Glu hs one extr egin stte. IRS( e ) = ll possile egin positions of words mtching e e.g. IRS( R (E G) (EX)* ) = { R } o check if Reg Expr e is llowed in DD we hve to construct prticulr finite utomton: the Glushkov utomton. Glu(e) must e deterministic. Glu(e) Note If Glu(e) is deterministic, then its size (# trnsitions) is liner in size(e)! Cn you explin why this is the cse? not correct: liner in size(e) * #letters(e)

3 4 position position position 5... ollowing slides from: IRS( e )

4 9 0 Glushkov utomton G(e) Ech position in the Reg Expr e ecomes one stte of G; plus, G hs one extr egin stte. IRS( e ) = ll possile egin positions of words mtching e e.g. IRS( R (E G) (EX)* ) = { R } OLLOW( e, x ) = ll possile positions following position x in e e.g. OLLOW( R (E G) (EX)*, R ) = { E, G } rom stte R : dd E-trnsition to E G-trnsition to G OLLOW( e, R ) = OLLOW( e, E ) = 4 OLLOW( e, E 4 ) = OLLOW( e, G ) = 4

5 5 6 Glushkov utomton G(e) Ech position in the Reg Expr e ecomes one stte of G; plus, G hs one extr egin stte. IRS( e ) = ll possile egin positions of words mtching e e.g. IRS( R (E G) (EX)* ) = { R } OLLOW( e, x ) = ll possile positions following position x in e e.g. OLLOW( R (E G) (EX)*, R ) = { E, G } OLLOW( e, X 5 ) = rom stte R : dd E-trnsition to E G-trnsition to G LAS( e ) = ll possile end positions of words mtching e e.g. LAS( R (E G) (EX)* ) = { E, G, X 5 } 7 8 Is this utomton deterministic?? 9 0 Glushkov utomton G(e) Glushkov utomton G(e) Another exmple (* )* his A is deterministic. Ech position in the Reg Expr e ecomes one stte of G; plus, G hs one extr egin stte. IRS( e ) = ll possile egin positions of words mtching e e.g. IRS( R (E G) (EX)* ) = { R } OLLOW( e, x ) = ll possile positions following position x in e LAS( e ) = ll possile end positions of words mtching e Which of these is deterministic? () (c) ( c) ( )*c Nïve implementtion: O(n^) time, where n = size( e ) (for ech position: computing OLLOW goes through every position t ech step, needs to compute union O( n*n*n ) 5

6 Glushkov utomton G(e) Glushkov utomton G(e) Ech position in the Reg Expr e ecomes one stte of G; plus, G hs one extr egin stte. Ech position in the Reg Expr e ecomes one stte of G; plus, G hs one extr egin stte. IRS( e ) = ll possile egin positions of words mtching e IRS( e ) = ll possile egin positions of words mtching e e.g. IRS( R (E G) (EX)* ) = { R } e.g. IRS( R (E G) (EX)* ) = { R } OLLOW( e, x ) = ll possile positions following position x in e OLLOW( e, x ) = ll possile positions following position x in e LAS( e ) = ll possile end positions of words mtching e LAS( e ) = ll possile end positions of words mtching e Nïve implementtion: O(n^) time, where n = size( e ) Nïve implementtion: O(n^) time, where n = size( e ) (for ech position: computing OLLOW goes through every position t ech step, needs to compute union O( n*n*n ) Not relly needed. Cn e improved to O(n^) (for ech position: computing OLLOW goes through every position t ech step, needs to compute union O( n*n*n ) Cn e improved to O( size(e) + size(g(e)) ) Not relly needed. Cn e improved to O(n^) Glushkov utomton G(e) o void these expensive running times DD requires tht A=G(e) must e deterministic! 4 Note If G(e) is deterministic, then its size (# trnsitions) is qudrtic in size(e)! Liner in size(e) * #letters(e), if G(e) is deterministic! O( size(e) * #letters(e) ) n = length(w) m = size(e) otl Running time O(n + m) If s = #letters(e) is ssumed fixed (not prt of the input) Otherwise: O(n + ms) Nïve implementtion: O(n^) time, where n = size( e ) (for ech position: computing OLLOW goes through every position t ech step, needs to compute union O( n*n*n ) Cn e improved to O( size(e) + size(g(e)) ) Not relly needed. Cn e improved to O(n^) How cn you implement regulr expression? Input: Reg Expr e, string w : Does w mtch e? deterministic A: run on w tkes time liner in length(w) Unrestricted Reg Expr e Algorithm A = BuildA(e); DA = BuildDA(A); Size of A is liner in size(e)=m Size of DA is exponentil in m otl Running time O(n + ^m) Other lterntive: O(nm) 5 6 Summry Deterministic (-unmiguous) content models give rise to efficient mtching lgorithms. (they void O(nm) or O(n+^m) complexities) Now tht we know how the check ll the different content-models (in prticulr det. Reg Expr s) how to uild full vlidtor for DD? elem-nme_ RegExpr_ elem-nme_ RegExpr_ elem-nme_k RegExpr_k Automt A_, A_,, A_k Disdvntges Hrd to know whether given reg expr is OK (deterministic) Det. reg exprs re NO closed under union. (not so nice..) Hint: Cn you see why? find det. reg. exprs. e nd e such tht their union is equl to ( )* ( ) he Vlidtion Prolem Given DD nd document D, is D vlid wrt? op-down Implementtion t element node w. lel elem-nme_i, run utomton A_i check ttriute constrints check ID/IDRE constrints otl Running time (Given A_, A_,, A_k) liner in the sum of sizes of the DD nd the document. O( size() + size(d) ) 6

7 DDs hve the 7 Beyond DDs 8 lel-gurded sutree exchnge property: t, t trees in DD lnguge v node in t, leled l v node in t, leled l trees otined y exchnging the sutrees rooted t v nd v re lso in k locl content model only depends on lel of prent Often, the expressive power of DDs is not sufficient. Prolem ech element nme hs precisely one content-model in DD. Would like to distuingish, depending on the context (prent). deler t t cr cr v l v l model yer model cr hs different structure, in different contexts Beyond DDs Specilized DDs Often, the expressive power of DDs is not sufficient. Prolem ech element nme hs precisely one content-model in DD. Would like to distuingish, depending on the context (prent). deler deler, (cr )* (cr )* cr model, yer cr model deler deler speciliztion cr cr model yer model cr cr cr cr cr hs different structure, in different contexts. model yer model model yer model Specilized DDs New nottion. Use cpitlized YPE Nmes 4 New nottion. Use cpitlized YPE Nmes 4 deler, (cr )* (cr )* cr model, yer cr model Deler deler [Used, New] Used [(Cr )*] New [(Cr )*] Cr cr [Model, Yer] Cr cr [Model] Deler deler [Used, New] Used [(Cr )*] New [(Cr )*] Cr cr [Model, Yer] Cr cr [Model] Not locl deler deler Let us cll this concept grmmr. the locl restriction A grmmr G is locl, if for ny lel[regexpr_], lel[regexpr_] present in G it holds tht RegExpr_ = RegExpr_. By definition: Every DD is locl grmmr, nd vice vers. cr cr cr cr model yer model model yer model 7

8 New nottion. Use cpitlized YPE Nmes 4 New nottion. Use cpitlized YPE Nmes 44 Deler deler [Used, New] Used [(Cr )*] New [(Cr )*] Cr cr [Model, Yer] Cr cr [Model] Not locl competing Deler deler [Used, New] Used [(Cr )*] New [(Cr )*] Cr cr [Model, Yer] Cr cr [Model] Let us cll this concept grmmr. the locl restriction A grmmr G is locl, if for ny lel[regexpr_], lel[regexpr_] present in G it holds tht RegExpr_ = RegExpr_. Alterntively: Cll two YPE Nmes nd competing if they hve the sme element nme (ut not identicl rules) By definition: Every DD is locl grmmr, nd vice vers. Clsses of Grmmrs A grmmr G is single-type, if for ny lel[regexpr_], lel[regexpr_] occurring in the sme rule of G it holds tht RegExpr_ = RegExpr_. WRONG the single-type restriction locl no competing YPE nmes! (DDs) single-type YPE nmes in the sme content model do not compete! regulr no restriction (RELAX NG) (XML Schem s) New nottion. Use cpitlized YPE Nmes 45 New nottion. Use cpitlized YPE Nmes 46 competing Deler deler [Used, New] Used [(Cr )*] New [(Cr )*] Cr cr [Model, Yer] Cr cr [Model] competing ut re not in sme content model! Person person [PersonNme, Gender, Spouse?, Pet*] PersonNme nme [irst, Lst] Pet pet [Kind, PetNme] PetNme nme [#PCDAA] Are there single-type grmmrs (XML Schems) which cnnot e expressed y locl grmmrs (DDs). Are there single-type grmmrs (XML Schems) which cnnot e expressed y locl grmmrs (DDs). YES! Clsses of Grmmrs Clsses of Grmmrs locl no competing YPE nmes! (DDs) locl no competing YPE nmes! (DDs) single-type YPE nmes in the sme content model do not compete! (XML Schem s) regulr no restriction (RELAX NG) single-type YPE nmes in the sme content model do not compete! (XML Schem s) regulr no restriction (RELAX NG) New nottion. Use cpitlized YPE Nmes 47 New nottion. Use cpitlized YPE Nmes 48 competing Deler deler [Used, New] Used [(Cr )*] New [(Cr )*] Cr cr [Model, Yer] Cr cr [Model] competing Deler deler [Used, New] Used [(Cr )*] New [(Cr )*] Cr cr [Model, Yer] Cr cr [Model] hrough the use of YPE Nmes (nonterminls / sttes) you cn distinguish deep context! deler hrough the use of YPE Nmes (nonterminls / sttes) you cn distinguish deep context! deler cr Cr cr cr speciliztion through prent Cn we model context tht is fr wy from the specilized node? cr Cr cr cr speciliztion through prent model yer model model yer model 8

9 New nottion. Use cpitlized YPE Nmes competing Deler deler [Used, New] Used [(Cr )*] New [(Cr )*] Cr cr [Model, Yer] Cr cr [Model] d root 49 DB d [Deler, User] Deler deler [Used, New] Used [(Cr )*] New [(Cr )*] Cr cr [Model, Yer] Cr cr [Model] Doc root [DB sdb] sdb d [sdeler, suser] sdeler deler [sused, New] sused [(scr )*] scr cr [Model, Own, Yer] sdeler d root sdb suser 50 hrough the use of YPE Nmes (nonterminls / sttes) you cn user distinguish deep context! deler specil hrough the use of YPE Nmes (nonterminls / sttes) you cn user distinguish deep context! deler specil Cn we model context tht is fr wy from the specilized node? Sure! cr cr My Cr cr speciliztion through following node Cn we model context tht is fr wy from the specilized node? Sure! scr sused cr cr My Cr cr model yer model model yer model #owners #owners DB d [Deler, User] Deler deler [Used, New] Used [(Cr )*] New [(Cr )*] Cr cr [Model, Yer] Cr cr [Model] Doc root [DB sdb] sdb d [sdeler, suser] sdeler deler [sused, New] sused [(scr )*] scr cr [Model, Own, Yer] sdeler d root sdb suser 5 root d root d 5 hrough the use of YPE Nmes (nonterminls / sttes) you cn user distinguish deep context! deler specil Cn we model context tht is fr wy from the specilized node? Sure! scr sused cr cr My Cr cr model yer model Sure this grmmr is not locl (DD). But, is it single-type? specil deler Is this grmmr single-type? deler #owners 5 54 prev. exmple: proly, not expressle in single-type (XML Schem). prev. exmple: proly, not expressle in single-type (XML Schem). Other exmple: Other exmple: competing Reg Expr ut not in content Person MPerson Person MPerson person[nme, gender[mle], Spouse?, Children?] Person person[nme, gender[emle], MSpouse?, Children?] Mle mle[] emle femle[] Spouse spouse[nme, gender[emle]] MSpouse spouse[nme, gender[mle]] Children children[person+] person s spouse must hve opposite gender. Person MPerson Person MPerson person[nme, gender[mle], Spouse?, Children?] Person person[nme, gender[emle], MSpouse?, Children?] Mle mle[] emle femle[] Spouse spouse[nme, gender[emle]] MSpouse spouse[nme, gender[mle]] Children children[person+] person s spouse must hve opposite gender. Note his exmple nd the Pet-exmple re tken from Hosoy s ook (see course we pge). BU, is this even grmmr in our sense? 9

10 prev. exmple: proly, not expressle Other exmple: in single-type (XML Schem). competing Reg Expr in content Person root[mperson Person] MPerson person[nme, gender[mle], Spouse?, Children?] Person person[nme, gender[emle], MSpouse?, Children?] Mle mle[] emle femle[] Spouse spouse[nme, gender[emle]] MSpouse spouse[nme, gender[mle]] Children children[person+] person s spouse must hve opposite gender. BU, is this even grmmr in our sense? NO! Reg Expr only llowed inside content ( under n element nme ). 55 Clsses XML ype ormlisms locl no competing YPE nmes! (DDs) single-type YPE nmes in the sme content model do not compete! regulr no restriction (RELAX NG) Incresing Expressivness of defining sets of trees ( tree lnguges ) s Given two DDs D nd D cn we check if ll documents vlid for D re lso vlid for D? D nd D descrie the sme set of documents? Given Relx NG grmmr G, cn we check if there exists ny document tht is vlid for G? there is document vlid for G nd vlid for G? 56 (XML Schem s) (DD inclusion prolem) (DD equlity prolem) (emptiness prolem) (intersection & emptiness) Clsses XML ype ormlisms locl no competing YPE nmes! (DDs) 57 All of the checks cn e done utomticlly, for regulr tree grmmrs! 58 single-type YPE nmes in the sme content model do not compete! regulr no restriction (RELAX NG) Incresing Expressivness of defining sets of trees ( tree lnguges ) (XML Schem s) ree Automt: very powerful frmework, Hve ll the good properties of string utomt! Yet, they re more expressive! equivlent to tree utomt s s Given two DDs D nd D cn we check if ll documents vlid for D re lso vlid for D? D nd D descrie the sme set of documents? (DD inclusion prolem) (DD equlity prolem) Given two DDs D nd D cn we check if ll documents vlid for D re lso vlid for D? D nd D descrie the sme set of documents? (DD inclusion prolem) (DD equlity prolem) Given Relx NG grmmr G, cn we check if there exists ny document tht is vlid for G? there is document vlid for G nd vlid for G? (emptiness prolem) (intersection & emptiness) Given Relx NG grmmr G, cn we check if there exists ny document tht is vlid for G? there is document vlid for G nd vlid for G? (emptiness prolem) (intersection & emptiness) If we cn do it for regulr tree grmmrs, then lso works for single-type/locl!! If we cn do it for regulr tree grmmrs, then lso works for single-type/locl!! All of the checks cn e done utomticlly, for regulr tree grmmrs! All of the checks cn e done utomticlly, for regulr tree grmmrs! ree Automt: very powerful frmework, Hve ll the good properties of string utomt! Yet, they re more expressive! equivlent to tree utomt constnt memory computtion inite-stte utomt re importnt: equivlent to tree utomt Note String utomt re not sufficient to check DDs / Schems! Even if we only consider well-rcketed strings! Exmple c c[, c, ] empty empty c empty Exmple [ c, ] [, ] / / c empty hink you re in mze, with only fixed memory nd you cn only red the mze (cnnot mrk nything). wll Model y finite utomton. In stte q, (to [N S E W], ) ( q, [N S E W] ) q, (to [N S E W], ) ( q, [N S E W] ) empty q Cn n utomton serch the mze? 0

11 6 6 All of the checks cn e done utomticlly, for regulr tree grmmrs! All of the checks cn e done utomticlly, for regulr tree grmmrs! constnt memory computtion equivlent to tree utomt constnt memory computtion equivlent to tree utomt inite-stte utomt re importnt: inite-stte utomt re importnt: hink you re in mze, with only fixed memory nd you cn only red the mze (cnnot mrk nything). wll Model y finite utomton. In stte q, (to [N S E W], ) ( q, [N S E W] ) q, (to [N S E W], ) ( q, [N S E W] ) empty q Cn n utomton serch the mze? No!! need mrkers ( peles ). How mny? 5?? In our context, e.g., for KMP (efficient string mtching) [Knuth/Morris/Prtt] generliztion using utomt. Used, e.g., in grep Compression Sttic nlysis of schems & queries (= everything you cn do *efore* efore running over the ctul dt ) 4. Sttic Methods, sed on ree Automt 6 4. Sttic Methods, sed on ree Automt 64 Person MPerson Person MPerson person [Nme, gender[mle], Spouse?] Person person [Nme, gender[emle], MSpouse?] Regulr ree Grmmr Rules of the form ypenme ree person Leves my e leled y ypenmes person Given grmmrs D nd D cn we check if ll documents vlid for D re lso vlid for D? (inclusion prolem) D nd D descrie the sme set of documents? (equlity prolem) does there exists ny document tht is vlid for D? (emptiness prolem) there is document vlid for D *nd* vlid for D? (intersection & emptiness) ALL these checks re possile for regulr tree grmmrs!! hence, they re lso solvle for DDs / XML Schems / RELAX NG s Nme gender Spouse Nme gender () use inry tree encodings () trnslte XML ype Definition to ree Grmmr (esy) Mle Mle Alterntively, regulr tree lnguges re defined y ree Automt. stte, element-nme stte, stte conventionlly, defined for inry/rnked trees. Alterntively, regulr tree lnguges re defined y ree Automt. stte, element-nme stte, stte conventionlly, defined for inry/rnked trees. 4. Sttic Methods, sed on ree Automt 65 Automt 66 Given grmmrs D nd D cn we check if ll documents vlid for D re lso vlid for D? D nd D descrie the sme set of documents? does there exists ny document tht is vlid for D? there is document vlid for D *nd* vlid for D? (inclusion prolem) (equlity prolem) (emptiness prolem) (intersection & emptiness) ALL these checks re possile for regulr tree grmmrs!! he checks ove give rise to very powerful optimiztion procedures for XML Dtses! or exmple: documents d_, d_,, d_n re vlid for your schem Smll_xhtml. Are they lso vlid for schem XHML? Check inclusion prolem for Smll_html nd XHML!

12 Automt 67 Automt 68 Automt 69 Automt 70 Expressiveness exp. low-up exp. more succinct deteterministic = nondeterministic left-to-right = right-to-left word is ccepted y the utomton Automt 7 Automt 7 Expressiveness exp. low-up exp. more succinct deteterministic = nondeterministic left-to-right = right-to-left word is ccepted y the utomton Expressiveness exp. low-up exp. more succinct deteterministic = nondeterministic left-to-right = right-to-left word is ccepted y the utomton Automt on trees. ottom-up LABEL( stte, stte ) stte Automt on trees. ottom-up LABEL( stte, stte ) stte AND 0 0 0() () (, ) (, ) AND 0 0 0() () (, ) (, )

13 Automt 7 Automt 74 Expressiveness exp. low-up exp. more succinct deteterministic = nondeterministic left-to-right = right-to-left word is ccepted y the utomton Expressiveness exp. low-up exp. more succinct deteterministic = nondeterministic left-to-right = right-to-left word is ccepted y the utomton Automt on trees AND 0 0. ottom-up LABEL( stte, stte ) stte 0() () (, ) (, ) AND(, ) Automt on trees 0 AND 0. ottom-up LABEL( stte, stte ) stte 0() () (, ) (, ) AND(, ) tree is ccepted y the utomton Accepting Sttes = { } Automt Expressiveness exp. low-up exp. more succinct deteterministic = nondeterministic left-to-right = right-to-left 75 word is ccepted y the utomton How much memory do you need exctly, to run such ottom-up tree utomton? 76 Automt on trees 0 AND 0. ottom-up LABEL( stte, stte ) stte 0() () (, ) (, ) AND(, ) tree is ccepted y the utomton Accepting Sttes = { } his utomton is deterministic. nondeterminism LABEL( st, st ) { st, } Automt on trees 0 AND 0. ottom-up LABEL( stte, stte ) stte 0() () (, ) (, ) AND(, ) tree is ccepted y the utomton Accepting Sttes = { } his utomton is deterministic. nondeterminism LABEL( st, st ) { st, } Similrly s for word utomt: Similrly s for word utomt: or every nondeterministic ottom-up tree utomton there is n equivlent deterministic ottom-up tree utomton. or every nondeterministic ottom-up tree utomton there is n equivlent deterministic ottom-up tree utomton. Agin, the construction cn cuse exponentil size low-up. Agin, the construction cn cuse exponentil size low-up.. top-down stte, LABEL (stte, stte). top-down stte, LABEL (stte, stte), = inry node lels must contin $-lef e,$ = lef node lels top-most -node on the left-most pth must hve right-sutree which contins $-node., = inry node lels must contin $-lef e,$ = lef node lels top-most -node on the left-most pth must hve right-sutree which contins $-node. egin, (ny, find$) egin, (egin, ny) find$, / { (find$, ny), (ny, find$) } find$, $ ACC find$, e REJ ny, / (ny,ny) ny, $/e ACC nondeterministic ccept tree if there exists n ccepting run (= ll leves go to ACC)

14 or every nondeterministic ottom-up tree utomton there is n equivlent deterministic ottom-up tree utomton. Cn you find n equivlent ottom-up utomton for this exmple? 79 or every nondeterministic ottom-up tree utomton there is n equivlent deterministic ottom-up tree utomton, nd there is n equivlent nondeterministic top-down tree utomton. Yes! you cn 80. top-down stte, LABEL (stte, stte), = inry node lels must contin $-lef e,$ = lef node lels top-most -node on the left-most pth must hve right-sutree which contins $-node. egin, (ny, find$) egin, (egin, ny) find$, / { (find$, ny), (ny, find$) } find$, $ ACC find$, e REJ ny, / (ny,ny) ny, $/e ACC nondeterministic ccept tree if there exists n ccepting run. top-down stte, LABEL (stte, stte) must contin $-lef, = inry node lels e,$ = lef node lels top-most -node on the left-most pth must hve right-sutree which contins $-node. egin, (ny, find$) egin, (egin, ny) find$, / { (find$, ny), (ny, find$) } find$, $ ACC find$, e REJ ny, / (ny,ny) ny, $/e ACC nondeterministic ccept tree if there exists n ccepting run or every nondeterministic ottom-up tree utomton there is n equivlent deterministic ottom-up tree utomton, nd there is n equivlent nondeterministic top-down tree utomton. Is there n equivlent deterministic top-down utomton?? 8 or every nondeterministic ottom-up tree utomton there is n equivlent deterministic ottom-up tree utomton, nd there is n equivlent nondeterministic top-down tree utomton. Is there n equivlent deterministic top-down utomton?? NO! 8. top-down stte, LABEL (stte, stte) must contin $-lef, = inry node lels e,$ = lef node lels top-most -node on the left-most pth must hve right-sutree which contins $-node. egin, (ny, find$) egin, (egin, ny) find$, / { (find$, ny), (ny, find$) } find$, $ ACC find$, e REJ ny, / (ny,ny) ny, $/e ACC nondeterministic ccept tree if there exists n ccepting run first nme lst lst nme first his set of two trees cnno e recognized y ny determinstic top-down tree utomton!! Why? or every nondeterministic ottom-up tree utomton there is n equivlent deterministic ottom-up tree utomton, nd there is n equivlent nondeterministic top-down tree utomton. Is there n equivlent deterministic top-down utomton?? NO! 8 or every nondeterministic ottom-up tree utomton there is n equivlent deterministic ottom-up tree utomton, nd there is n equivlent nondeterministic top-down tree utomton. Is there n equivlent deterministic top-down utomton?? NO! 84 s Wht out locl tree lnguges (defined y DDs). Cn they e ccepted y deterministic top-down utomt? Wht out single-type tree lnguges (defined y XML Schem s) Cn they e ccepted y deterministic top-down utomt? s Wht out locl tree lnguges (defined y DDs). Cn they e ccepted y deterministic top-down utomt? Wht out single-type tree lnguges (defined y XML Schem s) Cn they e ccepted y deterministic top-down utomt? Yes! Hence, there is no DD / Schem for { nme[first,lst], nme[lst,first] } 4

15 85 86 or every deterministic ottom-up tree utomton there exists miniml unique equivlent one! or every deterministic ottom-up tree utomton there exists miniml unique equivlent one! Equivlence is decidle Equivlence is decidle In fct, YOU hve lredy produced miniml ottom-up tree utomt! In fct, YOU hve lredy produced miniml ottom-up tree utomt! he miniml DAG of tree t cn e seen s the miniml unique tree utomton tht only ccepts the tree t. he miniml DAG of tree t cn e seen s the miniml unique tree utomton tht only ccepts the tree t. How expensive (complexity) to find mininml one? Sme s for word utomt? ree Automt re very useful concept in CS! 87 XML ree Automt ply crucil rule for 88 Hevily in verifiction Derive property of complex oject from the properties of its constituents h h g g = glue ( g, glue ( h, h ) ) g glue glue h h Do ll grphs / chip-lyouts produced in this wy, hve property P? Use the hierrchicl construction history of n oject, in order to work on prse tree insted of complex grph. rom there, use tree utomt. Mny NP-complete grph prolems ecome trctle on ounded-treewidth grphs! Efficient vlidtors ginst XML ypes Optimiztions If doc is of YPE, then no need to vlidte ginst YPE, if we know YPE included in YPE - if only slightly different then only need to vlidte there - incrementl vlidtion ginst updtes - etc, etc. Efficient query evlutors, use richer utomt which cn select nodes nd produce query nswers Optimiztions If nswer of QUERY is in cche, then no need to evlute QUERY, if included in QUERY. - if every possile nswer set to QUERY (of YPE X) is EMPY, then no need to evlute on the rel dt! XML ype Checking for Progrmming Lnguges he uture In 5-0 yers from now: You cn write function in Progrmming Lnguge X 89 he uture In 5-0 yers from now: You cn write function in Progrmming Lnguge X Experimentl PL s In this direction: CDuce XDuce 90 unction foo(xml document D: YPE): YPE { trverse D & compute output;... return output } unction foo(xml document D: YPE): YPE { trverse D & compute output;... return output } Compiler (XML ype Checker) will complin, if your function does not compute documents of YPE. Compiler (XML ype Checker) will complin, if your function does not compute documents of YPE. If no complint, then gurnteed: ALL outputs re ALWAYS of correct type!!) If no complint, then correct type gurnteed. Compilers will hve to e le to give sttic gurntees out input/output ehviour of progrm! 5

16 9 END Lecture 5 6