Formal Methods for XML: Algorithms & Complexity

Forml Methods for XML: Algorithms & Complexity S. Mrgherit di Pul August 2004 Thoms Shwentik

Shwentik XML: Algorithms & Complexity Introdution 1 XML Exmple Doument Composer Nme Clude Debussy /Nme Vit Born When August 22, 1862 /When Where Pris /Where /Born Mrried When Otober 1899 /When Whom Roslie /Whom /Mrried Mrried When Jnury 1908 /When Whom Emm /Whom /Mrried Died When Mrh 25, 1918 /When Where Pris /Where /Died /Vit Piee PTitle L Mer /PTitle PYer 1905 /PYer Instruments Lrge orhestr /Instruments Movements 3 /Movements... /Piee... /Composer...

Shwentik XML: Algorithms & Complexity Introdution 1 XML Exmple Tree Nme Clude Debussy Born When 1862 Where Pris Composer Vit Mrried Mrried When 1899 Whom Roslie When 1908 Whom Emm Died When 1918 Where Pris PTitle L Mer Piee PYer 1905 Instruments Lrge orhestr Movements 3

Shwentik XML: Algorithms & Complexity Introdution 2 XML Proessing Four importnt kinds of XML proessing Vlidtion Chek whether n XML doument is of given type Nvigtion Selet set of positions in n XML doument Querying Extrt informtion from n XML doument Trnsformtion Construt new XML doument from given one

Shwentik XML: Algorithms & Complexity Introdution 2 XML Proessing Four importnt kinds of XML proessing.......... nd their lnguges Vlidtion DTD, XML Shem Chek whether n XML doument is of given type Nvigtion XPth Selet set of positions in n XML doument Querying XQuery Extrt informtion from n XML doument Trnsformtion XSLT Construt new XML doument from given one

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 3 Vlidtion: DTD Exmple doument DTD DTDs desribe types of XML douments Composer Nme Clude Debussy /Nme Vit Born When August 22, 1862 /When Where Pris /Where /Born Mrried When Otober 1899 /When Whom Roslie /Whom /Mrried Mrried When Jnury 1908 /When Whom Emm /Whom /Mrried Died When Mrh 25, 1918 /When Where Pris /Where /Died /Vit Piee PTitle L Mer /PTitle PYer 1905 /PYer Instruments Lrge orhestr /Instruments Movements 3 /Movements... /Piee... /Composer...

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 3 Vlidtion: DTD Exmple doument Composer Nme Clude Debussy /Nme Vit Born When August 22, 1862 /When Where Pris /Where /Born Mrried When Otober 1899 /When Whom Roslie /Whom /Mrried Mrried When Jnury 1908 /When Whom Emm /Whom /Mrried Died When Mrh 25, 1918 /When Where Pris /Where /Died /Vit Piee PTitle L Mer /PTitle PYer 1905 /PYer Instruments Lrge orhestr /Instruments Movements 3 /Movements... /Piee... /Composer... DTD DTDs desribe types of XML douments Exmple <!DOCTYPE Composers [ <!ELEMENT Composers (Composer*)> <!ELEMENT Composer (Nme, Vit, Piee*)> <!ELEMENT Vit (Born, Mrried*, Died?)> <!ELEMENT Born (When, Where)> <!ELEMENT Mrried (When, Whom)> <!ELEMENT Died (When, Where)> <!ELEMENT Piee (PTitle, PYer, Instruments, Movements)> ]>

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 4 Nvigtion: XPth Exmple doument Composer Nme Clude Debussy /Nme Vit Born When August 22, 1862 /When Where Pris /Where /Born Mrried When Otober 1899 /When Whom Roslie /Whom /Mrried Mrried When Jnury 1908 /When Whom Emm /Whom /Mrried Died When Mrh 25, 1918 /When Where Pris /Where /Died /Vit Piee PTitle L Mer /PTitle PYer 1905 /PYer Instruments Lrge orhestr /Instruments Movements 3 /Movements... /Piee... /Composer...

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 4 Composer Nme Clude Debussy /Nme Vit Nvigtion: XPth XPth XPth expressions selet sets of nodes of Exmple doument XML douments by speifying nvigtionl ptterns Born When August 22, 1862 /When Where Pris /Where /Born Mrried When Otober 1899 /When Whom Roslie /Whom /Mrried Mrried When Jnury 1908 /When Whom Emm /Whom /Mrried Died When Mrh 25, 1918 /When Where Pris /Where /Died /Vit Piee PTitle L Mer /PTitle PYer 1905 /PYer Instruments Lrge orhestr /Instruments Movements 3 /Movements... /Piee... /Composer...

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 4 Composer Nme Clude Debussy /Nme Vit Nvigtion: XPth Born When August 22, 1862 /When Where Pris /Where /Born Mrried When Otober 1899 /When Whom Roslie /Whom /Mrried Mrried When Jnury 1908 /When Whom Emm /Whom /Mrried Died When Mrh 25, 1918 /When Where Pris /Where /Died /Vit Piee PTitle L Mer /PTitle PYer 1905 /PYer Instruments Lrge orhestr /Instruments Movements 3 /Movements... /Piee... /Composer... XPth XPth expressions selet sets of nodes of Exmple doument XML douments by speifying nvigtionl ptterns Exmple query //Vit/Died/*

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 4 Composer Nme Clude Debussy /Nme Vit Nvigtion: XPth Born When August 22, 1862 /When Where Pris /Where /Born Mrried When Otober 1899 /When Whom Roslie /Whom /Mrried Mrried When Jnury 1908 /When Whom Emm /Whom /Mrried Died When Mrh 25, 1918 /When Where Pris /Where /Vit Piee PTitle L Mer /PTitle PYer 1905 /PYer Instruments Lrge orhestr /Instruments Movements 3 /Movements... /Piee... /Composer... XPth XPth expressions selet sets of nodes of Exmple doument XML douments by speifying nvigtionl ptterns /Died Exmple query //Vit/Died/*

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 5 Querying: XQuery Exmple doument Composer Nme Clude Debussy /Nme Vit Born When August 22, 1862 /When Where Pris /Where /Born Mrried When Otober 1899 /When Whom Roslie /Whom /Mrried Mrried When Jnury 1908 /When Whom Emm /Whom /Mrried Died When Mrh 25, 1918 /When Where Pris /Where /Died /Vit Piee PTitle L Mer /PTitle PYer 1905 /PYer Instruments Lrge orhestr /Instruments Movements 3 /Movements... /Piee... /Composer...

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 5 Querying: XQuery Exmple doument XQuery XQuery is full-fledged XML query lnguge Composer Nme Clude Debussy /Nme Vit Born When August 22, 1862 /When Where Pris /Where /Born Mrried When Otober 1899 /When Whom Roslie /Whom /Mrried Mrried When Jnury 1908 /When Whom Emm /Whom /Mrried Died When Mrh 25, 1918 /When Where Pris /Where /Died /Vit Piee PTitle L Mer /PTitle PYer 1905 /PYer Instruments Lrge orhestr /Instruments Movements 3 /Movements... /Piee... /Composer...

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 5 Querying: XQuery Exmple doument Composer Nme Hetor Berlioz /Nme Nme Clude Debussy /Nme Nme Cmille Sint-Sëns /Nme Vit Nme Born Frédéri When August Chopin 22, /Nme 1862 /When Where Pris /Where /Born Nme Mrried When Murie Rvel Otober /Nme 1899 /When Whom Roslie /Whom /Mrried Mrried When Jnury 1908 /When Whom Emm /Whom /Mrried Nme Jim Morrison /Nme Died When Mrh 25, 1918 /When Where Pris /Where /Died Nme /Vit Césr Frnk /Nme Nme Piee Gbriel Furé /Nme Nme PTitle George L Mer Bizet /PTitle /Nme PYer 1905 /PYer Instruments Lrge orhestr /Instruments Movements 3 /Movements... /Piee... /Composer... Result Nme Clude Debussy /Nme Nme Eri Stie /Nme... XQuery XQuery is full-fledged XML query lnguge Exmple query for $x in do( omposers.xml )/Composer where $x/vit/died/where = Pris return $x/nme

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 6 Trnsformtion: XSLT Exmple doument Composer Nme Clude Debussy /Nme Vit Born When August 22, 1862 /When Where Pris /Where /Born Mrried When Otober 1899 /When Whom Roslie /Whom /Mrried Mrried When Jnury 1908 /When Whom Emm /Whom /Mrried Died When Mrh 25, 1918 /When Where Pris /Where /Died /Vit Piee PTitle L Mer /PTitle PYer 1905 /PYer Instruments Lrge orhestr /Instruments Movements 3 /Movements... /Piee... /Composer...

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 6 Trnsformtion: XSLT XSLT XSLT trnsforms douments by Exmple doument mens of templtes Composer Nme Clude Debussy /Nme Vit Born When August 22, 1862 /When Where Pris /Where /Born Mrried When Otober 1899 /When Whom Roslie /Whom /Mrried Mrried When Jnury 1908 /When Whom Emm /Whom /Mrried Died When Mrh 25, 1918 /When Where Pris /Where /Died /Vit Piee PTitle L Mer /PTitle PYer 1905 /PYer Instruments Lrge orhestr /Instruments Movements 3 /Movements... /Piee... /Composer...

Result Trnsformtion: XSLT XSLT XSLT trnsforms douments by Exmple doument mens of templtes PrisComposer Nme Clude Debussy /Nme Composer Born Nme When Clude August Debussy 22, 1862 /Nme /When Vit Where Pris /Where /Born Born When August 22, 1862 /When Where Pris /Where /Born /PrisComposer Mrried When Otober 1899 /When Whom Roslie /Whom /Mrried PrisComposer Mrried When Jnury 1908 /When Whom Emm /Whom /Mrried Nme Died When FrédériMrh Chopin 25, /Nme 1918 /When Where Pris /Where /Died /Vit Born Piee When Mrh 1, 1810 /When PTitle Where LŽelzow Mer /PTitle /Where /Born PYer 1905 /PYer Exmple /PrisComposer Instruments Lrge orhestr /Instruments PrisComposer Movements 3 /Movements Nme... Cmille Sint-Sëns /Nme PrisComposer /Piee Born xsl:opy-of selet= Nme /... When Otober 9, 1835 /When xsl:opy-of selet= Vit/Born / /Composer Where Pris /Where /PrisComposer... /Born /xsl:templte /PrisComposer xsl:templte mth= Composer[Vit//Where= Pris ] Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 6

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 7 A Shemti View DTD/ XML Shem XPth yes/no XQuery XSLT

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 8 Fous of this Tlk Topis Expressive power of XML lnguges Complexity of lgorithmi tsks relted to XML proessing Trdeoff between expressiveness nd omplexity Gols of this Reserh Understnd expressive power nd omplexity of XML lnguges Identify interesting frgments with good trdeoff

Algorithmi Tsks Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 9

Evlution Evlution (Combined) I: Tree t, Query q O: q(t) Algorithmi Tsks Stti Anlysis Stisfibility I: Query q Q: Is q(t) for some t? Evlution (Dt(q)) I: Tree t O: q(t) Inrementl Evl. (q) I: Tree t, Chnges of t O: q(t) Continment I: Queries q 1, q 2 Q: Is lwys q 1 (t) q 2 (t)? Type Cheking I: Types d 1, d 2, Trnsformtion T Q: Does t = d 1 imply T (t) = d 2? Equivlene I: Queries q 1, q 2 Q: Is lwys q 1 (t) = q 2 (t)? Type Inferene Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 10

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 11 Expressive power Question: How do we mesure expressive power? Remrks Clsses of logil formuls re good yrdstik They provide methods to prove tht query n not be expressed

Contents Introdution Bkground on Tree Automt nd Logi Shem Lnguges XPth nd Node-seleting Queries XSLT XQuery Conlusion

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 13 Deidble Bkground: Complexity Clsses Overview of Complexity Clsses EXPSPACE EXPTIME PSPACE...... Equivlene of reg. expressions with squring.......................... 2-Plyer Corridor Tiling..................... Quntified Boolen Formuls onp NP.................... Stisfibility of prop. formuls P NC LOGCFL NL LOGSPACE........................ Boolen iruit evlution................. Effiiently prllelizble problems....................... Ayli onjuntive queries................... Rehbility in direted grphs................... Rehbility in direted forests

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 14 XML, Trees nd Automt Question: Why is XML ppeling for Theory people?

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 14 XML, Trees nd Automt Question: Why is XML ppeling for Theory people? Yers go... Theoretil Computer Siene for Dtbse Theorists: Logis, Complexity, Algorithms,... Dtbse Theory for Theoretil Computer Sientists:

XML, Trees nd Automt Question: Why trees? Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 15

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 15 XML, Trees nd Automt Question: Why trees? A Nturl Answer Trees reflet the hierrhil struture of XML Underlying dt model of XML is tree bsed Limittions But trees n not model ll spets of XML (e.g., IDREFs, dt vlues) Sometimes extensions re needed E.g., direted grphs insted of trees

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 16 XML, Trees nd Automt Question: Why utomt? Ingredients of XML Conepts from forml lnguges re obviously present round XML: Lbelled trees DTD: ontext-free grmmrs DTD: regulr expressions XPth: regulr pth expressions We will see Automt turn out to be useful s: mens to define robust lsses with ler semntis tool for proofs n lgorithmi tool for stti nlysis tool for query evlution

Introdution Contents Bkground on Tree Automt nd Logi Prllel Rnked Tree Automt Sequentil Rnked Tree Automt Deision Problems for Rnked Tree Automt Prllel Unrnked Tree Automt Sequentil Unrnked Tree Automt Sequentil Doument Automt Shem Lnguges XPth nd Node-seleting Queries XSLT XQuery Conlusion

From Strings to Trees A String bb Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 18

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 18 String s Tree From Strings to Trees A String bb b b

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 18 String s Tree b From Strings to Trees A String bb A Rnked Tree b b b b b

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 18 String s Tree From Strings to Trees A String bb A Rnked Tree An Unrnked Tree b b e e e e b e e e e b b b e e e

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 19 From Strings to Trees (ont.) XML nd Trees XML trees re unrnked : the number of hildren of node is not restrited Automt hve first been onsidered on rnked trees, where eh symbol hs fixed number of hildren (rnk) Most importnt ides were lredy developed for rnked trees Let us tke look t this first

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 20 Trees s Terms Remrk Sometimes trees re viewed s terms b Exmple b Exmple Tree s Term 1 (b 2 ( 1 ( 2 (b, )), b 2 ( 2 (b, ), ))) b b

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 21 From String Automt to Tree Automt Question How do utomt generlize to trees? Sequentil e Prllel e e e e e e e

From String Automt to Tree Automt Question How do utomt generlize to trees? Sequentil e Prllel e e e e e e e Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 21

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 21 From String Automt to Tree Automt Question How do utomt generlize to trees? Sequentil e Prllel e e e e e e e

From String Automt to Tree Automt Question How do utomt generlize to trees? Sequentil e Prllel e e e e e e e Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 21

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 21 From String Automt to Tree Automt Question How do utomt generlize to trees? Sequentil e Prllel e e e e e e e

Bottom-Up Automt Exmple: Tree-strutured Boolen Ciruits 0 1 1 0 0 1 1 1 Ide Tree-strutured Boolen iruits Two sttes: q 0, q 1 Aepting t the root: q 1 Trnsitions δ(, q 1 ) = {(q 1, q 1 )} δ(, q 0 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 0, q 0 )} δ(, q 1 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 1, q 1 )} δ(, q 0 ) = {(q 0, q 0 )} δ(0, q 0 ) = {ɛ}; δ(0, q 1 ) = δ(1, q 1 ) = {ɛ}; δ(1, q 0 ) = Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 22

Bottom-Up Automt Exmple: Tree-strutured Boolen Ciruits 0 q 0 1 q 1 1 q 1 0 q 0 0 q 0 1 q 1 1 q 1 1 q 1 Ide Tree-strutured Boolen iruits Two sttes: q 0, q 1 Aepting t the root: q 1 Trnsitions δ(, q 1 ) = {(q 1, q 1 )} δ(, q 0 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 0, q 0 )} δ(, q 1 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 1, q 1 )} δ(, q 0 ) = {(q 0, q 0 )} δ(0, q 0 ) = {ɛ}; δ(0, q 1 ) = δ(1, q 1 ) = {ɛ}; δ(1, q 0 ) = Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 22

Bottom-Up Automt Exmple: Tree-strutured Boolen Ciruits q 0 q 1 q 1 q 1 0 q 0 1 q 1 1 q 1 0 q 0 0 q 0 1 q 1 1 q 1 1 q 1 Ide Tree-strutured Boolen iruits Two sttes: q 0, q 1 Aepting t the root: q 1 Trnsitions δ(, q 1 ) = {(q 1, q 1 )} δ(, q 0 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 0, q 0 )} δ(, q 1 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 1, q 1 )} δ(, q 0 ) = {(q 0, q 0 )} δ(0, q 0 ) = {ɛ}; δ(0, q 1 ) = δ(1, q 1 ) = {ɛ}; δ(1, q 0 ) = Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 22

Bottom-Up Automt Exmple: Tree-strutured Boolen Ciruits q 1 q 1 q 0 q 1 q 1 q 1 0 q 0 1 q 1 1 q 1 0 q 0 0 q 0 1 q 1 1 q 1 1 q 1 Ide Tree-strutured Boolen iruits Two sttes: q 0, q 1 Aepting t the root: q 1 Trnsitions δ(, q 1 ) = {(q 1, q 1 )} δ(, q 0 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 0, q 0 )} δ(, q 1 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 1, q 1 )} δ(, q 0 ) = {(q 0, q 0 )} δ(0, q 0 ) = {ɛ}; δ(0, q 1 ) = δ(1, q 1 ) = {ɛ}; δ(1, q 0 ) = Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 22

Bottom-Up Automt Exmple: Tree-strutured Boolen Ciruits q 1 q 1 q 1 q 0 q 1 q 1 q 1 0 q 0 1 q 1 1 q 1 0 q 0 0 q 0 1 q 1 1 q 1 1 q 1 Ide Tree-strutured Boolen iruits Two sttes: q 0, q 1 Aepting t the root: q 1 Trnsitions δ(, q 1 ) = {(q 1, q 1 )} δ(, q 0 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 0, q 0 )} δ(, q 1 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 1, q 1 )} δ(, q 0 ) = {(q 0, q 0 )} δ(0, q 0 ) = {ɛ}; δ(0, q 1 ) = δ(1, q 1 ) = {ɛ}; δ(1, q 0 ) = Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 22

Non-det. Top-Down Automt Exmple 0 1 1 0 0 1 1 1 Ide Guess the orret vlues strting from the root Chek t the leves Three sttes: q 0, q 1, Initil stte q 1 t the root Aepting if ll leves end in Trnsitions δ(, q 1 ) = {(q 1, q 1 )} δ(, q 0 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 0, q 0 )} δ(, q 1 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 1, q 1 )} δ(, q 0 ) = {(q 0, q 0 )} δ(0, q 0 ) = {}; δ(0, q 1 ) = δ(1, q 1 ) = {}; δ(1, q 0 ) = Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 23

Non-det. Top-Down Automt Exmple q 1 0 1 1 0 0 1 1 1 Ide Guess the orret vlues strting from the root Chek t the leves Three sttes: q 0, q 1, Initil stte q 1 t the root Aepting if ll leves end in Trnsitions δ(, q 1 ) = {(q 1, q 1 )} δ(, q 0 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 0, q 0 )} δ(, q 1 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 1, q 1 )} δ(, q 0 ) = {(q 0, q 0 )} δ(0, q 0 ) = {}; δ(0, q 1 ) = δ(1, q 1 ) = {}; δ(1, q 0 ) = Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 23

Non-det. Top-Down Automt Exmple q 1 q 1 q 1 0 1 1 0 0 1 1 1 Ide Guess the orret vlues strting from the root Chek t the leves Three sttes: q 0, q 1, Initil stte q 1 t the root Aepting if ll leves end in Trnsitions δ(, q 1 ) = {(q 1, q 1 )} δ(, q 0 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 0, q 0 )} δ(, q 1 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 1, q 1 )} δ(, q 0 ) = {(q 0, q 0 )} δ(0, q 0 ) = {}; δ(0, q 1 ) = δ(1, q 1 ) = {}; δ(1, q 0 ) = Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 23

Non-det. Top-Down Automt Exmple q 1 q 1 q 1 q 0 q 1 q 1 q 1 0 1 1 0 0 1 1 1 Ide Guess the orret vlues strting from the root Chek t the leves Three sttes: q 0, q 1, Initil stte q 1 t the root Aepting if ll leves end in Trnsitions δ(, q 1 ) = {(q 1, q 1 )} δ(, q 0 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 0, q 0 )} δ(, q 1 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 1, q 1 )} δ(, q 0 ) = {(q 0, q 0 )} δ(0, q 0 ) = {}; δ(0, q 1 ) = δ(1, q 1 ) = {}; δ(1, q 0 ) = Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 23

Non-det. Top-Down Automt Exmple q 1 q 1 q 1 q 0 q 1 q 1 q 1 0 q 0 1 q 1 1 q 1 0 q 0 0 q 0 1 q 1 1 q 1 1 q 1 Ide Guess the orret vlues strting from the root Chek t the leves Three sttes: q 0, q 1, Initil stte q 1 t the root Aepting if ll leves end in Trnsitions δ(, q 1 ) = {(q 1, q 1 )} δ(, q 0 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 0, q 0 )} δ(, q 1 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 1, q 1 )} δ(, q 0 ) = {(q 0, q 0 )} δ(0, q 0 ) = {}; δ(0, q 1 ) = δ(1, q 1 ) = {}; δ(1, q 0 ) = Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 23

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 23 Non-det. Top-Down Automt Exmple q 1 q 1 q 1 q 0 q 1 q 1 q 1 0 q 0 1 q 1 1 q 1 0 q 0 0 q 0 1 q 1 1 q 1 1 q 1 Ide Guess the orret vlues strting from the root Chek t the leves Three sttes: q 0, q 1, Initil stte q 1 t the root Aepting if ll leves end in Trnsitions δ(, q 1 ) = {(q 1, q 1 )} δ(, q 0 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 0, q 0 )} δ(, q 1 ) = {(q 0, q 1 ), (q 1, q 0 ), (q 1, q 1 )} δ(, q 0 ) = {(q 0, q 0 )} δ(0, q 0 ) = {}; δ(0, q 1 ) = δ(1, q 1 ) = {}; δ(1, q 0 ) =

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 24 Regulr Tree Lnguges Definition A bottom-up utomton is deterministi if for eh nd p q: δ(, p) δ(, q) = Theorem The following re equivlent for tree lnguge L: () L is epted by nondeterministi bottom-up utomton (b) L is epted by deterministi bottom-up utomton () L is epted by nondeterministi top-down utomton Proof ide () = (b): Powerset onstrution () (): Just the sme thing, viewed in two different wys

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 25 Automt s Tiling Systems Observtion (q 0, q 1 ) δ(, q 1 ) n be interpreted s n llowed pttern:, q 1 q 0 q 1 A tree is epted, iff there is lbelling with sttes suh tht ll lol ptterns re llowed the root is lbelled with q 1

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 26 Formuls tlk bout Regulr tree lnguges nd logi edges of the tree (E) node lbels (Q 0, Q 1, Q, Q ) the root of the tree (root) Definition: (MSO logi) First-order-vribles represent nodes Mondi seond-order (MSO) vribles represent sets of nodes Exmple: Boolen Ciruits Boolen iruit true X X(root) x (Q 0 (x) X(x)) ((Q (x) X(x)) ( y[e(x, y) X(y)])) ((Q (x) X(x)) ( y[e(x, y) X(y)])) Theorem [Doner 70; Thther, Wright 68] MSO Regulr Tree Lnguges

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 27 Regulr tree lnguges nd logi (ont.) Theorem MSO Regulr Tree Lnguges Proof ide Automt MSO: Formul expresses tht there exists orret tiling MSO Automt: more involved Bsi ide: Automton omputes for eh node v the set of formuls whih hold in the subtree rooted t v

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 28 Regulr tree lnguges nd logi (ont.) Formul utomton Let ϕ be n MSO-formul, k := quntifier-depth of ϕ k-type of tree t := (essentilly) set of MSO-formuls ψ of quntifier-depth k whih hold in t t 1 k t 2 : k-type(t 1 ) = k-type(t 2 ) Automton omputes k-type of tree nd onludes whether ϕ holds Cruil ft l l t 1 k t 1 = t 1 t 2 k t 1 t 2 t 2 k t 2

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 29 Det. Top-Down Automt Question Wht is the right notion for deterministi top-down utomt? 3 Possibilities Stte t node v might depend on stte nd symbol of prent q stte nd symbol of prent nd symbol of v v q stte nd symbol of prent nd symbols t v nd its sibling v q v e

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 30 Det. Top-Down Automt: Aeptne Question Wht is good eptne mehnism for deterministi top-down utomt? Severl possibilitites (1) At ll leves sttes hve to be epting (2) There is leve with n epting stte (2) is problemti for omplement nd intersetion (1) is problemti for omplement nd union

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 31 Det. Top-Down Automt: Aeptne (ont.) Definition: (Root-to-frontier utomt with regulr eptne ondition) Tree utomt A re equipped with n dditionl regulr string lnguge L over Q Σ A epts t if the (stte,symbol)-string t the leves (from left to right) is in L [Jurvnen, Potthoff, Thoms 93] Illustrtion (q 1, 1 ) (q n, n ) A robust lss The resulting lss is losed under Boolen opertions Good lgorithmi properties Does not pture ll regulr tree lnguges

Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 32 Summry Regulr tree lnguges Regulr tree lnguges re robust lss Chrterized by prllel tree utomt MSO logi severl other models They re the nturl nlog of regulr string lnguges Deterministi top-down utomt with regulr eptne onditions define weker but nevertheless robust lss

Tree-Wlk Automt Definition: (Tree-wlk utomt) Depending on urrent stte symbol of urrent node r TWA T e position of urrent node wrt its siblings e A u t the utomton moves to neighbor nd tkes new stte o s Question Wht is the expressive power of tree-wlk utomt? Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 34

Tree-Wlk Automt (ont.) Ft Tree-wlk utomt n evlute Boolen iruit trees 5 sttes Exmple Ide q 0 0 1 q 1 1 0 0 1 1 0 0 1 1 1 Shwentik XML: Algorithms & Complexity Introdution - XML Proessing 35