master informatique 10/9/2007 1 typing semistructured data serge abiteboul 2008 typing...

Master Informatique 10/9/2007 1

Typing semistructured data

Serge Abiteboul

2008


Master Informatique 10/9/2007 2Master Informatique Typing semistructured data

Organization

• Motivations• Automata

– Automata on words– Ranked tree automata– Unranked tree automata– Automata and monadic second-order logic– Automata – to compute

• XML typing: DTD, XML schema• Graphs and bisimulation


Motivation



XML typing

• Not compulsory• Simplify writing software for XML

– Improve interoperability between programs

• Improve storage and performance• Ease querying: data guide • Simplify data protection

– Reject illegal update – like relational dependencies


Improve storage

Root

Company Employee

string

company

person

works-for

c.e.o.

address

name

managed-by

name

o i d n a m e a d d r e s s c . e . o .… … … …… … … …

Company

o i d n a m e m a n a g e d - b y w o r k s - f o r… … … …… … … …

Employee

Store rest in overflow graph

Lower-bound schema



Improve performance

Bib

paper book

yearjournal

title

int string string

addressauthor

title

zip city street

lastname

firstname

string string string string string

string

select X.titlefrom Bib._ Xwhere X.*.zip = “12345”

select X.titlefrom Bib._ Xwhere X.*.zip = “12345”

select X.titlefrom Bib.book Xwhere X.address.zip = “12345”

select X.titlefrom Bib.book Xwhere X.address.zip = “12345”



Type checking

• Who checks– XML editor: check that the data conforms to its type– XML exchange, e.g., with Web service

• Server when delivering the data• Client/application: when receiving it

• Dynamic verification: after the data is produced• Static verification: verification of the program that

generates the data


Static verification

• Input: input type T and code of function f– f is Xquery, Xpath, XSLT, etc.

• Verification of T’– Is it true that d╞T, f(d)╞T’ ?

• Type inference– Find the smallest T’ such that d╞T, f(d)╞T’

• Rapidly undecidable because of “joins”

Examplefor $p in doc("parts.xml“)//part[color=“red"]return <part>

<name>$p/name</name><desc>$p/desc</desc>

</part>

Result type (part (name (string) desc (any) )*

If the type of parts.xml//part/desc is string(part (name (string) desc (string) )*

Difficultyfor $X in Input, $Y in Input do { print ( }

Input: <a/> <a/> Result: Problem: { bi i=n2 for n ≥ 0 } cannot be described in XML

schemaThere is no « best » result

– b*– + b2 b*

– + b2 + b4b*

– + b2 + b4 + b9b*

– …


Why tree automata?

• XML = unranked trees• No theory for XML • Rich theory for strings: Automata• Extend to

rich theory for ranked trees: Tree automata – Nice algorithms– Nice theorems– Can this carry to unranked trees and XML?

• Yes!


From strings to treesa

b

b

a

a

b

b a

b

b

a b

a

b

b

a

b

b

a b

a b

a b

Word Binary tree… Unranked tree automataFinite State Ranked tree automata no bound on number of childrenAutomata

a

b b b


Only unranked tree automata?

• Missing practical gadgets• Complexity of verification

– Goal: typing at reasonable cost

• Unranked tree automata + …


Automata

Automata on words



Finite state automata on words

),,,,( 0 FqQ

Alphabet

State

Initial state Accepting states

Transitions

Qq 0 QF

)(: QPQ



q0

Nondeterministic automaton: Example

33

32

21

01

100

,

,

,

,

,,

qq

qq

qq

qqb

qqqa

2

3210 ,,,

,

qF

qqqqQ

ba

a b a a b - a b a -q0

q1

q0 q0

q1

q0

q1

q0 q0

q1

q0 q0 q2

q1

q0

KO OK


• Deterministic– No transition– No alternative transitions such as

• Determinization – It is possible to obtain an equivalent deterministic automaton– State of new automaton = set of states of the original one– Possible exponential blow-up

• Minimization• Limitations – cannot do

– Context-free languages• Essential tool – e.g., lexical analysis

Reminder

Ν, nba nn

100 ,, qqqa 0, qq


Reminder (2)• L(A) = set of words accepted by automata A• Regular languages• Can be described by regular expressions, e.g. a(b+c)*d• Closed under complement

• Closed under union, intersection

– Product automata with states (s,s’) where s is from A and s’ is from A’

)(* AL

)()(

)()(

BLAL

BLAL


Automata on words versus trees

a b b a

a

b

b a

b

b

a b

a

Left to right

Right to left

No difference

Bottom up

Top down

Differences


Automata

Automata on ranked trees



Binary tree automata

• Parallel evaluation

• For leaves:

• For other nodes:

),,,( FQ

)(: QP

)(: QPQQ

a

b

b a

b

a b

a

Bottom up

q q’

bq”

q1q”

q2

qqq’



Bottom-up tree automata

• Bottom-up: if a node labeled a has its children in states q, q’ then the node moves nondeterministically to state r or r’

• Accepts is the root is in some state in F

• Not deterministic if alternatives or -transitions:

',',, rrqqa

}',{',, rrqqa ', rr


Example: deterministic bottom-up

1102012112

0002

0102012002

1112

,,,,,,,,

,,

,,,,,,,,

,,

qqqqqqq

qqq

qqqqqqq

qqq

1

10 ,

,,1,0

qF

qqQ

11

01

1

0

q

q


1102

1012

1112

0002

0102

0012

0002

1112

,,

,,

,,

,,

,,

,,

,,

,,

qqq

qqq

qqq

qqq

qqq

qqq

qqq

qqq

Boolean circuit evaluation

v

v

v

1v v1

10

v

0

11

11

01

1

0

q

q

0q 1q 0q

1q1q

1q1q1q

1q

1q

1q

1q

1q

OK


Regular tree language = set of trees accepted by a bottom-up tree automata



Regular tree languages

The following are equivalent– L is a regular tree language– L is accepted by a nondeterministic bottom-up

automata– L is accepted by a deterministic bottom-up

automata– L is accepted by a nondeterministic top-down

automata

Deterministic top-down is weaker


Top-down tree automata

• Top-down: if a node labeled a is in state q”, then its left child moves to state q (right to q’)

• Accepts is all leaves are is in states in F• Not deterministic if

',", qqqa

',,',", rrqqqa


Why deterministic top-down is weaker?

• Consider the language– L = { f(a,b), f(b,a) }

• It can be accepted by a bottom-up TA– Exercise: write a BUTA A such that L = L(A)

• Suppose that B is a deterministic top-down TA with L = L(B)– Exercise: Show that B also accepts {f(a,a)} – A contradiction

Fact: No deterministic top-down tree automata accepts L


Ranked trees automata: Properties

• Like for words only higher complexity• Determinization • Minimization• Closed under

– Complement– Intersection– Union


But…

• XML documents are unranked• The kind of things we want to do:

book (intro,section*,conclusion)


Automata

Automata on unranked tree



Unranked tree automata

...,,,,,,

...,,,,,

...,,,,,

...,,,,,,

222

222

222

222

fffffffff

ttftfttt

ftffftff

ttttttttt

Issue: represent an infinite set of transitionsSolution: a regular language


• Rule:• Meaning: if the states of the children of some

node labeled a form a word in L(Q), this node moves to some state in {r1,…,rm}

Unranked tree automata (2)

mrrQLa ,...,)(, 1

fOrwherefOr

fttftOrwheretOr

ftfftAndwherefAnd

tAndwheretAnd

00,

*)(*)(11,

*)(*)(00,

11,

2

2

2

2


Building on ranked trees

a

b

b

b

b

a b

a b

a

b

b

b

b

a b

a b

Ranked tree: FirstChild-NextSibling

F: encoding into a ranked tree• F is a bijection

F-1: decoding


Building on bottom-up ranked trees (2)

• For each Unranked TA A, there is a Ranked TA accepting F(L(A))

• For each Ranked TA A, there is an unranked TA accepting F-1(L(A))

• Both are easy to construct

Consequence: Unranked TA are closed under union, intersection, complement


• Determinization always possible for bottom-up

• Can we use the FirstChild-NextSibling encoding – No: it does not preserve determinism

Determinization

.such that

),( rule unique a exists there,, *

Lw

LQw


Top-down?

• This is more delicate• Transition (a,q)=A(a,q)

– The state of the automata A(a,q) when reading the labels of the children of a node labeled a determines the states of the children of that node

– Accepts if all the leaves are in accepting state


1q

Boolean circuit evaluation

v

v

v1 v

11

0q

1q

010v

1

111

10

0v

v

v

1q

1q

1q

1q

1q 1q0q

0q

0q0q 1q

0q

0q 0q

1q 0q

0q

1q

It is acceptedIt rejects by if

some state of a leaf is neither 0 with q0

nor 1 with q1


Automata

Automata and monadic second-order logic



Monadic second-order logic

• Representation of a tree as a logical structure

E(1,2), E(1,3)… E(3,9)S(2,3), S(3,4), S(4,5)…S(8,9)

a(1), a(4), a(8)b(2), b(3), b(5), b(6), b(7), b(9)

a

b

b

b

b

a b

a b

1

6

3 42

7 8 9

5


XxXX

x

xayxSyxEyx

)(

...)(),(),(::

Monadic second-order logic

E(1,2), E(1,3)… E(3,9)S(2,3), S(3,4), S(4,5)…S(8,9)a(1), a(4), a(8)b(2), b(3), b(5), b(6), b(7), b(9)

MSO syntax

Set variable

Quantification over a set variable


Example of MSO

• Each a node has a b-descendant• This corresponds to the formula

For each node x labeled a: each set X that ()contains x and that () is closed under descendant, X contains some y labeled b

))()((

))()(),((

)(

)(

ybyXy

zXyXzyEzy

xX

whereXxax


Bridge

Theorem: for a set L of trees, the following are equivalent

1.L = L(A) for some bottom-up tree automata Ai.e. L is definable with bottom-tree automata

2.L = {T | T satisfies } for some MSO formula i.e. L is definable in MSO


XML typing

DTDs


DTD

• Describe the children of a node of a label a by a regular expression

• Bizarre syntax<!ELEMENT populationdata (continent*) >

<!ELEMENT continent (name, country*) >

<!ELEMENT country (name, province*)>

<!ELEMENT province (name, city*) >

<!ELEMENT city (name, pop) >

<!ELEMENT name (#PCDATA) >

<!ELEMENT pop (#PCDATA) >

DTD and deterministism

• Regular expressions in DTD should be deterministic– Complicated definition

• Intuition: the corresponding automata should be deterministic– (a+b)*a is not– When reading <a>, one cannot tell whether it is an a

from (a+b) or if it is the a of the end– (b*a)(b*a)* is an equivalent expression that is

deterministic


Very efficient validation

• It suffices to verify for each node a that the word formed by the labels of its children is accepted by the finite state automata Aa

• Possible to type check the document while scanning it, e.g. with SAX parser

Very efficient validation (2)<!ELEMENT a ( b c ) >

<!ELEMENT b ( d+ ) >

a

b c

d d

s t ub c

Aa

s’ t’d

dAb

<a><d/><d/><c/></a>

s’

st

t’

Acceptu

Warning

• The previous example can be checked with a simple automata on words

• But not the following one <!ELEMENT part ( part* ) >

• The stack is needed for accepting<a>…<a></a>…</a>

n <a> n </a>

Some bad news for DTD

• Not closed under union DTD1 …

<!ELEMENT used( ad*) >

<!ELEMENT ad ( year, brand )>

DTD2 …

<!ELEMENT new( ad*) >

<!ELEMENT ad ( brand )>

• L(DTD1) L(DTD2) cannot be described by a DTD but can be described easily by a tree automata– Problem with the type of ad that depends of its parent

• Also not closed under complement• Limited expressive power

Car example continued

• The best DTD we can choose does not distinguish between ads for used and new cars– <!ELEMENT ad (year?, brand) >

Car

Used New

Brand Year Brand

“Renault” “2008” “BMW”


Decoupled types in XML schema

• Each type corresponds to a label, not converselycar: [car] ( used + new )*

used:[used] (ad1*)

new: [new] (ad2*)

ad1: [ad] (year, brand)

ad2: [ad] (brand)

• The tags are in green; type names in blue• Nice closure properties• Many other « gadgets » in XML schemas


XML typing

XML Schemas



XML Schema

• Often criticized & unnecessarily complicated• Boosted by Web services• Richer than DTD – decoupled types• Deterministic top-down tree automata (close to)• XML schemas are extensible• Many other useful functionalities

– Namespaces – Atomic types– Integrity constraints, etc.


An XML schema is an XML document

• Since it is an XML syntax, it can use XML tools– Editor– Type checker– Etc.

• The type of all XML schemas can be described with an XML schema

<?xml version="1.0" encoding="utf-8"?> <xs:schema xmlns:xs="http://www.w3.org/2001/XMLSchema" targetnamespace="http://www.net-language.com"> <xs:element name="book"> <xs:complexType> <xs:sequence> <xs:element name="title" type="xs:string"/> <xs:element name="author" type="xs:string"/> <xs:element name="character" minOccurs="0" maxOccurs="unbounded"> <xs:complexType> <xs:sequence> <xs:element name="name" type="xs:string"/> <xs:element name="friend-of" type="xs:string" minOccurs="0" maxOccurs="unbounded"/> <xs:element name="since" type="xs:date"/> <xs:element name="qualification" type="xs:string"/> </xs:sequence> </xs:complexType> </xs:element> </xs:sequence> <xs:attribute name="isbn" type="xs:string"/> </xs:complexType> </xs:element> </xs:schema>

Simple elements and atomic types

Definition: <xs:element name="xxx" type="yyy"/>with common types: xs:string; xs:decimal; xs:integer; xs:boolean; xs:date; xs:time

Examples<xs:element name="lastname" type="xs:string"/> <xs:element name="age" type="xs:integer"/> <xs:element name="dateborn" type="xs:date"/>

Instances of such elements<lastname>Refsnes</lastname> <age>34</age> <dateborn>1968-03-27</dateborn>

Attributs

Definition: <xs:attribute name="xxx" type="yyy"/>

Example<xs:attribute name="lang" type="xs:string"/>

Instance of such attribute<lastname lang="EN">Smith</lastname>

Complex elements

• Empty element<product pid="1345"/>

• Contains only other elements<employee> <firstname>John</firstname> <lastname>Smith</lastname> </employee>

• Contains only text<food type="dessert">Ice cream</food>

• Contains both elements and text<description> It happened on <date lang="norwegian"> 03.03.99</date> .... </description>

Restriction of simple elements<xs:element name="age">

<xs:simpleType> <xs:restriction base="xs:integer"> <xs:minInclusive value="0"/>

<xs:maxInclusive value="100"/> </xs:restriction>

</xs:simpleType></xs:element>

Other restrictions: enumerated types, patterns, etc.

Restriction on complex elements

<xs:element name="person"> <xs:complexType>

<xs:sequence> <xs:element name="firstname" type="xs:string"/>

<xs:element name="lastname" type="xs:string"/> </xs:sequence> </xs:complexType>

</xs:element>

Possible to name a type<xs:element name="employee">

<xs:complexType> <xs:sequence> <xs:element name="firstname" type="xs:string"/> <xs:element name="lastname" type="xs:string"/> </xs:sequence> </xs:complexType>

</xs:element>

Only the "employee" element can use the specified complex type(<sequence> indicates an order on child elements)

Alternative

<xs:element name="employee" type="personinfo" />

<xs:complexType name="personinfo"> <xs:sequence> <xs:element name="firstname" type="xs:string"/> <xs:element name="lastname" type="xs:string"/> </xs:sequence>

</xs:complexType>

Other gadgets

• Import of types associated to a namespace– <import nameSpace = "http:// ..."

schemaLocation = "http:// ..." />

• Possible to include an existing schema– <include schemaLocation="http:// ..."/>

• Possible to extend/redefine an existing schema– <redefine schemaLocation="http:// ..."/>

.... Extensions ...

</redefine>

Example: a DTD<!ELEMENT EMAIL (TO+, FROM, CC*, BCC*, SUBJECT?, BODY?)>

<!ATTLIST EMAIL

LANGUAGE (Western|Greek|Latin|Universal) "Western"

ENCRYPTED CDATA #IMPLIED

PRIORITY (NORMAL|LOW|HIGH) "NORMAL">

<!ELEMENT TO (#PCDATA)>

<!ELEMENT FROM (#PCDATA)>

<!ELEMENT CC (#PCDATA)>

<!ELEMENT BCC (#PCDATA)>

<!ATTLIST BCC

HIDDEN CDATA #FIXED "TRUE">

<!ELEMENT SUBJECT (#PCDATA)>

<!ELEMENT BODY (#PCDATA)>

<!ENTITY SIGNATURE "Bill">

The same in XML schema(more verbose)

<?xml version="1.0" ?>

<Schema name="email" xmlns="urn:schemas-microsoft-com:xml-data"

xmlns:dt="urn:schemas-microsoft-com:datatypes">

<AttributeType name="language"

dt:type="enumeration" dt:values="Western Greek Latin Universal" />

<AttributeType name="encrypted" />

<AttributeType name="priority" dt:type="enumeration" dt:values="NORMAL LOW HIGH" />

<AttributeType name="hidden" default="true" />

<ElementType name="to" content="textOnly" />

<ElementType name="from" content="textOnly" />

<ElementType name="cc" content="textOnly" />

<ElementType name="bcc" content="mixed">

<attribute type="hidden" required="yes" />

</ElementType>

<ElementType name="subject" content="textOnly" />

<ElementType name="body" content="textOnly" />

<ElementType name="email" content="eltOnly">

<attribute type="language" default="Western" />

<attribute type="encrypted" />

<attribute type="priority" default="NORMAL" />

<element type="to" minOccurs="1" maxOccurs="*" />

<element type="from" minOccurs="1" maxOccurs="1" />

<element type="cc" minOccurs="0" maxOccurs="*" />

<element type="bcc" minOccurs="0" maxOccurs="*" />

<element type="subject" minOccurs="0" maxOccurs="1" />

<element type="body" minOccurs="0" maxOccurs="1" />

</ElementType>

</Schema>


Where to place XML schemas

• Some bizarre restriction– Inside an element, no two types with the same tag

• Closer to DTDs than to tree automata• Efficient type validation

Tree automata

Deterministic .top-down tree automata

DTDXML schema


Exercise: coupled vs decoupled

• Write a realistic DTD1 for new cars– With make, model, engine…

• Write a realistic DTD2 for used cars– Also year, miles, zipcode

• Write an XML schema for L(DTD1) L(DTD2) – Using decoupled schema


Automata

Automata to compute



Another use of automata: XPATH $x in //a/b

a

b

a a b

ab

b$x $x

NFA DFA

(0)


Example: //a/b

a

b

a a b

ab

b$x $x

NFA DFA

(0)(01)


Example: //a/b

a

b

a a b

ab

b$x $x

NFA DFA

(0)(01)(01)


Example: //a/b

a

b

a a b

ab

b$x $x

NFA DFA

(0)(01)(01)(02)

$x


Example: //a/b

a

b

a a b

ab

b$x $x

NFA DFA

(0)(01)(01)

$x


Example: //a/b

a

b

a a b

ab

b$x $x

NFA DFA

(0)(01)

$x


Example: //a/b

a

b

a a b

ab

b$x $x

NFA DFA

(0)(01)

$x

(01)


Example: //a/b

a

b

a a b

ab

b$x $x

NFA DFA

(0)(01)

$x


Example: //a/b

a

b

a a b

ab

b$x $x

NFA DFA

(0)(01)

$x

(02)$x


Example: //a/b

a

b

a a b

ab

b$x $x

NFA DFA

(0)(01)

$x

(02)$x

(01)


Example: //a/b

a

b

a a b

ab

b$x $x

NFA DFA

(0)(01)

$x

(02)

$x

(01)(02)

$x


Example: //a/b

a

b

a a b

ab

b$x $x

NFA DFA

(0)(01)

$x

(02)

$x

(01)$x


Example: //a/b

a

b

a a b

ab

b$x $x

NFA DFA

(0)(01)

$x

(02)

$x

$x


Example: //a/b

a

b

a a b

ab

b$x $x

NFA DFA

(0)(01)

$x

$x

$x


Example: //a/b

a

b

a a b

ab

b$x $x

NFA DFA

(0)

$x

$x

$x


Determinization: exponential blow up

//a/*/*/b



Proposal : k-pebble transducers

stack

[milo,suciu,vianu]


k-pebble transducers: result

root

a c

b a a b

a b

Capture a core aspect of Xquery but not the data management part


Graphs and bisimulation



Graph

• Graph semistructured data• Graph simulation • Graph bisimulation• Data guides


Semistructured data

• With ID-IDREF, XML is a graph model as well

• OEM = Object Exchange ModelLabeled (rooted) graph (E,r)– Set N of nodes– A finite ternary relation E NNLabel

E(s,t,l) = there is an edge from s to t labeled l– Possibly a root r


&r

&p8&p1 &p2 &p3 &p4 &p5 &p6 &p7

&c

company

employee

employee

employeeemployee employee employee

employeeemployee

worksfor

worksfor

worksforworksforworksfor

worksforworksfor

worksfor

manages

manages

manages

manages

managedby

managedby

managedby

manages

managedby

managedby


Equality revisited

• {1,2,2,1,5} = {1,2,5}– Ignores the order

• For trees, if we ignore the order of siblings and use a “set” semantics

=a

b c

d d

b

d d

a

b c

d


Simulation

A simulation of (E,r) with (E’,r’) is a relation between the nodes of E and E’ such that

1.(r,r’)2. if (s,s’) and E(s,t,l) for some l then there

exists t’ with (t,t’) and E’(s’,t’,l’)

(we simulate a move in E by a move in E’)


Bisimulation

Given , E, E’, is a bisimulation if is a simulation of E with E’ and

-1 is a simulation of E’ with E


Examples

a a

a d

a a

a d

a

a d

G G’ G”

They all have the same paths from the root

bisimulation Not bisimulation


root

e2 e3 e4e1

p1 p2 p3 p4 p5 p6 p7 p8 p9

"exercise" "lecture""finance" "adminstr.""PR" "undergrad""grad" "postgrad" "web"

leads

workson leadsworkson

leadsworkson leads

workson consults

employee

consultsworkson

workson

c1 c2programmer statistician

project

workson

employee employee

t1 t2

programmer | statistician

STRING_

employee

projects

R


t1

Graph bisimulationroot

e2 e3 e4e1

p1 p2 p3 p4 p5 p6 p7 p8 p9


leads


leadsworkson leads

workson consults

employee

consultsworkson

workson


project

workson

employee employee

t1 t2


STRING_

employee

projects

R


t1


e2 e3 e4e1

p1 p2 p3 p4 p5 p6 p7 p8 p9


leads


leadsworkson leads

workson consults

employee

consultsworkson

workson


project

workson

employee employee

t1 t2


STRING_

employee

projects

R



e2 e3 e4e1

p1 p2 p3 p4 p5 p6 p7 p8 p9


leads


leadsworkson leads

workson consults

employee

consultsworkson

workson


project

workson

employee employee

t1 t2


STRING_

employee

projects

R



e2 e3 e4e1

p1 p2 p3 p4 p5 p6 p7 p8 p9


leads


leadsworkson leads

workson consults

employee

consultsworkson

workson


project

workson

employee employee

t1 t2


STRING_

employee

projects

R

R



e2 e3 e4e1

p1 p2 p3 p4 p5 p6 p7 p8 p9


leads


leadsworkson leads

workson consults

employee

consultsworkson

workson


project

workson

employee employee

t1 t2


STRING_

employee

projects

R



e2 e3 e4e1

p1 p2 p3 p4 p5 p6 p7 p8 p9


leads


leadsworkson leads

workson consults

employee

consultsworkson

workson


project

workson

employee employee

t1 t2


STRING_

employee

projects

R

R


Computing bisimulation in ptime

• Start with = N N’ (for N, N’ the set of nodes)

• While there exists (x,x’) in that violate the definition of simulation, remove (x,x’) from

• This computes the maximal bisimulation in ptime(Note: this maximal bisimulation exists because is

a bisimulation, and if 1, 2 are bisimulation, 1 2 is also one)


What does this have to do with typing?

• Take a very complex graph E• How do you describe it?• By a “smaller” graph T that is a bisimulation of

E• There may be several bisimulation with more

and more details


Rough bisimulation

Root&r

Bosses&p1,&p4,&p6

Regulars&p2,&p3,&p5,&p7,&p8

Company&c

company employee

manages

managedby

worksfor

worksfor

employee


More precise one

Root&r

Employees&p1,&p1,&p3,P4

&p5,&p6,&p7,&p8

Bosses&p1,&p4,&p6

Regulars&p2,&p3,&p5,&p7,&p8

Company&c

company

employee

managesmanagedby

manages

managedby

worksfor

worksfor

worksfor


Other “typing”: data guide

• See the graph as an automata with root as the start symbol and only accepting states

• This graph accepts all the paths from the root• Obtain an equivalent, minimal, deterministic

automata – This is the data guide for the graph– It can be used for describing the data– It can be used to support Graphical Query

Interfaces


Data guide

• Gives all the paths from the root• Automata minimization


{root}

{c1}

programmer

{c2}

statistician

{p1,p2,p3,p4,p5,p6,p7,p8,p9}

project

{e1,e2,e3,e4}

employee

{p1,p3} {p2,p4} {p1,p3,p5,p7} {p4,p6} {p4}

workson leads workson leadsconsults

{e1,e2} {e2,e3}{p1,p3,p5,p7,p9}

{p2,p4,p6,p8}

workson

{p4,p9}

leadsconsultsemployee employee

root

e2 e3 e4e1

p1 p2 p3 p4 p5 p6 p7 p8 p9

"exercise" "lecture""finance""adminstr.""PR""undergrad""grad""postgrad""web"

leads


leadsworkson leads

workson consults

employee

consultsworkson

workson


project

workson

employee employee

•


What you should remember

• Tree automata = theoretical foundation for XML• Bottom-up tree automata are nice• Top-down and determinism together limitations • XML documents do not have to be typed• Typing may be very useful for XML

– In particular for software managing XML data• DTD: simple but limited• XML Schema: more expressive but still limited• Graph data: bisimulation is the answer


Merci



Bibliography

• TATA: the book, Tree Automata Techniques and Applications, tata.gforge.inria.fr/– The book on the topic and it is free

• XML schema, see http://w3.org http://www.w3schools.com/schema/

master informatique 10/9/2007 1 typing semistructured data serge abiteboul 2008 typing...

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