XML Research for Formal Language Theorists
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1 XML Research for Formal Language Theorists Wim Martens TU Dortmund Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
2 Goal of this talk XML Research vs Formal Languages Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
3 Goal of this talk XML Research vs Formal Languages XML benefits from Formal Language Theory XML schemas tree automata XPath patterns regular expressions Formal Language Theory has a nice algorithmic toolbox Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
4 Goal of this talk XML Research vs Formal Languages Warning XML benefits from Formal Language Theory XML schemas tree automata XPath patterns regular expressions Formal Language Theory has a nice algorithmic toolbox Formal Language Theory benefits from XML XML motivates interesting Formal Language problems Rather informal strongly biased survey Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
5 Outline 1 Introduction to XML 2 An FLT Approach to XML Research Document Type Definitions XML Queries Extended Document Type Definitions and XML Schema Characterizations of single-type EDTDs 3 From XML to Formal Language Theory Complexity of Regular Expressions Constructions on Regular Expressions Automata Minimization Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
6 Outline 1 Introduction to XML 2 An FLT Approach to XML Research Document Type Definitions XML Queries Extended Document Type Definitions and XML Schema Characterizations of single-type EDTDs 3 From XML to Formal Language Theory Complexity of Regular Expressions Constructions on Regular Expressions Automata Minimization Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
7 Searching the Internet Enough with these sissy keyword searches! Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
8 Searching the Internet A real search Where can I buy a flatscreen-tv, in a store at most 20km from Dresden, that is open tomorrow until 18:00? Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
9 An Example Internet Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
10 An Example Internet XML <time> 1800 </time> XML <time> 1200 </time> XML <time> 1700 </time> XML <time> 1800 </time> Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
11 An Example Internet XML <time> 1800 </time> XML <time> 1200 </time> XML <time> 1700 </time> XML <time> 1800 </time> XML Schema Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
12 A self-describing data format <store> <normal> <guitar type="electric"> <maker> Tandler </maker> <price> 3500 </price> </guitar> <guitar type="electric"> <maker> Fender </maker> <price> 1000 </price> </guitar> </normal> <discount> <guitar type="electric"> <maker> Gibson </maker> <price> 2500 </price> <discount> 10% </discount> </guitar> </discount> </store> element: <title>...</title> start tag: <title> end tag: </title> Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
13 XML as a hierarchical structure Example store normal discount guitar (type= electric ) guitar (type= electric ) guitar (type= electric ) maker price maker price maker price discount Tandler 3500 Fender 1000 Gibson % Abstraction: ordered, unranked, labeled tree (with data-values) Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
14 XML schema languages Schema A schema defines the set of allowable labels and the way they can be structured. Advantages automatic validation automatic integration of data automatic translation query optimization provides a user with a concrete semantics of the document aids in the specification of meaningful queries over XML data Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
15 XML schema languages In formal language theoretic terms A schema defines a tree language. Example DTDs (W3C) XML Schema (W3C) Relax NG (Clark, Murata) several dozen others (DSD, Schematron,... ) CFGs with REs tree automata tree automata Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
16 Summary slide What to remember? XML is an international standard for data exchange XML documents or XML data are simply ordered unranked labeled trees with data values a schema defines a tree language (no data values in this talk) Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
17 Outline 1 Introduction to XML 2 An FLT Approach to XML Research Document Type Definitions XML Queries Extended Document Type Definitions and XML Schema Characterizations of single-type EDTDs 3 From XML to Formal Language Theory Complexity of Regular Expressions Constructions on Regular Expressions Automata Minimization Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
18 Outline 1 Introduction to XML 2 An FLT Approach to XML Research Document Type Definitions XML Queries Extended Document Type Definitions and XML Schema Characterizations of single-type EDTDs 3 From XML to Formal Language Theory Complexity of Regular Expressions Constructions on Regular Expressions Automata Minimization Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
19 Document Type Definitions (DTDs) Example <!DOCTYPE store [ <!ELEMENT store <!ELEMENT normal <!ELEMENT discount <!ELEMENT guitar <!ELEMENT maker <!ELEMENT price <!ELEMENT discount ]> (normal,discount)> (guitar*)> (guitar+)> (maker,price,discount?)> (#PCDATA)> (#PCDATA)> (#PCDATA)> Corresponding grammar (start symbol store) store normal discount normal guitar discount guitar + guitar maker price discount? maker DATA price DATA discount DATA Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
20 Document Type Definitions (DTDs) XML Document store normal discount guitar guitar guitar maker price maker price maker price discount Tandler 3500 Fender 1000 Gibson % Corresponding grammar (start symbol store) store normal discount normal guitar discount guitar + guitar maker price discount? maker DATA price DATA discount DATA Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
21 Extended Context-free grammars as a formal abstraction Definition A DTD is a triple (Σ,d,s d ) where Σ is a finite alphabet s d Σ is the start symbol d : Σ RE(Σ) maps every Σ-symbol to a regular expression over Σ Definition A tree t satisfies d (is valid) iff the root of t is labeled s d for every node v labeled a the string formed by the children of v belongs to d(a). Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
22 Optimization questions: from FLT to XML Schema containment ( ) Given: Schemas d 1, d 2 Question: Is L(d 1 ) L(d 2 )? Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
23 Optimization questions: from FLT to XML Schema containment ( ) Given: Schemas d 1, d 2 Question: Is L(d 1 ) L(d 2 )? DTD containment reduces to containment of regular expressions d 1 d 2 iff d 1 (a) d 2 (a), a Σ (when d 1 and d 2 are reduced). Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
24 Optimization questions: from FLT to XML Schema containment ( ) Given: Schemas d 1, d 2 Question: Is L(d 1 ) L(d 2 )? DTD containment reduces to containment of regular expressions d 1 d 2 iff d 1 (a) d 2 (a), a Σ (when d 1 and d 2 are reduced). Theorem (Meyer, Stockmeyer, 1973) Containment of regular expressions is PSPACE-complete. Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
25 Optimization questions: from FLT to XML Schema containment ( ) Given: Schemas d 1, d 2 Question: Is L(d 1 ) L(d 2 )? DTD containment reduces to containment of regular expressions d 1 d 2 iff d 1 (a) d 2 (a), a Σ (when d 1 and d 2 are reduced). Theorem (Meyer, Stockmeyer, 1973) Containment of regular expressions is PSPACE-complete. Corollary DTD containment is PSPACE-complete. Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
26 Outline 1 Introduction to XML 2 An FLT Approach to XML Research Document Type Definitions XML Queries Extended Document Type Definitions and XML Schema Characterizations of single-type EDTDs 3 From XML to Formal Language Theory Complexity of Regular Expressions Constructions on Regular Expressions Automata Minimization Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
27 Queries for XML Conjunctive Queries over Trees XPath Tree: a Pattern: a b e c d b c d d d Pattern Matching Tree matches Pattern if there is a homomorphism h : Pattern Tree Homomorphism doesn t have to be injective Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
28 Queries for XML Conjunctive Queries over Trees XPath Tree: a Pattern: a b e c d b d c d Pattern Matching Tree matches Pattern if there is a homomorphism h : Pattern Tree Homomorphism doesn t have to be injective Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
29 Queries for XML Conjunctive Queries over Trees Conjunctive Queries over Trees Tree: a Pattern: a b e c d b c d d d Pattern Matching Tree matches Pattern if there is a homomorphism h : Pattern Tree Homomorphism doesn t have to be injective Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
30 Queries for XML Conjunctive Queries over Trees Conjunctive Queries over Trees Tree: a Pattern: a b e c d b c d d Pattern Matching Tree matches Pattern if there is a homomorphism h : Pattern Tree Homomorphism doesn t have to be injective Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
31 Queries for XML Conjunctive Queries over Trees Conjunctive Queries over Trees Tree: a Pattern: a b e c d b c d d Pattern Matching Tree matches Pattern if there is a homomorphism h : Pattern Tree Homomorphism doesn t have to be injective Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
32 Queries for XML Conjunctive Queries over Trees Conjunctive Queries over Trees Tree: a Pattern: a b e b c c d d Pattern Matching Tree matches Pattern if there is a homomorphism h : Pattern Tree Homomorphism doesn t have to be injective Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
33 Query Optimization L(Q): the set of trees that match query Q Query Containment Given two queries Q 1 and Q 2, is L(Q 1 ) L(Q 2 )? Query Containment w.r.t. a DTD Given Q 1, Q 2, and a DTD d, is L(Q 1 ) L(d) L(Q 2 )? Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
34 XPath Query Optimization Formal Language Theory to the Rescue! XPath Query a b d c d Lemma For each XPath query Q there is an Alternating Tree Automaton A s.t. L(Q) = L(A) Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
35 XPath Query Optimization Formal Language Theory to the Rescue! XPath Query a b d c d Lemma For each XPath query Q there is an Alternating Tree Automaton A s.t. Moreover, A is polynomial in Q L(Q) = L(A) Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
36 XPath Query Optimization Formal Language Theory to the Rescue! XPath Query a b d c d Lemma For each XPath query Q there is an Alternating Tree Automaton A s.t. L(Q) = L(A) Moreover, A is polynomial in Q, even if Q uses disjunction and negation Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
37 XPath Query Optimization Formal Language Theory to the Rescue! Lemma For each XPath query Q there is an Alternating Tree Automaton A s.t. L(Q) = L(A) Moreover, A is polynomial in Q, even if Q uses disjunction and negation Theorem XPath Containment is in EXPTIME XPath Containment w.r.t. DTDs is in EXPTIME Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
38 XPath Query Optimization Formal Language Theory to the Rescue! Lemma For each XPath query Q there is an Alternating Tree Automaton A s.t. L(Q) = L(A) Moreover, A is polynomial in Q, even if Q uses disjunction and negation Theorem XPath Containment (tree pattern fragment) is NP-complete [Miklau, Suciu 2002] XPath Containment (with and ) is EXPTIME-complete [Marx 2004] XPath Containment w.r.t. DTDs is EXPTIME-complete [Neven, Schwentick 2003] Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
39 Conjunctive Query Optimization Formal Language Theory to the Rescue! Conjunctive Query a b c d Lemma (Björklund, Mar., Schwentick 2008) For each Conjunctive Query Q there is an Alternating Tree Automaton A s.t. L(Q) = L(A) Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
40 Conjunctive Query Optimization Formal Language Theory to the Rescue! Conjunctive Query a b c d Lemma (Björklund, Mar., Schwentick 2008) For each Conjunctive Query Q there is an Alternating Tree Automaton A s.t. L(Q) = L(A) But, A is exponential in Q Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
41 Conjunctive Query Optimization Formal Language Theory to the Rescue! Conjunctive Query a b c d Lemma (Björklund, Mar., Schwentick 2008) For each Conjunctive Query Q there is an Alternating Tree Automaton A s.t. L(Q) = L(A) But, A is exponential in Q and this is optimal Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
42 Conjunctive Query Optimization Formal Language Theory to the Rescue! Lemma (Björklund, Mar., Schwentick 2008) For each Conjunctive Query Q there is an Alternating Tree Automaton A s.t. L(Q) = L(A) But, A is exponential in Q and this is optimal Theorem CQ Containment w.r.t. DTDs is 2EXPTIME-complete [Björklund, Mar., Schwentick 2008] Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
43 Conjunctive Query Optimization Formal Language Theory to the Rescue! Lemma (Björklund, Mar., Schwentick 2008) For each Conjunctive Query Q there is an Alternating Tree Automaton A s.t. L(Q) = L(A) But, A is exponential in Q and this is optimal Theorem CQ Containment is Π P 2 -complete [Björklund, Mar., Schwentick 2007] CQ Containment w.r.t. DTDs is 2EXPTIME-complete [Björklund, Mar., Schwentick 2008] Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
44 Outline 1 Introduction to XML 2 An FLT Approach to XML Research Document Type Definitions XML Queries Extended Document Type Definitions and XML Schema Characterizations of single-type EDTDs 3 From XML to Formal Language Theory Complexity of Regular Expressions Constructions on Regular Expressions Automata Minimization Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
45 Extended DTDs Grammar based approach to unranked regular tree languages tree t guitar store guitar guitar maker price maker price maker price discount Tandler 3500 Fender 1000 Gibson % Example store (guitar 1 ) (guitar 2 ) + guitar 1 maker price guitar 2 maker price discount Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
46 Extended DTDs Grammar based approach to unranked regular tree languages Typed tree t store guitar guitar guitar maker price maker price maker price discount Tandler 3500 Fender 1000 Gibson % Example store (guitar 1 ) (guitar 2 ) + guitar 1 maker price guitar 2 maker price discount Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
47 Extended DTDs Grammar based approach to unranked regular tree languages Definition (Papakonstantinou, Vianu, 2000) Let Σ N := {σ n σ Σ,n N} be the alphabet of types. An extended DTD (EDTD) is a tuple D = (Σ,d,s d ), where (d,s d ) is a (finite) DTD over Σ Σ N. A tree t is valid w.r.t. D if there is an assignment of types such that the typed tree is a derivation tree of d. Example store (guitar 1 ) (guitar 2 ) + guitar 1 maker price guitar 2 maker price discount Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
48 EDTDs versus Tree Automata Theorem (Papakonstantinou, Vianu, 2000, BMW) Non-deterministic (unranked) tree automata and EDTDs define precisely the class of (homogeneous) regular unranked tree languages. Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
49 EDTDs versus Tree Automata Theorem (Papakonstantinou, Vianu, 2000, BMW) Non-deterministic (unranked) tree automata and EDTDs define precisely the class of (homogeneous) regular unranked tree languages. Example EDTD store (guitar 1 ) (guitar 2 ) + guitar 1 maker price guitar 2 maker price discount NTA δ(store,store) = (guitar 1 ) (guitar 2 ) + δ(guitar 1,guitar) = maker price δ(guitar 2,guitar) = maker price discount Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
50 Does XML Schema correspond to EDTDs? <xs:element name="store"> <xs:complextype> <xs:sequence> <xs:element name="guitar" type="1" minoccurs="0" maxoccurs="unbounded"/> <xs:element name="guitar" type="2" minoccurs="1" maxoccurs="unbounded"/> </xs:sequence> </xs:complextype> </xs:element> Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
51 Does XML Schema correspond to EDTDs? <xs:element name="store"> <xs:complextype> <xs:sequence> <xs:element name="guitar" type="1" minoccurs="0" Rejected by XML Schema validator maxoccurs="unbounded"/> Violates<xs:element the Element Declarations name="guitar" Consistent type="2" Constraint. minoccurs="1" maxoccurs="unbounded"/> </xs:sequence> </xs:complextype> </xs:element> Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
52 A formalization of XML Schema: single-type EDTDs XML Schema 1: Element Declarations Consistent constraint (Section 3.8.6) It is illegal to have two elements of the same name [... ] but different types in a content model [... ]. Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
53 A formalization of XML Schema: single-type EDTDs XML Schema 1: Element Declarations Consistent constraint (Section 3.8.6) It is illegal to have two elements of the same name [... ] but different types in a content model [... ]. Definition (Murata, Lee, Mani, 2001) A single-type EDTD is an EDTD for which in no regular expression two types b i and b j with i j occur. Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
54 A formalization of XML Schema: single-type EDTDs XML Schema 1: Element Declarations Consistent constraint (Section 3.8.6) It is illegal to have two elements of the same name [... ] but different types in a content model [... ]. Definition (Murata, Lee, Mani, 2001) A single-type EDTD is an EDTD for which in no regular expression two types b i and b j with i j occur. Not single-type store (guitar 1 ) (guitar 2 ) + guitar 1 maker price guitar 2 maker price discount Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
55 A formalization of XML Schema: single-type EDTDs Definition (Murata, Lee, Mani, 2001) A single-type EDTD is an EDTD in which in no regular expression two types b i and b j with i j occur. Example store normal discount normal (guitar 1 ) discount (guitar 2 ) + guitar 1 maker price guitar 2 maker price discount Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
56 A formalization of XML Schema: single-type EDTDs Formal abstraction XML Schema single-type EDTDs Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
57 A formalization of XML Schema: single-type EDTDs Formal abstraction XML Schema single-type EDTDs Immediate Questions What kind of languages can be defined by single-type EDTDs? Is it decidable whether an EDTD rewritten to an equivalent single-type EDTD? smart XML Schema validator Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
58 Outline 1 Introduction to XML 2 An FLT Approach to XML Research Document Type Definitions XML Queries Extended Document Type Definitions and XML Schema Characterizations of single-type EDTDs 3 From XML to Formal Language Theory Complexity of Regular Expressions Constructions on Regular Expressions Automata Minimization Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
59 Properties of single-type EDTDs Three properties 1 Single-type EDTDs admit unique top-down typing 2 Closure under a certain form of subtree exchange 3 Characterization as a pattern-based language Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
60 (1) Single-type EDTDs: simple top-down typing store normal discount guitar guitar guitar maker price maker price maker price discount Tandler 3500 Fender 1000 Gibson % Example store normal discount normal (guitar 1 ) discount (guitar 2 ) + guitar 1 maker price guitar 2 maker price discount Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
61 (1) Single-type EDTDs: simple top-down typing store guitar normal guitar 1 1 discount guitar maker price maker price maker price discount Tandler 3500 Fender 1000 Gibson % Example store normal discount normal (guitar 1 ) discount (guitar 2 ) + guitar 1 maker price guitar 2 maker price discount Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
62 (1) Single-type EDTDs: simple top-down typing store normal discount guitar guitar guitar maker price maker price maker price discount Tandler 3500 Fender 1000 Gibson % Example store normal discount normal (guitar 1 ) discount (guitar 2 ) + guitar 1 maker price guitar 2 maker price discount Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
63 (1) Single-type EDTDs: simple top-down typing Algorithm to validate and type a tree (Murata et al., 2001) Given: tree t and single-type EDTD D = (Σ,d,a 0 ) 1 Check if root of t is labeled with a, assign type a 0 2 for every interior node u with type b i, test whether the children of u match µ(d(b i )). If so, assign unique type to every child. Else fail. µ(a 1 + b 1 c 2 ) = a + bc Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
64 (1) Single-type EDTDs: simple top-down typing Algorithm to validate and type a tree (Murata et al., 2001) Given: tree t and single-type EDTD D = (Σ,d,a 0 ) 1 Check if root of t is labeled with a, assign type a 0 2 for every interior node u with type b i, test whether the children of u match µ(d(b i )). If so, assign unique type to every child. Else fail. µ(a 1 + b 1 c 2 ) = a + bc Corollary Single-typedness implies unique top-down typing. Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
65 (2) An exchange property of single-type EDTDs The Ancestor-String a Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
66 (2) An exchange property for single-type EDTDs Ancestor-Guarded Subtree Exchange T is a regular tree language T T T T Theorem (Mar., Neven, Schwentick 2005) A regular tree language is definable by a single-type EDTD iff it is closed under ancestor-guarded subtree exchange. Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
67 (2) Tool for proving inexpressibility At least one discount guitar is not single-type store store guitar guitar guitar guitar maker price maker price discount maker price discount maker price Tandler 3500 Gibson % Fender % Gibson 2500 Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
68 (2) Tool for proving inexpressibility At least one discount guitar is not single-type store store guitar guitar guitar guitar maker price maker price discount maker price discount maker price Tandler 3500 Gibson % Fender % Gibson 2500 store guitar guitar maker price maker price Tandler 3500 Gibson 2500 Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
69 (2) Tool for proving inexpressibility At least one discount guitar is not single-type store store guitar guitar guitar guitar maker price maker price discount maker price discount maker price Tandler 3500 Gibson % Fender % Gibson 2500 store guitar guitar maker price maker price Tandler 3500 Gibson 2500 Single-type EDTDs are not closed under union or complement. Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
70 (3) Pattern-based Language Making dependencies explicit Definition An ancestor-based DTD A is a set of rules r s where r and s are regular expressions over Σ. L(r) a L(s) Definition A tree t is valid w.r.t. A iff for every vertex v there is some r s such that v s ancestor string matches r and the children of v match s. Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
71 (3) Pattern-based Language Making dependencies explicit single-type EDTD Ancestor-guarded DTD store normal discount normal (guitar 1 ) discount (guitar 2 ) + guitar 1 maker price guitar 2 maker price discount store normal discount normal guitar discount guitar + normal guitar maker price discount guitar maker price discount Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
72 Smart XML Schema validator Theorem (Mar., Neven, Schwentick, 2005) Deciding whether an EDTD is equivalent to a single-type EDTD or a DTD is EXPTIME-complete. Upper bound Compute single-type closure D of given EDTD D: E.g, a 1 b 1 b 2, b 1 c 1, b 2 c 2 becomes a {1} b {1,2} b {1,2} b {1,2} c {1} + c {2} L(D ) = L(D) iff L(D) is single-type. We know that L(D) L(D ). So, only need to test L(D ) L(D): D D = /0. Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
73 Smart XML Schema validator Theorem (Mar., Neven, Schwentick, 2005) Deciding whether an EDTD is equivalent to a single-type EDTD or a DTD is EXPTIME-complete. Upper bound Compute single-type closure D of given EDTD D: E.g, a 1 b 1 b 2, b 1 c 1, b 2 c 2 becomes a {1} b {1,2} b {1,2} b {1,2} c {1,2} + c {1,2} L(D ) = L(D) iff L(D) is single-type. We know that L(D) L(D ). So, only need to test L(D ) L(D): D D = /0. Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
74 Summary slide What to remember? XML Schema single-type EDTDs regular tree languages single-type EDTDs admit top-down unique typing XML Schema can be simply characterized without using types Relax NG corresponds to unranked regular tree languages (EDTDs) Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
75 Outline 1 Introduction to XML 2 An FLT Approach to XML Research Document Type Definitions XML Queries Extended Document Type Definitions and XML Schema Characterizations of single-type EDTDs 3 From XML to Formal Language Theory Complexity of Regular Expressions Constructions on Regular Expressions Automata Minimization Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
76 Outline 1 Introduction to XML 2 An FLT Approach to XML Research Document Type Definitions XML Queries Extended Document Type Definitions and XML Schema Characterizations of single-type EDTDs 3 From XML to Formal Language Theory Complexity of Regular Expressions Constructions on Regular Expressions Automata Minimization Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
77 Complexity of basic decision problems Theorem (Mar., Neven, Schwentick 2004) Let R be a class of regular expressions and C a complexity class. Then the following are equivalent: CONTAINMENT for R is in C ; CONTAINMENT for DTD(R) is in C ; CONTAINMENT for single-type EDTD(R) is in C ; Theorem (Seidl 1990, 1994) CONTAINMENT and EQUIVALENCE are EXPTIME-complete for EDTDs (even with deterministic REs). Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
78 Complexity of basic decision problems INTERSECTION: Given a number of schemas S 1,...,S n, decide if ni=1 L(S i ) /0. Theorem (Mar., Neven, Schwentick 2004) Let R be a class of regular expressions and C a complexity class. Then the following are equivalent: INTERSECTION for R is in C ; INTERSECTION for DTD(R) is in C. Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
79 Complexity of basic decision problems INTERSECTION: Given a number of schemas S 1,...,S n, decide if ni=1 L(S i ) /0. Theorem (Mar., Neven, Schwentick 2004) Let R be a class of regular expressions and C a complexity class. Then the following are equivalent: INTERSECTION for R is in C ; INTERSECTION for DTD(R) is in C. Theorem (Mar., Neven, Schwentick 2004) There is a class of regular expressions X such that INTERSECTION for X is NP-complete; INTERSECTION for single-type EDTD(X ) is EXPTIME-complete. Remark: INTERSECTION for deterministic REs is PSPACE-complete. Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
80 Focus on Regular Expressions What to remember? Decision problems for XML Schema translate to decision problems for regular expressions. Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
81 Focus on Regular Expressions What to remember? Decision problems for XML Schema translate to decision problems for regular expressions. What regular expression classes are interesting? Regular expressions that occur in schemas! A base symbol is a regular expression w, w?, or w where w is a non-empty string; A factor is of the form e, e?, e +, or e where e is a disjunction of base symbols. A CHAin Regular Expression (CHARE) is ε, /0, or a sequence f 1 f k of factors. [Bex,Neven,Van den Bussche 2004]: > 90% of expressions in practical DTDs or XSDs are CHAREs Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
82 Regular Expression Analysis Revisited Fragment CONTAINMENT EQUIVALENCE INTERSECTION a,a + in PTIME (DFA!) in PTIME in PTIME a,a conp in PTIME NP a, a? conp in PTIME NP a,(+a) PSPACE in PSPACE NP all {(+w),(+w) + } PSPACE in PSPACE NP a,(+w) PSPACE in PSPACE PSPACE [Bala 2002] RE PSPACE PSPACE PSPACE Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
83 Regular Expression Analysis Revisited Fragment CONTAINMENT EQUIVALENCE INTERSECTION a,a + in PTIME (DFA!) in PTIME in PTIME a,a conp in PTIME NP a, a? conp in PTIME NP a,(+a) PSPACE in PSPACE NP all {(+w),(+w) + } PSPACE in PSPACE NP a,(+w) PSPACE in PSPACE PSPACE [Bala 2002] RE PSPACE PSPACE PSPACE Observation Not many PTIME results... Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
84 What Regular Expressions are Allowed in Schemas? Counting and shuffle Numerical occurrence operator (#): (a [4,5] (b + c ) 7 ) shuffle operator (a&b = {ab, ba}) Theorem (Mayer, Stockmeyer 1994) CONTAINMENT and EQUIVALENCE for RE(&) is EXPSPACE-complete Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
85 What Regular Expressions are Allowed in Schemas? Counting and shuffle Numerical occurrence operator (#): (a [4,5] (b + c ) 7 ) shuffle operator (a&b = {ab, ba}) Theorem (Mayer, Stockmeyer 1994) CONTAINMENT and EQUIVALENCE for RE(&) is EXPSPACE-complete Theorem (Gelade, Mar., Neven 2007) CONTAINMENT and EQUIVALENCE is EXPSPACE-complete for RE(#) and RE(#,&) Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
86 On the Search for more PTIME fragments Theorem (Ghelli, Colazzo, Sartiani 2007) CONTAINMENT is in PTIME for conflict-free regular expressions Conflict-free counting and interleaving allowed! Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
87 On the Search for more PTIME fragments Theorem (Ghelli, Colazzo, Sartiani 2007) CONTAINMENT is in PTIME for conflict-free regular expressions Conflict-free counting and interleaving allowed! single occurrence Kleene star only applied to disjunctions single symbols Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
88 Outline 1 Introduction to XML 2 An FLT Approach to XML Research Document Type Definitions XML Queries Extended Document Type Definitions and XML Schema Characterizations of single-type EDTDs 3 From XML to Formal Language Theory Complexity of Regular Expressions Constructions on Regular Expressions Automata Minimization Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
89 Complementing schemas Schema Complementation I have a schema S which I update to S What are the documents I admitted in S, but not in S anymore? This should be L(S) L(S ) = L(S) L(S ) Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
90 Complementing regular expressions Given a regular expression r, define a regexp for L(r). Naive approach: transform to an NFA, determinize, complement, and transform again to a regular expression (2EXPTIME) Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
91 Complementing regular expressions Given a regular expression r, define a regexp for L(r). Naive approach: transform to an NFA, determinize, complement, and transform again to a regular expression (2EXPTIME) Lemma [Gelade and Neven 2008] For every n, there is a regular expression r of size O(n), such that any regular expression defining L(r) must be of size Ω(2 2n ) Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
92 Complementing regular expressions Given a regular expression r, define a regexp for L(r). Naive approach: transform to an NFA, determinize, complement, and transform again to a regular expression (2EXPTIME) Lemma [Gelade and Neven 2008] For every n, there is a regular expression r of size O(n), such that any regular expression defining L(r) must be of size Ω(2 2n ) Idea Ehrenfeucht, Zeiger (1974): There is a class of DFAs K n whose smallest equivalent regular expression is at least 2 n. (States = {1,...,n}, edges between i and j labeled with a i,j ) Generalize this theorem to four-letter alphabets Construct r of size O(n) for K 2 n Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
93 Outline 1 Introduction to XML 2 An FLT Approach to XML Research Document Type Definitions XML Queries Extended Document Type Definitions and XML Schema Characterizations of single-type EDTDs 3 From XML to Formal Language Theory Complexity of Regular Expressions Constructions on Regular Expressions Automata Minimization Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
94 Schema Minimization Schema Minimization Given a schema D, compute the smallest equivalent schema D Why relevant? Recall: Query Optimization Input: Queries Q 1, Q 2, and a schema D Smaller schema improves the run-time of the query optimization problems! Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
95 Schema Minimization Minimization is typically studied on automata models Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
96 Schema Minimization Minimization is typically studied on automata models and the results look prettier on deterministic automata Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
97 Schema Minimization Minimization is typically studied on automata models and the results look prettier on deterministic automata Question What s the deterministic automata model for XML? single-type EDTDs with DFAs? deterministic unranked tree automata? Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
98 Schema Minimization Minimization is typically studied on automata models and the results look prettier on deterministic automata Question What s the deterministic automata model for XML? single-type EDTDs with DFAs? deterministic unranked tree automata? top-down det. bottom-up det. Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
99 Single-type EDTD Minimization Theorem (Mar., Niehren 2005) Single-type EDTD with DFA Minimization is in PTIME Minimal models are unique Minimization Algorithm Reduce the input single-type EDTD For every pair of states q 1, q 2, decide equivalence If equivalent, merge q 1 and q 2 In the resulting EDTD, minimize each DFA Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
100 Unranked Tree Automaton Minimization (Brüggemann-Klein, Murata, Wood 2001) A bottom-up unranked tree automaton is deterministic if for every pair of rules a(l 1 ) q 1 and a(l 2 ) q 2, L 1 L 2 = /0 Additional requirement: L 1, L 2 represented by DFAs Theorem (Mar., Niehren 2005) MINIMIZATION is NP-complete for deterministic unranked tree automata Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
101 Unranked Tree Automaton Minimization For the right definition of bottom-up determinism: Theorem (Mar., Niehren 2005) MINIMIZATION is in P for bottom-up deterministic tree automata the Myhill-Nerode theorem for unranked tree languages holds Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
102 Myhill-Nerode for Unranked Tree Automata For tree language L, define relation L on trees Definition t 1 L t 2 if E : E L iff E L t 1 t 2 L is an equivalence relation on unranked trees Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
103 Myhill-Nerode for Unranked Tree Automata Theorem (Myhill-Nerode for Unranked Trees (Mar., Niehren 2005)) Let L be an unranked tree language. The following are equivalent: L is regular L has finitely many equivalence classes Moreover, the equivalence classes of L correspond to states of minimal (new) bottom-up deterministic unranked TA for L Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
104 Back to the Basics NFA Minimization Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
105 Back to the Basics NFA Minimization Question How much non-determinism can be admitted for PTIME minimization? Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
106 Back to the Basics NFA Minimization Question How much non-determinism can be admitted for PTIME minimization? Theorem (Jiang, Ravikumar 1993) DFA unambiguous FA MINIMIZATION is NP-complete Theorem (Malcher 2003) MINIMIZATION is NP-complete for NFAs with fixed branching ( 3) NFAs with at least two start states Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
107 Back to the Basics NFA Minimization Question How much non-determinism can be admitted for PTIME minimization? Theorem (Jiang, Ravikumar 1993) DFA unambiguous FA MINIMIZATION is NP-complete Theorem (Malcher 2003) MINIMIZATION is NP-complete for NFAs with fixed branching ( 3) NFAs with at least two start states Question Revisited Can there be any non-determinism at all? Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
108 Back to the Basics Finite State Automata Minimization Definition (δ NFA) The class of NFAs that have at most one pair (q,a) such that (q,a) q 1 and (q,a) q 2 are unambiguous do not loop Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
109 Back to the Basics Finite State Automata Minimization Definition (δ NFA) The class of NFAs that have at most one pair (q,a) such that (q,a) q 1 and (q,a) q 2 are unambiguous do not loop Theorem (Björklund, Mar., ICALP 2008) For every class C of NFAs such that δnfa C : DFA C MINIMIZATION is NP-hard Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
110 Conclusion and Outlook XML and Formal Languages are great for cross-fertilization Many problems in XML research are solved through FLT techniques XML research poses interesting questions for FLT Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
111 Conclusion and Outlook XML and Formal Languages are great for cross-fertilization So,... Many problems in XML research are solved through FLT techniques XML research poses interesting questions for FLT if you like formal language theory, but also want a PODS/ICDT paper have a look at XML if you like formal language theory, and you want more formal language theory have a look at XML Wim Martens (TU Dortmund) XML for Formal Language Theorists May 14, / 65
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