On Verifying Consistency of XML Specifications

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On Verifying Consistency of XML Specifications

• Wenfei Fan
• Internet Management Research Dept., Bell Labs
• Dept. of CIS, Temple University

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Overview

• XML Specifications
• types DTDs (Document Type Definitions)
• integrity constraints keys and foreign keys
• Interaction between DTDs and constraints
• Consistency analysis of XML specifications
• absolute constraints
• relative constraints
• regular constrains
• Implication analysis of XML constraints
• Joint work with L. Libkin and M. Arenas, Univ. of
  Toronto PODS01, PODS02, JACM

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• 1. XML specifications introduction

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XML data - an example

• Rooted, node-labeled tree
• elements db, province, capital, city,
  subtrees/sub-document
• subelements, e.g., the capital child of province
• _at_attributes _at_name, _at_inProvince, carrying text
• text nodes, e.g., Hasselt

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XML specifications with DTDs

• Production constrains the subelement list of
  each element lt!ELEMENT db (province,
  capital)gt
• lt!ELEMENT province (city, capital)gt
• Attributes uniquely identified by name for each
  element, unordered
• province _at_name, capital _at_inProvince

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Document Type Definition a formalism

• DTD D (E, A, P, R, r)
• E a set of element types, e.g., db, province,
  capital, city
• A a set of attributes, e.g., _at_name,
  _at_inProvince
• P element type definitions in terms of regular
  expressions, e.g., db ? province, capital
• R attribute definitions,
• e.g., province._at_name, capital._at_inProvince
• r the element type of the root, e.g., db.
• ECFG nonterminals (E, A), productions (P, R),
  start symbol (r)

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XML specifications with constraints

• Keys and foreign keys (vs. relational
  constraints)
• key the value of a _at_name uniquely identifies a
  province
• province._at_name ? province
• capital._at_inProvince ? capital
• FK _at_inProvince of a capital references _at_name of
  a province
• capital._at_inProvince ? province._at_name

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Why keys and foreign keys?

• Supported by the XML standard, XML Schema, XML
  Data
• In databases (supported by SQL standard)
• essential part of the semantics of data,
• fundamental to conceptual design,
• useful for choosing efficient storage and access
  methods,
• central to update anomaly prevention,
• In the XML setting have proved useful in
• database storage of XML data (query and update),
• database publishing in XML,
• data integration,
• XML query optimization and formulation,
• design theory for XML specifications,

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XML specification

• A DTD D
• A set of keys and foreign keys, ?
• Example
• DTD D structure of the document
• lt!ELEMENT db (province, capital)gt
• lt!ELEMENT province (city, capital)gt
• province._at_name, capital._at_inProvince
• Constraints ? fundamental semantics of the data
• province._at_name ? province
• capital._at_inProvince ? capital
• capital._at_inProvince ? province._at_name

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• 2. Interaction between DTDs and constraints

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Consistency of XML specifications

• Given D a DTD
• ? a set of keys and foreign keys
  over D
• Consistency Is there an XML document that both
  conforms to D and satisfies ??
• One wants to know whether XML specifications make
  sense!
• Run-time check attempts to validate documents
  with (D, ?).
• This would not tell us whether repeated failures
  are due to a bad specification or problems with
  the documents
• ? static analysis is a better approach

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An inconsistent specification

• The specification with D and ? is inconsistent!
• DTD D
• lt!ELEMENT db (province, capital)gt
• lt!ELEMENT province (city, capital)gt
• province._at_name, capital._at_inProvince
• Constraints ?
• province._at_name ? province
• capital._at_inProvince ? capital
• capital._at_inProvince ? province._at_name
• In contrast, one can specify keys and foreign
  keys in SQL without worrying about their
  consistency with schema.

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Cardinality constraints by keys, foreign keys

• Constraints ?
• province._at_name ? province
• capital._at_inProvince ? capital
• capital._at_inProvince ? province._at_name
• Notation
• ext(?) the set of ? elements in an XML document
• ext(?.l) the set of l attribute values of all ?
  elements
• ?
• ext(province._at_name)
  ext(province)
• ext(capital._at_inProvince) ext(capital)
• ext(capital._at_inProvince) ?
  ext(province._at_name)
• ?
• ext(capital) ? ext(province)

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Cardinality constraints imposed by DTDs

• DTD D lt!ELEMENT db (province, capital)gt
• lt!ELEMENT province (city,
  capital)gt
• Variables
• Xprovince the number of province elements under
  the root
• Xcapital the number of capital subelements of
  the root
• Ycapital the number of capital subelements of
  provinces
• ?
• Xprovince ? 1, Xcapital ? 1
• ext(province) Xprovince,
  Xprovince Ycapital
• ext(capital) Xcapital Ycapital
• ?
• ext(capital) gt ext(province)

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The interaction

• Contradiction
• From the constraints ? ext(capital) ?
  ext(province)
• From the DTD D ext(capital) gt
  ext(province)
• Thus there exists NO XML document that both
  conforms to D and satisfies ?.

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• 3. Consistency analysis of XML specifications

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The consistency problem

• Given D a DTD
• ? a set of keys and foreign keys
  expressed in C
• Consistency (C ) Is there an XML document that
  both conforms to D and satisfies ??
• C a constraint language, ranges over
• absolute constraints
• relative constraints
• regular constraints
• These constraint languages are important for
  hierarchically structured data, including but not
  limited to XML.

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Absolute keys and foreign keys

• key ??X ? ?. A document satisfies the key
  iff
• ? x y ? ext(?) (?l ?X (x.l y.l) ? x y)
  
• foreign key (FK) a combination of an inclusion
  constraint ??1X ?? ??2Y, and a key ?
  ?2Y ? ? ??2 .
• A document satisfies the FK iff it satisfies the
  key and
• ? x ? ext(??1 ) ? y ? ext(??2 ) (xX yY)
• where ??, ?1 ,??2 element types X, Y sets
  (lists) of attributes
• ext(?) the set of ? elements in an XML document.
• Equality issue
• value equality when comparing attributes
• node identify when comparing XML elements

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More on absolute constraints

• Absolute constraints are to hold on the entire
  document.
• Unary constraints keys and foreign keys defined
  in terms of single-attribute.
• Example of unary constraints
• province._at_name ? province
• capital._at_inProvince ? capital
• capital._at_inProvince ? province._at_name

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Consistency analysis

• Trivial for relational databases given any
  schema and keys, foreign keys, one can always
  find a nonempty instance of the schema satisfying
  the constraints.
• Hard for XML XML specifications may not be
  consistent!
• Both DTDs and constraints impose cardinality
  constraints
• The interaction between these two classes of
  cardinality constraints is rather complicated.

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Consistency analysis of absolute constraints

• Theorem The consistency problem is undecidable
  for multi-attribute keys and foreign keys.
• Theorem It becomes NP-complete for unary
  constraints.
• Primary key restriction at most one key for each
  element type.
• Theorem It remains intractable for unary
  constraints under the primary key restriction.
• Theorem For primary multi-attribute keys and
  unary foreign keys, the consistency problem is
  decidable in NEXPTIME.
• As opposed to the trivial analysis of the
  relational counterpart.

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Proof ideas

• Multi-attribute constraints reduction from the
  implication problem for functional and inclusion
  dependencies in RDBs.
• Unary keys and foreign keys
• a nontrivial encoding of DTDs and unary
  constraints in terms of linear integer
  constraints (O(n2 log n)-time)
• polynomially equivalent to LIP, linear integer
  programming
• Multi-attribute primary keys and unary foreign
  keys
• polynomially equivalent to Prequadratic
  Diophantine Problem (PDE) satisfiability of
  linear integer constraints and prequadratic
  constraints of the form x ? y z
• the precise complexity of PDE, a restriction to
  the Hilberts 10th problem, is open --
  nontrivial.

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Introduction to relative constraints

• An XML tree specifies countries, provinces,
  province capitals.
• What is a key for a province?
• What does _at_inProvince of a capital reference?

db
...
country
country
...
...
province
capital
capital
province
_at_name
_at_name
Holland
Belgium
capital
_at_name
_at_name
capital
_at_inProvince
Hasselt
_at_inProvince
Maastricht
Limburg
Limburg
Limburg
Limburg
_at_inProvince
Hasselt
_at_inProvince
Hasselt
Limburg
Limburg
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Examples of relative constraints

• Relative constraints on a subdocument rooted at
  a country
• key country (province._at_name ?
  province)
• country (capital._at_inProvince ? capital)
• FK country (capital._at_inProvince ?
  province._at_name)
• Absolute on the entire document country._at_name
  ? country

db
...
country
country
...
...
province
capital
capital
province
_at_name
_at_name
Belgium
Holland
capital
_at_name
Hasselt
capital
_at_name
_at_inProvince
_at_inProvince
Maastricht
Limburg
Limburg
Limburg
Limburg
_at_inProvince
Hasselt
_at_inProvince
Hasselt
Limburg
Limburg
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Relative keys and foreign keys

• key ??(??1X ? ??1). A document satisfies the
  key iff
• ? c ? ext(?) ? y, z ? ext(?1)
• ( (y ?? c) ? (z ?? c) ? ?l ?X (y.l z.l) ?
  y z)
• foreign key (FK) ??( ?1X ?? ??2Y ) and a key
  ?( ?2Y ? ??2) .
• A document satisfies the FK iff it satisfies the
  key and
• ? c ? ext(?) ? y ? ext(?1) (( y ?? c) ?
• ? z ? ext(??2 ) ((z ?? c) ? yX zY
  ))
• where ?
• (y ?? c) y is a descendant of c (y in the
  subtree rooted at c)
• ? context type
• ext(?) the set of ? elements in an XML document.

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Relative vs. Absolute

• Absolute constraints are a special case of
  relative ones
• country._at_name ? country ?
• db ( country._at_name ? country )
• absolute a fixed context type r
• Absolute constraints are scoped within the entire
  document whereas relative ones within the
  context of a subdocument.
• country (province._at_name ? province)
• country (capital._at_inProvince ? capital)
• country (capital._at_inProvince ?
  province._at_name)
• country._at_name ? country
• Together they specify constraints on the entire
  document
• Important for hierarchically structured data
  XML, scientific databases, biomedical data, ...

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Consistency analysis of relative constraints

• Theorem The consistency problem is undecidable
  for relative keys and foreign keys, even when all
  the constraints are unary and are under the
  primary key restriction.
• As opposed to the NP complexity of its absolute
  counterpart.
• Proof ideas reduction from the Hilberts 10th
  problem.
• Diophantine equation problem
• P1 (x1, , xk) Q1 (x1, , xk) c1
• . . .
• Pn (x1, , xk) Qn (x1, , xk) cn

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Introduction to regular constraints

• XML data is hierarchically structured
• define _at_eid as a key of employees of companies
  and schools
• define _at_taughtBy as a foreign key of students
  referencing _at_eid of school employees.

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Examples of regular constraints

• Key (university._ company._).employee._at_eid
  ?
• (university._ company._).employee
• FK _.student._at_taughtBy ? university._.employe
  e._at_eid
• _ wildcard that matches any label
• _ the Kleene closure of _

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Regular path expression

• Vertical regular expressions
• ? ? ? _ ?.? ??
  ?
• ? empty word ? element type _
  wildcard
• ., , concatenation, disjunction, Kleene
  star
• Example (university._ company._).employee
• university._.employee
• nodes(?. ?) the set of ? elements in an XML
  document that are reachable from the root by
  following ?

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Regular expression constraints

• key ? ?.?X ? ? ?.?. A document satisfies
  the key iff
• ? x y ? nodes( ?.? ) (?l ?X (x.l y.l)
  ? x y)
• foreign key ? ?1.?1X ?? ?2.?2Y, and a key
  ??2.?2Y ? ??2.?2
• A document satisfies the FK iff it satisfies
  the key and
• ? x ? nodes(? ?1.?1 ) ? y ? nodes(? ?2.?2 )
  (xX yY)
• where nodes(?.?) the set of ? elements reachable
  from the root by following ?.

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Regular an extension of absolute constraints

• Example
• Key (university._ company._).employee._at_eid
  ?
• (university._ company._).employee
• FK _.student._at_taughtBy ? university._.employe
  e._at_eid
• Observation nodes( _. ? ) ext(?)
• Recall absolute constraints
• key ??X ? ? ? ? ? _. ? X ? ? _. ?
• foreign key ??1X ?? ??2Y, ??2Y ? ? ??2
  ?
• ? _. ?1 X ?? _.??2 Y, ?
  _. ?2 Y ? ?_.??2

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Consistency analysis of regular constraints

• Corollary The consistency problem is undecidable
  for multi-attribute regular keys and foreign
  keys.
• Theorem It is decidable in NEXPTIME and is
  PSPACE-hard for unary regular constraints.
• NEXPTIME an involved encoding in terms of LIP
• regular expressions in a DTD interact with
  (vertical) regular path expressions reduce DTD
  to a simple normal form
• regular path expressions interact with each
  other introduce exponentially many variables for
  all boolean combinations
• encoding reachability (nodes(?.?)) of a path
  expression tag variables with states of finite
  state automata

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Some tractable cases

• Restrictions on constraints unary, primary.
• Theorem For multi-attribute relative keys only,
  the consistency problem is in linear time for
  arbitrary DTDs.
• Recall relative keys country (province._at_name
  ? province)
• Restrictions on DTDs
• Theorem When DTD is fixed, the consistency
  problem is in PTIME for absolute unary
  constraints.
• In practice, DTD is designed at one time, but
  constraints are written in stages constraints
  are incrementally added.

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Other restricted cases

• Theorem In the absence of recursion and Kleene
  star in the DTD involved, the consistency problem
  remains
• undecidable for multi-attribute absolute
  constraints,
• intractable for unary absolute constraints,
• PSPACE-hard for unary regular constraints.
• Recall absolute ? regular unary
  single-attribute
• Other severely restricted cases
• nonrecursive DTD of a bounded depth, a set of
  absolute unary constraints of a bounded size
  NLOGSPACE
• nonrecursive DTD, unary relative constraints that
  can be partitioned into sets local to each
  other with respect to the DTD (without
  interaction) PSPACE-complete

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• 4. Implication analysis of XML constraints

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Implication of XML constraints

• Given D a DTD
• ? a set of keys and foreign keys
  expressed in C
• ? a property (a key or foreign key of C)
• Implication (C ) Is it the case that for any
  XML document, if it conforms to D and satisfies
  ?, then it must satisfy ??
• C a constraint language
• The need for studying implication
• data integration constraints checking at virtual
  views
• optimization of XML queries and XML relational
  storage
• design theory for XML specifications
  normalization

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Some complexity results for implication

• Proposition For any class of XML constraints, if
  its consistency problem is K-hard, then its
  implication problem is coK-hard, where K is some
  complexity class that contains DLOGSPACE.
• Corollary The implication problem is
• undecidable for multi-attribute absolute
  constraints and for unary relative constraints
• PSPACE-hard for unary regular constraints
• coNP-hard for unary absolute constraints.
• Recall relative country (province._at_name ?
  province)
• regular _.student._at_taughtBy ?
  _.professor._at_id
• absolute country._at_name ? country

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Upper bounds

• Theorem The implication problem is
  coNP-complete for unary absolute constraints.
• Proof idea a nontrivial encoding in terms of
  LIP and the Set Intersection Pattern Problem
• Theorem The implication problem is decidable in
• linear time for absolute multi-attribute keys,
  and
• in PTIME for arbitrary absolute constraints when
  the DTD is fixed.

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Summary

• XML specification DTD and constraints (keys,
  foreign keys)
• Consistency and implication analysis of XML
  constraints
• DTDs interact with XML constraints
• The analysis is far more intricate than its
  relational counterpart
• The negative results carry over to XML Schema,
  XML Data and other more expressive specifications