Satisfiability of XPath Expressions

1
Satisfiability of XPath Expressions

• Jan Hidders

2
Motivation

• Satisfiability
• Result of XPath expression not always empty?
• Related to
• Avoiding duplicate elimination and ordering
• Subsumption problem
• Examples
• selfa/selfb
• childa/child/parentb
• /child/parent/parent
• /preceding
• /following

3
Problem Definition

• Data Model
• Alphabet of tag names S
• XML tree ordered node-labeled tree T
• Document order preorder tree walk

XPath Expressions (subset XPath 2.0) P
A S A P/P / PP P ? P P ? P P
P. A self parent child anc-or-self
desc-or-self foll-sibl prec-sibl. Note
desc ? desc-or-self/child foll ?
anc-or-self/foll-sibl/desc-or-self Fragments
P/,, ? ,P?, ?,
4
Tree Description Graphs

• Meaning
• Special r node must map to the root.
• Node with label must map to a node with that
  label.
• Soled edge with descendent relationship.
• Solid edge with desc-or-self relationship.
• Solid edge with no label child relationship.
• Dashed edge follows in document order.

• Studied in Computational Linguistics
• Generalization of Tree Patterns.
• Defines an existential statement about an XML
  tree.
• Satisfiability problem.
• Def.s binary relations if input/output nodes are
  indicated.

5
TDGs and XPath

• (desc-or-selfa/parentb/anc-or-self)
• ?
• (descc/foll-sibl)

Theorem For every path expression in P/,, ?
there is an equivalent TDG. Reverse? Lemma If a
TDG with n nodes has a model then there is a
satisfying XML tree with at most n
nodes. Corollary Satisfiability of TDGs and
P/,, ? is in NP.
6
String Matching

• BMS The Bounded Multiple String matching
  problem.
• Given a finite set of patterns that are strings
  over 1,0, is there a string over 1,0 such
  that
• 1. all the patterns can be matched and
• 2. this string is not longer than the longest
  pattern?
• Examples
• 11, 0 is satisfiable 1010
• 11, 010 is satisfiable 10100
• 111, 00 is not satisfiable
• Theorem Deciding BMS is NP-complete. (red. of
  SAT3)

7
Lowerbounds (1/3)

• Theorem Deciding satisf. for P? is
  NP-hard.(red. of BMS)

?
aba, ba, bbb
Corollary Deciding satisf. for TDGs is
NP-hard. Corollary Deciding satisf. for P- is
NP-hard.
8
Lowerbounds (2/3)

• Theorem Deciding satisf. for P,? is
  NP-hard.(red. of SAT3)
• Intuition ith parent represents Xi

(para/par/par/par) ? ( X1 ? ?X2
? X4 ) ? (par/parb/par/par)
? (para/par/par/para)
( C1 ? C2 ? C3 ) ? selfpC1pC2pC3

9
Lowerbounds (3/3)

• Theorem Deciding satisf. for P/, is
  NP-hard.(red. of BMS)

?
aba, ba, bbb

• /selfa/childb/child/child/childa
• anc-or-selfa/parentb/parent
• anc-or-selfb/parentb/parentb

10
Upperbounds

• Theorem Deciding satisf. for P is in PTIME
• Sketch of Algorithm
• Transform path-expression to TDG.
• Merge all parents of the same node.
• Repeat previous step until no more merges.
• If no label-conflict while merging then the
  answer is yes else no.
• Theorem Deciding satisf. for P/ is in PTIME
• Algorithm Similar

11
Summary

• Sat. of TDGs is NP-complete.
• Xpath fragments

• Open problems
• upperbound P-
• complexity P? and P/,?
• relationship TDGs and P/,,?