XPath, the best known modal logic ever.
This presentation is the property of its rightful owner.
Sponsored Links
1 / 18

XPath, the best known modal logic ever. And . . . made in Amsterdam! Maarten Marx Information and Language Processing Systems (ILPS) PowerPoint PPT Presentation


  • 82 Views
  • Uploaded on
  • Presentation posted in: General

XPath, the best known modal logic ever. And . . . made in Amsterdam! Maarten Marx Information and Language Processing Systems (ILPS) Informatics Institute, University of Amsterdam, The Netherlands. XPath, what is that? • A standard language proposed by the W3C in November 1999.

Download Presentation

XPath, the best known modal logic ever. And . . . made in Amsterdam! Maarten Marx Information and Language Processing Systems (ILPS)

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Xpath the best known modal logic ever and made in amsterdam maarten marx information and language processing systems ilps

XPath, the best known modal logic ever.

And . . . made in Amsterdam!

Maarten Marx

Information and Language Processing Systems (ILPS)

Informatics Institute,

University of Amsterdam, The Netherlands


Xpath the best known modal logic ever and made in amsterdam maarten marx information and language processing systems ilps

XPath, what is that?

• A standard language proposed by the W3C in November 1999.

• XPath is a language for addressing parts of an XML document.

• XPath beats temporal logic as the best known modal logic:

• Google: Resultaten 1 - 10 van circa 1.870.000 voor XPath

• Google: Resultaten 1 - 10 van circa 242.000 voor ”temporal logic”


Xpath the best known modal logic ever and made in amsterdam maarten marx information and language processing systems ilps

Research aim behind this talk

• Create an expressively complete navigational query language for XML

documents.

Aim of this talk

• Show that modal logic is the right paradigm for such a task.

• One can get remarkable results with (for modal logicians) simple proofs.

• The modal logic literature is full of hints and almost-results.


Xpath the best known modal logic ever and made in amsterdam maarten marx information and language processing systems ilps

  • FO logic of “XML documents”

  • • An XML document can be seen as a finite, node labeled, sibling ordered, unbounded tree.

  • • Nb. We abstract away from the “data details” and only focus on the skeleton of an XML

  • document.

  • • The first order language for these models has

  • descendant relation

  • 2. following sibling relation

  • 3. unary predicates corresponding to node labels and attribute–value

  • pairs.

  • • Nb. we cannot express joins on attribute values!


Xpath the best known modal logic ever and made in amsterdam maarten marx information and language processing systems ilps

Known results

• For binary relations, very little is known. Immerman Kozen:

1. strings have the 3 variable property;

2. bounded trees have a k variable property.

• For unary relations, more is known:

1. Kamp’s theorem, strings have H-dimension 3;

2. unbounded unordered trees have no finite H-dimension (Schlingloff)

3. unbounded ordered trees have H-dimension 3 (PODS 2004).

• Nb.

1. k-variable property is stronger than H-dimension;

2. k-variable property is independent from “finite complete set of

operators property” (Hodkinson–Simon, JPhL).

• Thus we cannot answer our research goal by known results.


Xpath the best known modal logic ever and made in amsterdam maarten marx information and language processing systems ilps

  • Conditional XPath

  • The syntax is based on

  • XPath 1.0 (W3C)

  • 2. Kleene algebras (= regular path queries) with tests (Kozen)

  • 3. Propositional Dynamic Logic (Pratt, Harel)

  • step ::= child | parent | right | left.

  • path wff ::= step | (step?node wff) +

  • ?node wff

  • |path wff/path wff | path wff [ path wff.

  • node wff ::= p | hh path wff ii | ¬ node wff | node wff ^ node wff.


Xpath the best known modal logic ever and made in amsterdam maarten marx information and language processing systems ilps

Semantics

• Given an ordered tree,

– each path wff denotes a set of pairs of nodes, and

– each node wff denotes a set of nodes.

• All set theoretical operations have their standard meaning.

• hh p wff ii is true at a node n iff n is in the domain of the relation p wff.

Note! Every path wff (node wff) defines a first order definable binary

(unary) relation.


Xpath the best known modal logic ever and made in amsterdam maarten marx information and language processing systems ilps

Example expressions

child :: pi child/?pi

child :: pi[descendant :: ] child/?pi/? hh child +

ii /descendant :: pi ? ¬hh parent ii /child + /?pi

child :: child

self :: pi[child] ?(pi ^ hh child ii )

preceding :: pi parent/left + /child/?pi.

Equivalent XPath 1.0 and Conditional XPath expressions.


Xpath the best known modal logic ever and made in amsterdam maarten marx information and language processing systems ilps

Conditional XPath fulfills our research goal

• Theorem 1 (Kamp/PODS 2004) Every FO definable set of nodes is

definable by a Conditional XPath node wff.

• Theorem 2 Every first order definable binary relation is definable by a

Conditional XPath path wff.

• Corollary Every FO relation °(x1, . . . , xn) is equivalent to a union of

conjunctive queries consisting of atoms of the form xi path wff xj.


Xpath the best known modal logic ever and made in amsterdam maarten marx information and language processing systems ilps

Difference between the two theorems

• Theorem 1 is about node wffs and unary relations. Theorem 2 about

path wffs and binary relations.

• Theorem 2 implies theorem 1, but not conversely.

• Node wffs have much stronger operators (and, not, bounded quantification).

• Path wffs only have “until”, concatenation and union.


Xpath the best known modal logic ever and made in amsterdam maarten marx information and language processing systems ilps

XML document

An XML document can be seen as a

finite, node labelled, sibling ordered unbounded tree.

(Nb. We abstract away from the “data details” and only focus on the skeleton of an XML document.)


Xpath the best known modal logic ever and made in amsterdam maarten marx information and language processing systems ilps

Design Constraints

• Stay as close as possible to the existing W3C standard XPath.

• This means:

– no (first or second order) variables.

– express sets of nodes (answer sets) and relations between nodes

(paths).

– relations should be “drawable” (use only the regular expression

operators)


Xpath the best known modal logic ever and made in amsterdam maarten marx information and language processing systems ilps

Navigational XPath

We can give W3C XPath 1.0 a PDL like definition:

step ::= child | parent | right | left.

path wff ::= step | step +

| ?node wff

| path wff ; path wff | path wff [ path wff.

node wff ::= p | hh path wff ii | ¬ node wff | node wff ^ node wff.

• Note the very restricted use of ( · ) + !

• We use hh path wff ii to mean “I start a path wff”.

Modal Logic, XPath and XML , ten Cate Workshop, Februari, 2005. 8


Xpath the best known modal logic ever and made in amsterdam maarten marx information and language processing systems ilps

  • Examples of Navigational XPath expressions

  • ¬hh parent ii

  • (2) ¬hh child ii

  • (3) hh child ;?first ; right ;?last ii

  • (4) root ^ ¬hh child ;?( ¬ leaf ^ ¬hh child ;?first ; right ;?last ii ) ii

  • Not expressible in this version of XPath are

  • (child ;?q) ; child ;?p until p, q holds (as a relation)

  • (child ; child) ;?leaf the relation of being an even number

  • of steps above a leaf

  • Modal Logic, XPath and XML , ten Cate Workshop, Februari, 2005. 9


Xpath the best known modal logic ever and made in amsterdam maarten marx information and language processing systems ilps

  • Results (from the modal logic literature)

  • The node wffs form a modal language, created by Blackburn, Meyer-Viol, de Rijke in the 90’s. The logic is finitely axiomatizable.

  • 2. SAT problem is hard for EXPTIME (from Fisher-Ladner 79 for PDL).

  • 3. SAT problem is decidable in EXPTIME (from Vardi, Wolper 86 for PDL with converse).

  • 4. This language is expressively complete w.r.t. first order logic in two variables (use the result for the line of Etessami, Vardi, Wilke 97).

  • 5. Cf. ACM SIGMOD Record March 2005.


Xpath the best known modal logic ever and made in amsterdam maarten marx information and language processing systems ilps

Conclusion

• The W3C standard is a well designed language. They have reinvented a wheel which has been shown to possess very good properties.

• Still, the expressive completeness is not completely satisfactory. (Note that W3C XPath is not complete for two variable FO for paths.)

• Conditional XPath is excellent for expressing first order queries.

• Implementing the conditional axis is still open (special staircase joins?)


  • Login