Complete axiomatizations for XPath fragments

We provide complete axiomatizations for several fragments of Core XPath, the navigational core of XPath 1.0 introduced by Gottlob, Koch and Pichler. A complete axiomatization for a given fragment is a set of equivalences from which every other valid equivalence is derivable; equivalences can be thought of as (undirected) rewrite rules. Specifically, we axiomatize single axis fragments of Core XPath as well as full Core XPath. Our completeness proofs use results and techniques from modal logic.

[1]  A. P. Bowran.  A Boolean Algebra , 1965 .

[2]  M. de Rijke,et al.  A Proof System for Finite Trees , 1995, CSL.

[3]  Chun Zhang,et al.  Storing and querying ordered XML using a relational database system , 2002, SIGMOD '02.

[4]  Maarten Marx,et al.  Navigational XPath: calculus and algebra , 2007, SGMD.

[5]  Dexter Kozen.  A Completeness Theorem for Kleene Algebras and the Algebra of Regular Events , 1994, Inf. Comput..

[6]  Alfred Tarski,et al.  Relational selves as self-affirmational resources , 2008 .

[7]  E. V. Huntington.  A New Set of Independent Postulates for the Algebra of Logic with Special Reference to Whitehead and Russell's Principia Mathematica. , 1932, Proceedings of the National Academy of Sciences of the United States of America.

[8]  Gabriel M. Kuper,et al.  Structural properties of XPath fragments , 2003, Theor. Comput. Sci..

[9]  Andrey Balmin,et al.  Grouping and optimization of XPath expressions in DB2® pureXML , 2008, SIGMOD Conference.

[10]  Yde Venema.  COMPLETENESS VIA COMPLETENESS: SINCE AND UNTIL , 2003 .

[11]  Georg Gottlob,et al.  Monadic queries over tree-structured data , 2002, Proceedings 17th Annual IEEE Symposium on Logic in Computer Science.

[12]  S C Kleene,et al.  Representation of Events in Nerve Nets and Finite Automata , 1951 .

[13]  Thomas Hillenbrand,et al.  WALDMEISTER - High-Performance Equational Deduction , 1997, Journal of Automated Reasoning.

[14]  maarten marx,et al.  Arrow logic and multi-modal logic , 1997 .

[15]  Lawrence S. Moss,et al.  FINITE MODELS CONSTRUCTED FROM CANONICAL FORMULAS , 2007, J. Philos. Log..

[16]  Marco Hollenberg.  An Equational Axiomatization of Dynamic Negation and Relational Composition , 1997, J. Log. Lang. Inf..

[17]  B. Jónsson,et al.  Computer-aided investigations of relation algebras , 1992 .

[18]  R. Goodstein.  Boolean algebra , 1963 .

[19]  Vikas Arora,et al.  Query Rewrite for XML in Oracle XML DB , 2004, VLDB.

[20]  Johan van Benthem,et al.  Modal Frame Correspondences and Fixed-Points , 2006, Stud Logica.

[21]  A. Tarski,et al.  A Formalization Of Set Theory Without Variables , 1987 .

[22]  Johan van Benthem,et al.  Exploring logical dynamics , 1996, Studies in logic, language and information.

[23]  M. de Rijke,et al.  Semantic characterizations of navigational XPath , 2005, SGMD.

[24]  M. de Rijke,et al.  Modal Logic , 2001, Cambridge Tracts in Theoretical Computer Science.

[25]  Y. Venema.  A crash course in arrow logic , 1994 .

[26]  E. V. Huntington.  Boolean Algebra. A Correction , 1933 .

[27]  Balder ten Cate,et al.  Some modal aspects of XPath , 2010, J. Appl. Non Class. Logics.

[28]  Kit Fine,et al.  Normal forms in modal logic , 1975, Notre Dame J. Formal Log..

[29]  R. Goldblatt.  Logics of Time and Computation , 1987 .

[30]  Mark Reynolds,et al.  An axiomatization for until and since over the reals without the IRR rule , 1992, Stud Logica.

[31]  Vaughan R. Pratt,et al.  Dynamic algebras: Examples, constructions, applications , 1991, Stud Logica.

[32]  William McCune,et al.  Solution of the Robbins Problem , 1997, Journal of Automated Reasoning.

[33]  Maarten Marx,et al.  XPath with Conditional Axis Relations , 2004, EDBT.

[34]  Zoltán Ésik,et al.  A Semiring-Semimodule Generalization of omega-Context-Free Languages , 2004, Theory Is Forever.

[35]  Carsten Lutz,et al.  The complexity of query containment in expressive fragments of XPath 2.0 , 2007, PODS.

[36]  Robin Hirsch,et al.  Peirce Algebras and Boolean Modules , 2007, J. Log. Comput..

[37]  Yde Venema.  Completeness via Completeness , 1993 .

[38]  Maarten Marx,et al.  Axiomatizing the Logical Core of XPath 2.0 , 2008, Theory of Computing Systems.