Temporal logics over unranked trees

We consider unranked trees that have become an active subject of study recently due to XML applications, and characterize commonly used fragments of first-order (FO) and monadic second-order logic (MSO) for them via various temporal logics. We look at both unordered trees and ordered trees (in which children of the same node are ordered by the next-sibling relation), and characterize Boolean and unary FO and MSO queries. For MSO Boolean queries, we use extensions of the /spl mu/-calculus: with counting for unordered trees, and with the past for ordered. For Boolean FO queries, we use similar extensions of CTL*. We then use composition techniques to transfer results to unary queries. For the ordered case, we need the same logics as for Boolean queries, but for the unordered case, we need to add both past and counting to the /spl mu/-calculus and CTL*. We also consider MSO sibling-invariant queries, that can use the sibling ordering but do not depend on the particular one used, and capture them by a variant of the /spl mu/-calculus with modulo quantifiers.

[1]  Michael Benedikt,et al.  XPath satisfiability in the presence of DTDs , 2008, JACM.

[2]  Thomas Schwentick,et al.  Expressive and efficient pattern languages for tree-structured data (extended abstract) , 2000, PODS '00.

[3]  Radu Mateescu,et al.  Local Model-Checking of Modal Mu-Calculus on Acyclic Labeled Transition Systems , 2002, TACAS.

[4]  Igor Walukiewicz,et al.  On the Expressive Completeness of the Propositional mu-Calculus with Respect to Monadic Second Order Logic , 1996, CONCUR.

[5]  Anuj Dawar,et al.  On the Bisimulation Invariant Fragment of Monadic S1 in the Finite , 2004, FSTTCS.

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

[7]  Frank Neven,et al.  Automata, Logic, and XML , 2002, CSL.

[8]  Alexander Moshe Rabinovich,et al.  Expressive Power of Temporal Logics , 2002, CONCUR.

[9]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs V: On Closing the Gap Between Definability and Recognizability , 1991, Theor. Comput. Sci..

[10]  Michael Benedikt,et al.  Regular Tree Languages Definable in FO , 2005, STACS.

[11]  E. Allen Emerson,et al.  Tree automata, mu-calculus and determinacy , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[12]  Johann A. Makowsky,et al.  Algorithmic uses of the Feferman-Vaught Theorem , 2004, Ann. Pure Appl. Log..

[13]  Thomas Schwentick,et al.  Counting in Trees for Free , 2004, ICALP.

[14]  Stephan Kreutzer,et al.  Inflationary Fixed Points in Modal Logic , 2001, CSL.

[15]  Victor Vianu,et al.  A Web Odyssey: from Codd to XML , 2001, PODS.

[16]  Thomas Schwentick,et al.  Query automata over finite trees , 2002, Theor. Comput. Sci..

[17]  Igor Walukiewicz,et al.  Characterizing EF and EX tree logics , 2006, Theor. Comput. Sci..

[18]  Gabriel M. Kuper,et al.  Structural Properties of XPath Fragments , 2003, ICDT.

[19]  Faron Moller,et al.  Counting on CTL*: on the expressive power of monadic path logic , 2003, Inf. Comput..

[20]  Girish Bhat,et al.  Efficient model checking via the equational /spl mu/-calculus , 1996, Proceedings 11th Annual IEEE Symposium on Logic in Computer Science.

[21]  Moshe Y. Vardi.  Reasoning about The Past with Two-Way Automata , 1998, ICALP.

[22]  M. de Rijke,et al.  CTL model checking for processing simple XPath queries , 2004, Proceedings. 11th International Symposium on Temporal Representation and Reasoning, 2004. TIME 2004..

[23]  Dan Suciu,et al.  Containment and equivalence for a fragment of XPath , 2004, JACM.

[24]  Damian Niwinski,et al.  Fixed points vs. infinite generation , 1988, [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science.

[25]  Maarten Marx,et al.  Conditional XPath, the first order complete XPath dialect , 2004, PODS.

[26]  J.F.A.K. van Benthem,et al.  Modal logic and classical logic , 1983 .

[27]  Igor Walukiewicz.  Monadic second-order logic on tree-like structures , 2002, Theor. Comput. Sci..

[28]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs , 1990, Inf. Comput..

[29]  Johan Anthory Willem Kamp,et al.  Tense logic and the theory of linear order , 1968 .

[30]  Mads Dam.  CTL* and ECTL* as Fragments of the Modal mu-Calculus , 1994, Theor. Comput. Sci..

[31]  Luca Cardelli,et al.  A Query Language Based on the Ambient Logic , 2001, SEBD.

[32]  Holger Schlingloff.  Expressive completeness of temporal logic of trees , 1992, J. Appl. Non Class. Logics.

[33]  Wolfgang Thomas,et al.  Computation Tree Logic CTL* and Path Quantifiers in the Monadic Theory of the Binary Tree , 1987, ICALP.

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

[35]  Frank Neven,et al.  Logical definability and query languages over unranked trees , 2003, 18th Annual IEEE Symposium of Logic in Computer Science, 2003. Proceedings..

[36]  Martin Grohe,et al.  The complexity of first-order and monadic second-order logic revisited , 2002, Proceedings 17th Annual IEEE Symposium on Logic in Computer Science.

[37]  Giacomo Lenzi,et al.  Relating levels of the mu-calculus hierarchy and levels of the monadic hierarchy , 2001, Proceedings 16th Annual IEEE Symposium on Logic in Computer Science.

[38]  Nicole Schweikardt,et al.  Comparing the Succinctness of Monadic Query Languages over Finite Trees , 2003, CSL.