Counting in trees

We consider automata and logics that allow to reason about numerical properties of unranked trees, expressed as Presburger constraints. We characterize non-deterministic automata by Presburger Monadic Second-Order logic, and deterministic automata by Presburger Fixpoint logic. We show how our results can be used in order to obtain efficient querying algorithms on XML trees.

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

[2]  Stéphane Demri,et al.  Complexity of modal logics with Presburger constraints , 2010, J. Appl. Log..

[3]  I. Walukiewicz.  Pushdown Processes: Games and Model Checking , 1996 .

[4]  M. Fischer,et al.  SUPER-EXPONENTIAL COMPLEXITY OF PRESBURGER ARITHMETIC , 1974 .

[5]  Jean-Marc Talbot,et al.  Automata and Logics for Unranked and Unordered Trees , 2005, RTA.

[6]  Wolfgang Thomas,et al.  On Intersection Problems for Polynomially Generated Sets , 2006, ICALP.

[7]  Toshinori Takai,et al.  Decidability and Closure Properties of Equational Tree Languages , 2002, RTA.

[8]  Rohit Parikh,et al.  On Context-Free Languages , 1966, JACM.

[9]  Luca Cardelli,et al.  Anytime, anywhere: modal logics for mobile ambients , 2000, POPL '00.

[10]  Christos H. Papadimitriou,et al.  On the complexity of integer programming , 1981, JACM.

[11]  N. Sloane.  A Handbook Of Integer Sequences , 1973 .

[12]  Kenneth H. Rosen,et al.  Catalan Numbers , 2002 .

[13]  Joachim Niehren,et al.  Feature Automata and Recognizable Sets of Feature Trees , 1993, TAPSOFT.

[14]  Giorgio Ghelli,et al.  TQL ALGEBRA AND ITS IMPLEMENTATION (EXTENDED ABSTRACT) , 2002 .

[15]  Thomas Schwentick,et al.  On the Complexity of Equational Horn Clauses , 2005, CADE.

[16]  Hitoshi Ohsaki,et al.  Beyond Regularity: Equational Tree Automata for Associative and Commutative Theories , 2001, CSL.

[17]  Silvano Dal-Zilio,et al.  A logic you can count on , 2004, POPL.

[18]  Frank Neven,et al.  Expressiveness of structured document query languages based on attribute grammars , 1998, JACM.

[19]  Silvano Dal-Zilio,et al.  XML Schema, Tree Logic and Sheaves Automata , 2003, RTA.

[20]  Paolo Manghi,et al.  The Query Language TQL , 2002, WebDB.

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

[22]  F. Klaedtke.  On the automata size for Presburger arithmetic , 2004, LICS 2004.

[23]  Joachim Niehren,et al.  Minimizing Tree Automata for Unranked Trees , 2005, DBPL.

[24]  S. Ginsburg,et al.  Semigroups, Presburger formulas, and languages. , 1966 .

[25]  Frank Neven,et al.  Expressiveness of structured document query languages based on attribute grammars , 2002, J. ACM.

[26]  Ulrike Sattler,et al.  The Complexity of the Graded µ-Calculus , 2002, CADE.

[27]  P. Hilton,et al.  Catalan Numbers, Their Generalization, and Their Uses , 1991 .

[28]  Helmut Seidl,et al.  On Guarding Nested Fixpoints , 1999, CSL.

[29]  Denis Lugiez,et al.  Multitrees Automata , Presburger ’ s Constraints and Tree Logics , 2022 .

[30]  Mojżesz Presburger,et al.  On the completeness of a certain system of arithmetic of whole numbers in which addition occurs as the only operation , 1991 .

[31]  Helmut Seidl,et al.  Haskell Overloading is DEXPTIME-Complete , 1994, Inf. Process. Lett..

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

[33]  Derick Wood,et al.  Regular Tree Languages Over Non-Ranked Alphabets , 1998 .

[34]  M. Minsky.  Recursive Unsolvability of Post's Problem of "Tag" and other Topics in Theory of Turing Machines , 1961 .

[35]  Helmut Seidl,et al.  Locating Matches of Tree Patterns in Forests , 1998, FSTTCS.

[36]  Albert R. Meyer,et al.  Word problems requiring exponential time(Preliminary Report) , 1973, STOC.

[37]  Frank Harary,et al.  Graph Theory , 2016 .

[38]  Javier Esparza.  Petri Nets, Commutative Context-Free Grammars, and Basic Parallel Processes , 1997, Fundam. Informaticae.

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

[40]  Denis Lugiez,et al.  A Good Class of Tree Automata and Application to Inductive Theorem Proving , 1998, ICALP.

[41]  Thomas Schwentick,et al.  Numerical document queries , 2003, PODS.

[42]  Igor Walukiewicz,et al.  Pushdown Processes: Games and Model-Checking , 1996, Inf. Comput..

[43]  Christof Löding,et al.  Deterministic Automata on Unranked Trees , 2005, FCT.

[44]  James W. Thatcher,et al.  Generalized finite automata theory with an application to a decision problem of second-order logic , 1968, Mathematical systems theory.

[45]  Frank Neven,et al.  Typechecking Top-Down Uniform Unranked Tree Transducers , 2003, ICDT.