Multitrees Automata , Presburger ’ s Constraints and Tree Logics

Résumé We describe multitree automata and a related logic on multitrees. Multitrees are extensions of trees with both associative and associative-commutative symbols that may bear arbitrary numbers of sons. An originality of our approach is that transitions of an automaton are restricted using Presburger’s constraints. The benefit of this extension is that we generalize, all together, automata with equality and disequalities constraints as well as counting constraints. This new class of automata appears very general as it may encompass hedge automata, a simple yet effective model for XML schemata, feature tree automata, automata with constraints between brothers and automata with arithmetical constraints. Moreover, the class of recognizable languages enjoys all the typical good properties of traditional regular languages: closure under boolean operations and composition by associative and associative-commutative operators, determinisation, decidability of the test for emptiness, ... We apply our automata to query languages for XML-like documents and to automated inductive theorem proving based on rewriting, obtaining each time new results. Using a classical connection between logic and automata, we design a decidable logic for (multi)trees that can be used as a foundation for querying XML-like document. This proposition has the same flavour as a query language for semi-structured recently proposed by Cardelli and Ghelli. The same tree logic is used to yield decidable cases of inductive reducibility modulo associativity-commutativity, a key property in inductive theorem proving based on rewriting.

[1]  M. Murata.  Extended Path Expressions for XML [ Extended , 2001 .

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

[3]  Florent Jacquemard,et al.  Ground Reducibility and Automata with Disequality Constraints , 1994, STACS.

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

[5]  Leonid Libkin.  Monadic Second-Order Logic and Automata , 2004 .

[6]  Denis Lugiez,et al.  Tree Automata Help One To Solve Equational Formulae In AC-Theories , 1994, J. Symb. Comput..

[7]  David A. Plaisted,et al.  Semantic Confluence Tests and Completion Methods , 1985, Inf. Control..

[8]  Hubert Comon-Lundh.  Sequentiality, Monadic Second-Order Logic and Tree Automata , 2000, Inf. Comput..

[9]  Denis Lugiez.  A Good Class of Tree Automata , 1998 .

[10]  Alain Quéré,et al.  Définition et Etude des Bilangages Réguliers , 1968, Inf. Control..

[11]  Max Dauchet,et al.  Encompassment Properties and Automata with Constraints , 1993, RTA.

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

[13]  Daniel J. Rosenkrantz,et al.  Sufficient-completeness, ground-reducibility and their complexity , 1991, Acta Informatica.

[14]  Jean Goubault-Larrecq,et al.  Alternating two-way AC-tree automata , 2007, Inf. Comput..

[15]  Sophie Tison,et al.  Equality and Disequality Constraints on Direct Subterms in Tree Automata , 1992, STACS.

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

[17]  John Doner,et al.  Tree Acceptors and Some of Their Applications , 1970, J. Comput. Syst. Sci..

[18]  Moshe Y. Vardi.  Alternating Automata: Unifying Truth and Validity Checking for Temporal Logics , 1997, CADE.

[19]  Jean-Pierre Jouannaud,et al.  Automatic Proofs by Induction in Theories without Constructors , 1989, Inf. Comput..

[20]  John Mitchell,et al.  Tree Automata with One Memory, Set Constraints, and Ping-Pong Protocols , 2001, ICALP.

[21]  Dan Suciu,et al.  Data on the Web: From Relations to Semistructured Data and XML , 1999 .

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

[23]  Florent Jacquemard,et al.  Pumping, Cleaning and Symbolic Constraints Solving , 1994, ICALP.

[24]  Peter W. O'Hearn,et al.  The Logic of Bunched Implications , 1999, Bulletin of Symbolic Logic.

[25]  Alexander Aiken,et al.  Solving systems of set constraints , 1992, [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science.