research-article

Efficient reasoning about data trees via integer linear programming

Authors:

Tony TanAuthors Info & Claims

ACM Transactions on Database Systems (TODS), Volume 37, Issue 3

Article No.: 19, Pages 1 - 28

https://doi.org/10.1145/2338626.2338632

Published: 06 September 2012 Publication History

Abstract

Data trees provide a standard abstraction of XML documents with data values: they are trees whose nodes, in addition to the usual labels, can carry labels from an infinite alphabet (data). Therefore, one is interested in decidable formalisms for reasoning about data trees. While some are known—such as the two-variable logic—they tend to be of very high complexity, and most decidability proofs are highly nontrivial. We are therefore interested in reasonable complexity formalisms as well as better techniques for proving decidability.

Here we show that many decidable formalisms for data trees are subsumed—fully or partially—by the power of tree automata together with set constraints and linear constraints on cardinalities of various sets of data values. All these constraints can be translated into instances of integer linear programming, giving us an NP upper bound on the complexity of the reasoning tasks. We prove that this bound, as well as the key encoding technique, remain very robust, and allow the addition of features such as counting of paths and patterns, and even a concise encoding of constraints, without increasing the complexity. The NP bound is tight, as we also show that the satisfiability of a single set constraint is already NP-hard.

We then relate our results to several reasoning tasks over XML documents, such as satisfiability of schemas and data dependencies and satisfiability of the two-variable logic. As a final contribution, we describe experimental results based on the implementation of some reasoning tasks using the SMT solver Z3.

References

[1]

Alon, N., Milo, T., Neven, F., Suciu, D., and Vianu, V. 2003. XML with data values: Typechecking revisited. J. Comput. Syst. Sci. 66, 4, 688--722.

Digital Library

[2]

Arenas, M., Fan, W., and Libkin, L. 2008. On the complexity of verifying consistency of XML specifications. SIAM J. Comput. 38, 3, 841--880.

Digital Library

[3]

Arenas, M. and Libkin, L. 2008. XML data exchange: consistency and query answering. J. ACM 55, 2.

Digital Library

[4]

Björklund, H., Martens, W., and Schwentick, T. 2008. Optimizing conjunctive queries over trees using schema information. In Mathematical Foundations of Computer Science. Springer, 132--143.

Digital Library

[5]

Bojanczyk, M., David, C., Muscholl, A., Schwentick, T., and Segoufin, L. 2011. Two-variable logic on data words. ACM Trans. Comput. Logic 12, 4.

Digital Library

[6]

Bojanczyk, M., Muscholl, A., Schwentick, T., and Segoufin, L. 2009. Two-Variable logic on data trees and XML reasoning. J. ACM 56, 3.

Digital Library

[7]

Bouyer, P., Petit, A., and Thérien, D. 2001. An algebraic characterization of data and timed languages. In CONCUR. Springer, 248--261.

Digital Library

[8]

Buneman, P., Davidson, S., Fan, W., Hara, C., and Tan, W.-C. 2002. Keys for XML. Comput. Netw. 39, 5.

[9]

Calvanese, D., Giacomo, G. D., Lenzerini, M., and Vardi, M. 2009. An automata-theoretic approach to regular XPath. In Database Programming Languages. 18--35.

Digital Library

[10]

Comon, H., Dauchet, M., Gilleron, R., Jacquemard, F., Löding, C., Lugiez, D., Tison, S., and Tommasi, M. 2007. Tree Automata: Techniques and Applications. http://www.grappa.univ-lille3.fr/tata

[11]

David, C., Libkin, L., and Tan, T. 2011. Efficient reasoning about data trees via integer linear programming. In International Conference on Database Theory.

Digital Library

[12]

de Moura, L. M. and Bjørner, N. 2008. Z3: An efficient SMT solver. In Tools and Algorithms for the Construction and Analysis of Systems. 337--340.

Digital Library

[13]

Demri, S. and Lazic, R. 2009. Ltl with the freeze quantifier and register automata. ACM Trans. Comput. Logic 10, 3.

Digital Library

[14]

Fan, W. and Libkin, L. 2002. On XML integrity constraints in the presence of dtds. J. ACM 49, 3.

Digital Library

[15]

Figueira, D. 2009. Satisfiability of downward xpath with data equality tests. In Proceedings of the Symposium on Principles of Database Systems. ACM, 197--206.

Digital Library

[16]

Genevés, P. and Layaida, N. 2006. A system for the static analysis of XPath. ACM Trans. Inf. Syst. 24, 4, 475--502.

Digital Library

[17]

Givan, R., McAllester, D. A., Witty, C., and Kozen, D. 2002. Tarskian set constraints. Inf. Comput. 174, 105--131.

Digital Library

[18]

Jurdzinski, M. and Lazic, R. 2007. Alternation-Free modal mu-calculus for data trees. In Proceedings of the Symposium on Logic in Computer Science. IEEE Computer Society, 131--140.

Digital Library

[19]

Kaminski, M. and Tan, T. 2008. Tree automata over infinite alphabets. In Pillars of Computer Science. Springer, 386--423.

Digital Library

[20]

Kopczynski, E. and To, A. W. 2010. Parikh images of grammars: Complexity and applications. In Proceedings of the Symposium on Logic in Computer Science. IEEE Computer Society.

Digital Library

[21]

Libkin, L. and Sirangelo, C. 2010. Reasoning about XML with temporal logics and automata. J. Appl. Logic 8, 2, 210--232.

[22]

Malik, S. and Zhang, L. 2009. Boolean satisfiability: From theoretical hardness to practical success. Comm. ACM 52, 8, 76--82.

Digital Library

[23]

Martens, W., Neven, F., and Schwentick, T. 2007. Simple off the shelf abstractions for XML schema. SIGMOD Rec. 36, 3, 15--22.

Digital Library

[24]

Marx, M. 2005. Conditional XPath. ACM Trans. Database Syst. 30, 4, 929--959.

Digital Library

[25]

Milo, T., Suciu, D., and Vianu, V. 2003. Typechecking for XML transformers. J. Comput. Syst. Sci. 66, 1, 66--97.

Digital Library

[26]

Neven, F. 2002. Automata, logic, and XML. In Computer Science Logic. Springer, 2--26.

Digital Library

[27]

Neven, F. and Schwentick, T. 2002. Query automata over finite trees. Theor. Comput. Sci. 275, 1-2, 633--674.

Digital Library

[28]

Neven, F., Schwentick, T., and Vianu, V. 2004. Finite state machines for strings over infinite alphabets. ACM Trans. Comput. Logic 5, 3, 403--435.

Digital Library

[29]

Pacholski, L. and Podelski, A. 1997. Set constraints: A pearl in research on constraints. In Principles and Practice of Constraint Programming. 549--562.

[30]

Papadimitriou, C. 1981. On the complexity of integer programming. J. ACM 28, 765--768.

Digital Library

[31]

Robinson, A. and Voronkov, A. 2001. Handbook of Automated Reasoning. MIT Press.

Digital Library

[32]

Schwentick, T. 2004. XPath query containment. SIGMOD Rec. 33, 1, 101--109.

Digital Library

[33]

Thatcher, J. 1967. Characterizing derivation trees of context-free grammars through a generalization of finite automata theory. J. Comput. Syst. Sci. 1, 317--322.

Digital Library

[34]

Verma, K., Seidl, H., and Schwentick, T. 2005. On the complexity of equational horn clauses. In Conference on Automated Deduction. Springer, 337--352.

Digital Library

[35]

Vianu, V. 2001. A Web odyssey: From Codd to XML. In Proceedings of the Symposium on Principles of Database Systems. ACM, 1--15.

Digital Library

[36]

West, D. 2001. Introduction to Graph Theory. Prentice Hall.

Cited By

Hague MJeż ALin A(2024)Parikh’s Theorem Made SymbolicProceedings of the ACM on Programming Languages10.1145/36329078:POPL(1945-1977)Online publication date: 5-Jan-2024
https://dl.acm.org/doi/10.1145/3632907
Hague MLin AHong C(2019)CSS Minification via Constraint SolvingACM Transactions on Programming Languages and Systems10.1145/331033741:2(1-76)Online publication date: 19-Jun-2019
https://dl.acm.org/doi/10.1145/3310337
Figueira D(2018)Satisfiability of Xpath on data treesACM SIGLOG News10.1145/3212019.32120215:2(4-16)Online publication date: 30-Apr-2018
https://dl.acm.org/doi/10.1145/3212019.3212021
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Efficient reasoning about data trees via integer linear programming

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Efficient reasoning about data trees via integer linear programming
ICDT '11: Proceedings of the 14th International Conference on Database Theory

Data trees provide a standard abstraction of XML documents with data values: they are trees whose nodes, in addition to the usual labels, can carry labels from an infinite alphabet (data). Therefore, one is interested in decidable formalisms for ...
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cover image ACM Transactions on Database Systems

ACM Transactions on Database Systems Volume 37, Issue 3

August 2012

191 pages

ISSN:0362-5915

EISSN:1557-4644

DOI:10.1145/2338626

Issue’s Table of Contents

Copyright © 2012 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 06 September 2012

Accepted: 01 June 2012

Revised: 01 April 2012

Received: 01 October 2011

Published in TODS Volume 37, Issue 3

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Cited By

Hague MJeż ALin A(2024)Parikh’s Theorem Made SymbolicProceedings of the ACM on Programming Languages10.1145/36329078:POPL(1945-1977)Online publication date: 5-Jan-2024
https://dl.acm.org/doi/10.1145/3632907
Hague MLin AHong C(2019)CSS Minification via Constraint SolvingACM Transactions on Programming Languages and Systems10.1145/331033741:2(1-76)Online publication date: 19-Jun-2019
https://dl.acm.org/doi/10.1145/3310337
Figueira D(2018)Satisfiability of Xpath on data treesACM SIGLOG News10.1145/3212019.32120215:2(4-16)Online publication date: 30-Apr-2018
https://dl.acm.org/doi/10.1145/3212019.3212021
Tan T(2014)Extending two-variable logic on data trees with order on data values and its automataACM Transactions on Computational Logic10.1145/255994515:1(1-39)Online publication date: 6-Mar-2014
https://dl.acm.org/doi/10.1145/2559945
Fülöp ZVogler H(2014)Forward and backward application of symbolic tree transducersActa Informatica10.1007/s00236-014-0197-751:5(297-325)Online publication date: 1-Aug-2014
https://dl.acm.org/doi/10.1007/s00236-014-0197-7
Figueira DHull RFan W(2013)On XPath with transitive axes and data testsProceedings of the 32nd ACM SIGMOD-SIGACT-SIGAI symposium on Principles of database systems10.1145/2463664.2463675(249-260)Online publication date: 22-Jun-2013
https://dl.acm.org/doi/10.1145/2463664.2463675
Gheerbrant ALibkin LReutter J(2013)Static Analysis and Query Answering for Incomplete Data Trees with ConstraintsIn Search of Elegance in the Theory and Practice of Computation10.1007/978-3-642-41660-6_15(273-290)Online publication date: 2013
https://doi.org/10.1007/978-3-642-41660-6_15

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