. Afanasiev, PDL for ordered trees, Journal of Applied Non-Classical Logics, vol.32, issue.2, pp.115-135, 2005.
DOI : 10.3166/jancl.15.115-135

URL : https://hal.archives-ouvertes.fr/inria-00001206

H. Alur, R. Alur, and T. A. Henzinger, A really temporal logic, FOCS, pp.164-169, 1989.
DOI : 10.1145/174644.174651

N. Arnold, A. Arnold, and D. Niwinski, Fixed point characterization of weak monadic logic definable sets of trees, Tree Automata and Languages, pp.159-188, 1992.

. Bárány, Cardinality Quantifiers in MLO over Trees, Lecture Notes in Computer Science, vol.102, issue.303, pp.117-131, 2009.
DOI : 10.1007/3-540-63246-8_8

. Barceló, . Libkin, P. Barceló, and L. Libkin, Temporal Logics over Unranked Trees, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05), pp.31-40, 2005.
DOI : 10.1109/LICS.2005.51

A. P. Beltyukov, Decidability of the universal theory of natural numbers with addition and divisibility, Journal of Soviet Mathematics, vol.165, issue.No. 1, pp.15-28, 1976.
DOI : 10.1007/BF01693974

. Benedikt, Structural properties of XPath fragments, ICDT, volume 2572 of Lecture Notes in Computer Science, pp.79-95, 2003.

A. Bés, A survey of arithmetical definability, p.24, 2002.

. Bianco, Graded computation tree logic Mona: decidable arithmetic in practice (demo), LICS Formal Techniques in Real-Time and Fault-Tolerant Systems, 4th International Symposium, pp.342-351, 1996.

. Bojanczyk, Two-variable logic on data trees and XML reasoning, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00151833

. Bonatti, The Complexity of Enriched ??-Calculi, Lecture Notes in Computer Science, vol.4052, issue.56, pp.540-551, 2006.
DOI : 10.1007/11787006_46

. Bonatti, . Peron, P. A. Bonatti, and A. Peron, On the undecidability of logics with converse, nominals, recursion and counting, Artificial Intelligence, vol.158, issue.1, pp.75-96, 2004.
DOI : 10.1016/j.artint.2004.04.012

R. Iosif, On decidability within the arithmetic of addition and divisibility, Lecture Notes in Computer Science, vol.3441, pp.425-439, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00374872

. Bray, Extensible markup language (XML) W3C recommendation, p.63, 2004.

R. E. Bryant, Graph-Based Algorithms for Boolean Function Manipulation, IEEE Transactions on Computers, vol.35, issue.8, pp.677-691, 1986.
DOI : 10.1109/TC.1986.1676819

J. R. Büchi, Weak Second-Order Arithmetic and Finite Automata, Zeitschrift f??r Mathematische Logik und Grundlagen der Mathematik, vol.9, issue.1-6, pp.66-92, 1960.
DOI : 10.1002/malq.19600060105

J. R. Büchi, On a Decision Method in Restricted Second Order Arithmetic, Proc. Int. Congr. for Logic Methodology and Philosophy of Science, pp.1-11, 1962.
DOI : 10.1007/978-1-4613-8928-6_23

. Calvanese, An Automata-Theoretic Approach to Regular XPath, Lecture Notes in Computer Science, vol.98, issue.2, pp.18-35, 2009.
DOI : 10.1007/3-540-46093-4_3

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.147.7809

. Cardelli, . Gordon, L. Cardelli, and A. D. Gordon, Anytime, anywhere, Proceedings of the 27th ACM SIGPLAN-SIGACT symposium on Principles of programming languages , POPL '00, pp.365-377, 2000.
DOI : 10.1145/325694.325742

A. Church, A note on the Entscheidungsproblem, The Journal of Symbolic Logic, vol.166, issue.01, pp.40-41, 1936.
DOI : 10.1007/BF01475439

A. Church, An Unsolvable Problem of Elementary Number Theory, American Journal of Mathematics, vol.58, issue.2, pp.345-363, 1936.
DOI : 10.2307/2371045

A. Church, Applications of recursive arithmetic to the problem of circuit synthesis, Summaries of the Summer Institute of Symbolic Logic, pp.3-50, 1957.

C. , D. Clark, J. Derose, and S. , XML path language (XPath) version 1.0. W3C recommendation, p.58, 1999.

C. , M. Clark, J. Murata, and M. , RELAX NG specification . OASIS committee specification, p.63, 2001.

E. Clarke, E. M. Clarke, and E. A. Emerson, Design and synthesis of synchronization skeletons using branching time temporal logic, Logic of Programs, pp.52-71, 1981.
DOI : 10.1007/BFb0025774

. Colazzo, Efficient asymmetric inclusion between regular expression types, Proceedings of the 12th International Conference on Database Theory, ICDT '09, pp.174-182, 2009.
DOI : 10.1145/1514894.1514916

URL : https://hal.archives-ouvertes.fr/inria-00536884

B. Courcelle, The monadic second-order logic of graphs. I. Recognizable sets of finite graphs, Information and Computation, vol.85, issue.1, pp.12-75, 1990.
DOI : 10.1016/0890-5401(90)90043-H

URL : https://hal.archives-ouvertes.fr/hal-00353765

L. Dal-zilio, S. Dal-zilio, and D. Lugiez, XML Schema, Tree Logic and Sheaves Automata, Lecture Notes in Computer Science, vol.2706, pp.246-263, 2003.
DOI : 10.1007/3-540-44881-0_18

URL : https://hal.archives-ouvertes.fr/hal-00109288

. Dal-zilio, A logic you can count on, POPL, pp.135-146, 2004.
URL : https://hal.archives-ouvertes.fr/inria-00071562

. Demri, . Lazic, S. Demri, and R. Lazic, LTL with the freeze quantifier and register automata, LICS, pp.17-26, 2006.

. Demri, . Lugiez, S. Demri, and D. Lugiez, Presburger Modal Logic Is PSPACE-Complete, Lecture Notes in Computer Science, vol.4130, issue.35, pp.541-556, 2006.
DOI : 10.1007/11814771_44

J. Doner, Tree acceptors and some of their applications, Journal of Computer and System Sciences, vol.4, issue.5, pp.406-451, 1970.
DOI : 10.1016/S0022-0000(70)80041-1

. Elgaard, MONA 1.x: new techniques for WS1S and WS2S Decision problems of finite automata design and related arithmetics. Transactions of the, Proc. 10th International Conference on Computer-Aided Verification, CAV '98, pp.516-52021, 1960.

. Emerson, E. A. Halpern-]-emerson, and J. Y. Halpern, ???Sometimes??? and ???not never??? revisited: on branching versus linear time temporal logic, Journal of the ACM, vol.33, issue.1, pp.151-178, 1986.
DOI : 10.1145/4904.4999

. Emerson, E. A. Emerson, and C. S. Jutla, The complexity of tree automata and logics of programs (extended abstract), FOCS, pp.328-337, 1988.

W. Fallside, D. C. Fallside, and P. Walmsley, XML schema. W3C recommendation, p.63, 2004.

. Graded-ctl, Satisfiability and symbolic model checking, Lecture Notes in Computer Science, vol.5885, pp.306-325

K. Fine, In so many possible worlds., Notre Dame Journal of Formal Logic, vol.13, issue.4, pp.516-520, 1972.
DOI : 10.1305/ndjfl/1093890715

L. Fischer, M. J. Fischer, and R. E. Ladner, Propositional dynamic logic of regular programs, Journal of Computer and System Sciences, vol.18, issue.2, pp.194-211, 1979.
DOI : 10.1016/0022-0000(79)90046-1

R. Fischer, M. J. Fischer, and M. O. Rabin, Superexponential complexity of Presburger Arithmetic, SIAM-AMS Symposium in Applied Mahtematics, p.24, 1974.

. Gelade, Regular Expressions with Counting: Weak versus Strong Determinism, MFCS, pp.369-381, 2009.
DOI : 10.1142/S012905410200100X

. Gelade, Optimizing Schema Languages for XML: Numerical Constraints and Interleaving, SIAM Journal on Computing, vol.38, issue.5, 2009.
DOI : 10.1137/070697367

P. Genevès, Logics for XML, p.64, 2006.

L. Genevès, P. Genevès, and N. Laya¨?dalaya¨?da, A system for the static analysis of XPath, ACM Transactions on Information Systems, vol.24, issue.4, pp.475-502, 2006.
DOI : 10.1145/1185877.1185882

L. Genevès, P. Genevès, and N. Laya¨?dalaya¨?da, Deciding XPath containment with MSO, Data & Knowledge Engineering, vol.63, issue.1, pp.108-136, 2007.
DOI : 10.1016/j.datak.2006.11.003

L. Genevès, P. Genevès, and N. Laya¨?dalaya¨?da, Eliminating dead-code from XQuery programs, Proceedings of the 32nd ACM/IEEE International Conference on Software Engineering, ICSE '10, pp.305-306, 2010.
DOI : 10.1145/1810295.1810363

L. Genevès, P. Genevès, N. F. Laya¨?dalaya¨?da, M. M. Moro, S. Ghandeharizadeh et al., XML reasoning made practical, 2010 IEEE 26th International Conference on Data Engineering (ICDE 2010), pp.1169-1172, 2010.
DOI : 10.1109/ICDE.2010.5447786

. Genevès, Identifying query incompatibilities with evolving XML schemas, pp.221-230, 2009.

. Genevès, Efficient static analysis of XML paths and types, PLDI, pp.342-351, 2007.

R. Genevès, P. Genevès, and K. H. Rose, Compiling XPath for streaming access policy, Proceedings of the 2005 ACM symposium on Document engineering , DocEng '05, pp.52-54, 2005.
DOI : 10.1145/1096601.1096617

L. F. Goble, Grades of modality, Logique et Analyse, vol.13, pp.323-334, 1970.

. Gottlob, Efficient algorithms for processing XPath queries, ACM Transactions on Database Systems, vol.30, issue.2, pp.444-491, 2005.
DOI : 10.1145/1071610.1071614

. Grädel, . Otto, E. Grädel, and M. Otto, On logics with two variables, Theoretical Computer Science, vol.224, issue.1-2, pp.73-113, 1999.
DOI : 10.1016/S0304-3975(98)00308-9

. Habermehl, Tight bounds on the descriptional complexity of regular expressions, Developments in Language Theory, pp.276-287, 2009.

. Henriksen, Mona: Monadic secondorder logic in practice, Tools and Algorithms for the Construction and Analysis of Systems, First International Workshop, TACAS '95, p.27, 1995.

. Hollunder, . Baader, B. Hollunder, and F. Baader, Qualifying number restrictions in concept languages, In KR, pp.335-346, 1991.

H. Hosoya and B. C. Pierce, XDuce, ACM Transactions on Internet Technology, vol.3, issue.2, pp.117-148, 2003.
DOI : 10.1145/767193.767195

. Hosoya, Regular expression types for XML, ACM Transactions on Programming Languages and Systems, vol.27, issue.1, pp.46-90, 2005.
DOI : 10.1145/1053468.1053470

URL : https://hal.archives-ouvertes.fr/hal-00009271

W. Janin, D. Janin, and I. Walukiewicz, On the expressive completeness of the propositional mu-calculus with respect to monadic second order logic, Lecture Notes in Computer Science, vol.1119, pp.263-277, 1996.
DOI : 10.1007/3-540-61604-7_60

. Jensen, Automatic verification of pointer programs using monadic second-order logic, PLDI '97, p.27, 1997.

. Jurdzinski, . Lazic, M. Jurdzinski, and R. Lazic, Alternationfree modal mu-calculus for data trees, LICS, pp.131-140, 2007.
DOI : 10.1109/lics.2007.11

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.110.9659

. Kieronski, . Otto, E. Kieronski, and M. Otto, Small Substructures and Decidability Issues for First-Order Logic with Two Variables, 20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05), pp.448-457, 2005.
DOI : 10.1109/LICS.2005.49

P. Kilpeläinen and R. Tuhkanen, Regular expressions with numerical occurrence indicators -preliminary results, SPLST, pp.163-173, 2003.

S. C. Kleene, Representation of Events in Nerve Nets and Finite Automata, Automata Studies, pp.3-41, 1956.
DOI : 10.1515/9781400882618-002

D. Kozen, Results on the propositional ??-calculus, Theoretical Computer Science, vol.27, issue.3, pp.333-354, 1983.
DOI : 10.1016/0304-3975(82)90125-6

M. Kracht, Syntactic codes and grammar refinement, Journal of Logic, Language and Information, vol.4, issue.1, pp.41-60, 1995.
DOI : 10.1007/BF01048404

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.4252

S. A. Kripke, A semantical analysis of modal logic I: normal modal propositional calculi, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, pp.67-96, 1963.

. Kupferman, The Complexity of the Graded ??-Calculus, Lecture Notes in Computer Science, vol.2392, issue.32, pp.423-437, 2002.
DOI : 10.1007/3-540-45620-1_34

R. E. Ladner, The Computational Complexity of Provability in Systems of Modal Propositional Logic, SIAM Journal on Computing, vol.6, issue.3, pp.467-480, 1977.
DOI : 10.1137/0206033

F. Laroussinie and P. Schnoebelen, Specification in CTL+Past for Verification in CTL, Information and Computation, vol.156, issue.1-2, pp.236-263, 2000.
DOI : 10.1006/inco.1999.2817

N. Laya¨?dalaya¨?da and P. Genevès, Debugging standard document formats, pp.1269-1272, 2010.

R. Lazic, Safely Freezing LTL, Lecture Notes in Computer Science, vol.4337, pp.381-392, 2006.
DOI : 10.1007/11944836_35

L. Libkin and C. Sirangelo, Reasoning about XML with temporal logics and automata, Journal of Applied Logic, vol.8, issue.2, pp.210-232, 2010.
DOI : 10.1016/j.jal.2009.09.005

L. Lipshitz, The diophantine problem for addition and divisibility. Transaction of the, pp.271-283, 1976.

. Manna, Verifying Balanced Trees, Lecture Notes in Computer Science, vol.4514, issue.24, pp.363-378, 2007.
DOI : 10.1007/978-3-540-72734-7_26

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.129.1385

D. Marx, M. Marx, M. De-rijke, and E. Mendelson, Semantic characterizations of navigational XPath, ACM SIGMOD Record, vol.34, issue.2, pp.41-46, 1964.
DOI : 10.1145/1083784.1083792

A. R. Meyer, Weak monadic second order theory of succesor is not elementary-recursive, Proceedings of Logic Colloquium, pp.132-154, 1975.
DOI : 10.1007/BFb0064872

. Meyer, . Stockmeyer, A. R. Meyer, and L. J. Stockmeyer, The equivalence problem for regular expressions with squaring requires exponential space, 13th Annual Symposium on Switching and Automata Theory (swat 1972), pp.125-129, 1972.
DOI : 10.1109/SWAT.1972.29

S. Miklau, G. Miklau, and D. Suciu, Containment and equivalence for a fragment of XPath, Journal of the ACM, vol.51, issue.1, pp.2-45, 2004.
DOI : 10.1145/962446.962448

A. Møller and M. I. Schwartzbach, The pointer assertion logic engine, Proc. ACM SIGPLAN Conference on Programming Language Design and Implementation, PLDI '01, p.27, 2001.

F. Moller and A. M. Rabinovich, Counting on CTL*: on the expressive power of monadic path logic, Information and Computation, vol.184, issue.1, pp.147-159, 2003.
DOI : 10.1016/S0890-5401(03)00104-4

M. Mortimer, On languages with two variables, Zeitschrift fur mathematische Logik und Grundlagen der Mathematik, 1975.
DOI : 10.1002/malq.19750210118

. Murata, Taxonomy of XML schema languages using formal language theory, ACM Transactions on Internet Technology, vol.5, issue.4, pp.660-704, 2005.
DOI : 10.1145/1111627.1111631

F. Neven, Automata theory for XML researchers, ACM SIGMOD Record, vol.31, issue.3, pp.39-46, 2002.
DOI : 10.1145/601858.601869

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.12.907

. Nguyen, Automated Verification of Shape and Size Properties Via Separation Logic, Lecture Notes in Computer Science, vol.4349, pp.251-266, 2007.
DOI : 10.1007/978-3-540-69738-1_18

D. C. Oppen, A 222pn upper bound on the complexity of Presburger Arithmetic, Journal of Computer and System Sciences, vol.16, issue.3, pp.323-332, 1978.
DOI : 10.1016/0022-0000(78)90021-1

. Pacholski, Complexity of two-variable logic with counting, Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science, pp.318-327, 1997.
DOI : 10.1109/LICS.1997.614958

. Pnueli and A. Pnueli, The temporal logic of programs, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977), pp.46-57, 1977.
DOI : 10.1109/SFCS.1977.32

A. Pnueli, Linear and branching structures in the semantics and logics of reactive systems, Lecture Notes in Computer Science, vol.194, pp.15-32, 1985.
DOI : 10.1007/BFb0015727

V. R. Pratt, Semantical considerations on Floyd-Hoare logic, FOCS, pp.109-121, 1976.
DOI : 10.1109/sfcs.1976.27

V. R. Pratt, A practical decision method for propositional dynamic logic (Preliminary Report), Proceedings of the tenth annual ACM symposium on Theory of computing , STOC '78, pp.326-337, 1978.
DOI : 10.1145/800133.804362

V. R. Pratt, A near-optimal method for reasoning about action, Journal of Computer and System Sciences, vol.20, issue.2, pp.231-254, 1980.
DOI : 10.1016/0022-0000(80)90061-6

. Pratt-hartmann and I. Pratt-hartmann, Complexity of the Two-Variable Fragment with Counting Quantifiers, Journal of Logic, Language and Information, vol.28, issue.4, pp.369-395, 2005.
DOI : 10.1007/s10849-005-5791-1

M. Presburger, ¨ Uber die Vollstandigkeit eines gewissen Systems der Arithmetik, Comptes rendus du I Congrés des Pays Slaves, p.24, 1929.

S. Queille, J. Queille, and J. Sifakis, Specification and verification of concurrent systems in CESAR, Symposium on Programming, pp.337-351, 1982.
DOI : 10.1007/3-540-11494-7_22

M. O. Rabin, Decidability of second order theories and automata on infinite trees, Trans. Amer. Math. Soc, vol.141, pp.1-35, 1969.

J. B. Rosser, Extensions of some theorems of G??del and Church, The Journal of Symbolic Logic, vol.38, issue.03, pp.87-91, 1936.
DOI : 10.1007/BF01565439

K. Schneider, Verification of Reactive Systems: Formal Methods and Algorithms. SpringerVerlag, p.47, 2004.
DOI : 10.1007/978-3-662-10778-2

. Seidl, Numerical document queries, Proceedings of the twenty-second ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems , PODS '03, pp.155-166, 2003.
DOI : 10.1145/773153.773169

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.109.4976

. Seidl, Counting in Trees for Free, Lecture Notes in Computer Science, vol.3142, issue.93, pp.1136-1149, 2004.
DOI : 10.1007/978-3-540-27836-8_94

URL : https://hal.archives-ouvertes.fr/hal-00159525

L. J. Stockmeyer and . Tanabe, The complexity of decision problems in automata theory and logic A decision procedure for the alternation-free twoway modal µ-calculus, Lecture Notes in Computer Science, vol.3702, issue.21, pp.28-277, 1974.

A. Tarski, A lattice-theoretical fixpoint theorem and its applications, Pacific Journal of Mathematics, vol.5, issue.2, pp.285-309, 1955.
DOI : 10.2140/pjm.1955.5.285

C. , M. Cate, B. Marx, and M. , Axiomatizing the logical core of XPath 2.0, Theory Comput. Syst, vol.44, issue.20, pp.561-589, 2009.

W. Thatcher, J. W. Thatcher, and J. B. Wright, Generalized finite automata theory with an application to a decision problem of second-order logic, Mathematical Systems Theory, vol.12, issue.1, pp.57-81, 1968.
DOI : 10.1007/BF01691346

S. Tobies, A PSpace Algorithm for Graded Modal Logic, Lecture Notes in Computer Science, vol.1632, pp.52-66, 1999.
DOI : 10.1007/3-540-48660-7_4

B. A. Trakhtenbrot, On operators realizable in logical nets, Dokl. Akad. Naut. SSSR, vol.112, issue.26, pp.1005-1007, 1957.

A. M. Turing, On computable numbers, with an application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, pp.230-265, 1936.

. Van-benthem and J. Van-benthem, Modal Correspondance Theory, p.28, 1976.

. Zee, Full functional verification of linked data structures, PLDI, pp.349-361, 2008.