R. Alur and &. P. Madhusudan, Adding nesting structure to words, J. ACM, vol.56, issue.3, p.43, 2009.

P. Blackburn and &. Van-benthem, Modal logic: a semantic perspective, Handbook of Modal Logic, vol.3, pp.80004-80012, 2007.
URL : https://hal.archives-ouvertes.fr/inria-00119856

A. Balliu, D. Gianlorenzo, P. Angelo, . Fraigniaud-&-dennis, and . Olivetti, What Can Be Veriied Locally?, 34th Symposium on Theoretical Aspects of Computer Science, STACS 2017, vol.66, pp.1-8, 2017.

J. Van-benthem, Modal Logic and Classical Logic. Indices: Monographs in Philosophical Logic and Formal Linguistics, vol.3, 1983.

P. Blackburn, . Maarten-de-rijke-&-yde, and . Venema, Modal logic, Cambridge Tracts in Theoretical Computer Science, vol.53, 2002.
URL : https://hal.archives-ouvertes.fr/inria-00100503

J. Braddeld and &. Stirling, Modal mu-calculi, Handbook of Modal Logic, vol.3, pp.80015-80017, 2007.

J. and R. Büchi, Weak Second-Order Arithmetic and Finite Automata. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 6, pp.66-92, 1960.

C. Balder-ten, Expressivity of Second Order Propositional Modal Logic, J. Philosophical Logic, vol.35, issue.2, pp.209-223, 2006.

H. Comon, M. Dauchet, R. Gilleron, F. Jacquemard, C. Löding et al., Bibliography Tree Automata Techniques and Applications. Available at hhp://tata.gforge. inria.fr. Release, 2008.

B. Engelfriet, Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach. Encyclopedia of mathematics and its applications 138, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00646514

N. Chomsky, Three models for the description of language, IRE Trans. Information Theory, vol.2, issue.3, pp.113-124, 1956.

K. Ashok, D. Chandra, . Kozen, J. Larry, and . Stockmeyer, Alternation, J. ACM, vol.28, issue.1, pp.114-133, 1981.

B. Courcelle, The Monadic Second-Order Logic of Graphs. I. Recognizable Sets of Finite Graphs, Inf. Comput, vol.85, issue.1, pp.12-75, 1990.
URL : https://hal.archives-ouvertes.fr/hal-00353765

. Volker-diekert-&-yves and . Métivier, Partial Commutation and Traces, Grzegorz Rozenberg & Arto Salomaa, vol.3, pp.457-533, 1997.

J. Doner, Tree Acceptors and Some of Their Applications, J. Comput. Syst. Sci, vol.4, issue.5, pp.80041-80042, 1970.
DOI : 10.1016/s0022-0000(70)80041-1

URL : https://doi.org/10.1016/s0022-0000(70)80041-1

C. C. Elgot, Decision Problems of Finite Automata Design and Related Arithmetics, Transactions of the American Mathematical Society, vol.98, issue.1, pp.21-51, 1961.
DOI : 10.1090/s0002-9947-1961-0139530-9

URL : https://www.ams.org/tran/1961-098-01/S0002-9947-1961-0139530-9/S0002-9947-1961-0139530-9.pdf

R. Fagin, Generalized First-Order Spectra and Polynomial-Time Recognizable Sets, Complexity of Computation, SIAM-AMS Proceedings, vol.7, pp.43-73, 1974.

R. Fagin, Monadic generalized spectra, Math. Log. Q, vol.21, issue.1, pp.89-96, 1975.
DOI : 10.1002/malq.19750210112

L. Feuilloley-&-pierre-fraigniaud, Survey of Distributed Decision, 2016.

L. Feuilloley and P. Fraigniaud-&-juho-hirvonen, A Hierarchy of Local Decision, 43rd International Colloquium on Automata, Languages, and Programming, vol.55, p.15, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01423644

E. Grädel, G. Phokion, L. Kolaitis, M. Libkin, J. Marx et al., Finite Model Theory and Its Applications, 2007.

D. Giammarresi-&-antonio-restivo, Recognizable Picture Languages. IJPRAI 6(2&3), pp.241-256, 1992.

D. Giammarresi, A. Restivo, S. Seibert-&-wolfgang, and . Thomas, Monadic Second-Order Logic Over Rectangular Pictures and Recognizability by Tiling Systems, Inf. Comput, vol.125, issue.1, pp.32-45, 1996.
DOI : 10.1006/inco.1996.0018

URL : https://doi.org/10.1006/inco.1996.0018

M. Suomela, Locally checkable proofs, Proceedings of the 30th Annual ACM Symposium on Principles of Distributed Computing, PODC 2011, pp.159-168, 2011.

M. Suomela, Locally Checkable Proofs in Distributed Computing, vol.12, pp.1-33, 2016.

L. Hella, M. Järvisalo, A. Kuusisto, J. Laurinharju, T. Lempiäinen et al., Weak models of distributed computing, with connections to modal logic, ACM Symposium on Principles of Distributed Computing, PODC '12, pp.185-194, 2012.
DOI : 10.1007/s00446-013-0202-3

URL : https://helda.helsinki.fi/bitstream/10138/37409/2/weak_models.pdf

L. Hella, M. Järvisalo, A. Kuusisto, J. Laurinharju, T. Lempiäinen et al., Weak models of distributed computing, with connections to modal logic, Distributed Computing, vol.28, issue.1, pp.31-53, 2015.
DOI : 10.1007/s00446-013-0202-3

URL : https://helda.helsinki.fi/bitstream/10138/37409/2/weak_models.pdf

N. Immerman, Descriptive complexity. Graduate texts in computer science, 1999.

A. Korman and &. Shay-kutten, Distributed veriication of minimum spanning trees, Distributed Computing, vol.20, issue.4, pp.253-266, 2007.
DOI : 10.1007/s00446-007-0025-1

C. Stephen and . Kleene, Representation of events in nerve nets and nite automata, Automata studies, pp.3-41, 1956.

A. Kuusisto-&-fabian and . Reiter, Emptiness Problems for Distributed Automata, Proceedings Eighth International Symposium on Games, Automata, Logics and Formal Veriication, vol.256, pp.210-222, 2017.

M. Kutrib, Cellular Automata-A Computational Point of View, New Developments in Formal Languages and Applications, vol.113, pp.183-227, 2008.
DOI : 10.1007/978-3-540-78291-9_6

A. Kuusisto, A modal perspective on monadic second-order alternation hierarchies, Advances in Modal Logic, vol.7, pp.231-247, 2008.

A. Kuusisto, Modal Logic and Distributed Message Passing Automata, LIPIcs 23, Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, pp.452-468, 2013.

A. Kuusisto, Available at hhp://urn, Modal Logics and Deenability. TamPub, pp.1-6, 2013.

A. Kuusisto, Innnite Networks, Halting and Local Algorithms, Proceedings Fifth International Symposium on Games, Automata, Logics and Formal Veriication, vol.161, pp.147-160, 2014.
DOI : 10.4204/eptcs.161.14

URL : https://arxiv.org/pdf/1408.5963

A. Kuusisto, Second-order propositional modal logic and monadic alternation hierarchies, Ann. Pure Appl. Logic, vol.166, issue.1, pp.1-28, 2015.
DOI : 10.1016/j.apal.2014.08.003

G. Lenzi, The Modal µ-Calculus: a Survey, TASK Quarterly-Scientiic Bulletin of the Academic Computer Centre in Gdansk, vol.9, issue.3, pp.293-316, 2005.

L. Libkin, Elements of Finite Model Theory. Texts in Theoretical Computer Science, An EATCS Series, 2004.

C. Löding, Modern Applications of Automata Theory, vol.2, pp.79-109, 2012.

C. Löding and &. Thomas, Alternating Automata and Logics over Innnite Words, Theoretical Computer Science, Exploring New Frontiers of Theoretical Informatics, International Conference IFIP TCS, pp.521-535, 2000.

N. A. Lynch, Distributed Algorithms, 1996.

O. Matz, Dot-depth, monadic quantiier alternation, and rstorder closure over grids and pictures, Theor. Comput. Sci, vol.270, issue.1-2, pp.1-70, 2002.
DOI : 10.1016/s0304-3975(01)00277-8

URL : https://doi.org/10.1016/s0304-3975(01)00277-8

O. Matz, N. Schweikardt, and W. Thomas, The Monadic Quantiier Alternation Hierarchy over Grids and Graphs, Inf. Comput, vol.179, issue.2, pp.356-383, 2002.
DOI : 10.1006/inco.2002.2955

URL : https://doi.org/10.1006/inco.2002.2955

O. Matz and &. Thomas, The Monadic Quantiier Alternation Hierarchy over Graphs is Innnite, Proceedings, 12th Annual IEEE Symposium on Logic in Computer Science, pp.236-244, 1997.
DOI : 10.1109/lics.1997.614951

A. Nerode, Linear automaton transformations, Proc. Amer. Math. Soc, vol.9, pp.541-544, 1958.
DOI : 10.2307/2033204

URL : https://www.ams.org/proc/1958-009-04/S0002-9939-1958-0135681-9/S0002-9939-1958-0135681-9.pdf

D. Peleg, Distributed Computing: A Locality-Sensitive Approach. SIAM Monographs on Discrete Mathematics and Applications 5, 2000.
DOI : 10.1137/1.9780898719772

F. Reiter, Distributed Graph Automata, 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, pp.192-201, 2015.
DOI : 10.1109/lics.2015.27

URL : http://arxiv.org/pdf/1404.6503

F. Reiter, Alternating Set Quantiiers in Modal Logic, 2016.

F. Reiter, Asynchronous Distributed Automata: A Characterization of the Modal Mu-Fragment, vol.80, p.14, 2017.

T. Schwentick-&-klaus-barthelmann, Local Normal Forms for First-Order Logic with Applications to Games and Automata, Discrete Mathematics & Theoretical Computer Science, vol.3, issue.3, pp.109-124, 1999.

N. Schweikardt, The Monadic Quantiier Alternation Hierarchy over Grids and Pictures, Computer Science Logic, 11th International Workshop, CSL '97, Annual Conference of the EACSL, pp.441-460, 1414.
DOI : 10.1007/bfb0028030

J. Suomela, Survey of local algorithms, ACM Comput. Surv, vol.45, issue.2, 2013.

V. Terrier, Language Recognition by Cellular Automata, Handbook of Natural Computing, pp.123-158, 2012.
DOI : 10.1007/978-3-540-92910-9_4

URL : https://hal.archives-ouvertes.fr/hal-01086432

W. Thomas, Automata, Languages and Programming, 18th International Colloquium, ICALP91, vol.510, pp.441-454, 1991.

W. Thomas, Elements of an automata theory over partial orders, Partial Order Methods in Veriication, Proceedings of a DIMACS Workshop, vol.29, pp.25-40, 1996.

W. Thomas, Automata Theory on Trees and Partial Orders, TAPSOFT'97: Theory and Practice of Software Development, 7th International Joint Conference CAAP/FASE, pp.20-38, 1997.

W. Thomas, Grzegorz Rozenberg & Arto Salomaa, vol.3, pp.389-455, 1997.

B. A. Trakhtenbrot, Finite automata and the logic of single-place predicates, Soviet Physics Dokl, vol.6, pp.753-755, 1961.

J. W. Thatcher-&-jesse and B. Wright, Generalized Finite Automata Theory with an Application to a Decision Problem of Second-Order Logic, Mathematical Systems Theory, vol.2, issue.1, p.51, 1968.

V. , , vol.24, pp.85-89

G. , , vol.12, p.81

?. , , vol.17, p.83

?. , , vol.16, p.46

R. , , vol.16, pp.84-86

R. , , vol.18, p.89

, enriched run, 44-48 enriched run segment, 45, 46 existential connguration, 24-26, 30 existential monadic second-order logic, vol.14, p.68