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A Hierarchy of Automatic ω-Words having a Decidable MSO Theory

  • Vince Bárány (a1)

Abstract

We investigate automatic presentations of ω-words. Starting points of our study are the works of Rigo and Maes, Caucal, and Carton and Thomas concerning lexicographic presentation, MSO-interpretability in algebraic trees, and the decidability of the MSO theory of morphic words. Refining their techniques we observe that the lexicographic presentation of a (morphic) word is in a certain sense canonical. We then generalize our techniques to a hierarchy of classes of ω-words enjoying the above mentioned definability and decidability properties. We introduce k-lexicographic presentations, and morphisms of level k stacks and show that these are inter-translatable, thus giving rise to the same classes of k-lexicographic or level k morphic words. We prove that these presentations are also canonical, which implies decidability of the MSO theory of every k-lexicographic word as well as closure of these classes under MSO-definable recolorings, e.g. closure under deterministic sequential mappings. The classes of k-lexicographic words are shown to constitute an infinite hierarchy.

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[1] J.-P. Allouche and J. Shallit, Automatic Sequences, Theory, Applications, Generalizations. Cambridge University Press (2003).
[2] J.-M. Autebert and J. Gabarró, Iterated GSMs and Co-CFL. Acta Informatica 26, 749–769 (1989).
[3] V. Bárány, Invariants of automatic presentations and semi-synchronous transductions. In STACS '06. Lect. Notes Comput. Sci. 3884, 289 (2006).
[4] V. Bárány, Automatic Presentations of Infinite Structures. Ph.D. thesis, RWTH Aachen (2007).
[5] J. Berstel, Transductions and Context-Free Languages. Teubner, Stuttgart (1979).
[6] D. Berwanger and A. Blumensath, The monadic theory of tree-like structures. In Automata, Logics, and Infinite Games. Lect. Notes Comput. Sci. 2500, 285–301 (2002).
[7] A. Bès, Undecidable extensions of Büchi arithmetic and Cobham-Semënov theorem. Journal of Symbolic Logic 62, 1280–1296 (1997).
[8] A. Blumensath, Automatic Structures. Diploma thesis, RWTH-Aachen (1999).
[9] A. Blumensath, Axiomatising Tree-interpretable Structures. In STACS. Lect. Notes Comput. Sci. 2285, 596–607 (2002).
[10] A. Blumensath and E. Grädel, Finite presentations of infinite structures: Automata and interpretations. Theor. Comput. Syst. 37, 641–674 (2004).
[11] L. Braud, Higher-order schemes and morphic words. Journées Montoises, Rennes (2006).
[12] V. Bruyère and G. Hansel, Bertrand numeration systems and recognizability. Theoretical Computer Science 181, 17–43 (1997).
[13] V. Bruyère, G. Hansel, Ch. Michaux and R. Villemaire, Logic and p-recognizable sets of integers. Bull. Belg. Math. Soc. Simon Stevin 1, 191–238 (1994).
[14] J.W. Cannon, D.B.A. Epstein, D.F. Holt, S.V.F. Levy, M.S. Paterson and W.P. Thurston, Word processing in groups. Jones and Barlett Publ., Boston, MA (1992).
[15] A. Carayol and A. Meyer, Context-sensitive languages, rational graphs and determinism (2005).
[16] A. Carayol and S. Wöhrle, The Caucal hierarchy of infinite graphs in terms of logic and higher-order pushdown automata. In FSTTCS. Lect. Notes Comput. Sci. 2914, 112–123 (2003).
[17] O. Carton and W. Thomas, The monadic theory of morphic infinite words and generalizations. Information and Computation 176, 51–65 (2002).
[18] D. Caucal, Monadic theory of term rewritings. In LICS, pp. 266–273. IEEE Computer Society (1992).
[19] D. Caucal, On infinite transition graphs having a decidable monadic theory. In ICALP'96. Lect. Notes Comput. Sci. 1099, 194–205 (1996).
[20] D. Caucal, On infinite terms having a decidable monadic theory. In MFCS, pp. 165–176 (2002).
[21] Th. Colcombet, A combinatorial theorem for trees. In ICALP'07. Lect. Notes Comput. Sci. 4596, 901–912 (2007).
[22] Th. Colcombet, On factorisation forests and some applications. arXiv:cs.LO/0701113v1 (2007).
[23] B. Courcelle, The monadic second-order logic of graphs ix: Machines and their behaviours. Theoretical Computer Science 151, 125–162 (1995).
[24] B. Courcelle and I. Walukiewicz, Monadic second-order logic, graph coverings and unfoldings of transition systems. Annals of Pure and Applied Logic 92, 35–62 (1998).
[25] C.C. Elgot and M.O. Rabin, Decidability and undecidability of extensions of second (first) order theory of (generalized) successor. Journal of Symbolic Logic 31, 169–181 (1966).
[26] S. Fratani and G. Sénizergues, Iterated pushdown automata and sequences of rational numbers. Annals of Pure and Applied Logic 141, 363–411, (2006).
[27] Ch. Frougny and J. Sakarovitch, Synchronized rational relations of finite and infinite words. Theoretical Computer Science 108, 45–82 (1993).
[28] E. Grädel, W. Thomas and T. Wilke, Eds. Automata, Logics, and Infinite Games. Lect. Notes Comput. Sci. 2500, (2002).
[29] B.R. Hodgson, Décidabilité par automate fini. Ann. Sci. Math. Québec 7, 39–57 (1983).
[30] J. Honkala and M. Rigo, A note on decidability questions related to abstract numeration systems. Discrete Math. 285, 329–333 (2004).
[31] K. Culik II and J. Karhumäki, Iterative devices generating infinite words. In STACS '92. Lect. Notes Comput. Sci. 577, 529–543 (1992).
[32] L. Kari, G. Rozenberg and A. Salomaa, L systems. In Handbook of Formal Languages, G. Rozenberg and A. Salomaa Eds., volume I, pp. 253–328. Springer, New York (1997).
[33] B. Khoussainov and A. Nerode, Automatic presentations of structures. In LCC '94. Lect. Notes Comput. Sci. 960, 367–392 (1995).
[34] B. Khoussainov and S. Rubin, Automatic structures: Overview and future directions. J. Autom. Lang. Comb. 8, 287–301 (2003).
[35] B. Khoussainov, S. Rubin and F. Stephan, Definability and regularity in automatic structures. In STACS '04. Lect. Notes Comput. Sci. 2996, 440–451 (2004).
[36] T. Lavergne, Prédicats algébriques d'entiers. Rapport de stage, IRISA: Galion (2005).
[37] O. Ly, Automatic Graph and D0L-Sequences of Finite Graphs. Journal of Computer and System Sciences 67, 497–545 (2003).
[38] A. Maes, An automata theoretic decidability proof for first-order theory of $\langle \mathbb{N}, <, P \rangle$ with morphic predicate P. J. Autom. Lang. Comb. 4, 229–245 (1999).
[39] Ch. Morvan and Ch. Rispal, Families of automata characterizing context-sensitive languages. Acta Informatica 41, 293–314 (2005).
[40] D.E. Muller and P.E. Schupp, The theory of ends, pushdown automata, and second-order logic. Theor. Comput. Sci. 37, 51–75 (1985).
[41] J.-J. Pansiot, On various classes of infinite words obtained by iterated mappings. In Automata on Infinite Words, pp. 188–197 (1984).
[42] J.-E. Pin and P.V. Silva, A topological approach to transductions. Theoretical Computer Science 340, 443–456 (2005).
[43] A. Rabinovich, On decidability of monadic logic of order over the naturals extended by monadic predicates. Unpublished note (2005).
[44] A. Rabinovich and W. Thomas, Decidable theories of the ordering of natural numbers with unary predicates. In CSL 2006. Lect. Notes Comput. Sci. 4207, 562–574 (2006).
[45] M. Rigo and A. Maes, More on generalized automatic sequences. J. Autom. Lang. Comb. 7, 351–376 (2002).
[46] Ch. Rispal, The synchronized graphs trace the context-sensistive languages. Elect. Notes Theoret. Comput. Sci. 68 (2002).
[47] G. Rozenberg and A. Salomaa, The Book of L. Springer Verlag (1986).
[48] S. Rubin, Automatic Structures. Ph.D. thesis, University of Auckland, NZ (2004).
[49] S. Rubin, Automata presenting structures: A survey of the finite-string case. Manuscript.
[50] G. Sénizergues, Sequences of level 1, 2, 3,..., k,... In CSR'07. Lect. Notes Comput. Sci. 4649, 24–32 (2007).
[51] S. Shelah, The monadic theory of order. Annals of Mathematics 102, 379–419 (1975).
[52] L. Staiger, Rich omega-words and monadic second-order arithmetic. In CSL, pp. 478–490 (1997).
[53] W. Thomas, Languages, automata, and logic. In Handbook of Formal Languages, G. Rozenberg and A. Salomaa, Eds., Vol. III, pp. 389–455. Springer, New York (1997).
[54] W. Thomas, Constructing infinite graphs with a decidable mso-theory. In MFCS'03. Lect. Notes Comput. Sci. 2747, 113–124 (2003).
[55] I. Walukiewicz, Monadic second-order logic on tree-like structures. Theoretical Computer Science 275, 311–346 (2002).

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A Hierarchy of Automatic ω-Words having a Decidable MSO Theory

  • Vince Bárány (a1)

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