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AN ALGEBRAIC APPROACH TO MSO-DEFINABILITY ON COUNTABLE LINEAR ORDERINGS

Published online by Cambridge University Press:  23 October 2018

OLIVIER CARTON ,
THOMAS COLCOMBET  and
GABRIELE PUPPIS
OLIVIER CARTON
Affiliation:
INSTITUT DE RECHERCHE EN INFORMATIQUE FONDAMENTALE (IRIF) UNIVERSITÉ PARIS DIDEROT PARIS, FRANCE E-mail: olivier.carton@irif.fr
THOMAS COLCOMBET
Affiliation:
CNRS / INSTITUT DE RECHERCHE EN INFORMATIQUE FONDAMENTALE (IRIF) UNIVERSITÉ PARIS DIDEROT PARIS, FRANCE E-mail: thomas.colcombet@irif.fr
GABRIELE PUPPIS
Affiliation:
CNRS / LABORATOIRE BORDELAIS DE RECHERCHE EN INFORMATIQUE (LABRI) UNIVERSITÉ BORDEAUX BORDEAUX, FRANCE E-mail: gabriele.puppis@labri.fr
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Abstract

We develop an algebraic notion of recognizability for languages of words indexed by countable linear orderings. We prove that this notion is effectively equivalent to definability in monadic second-order (MSO) logic. We also provide three logical applications. First, we establish the first known collapse result for the quantifier alternation of MSO logic over countable linear orderings. Second, we solve an open problem posed by Gurevich and Rabinovich, concerning the MSO-definability of sets of rational numbers using the reals in the background. Third, we establish the MSO-definability of the set of yields induced by an MSO-definable set of trees, confirming a conjecture posed by Bruyère, Carton, and Sénizergues.

Keywords

03-XXgeneralized wordsrecognizabilitymonoidslinear orderingsmondaic second-order logic
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 3 , September 2018 , pp. 1147 - 1189
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

REFERENCES

Bedon, N., Bès, A., Carton, O., and Rispal, C., Logic and rational languages of words indexed by linear orderings. Theoretical Computer Science, vol. 46 (2010), no. 4, pp. 737–760.Google Scholar
Bruyère, V. and Carton, O., Automata on linear orderings. Journal of Computer and System Sciences, vol. 73 (2007), no. 1, pp. 1–24.CrossRefGoogle Scholar
Bruyère, V. and Carton, O., Automata on linear orderings, Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science (MFCS) (Sgall, J., Pultr, A., and Kolman, P., editors), Lecture Notes in Computer Science, vol. 2136, Springer, London, 2001, pp. 236–247.Google Scholar
Bruyère, V., Carton, O., and Sénizergues, G., Tree automata and automata on linear orderings. RAIRO Theoretical Informatics and Applications, vol. 43 (2009), pp. 321–338.CrossRefGoogle Scholar
Büchi, J. R., On a decision method in restricted second order arithmetic, Proceedings of the International Congress for Logic, Methodology and Philosophy of Science (Nagel, E., editor), Stanford University Press, Redwood City, CA, 1962, pp. 1–11.Google Scholar
Büchi, J. R., Transfinite automata recursions and weak second order theory of ordinals, Proceedings of the 3rd International Congress of Logic, Methodology, and Philosophy of Science (Rootselaar, B. and Staal, F., editors), North Holland, Amsterdam, 1967, pp. 2–23.Google Scholar
Carton, O., Colcombet, T., and Puppis, G., Regular languages of words over countable linear orderings, Proceedings of the 38th International Colloquium on Automata, Languages and Programming (ICALP) (Aceto, L., Henzinger, M., and Sgall, J., editors), Lecture Notes in Computer Science, vol. 6756, Springer, London, 2011, pp. 125–136.CrossRefGoogle Scholar
Colcombet, T., Factorisation forests for infinite words and applications to countable scattered linear orderings.Theoretical Computer Science, vol. 411 (2010), pp. 751–764.CrossRefGoogle Scholar
Colcombet, T., Composition with algebra at the background – on a question by Gurevich and Rabinovich on the monadic theory of linear orderings, Proceedings of the 8th International Computer Science Symposium in Russia (CSR) (Bulatov, A. A. and Shur, A. M., editors), Lecture Notes in Computer Science, vol. 7913, Springer, London, 2013, pp. 391–404.Google Scholar
Courcelle, B., Frontiers of infinite trees. RAIRO Theoretical Informatics and Applications, vol. 12 (1978), no. 4, pp. 319–337.Google Scholar
Dauchet, M. and Timmerman, E., Continuous monoids and yields of infinite trees. RAIRO Theoretical Informatics and Applications, vol. 20 (1986), no. 3, pp. 251–274.CrossRefGoogle Scholar
Feferman, S. and Vaught, R., The first-order properties of products of algebraic systems. Fundamenta Mathematicae, vol. 47 (1959), pp. 57–103.CrossRefGoogle Scholar
Gécseg, F. and Steinby, M., Tree languages, Handbook of Formal Languages, vol. 3: Beyond Words (Rozenberg, G. and Salomaa, A., editors), Springer, New York, 1997, pp. 1–68.Google Scholar
Gurevich, Y. and Rabinovich, A. M., Definability and undefinability with real order at the background, this JOURNAL, vol. 65 (2000), no. 2, pp. 946–958.Google Scholar
Gurevich, Y. and Shelah, S., Monadic theory of order and topology in ZFC. Annals of Mathematical Logic, vol. 23 (1982), no. 2–3, pp. 179–198.CrossRefGoogle Scholar
Hodges, W., Model Theory, vol. 42, Cambridge University Press, Cambridge, 1993.CrossRefGoogle Scholar
Läuchli, H. and Leonard, J., On the elementary theory of linear order. Fundamenta Mathematicae, vol. 59 (1966), no. 1, pp. 109–116.CrossRefGoogle Scholar
Läuchli, H. and Leonard, J., A decision procedure for the weak second order theory of linear order, Contributions to Mathematical Logic, Proceedings of Logic Colloquium (Arnold Schmidt, H., Schütte, K., and Thiele, H.-J., editors), Studies in Logic and the Foundations of Mathematics, vol. 50, North-Holland, Amsterdam, 1968, pp. 189–197.Google Scholar
Pin, J. E. and Perrin, D., Infinite Words: Automata, Semigroups, Logic and Games, Elsevier Science Publishers, Amsterdam, 2004.Google Scholar
Rabin, M. O., Decidability of second-order theories and automata on infinite trees. Transactions of the American Mathematical Society, vol. 141 (1969), pp. 1–35.Google Scholar
Ramsey, F. P., On a problem of formal logic. Proceedings of the London Mathematical Society, vol. 30 (1929), pp. 264–286.Google Scholar
Rispal, C. and Carton, O., Complementation of rational sets on countable scattered linear orderings. International Journal of Foundations of Computer Science, vol. 16 (2005), no. 4, pp. 767–786.CrossRefGoogle Scholar
Rosenstein, J. G., Linear Orderings, Academic Press, New York, 1982.Google Scholar
Shelah, S., The monadic theory of order. Annals of Mathematics, vol. 102 (1975), pp. 379–419.CrossRefGoogle Scholar
Thatcher, J. W., Characterizing derivation trees of context-free grammars through a generalization of finite automata theory. Journal of Computer and System Sciences, vol. 1 (1967), no. 4, pp. 317–322.CrossRefGoogle Scholar
Thomas, W., On the Ehrenfeucht-Fraïssé game in theoretical computer science, Theory and Practice of Software Development (TAPSOFT) (Gaudel, M.-C. and Jouannaud, J.-P., editors), Lecture Notes in Computer Science, vol. 668, Springer, London, 1993, pp. 559–568.CrossRefGoogle Scholar
Thomas, W., Languages, automata, and logic, Handbook of Formal Languages (Rozenberg, G. and Salomaa, A., editors), vol. 3, Springer, New York, 1997, pp. 389–455.CrossRefGoogle Scholar
Wilke, T., An algebraic theory for regular languages of finite and infinite words. International Journal of Algebra and Computation, vol. 3 (1993), no. 4, pp. 447–489.CrossRefGoogle Scholar