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Abstract

The purpose of this chapter is to give an introduction into languages of infinite strings (of order type ω), so-called ω-languages. The set of all infinite strings over a finite alphabet may be considered, as we shall see below, in a natural way as a metric space.

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References

  1. A. Arnold, A syntactic congruence for rational ω-languages, Theoret. Comput. Sci. 39(1985)2/3, 333–335.

    Google Scholar 

  2. R. Barua, The Hausdorff—Kuratowski hierarchy of ω-regular languages and a hierarchy of Muller automata. Theoret. Comput. Sci. 96 (1992), 345–360.

    Article  MathSciNet  MATH  Google Scholar 

  3. L. Boasson and M. Nivat, Adherences of languages, J. Comput. System Sci. 20(1980) 3, 285–309

    Article  MathSciNet  MATH  Google Scholar 

  4. J. R. Büchi, On a decision method in restricted second order arithmetic. Proc. 1960 Int. Congr. for Logic, Stanford Univ. Press, Stanford 1962, 1–11.

    Google Scholar 

  5. J. R. Büchi and L. H. Landweber, Solving sequential conditions by finitestate strategies. Trans. Amer. Math. Soc. 138 (1969)

    Google Scholar 

  6. Y. Choueka, Theories of automata on ω-tapes: A simplified approach, J. Comput. System Sci. 8(1974), 117–141.

    Article  MathSciNet  MATH  Google Scholar 

  7. E. M. Clarke, I. A. Draghicescu and R. P. Kurshan, A unified approach for showing language containment and equivalence between various types of ω-automata. Inform. Process. Letters 46 (1993) 6, 301–308.

    Article  MathSciNet  MATH  Google Scholar 

  8. R. S. Cohen and A. Y. Gold, Theory of ω-languages I: Characterizations of ω-context-free languages, and II: A study of various models of ω-type generation and recognition, J. Comput. System Sci. 15(1977)2, 169–184 and 185–208.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. S. Cohen and A. Y. Gold, ω-computations on Turing machines, Theoret. Comput. Sci. 6(1978), 1–23.

    Article  MathSciNet  MATH  Google Scholar 

  10. R. S. Cohen and A. Y. Gold, ω-computations on deterministic push down machines, J. Comput. System Sci. (1978) 3, 257–300.

    MathSciNet  Google Scholar 

  11. R. S. Cohen and A. Y. Gold, On the complexity of ω-type Turing acceptors, Theoret. Comput. Sci. 10(1980), 249–272.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Devolder, M. Latteux, I. Litovsky and L. Staiger (1994), Codes and infinite words. Acta Cybernetica, 11 (1995) 4, 241–256.

    MathSciNet  Google Scholar 

  13. L. V. Do, B. LeSaëc and I. Litovsky, Characterizations of rational ω-languages by means of right congruences, Theoret. Comput. Sci. 143 (1995) 1, 1–21.

    MathSciNet  MATH  Google Scholar 

  14. A. J. M. van Engelen, Homogeneous zero-dimensional absolute Borel sets, CWI Tract 27, Amsterdam 1986.

    MATH  Google Scholar 

  15. S. Eilenberg, Automata, Languages and Machines. Vol. A, Academic Press, NewYork 1974.

    MATH  Google Scholar 

  16. J. Engelfriet, H. J. Hoogeboom: X-automata on ω-words. Theoret. Comput. Sci. 110 (1993) 1, 1–51.

    Article  MathSciNet  MATH  Google Scholar 

  17. (A preliminary version appeared as: J. Engelfriet, H. J. Hoogeboom: Automata with storage on infinite words. In: Proc. Automata, Languages and Programming, 16th Coll., (G. Ausiello, M. Dezani-Ciancaglini, S. Ronchi Della Rocco, eds.), Lect. Notes Comput. Sci. 372, Springer-Verlag, Berlin 1989, 289–303.)

    Google Scholar 

  18. S. Ginsburg, Automata-theoretic Properties of Formal Languages, North-Holland, Amsterdam 1975.

    MATH  Google Scholar 

  19. F. Gire, Une extension aux mots infinis de la notion de transduction rationelle, in: Theoret. Comput. Sci., Proc. 6th GI conference (A. B. Cremers and H. P. Kriegel, eds.), Lect. Notes Comput. Sci. 145, Springer-Verlag, Berlin 1984, 123–139.

    Google Scholar 

  20. F. Gire and M. Nivat, Relations rationelles infinitaires, Calcolo 21 (1984), 91–125.

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Hartmanis and R. Stearns, Sets of numbers defined by finite automata. Amer. Math. Monthly 74(1967), 539–542.

    Article  MathSciNet  MATH  Google Scholar 

  22. H. J. Hoogeboom and G. Rozenberg, Infinitary languages: Basic theory and applications to concurrent systems. In: Current Trends in Concurrency. Overviews and Tutorials (J. W. de Bakker, W.-P. de Roever and G. Rozenberg, eds.), Lect. Notes Comput. Sci. 224, Springer-Verlag, Berlin 1986,266–342.

    Chapter  Google Scholar 

  23. J. E. Hoperoft and J. D. Ullman, Introduction to Automata Theory, Languages and Computation, Addison-Wesley, Reading, MA 1979.

    Google Scholar 

  24. T. Head, The topological structure of adherences of regular languages. Infor. Théor. et Appl. 20(1986)1, 31–41.

    Google Scholar 

  25. H. Jürgensen and G. Thierrin, On ω-languages whose syntactic monoid is trivial. Intern. J. Comput. Inform Sci. 12 (1983) 5, 359–365.

    Article  MATH  Google Scholar 

  26. H. Jürgensen and G. Thierrin, Varieties of monoids and classes of ω-languages. In: Proc. of the 1984 Conf. Math. Gesellsch. d. DDR, Berlin 1985, 62–67.

    Google Scholar 

  27. H. Jürgensen and G. Thierrin, Which monoids are syntactic monoids of ω-languages? Elektron. Informationsverarb. Kybernetik EIK 22 (1986) 10/11,513–526.

    MATH  Google Scholar 

  28. M. Kaminski, A classification of ω-regular languages, Theoret. Comput. Sci. 36 (1985), 217–229.

    Article  MathSciNet  MATH  Google Scholar 

  29. K. Kobayashi, M. Takahashi and H. Yamasaki, Characterization of ω-regular languages by first order formulas, Theoret. Comput. Sci. 28 (1984) 3,315–327.

    Article  MathSciNet  MATH  Google Scholar 

  30. K. Kuratowski, Topology I, Academic Press, New York 1966.

    Google Scholar 

  31. L.H. Landweber, Decision problems for ω-automata, Math. Syst. Theory 3(1969) 4, 376–384.

    Google Scholar 

  32. M. Latteux and E Timmermann, Two characterizations of rational adherences, Theoret. Comput. Sci. 46 (1986) 1, 101–106.

    Article  MathSciNet  MATH  Google Scholar 

  33. M. Latteux and E. Timmermann, Finitely generated ω-languages, Inform. Process. Letters 23 (1986) 4, 171–175.

    Article  MATH  Google Scholar 

  34. M. Latteux and E. Timmermann, Rational ω-transductions. in: Math. Found. of Comput. Sci. 1990, Lect. Notes Comput. Sci. 542, Springer-Verlag, Berlin 1990, 407–415.

    Chapter  Google Scholar 

  35. B. LeSaëc, Saturating right congruences, RAIRO Infor. théor. et Appl., 24 (1990), 545–560.

    Google Scholar 

  36. B. LeSaëc and I. Litovsky, On the minimization problem for ω-automata. In: Math. Found. of Comput. Sci. 1994 (I. Prívara, B. Rovan and P. Ružička, eds.), Lect. Notes Comput. Sci. 542, Springer-Verlag, Berlin 1994, 504–514.

    Google Scholar 

  37. R. Lindner and L. Staiger, Algebraische Codierungstheorie — Theorie der sequentiellen Codierungen. Akademie-Verlag, Berlin 1977.

    MATH  Google Scholar 

  38. M. Linna, On ω-sets associated with context-free languages, Inform. Control 31 (1976) 3, 272–293.

    Article  MathSciNet  MATH  Google Scholar 

  39. M. Linna, A decidability result for deterministic ω-context-free languages,Theoret. Comput. Sci. 4 (1977), 83–98.

    Article  MathSciNet  MATH  Google Scholar 

  40. I. Litovsky, Prefix-free languages as ω-generators, Inform. Process. Letters 37 (1991) 1, 61–65.

    Article  MathSciNet  MATH  Google Scholar 

  41. I. Litovsky, Free submonoids and minimal ω-generators of R ω, Acta Cybenetica 10 (1991) 1–2, 35–43.

    MathSciNet  MATH  Google Scholar 

  42. I. Litovsky and E. Timmerman, On generators of rational ω-power languages. Theoret. Comput Sci. 53 (1987) 2/3, 187–200.

    Article  MathSciNet  MATH  Google Scholar 

  43. R. McNaughton, Testing and generating infinite sequences by a finite automaton, Inform. Control 9 (1966), 521–530.

    Article  MathSciNet  MATH  Google Scholar 

  44. O. Maler and A. Pnueli, On the learnability of infinitary regular sets, Information and Computation,118, 316–326, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  45. A preliminary version appeared in: Proc. 4th Annual Workshop on Computational Learning Theory, (L. G. Valiant and M.K. Warmuth, eds.), Santa Cruz, CA, 128–136, Morgan Kaufman, 1991).

    Google Scholar 

  46. O. Maler and L. Staiger, On syntactic congruences for ω-languages. In: STACS ‘83 Proc. 10 th Annual Symposium on Theoretical Computer Science (P. Enjalbert, A. Finkel and K. W. Wagner, eds.), Lect. Notes Comput. Sci. 665, Springer-Verlag, Berlin

    Google Scholar 

  47. T. Moriya and H. Yamasaki, Accepting conditions for automata on ω-words, Theoret. Comput. Sci. 61 (1988), 137–147.

    Article  MathSciNet  MATH  Google Scholar 

  48. D. E. Muller, Infinite sequences and finite machines, in: Proc. 4th Ann. IEEE Symp. Switching Theory and Logical Design, Chicago 1963, 3–16.

    Google Scholar 

  49. Y. N. Moschovacis, Descriptive Set Theory, North-Holland, Amsterdam 1980.

    Google Scholar 

  50. M. Nivat, Mots infinis engendres par une grammaire algebrique, RAIRO Infor. théor. 11 (1977), 311–327.

    MathSciNet  MATH  Google Scholar 

  51. M. Nivat, Sur les ensembles de mots infinis par une grammaire algebraique, RAIRO Infor. théor. 12 (1978), 259–278.

    MathSciNet  MATH  Google Scholar 

  52. M. Nivat, Infinite words, infinite trees, infinite computations, Math. Centre Tracts 109 (1979), 1–52.

    MathSciNet  Google Scholar 

  53. Automata on Infinite Words (M. Nivat and D. Perrin, eds.), Lect. Notes Comput. Sci. 192, Springer-Verlag, Berlin 1985.

    Google Scholar 

  54. D. Park, Concurrency and automata on infinite sequences, in: Theoret. Comput. Sci. (P. Deussen ed.), Lect. Notes Comput. Sci. 104, Springer-Verlag, Berlin 1981, 167–183.

    Chapter  Google Scholar 

  55. D. Perrin, An introduction to finite automata on infinite words. In: [NP85], 1–17.

    Google Scholar 

  56. D. Perrin and J.-E. Pin, Mots Infinis, Report LITP 93.40, Institut Blaise Pascal, Paris, 1993.

    Google Scholar 

  57. T. H. Phan, I. Litovsky and L. V. Do, Which monoids are syntactic monoids of rational ω-languages, Inform. Process. Letters 42(1992), 127–132.

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  59. R. R. Redziejowski, Infinite word languages and continuous mappings. Theoret. Comput. Sci. 43 (1986) 1, 59–79.

    Article  MathSciNet  MATH  Google Scholar 

  60. H. Rogers, Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York 1967.

    MATH  Google Scholar 

  61. S. Safra, On the complexity of ω-automata. In: Proc. 29th IEEE Symp. on Found. of Comput. Sci., 1988, 319–327.

    Google Scholar 

  62. A. Salomaa, Formal Languages, Academic Press, New York 1973.

    MATH  Google Scholar 

  63. V. Selivanov, Fine hierarchy of regular ω-languages. In: TAPSOFT ’95: Theory and Practice of Software Development (P. D. Mosses, M. Nielsen and M. I. Schwartzbach, eds.), Lect. Notes Comput. Sci. 915, Springer-Verlag, Berlin 1995, 277–287.

    Chapter  Google Scholar 

  64. V. Selivanov, Fine hierarchies and Boolean terms. J. Symbolic Logic 80 (1995) 1, 289–317

    Article  MathSciNet  Google Scholar 

  65. L. Staiger, Empty-storage-acceptance of ω-languages. In: Proc. Funda- mentals of Computation Theory ’77 (M. Karpinski, ed.), Lect. Notes Comput. Sci. 56, Springer-Verlag, Berlin 1977, 516–521.

    Chapter  Google Scholar 

  66. L. Staiger, Finite-state ω-languages. J. Comput. System Sci. 27(1983) 3, 434–448.

    Article  MathSciNet  MATH  Google Scholar 

  67. L. Staiger, Hierarchies of recursive ω-languages. J. Inform. Process. Cybernetics EIK 22 (1986) 5/6, 219–241.

    MathSciNet  MATH  Google Scholar 

  68. L. Staiger, ω-computations on Turing machines and the accepted languages. In: Theory of Algorithms (L. Lovász and E. Szemerédi, eds.), Coll. Math. Soc. Janos Bolyai No.44, North-Holland, Amsterdam 1986, 393–403.

    Google Scholar 

  69. L. Staiger, On infinitary finite length codes. RAIRO Infor. théor. et Appl. 20 (1986) 4, 483–494.

    MathSciNet  MATH  Google Scholar 

  70. L. Staiger, Sequential mappings of w-languages. RAIRO Infor. théor. et Appl. 21 (1987) 2, 147–173.

    MathSciNet  MATH  Google Scholar 

  71. L. Staiger, Research in the theory of ω-languages. J. Inform. Process. Cybernetics EIK 23 (1987) 8/9, 415–439.

    MathSciNet  MATH  Google Scholar 

  72. L. Staiger and W. Nehrlich, The centers of context-sensitive languages. In: Proc. Mathematical Foundations of Computer Science’86 (J. Gruska, B. Rovan and J. Wiedermann,eds.), Lect. Notes Comput. Sci. 233, Springer-Verlag, Berlin 1986, 594–601.

    Google Scholar 

  73. L. Staiger and K. Wagner, Automatentheoretische und automatenfreie Charakterisierungen topologischer Klassen regulärer Folgenmengen. Elektron. Informationsverarb. Kybernetik EIK 10 (1974) 7, 379–392.

    MathSciNet  Google Scholar 

  74. L. Staiger and K. Wagner, Rekursive Folgenmengen I. Zeitschr. Math. Logik u. Grundl. Mathematik 24 (1978) 6, 523–538.

    Article  MathSciNet  Google Scholar 

  75. (A preliminary version appeared as: K. Wagner and L. Staiger, Recursive ω-languages. In: Proc. Fundamentals of Computation Theory ’77 (M. Karpinski, ed.), Lect. Notes Comput. Sci. 56, Springer-Verlag, Berlin 1977, 532–537.)

    Google Scholar 

  76. R.S. Streett, Propositional dynamic logic of looping and converse, Inform. Control 54 (1982), 121–141.

    Article  MathSciNet  MATH  Google Scholar 

  77. M. Takahashi and H. Yamasaki, A note on ω-regular languages. Theoret. Comput. Sci. 23 (1983), 217–225.

    Article  MathSciNet  MATH  Google Scholar 

  78. W. Thomas, A combinatorial approach to the theory of ω-automata, Inform. Control 48 (1981) 3, 261–283.

    Article  MATH  Google Scholar 

  79. W. Thomas, Automata on Infinite Objects. In P. J. Van Leeuwen, ed.), Handbook of Theoretical Computer Science, Vol. B, 133–191, Elsevier, Amsterdam, 1990.

    Google Scholar 

  80. W. Thomas, Infinite trees and automaton-definable relations over ω-words. Theoret. Comput. Sci. 103 (1992), 143–159.

    Article  MathSciNet  MATH  Google Scholar 

  81. W. Thomas, Languages, automata, and logic, this volume.

    Google Scholar 

  82. E. Timmerman, The three subfamilies of rational u-languages closed un- der ω-transduction. Theoret. Comput. Sci. 76 (1990), 243–250.

    Article  MathSciNet  MATH  Google Scholar 

  83. B. A. Trakhtenbrot, Finite automata and monadic second order logic. Siberian Math. J. 3 (1962), 103–131.

    MATH  Google Scholar 

  84. (Russian; English translation in: AMS Transl. 59 (1966), 23–55.)

    Google Scholar 

  85. B. A. Trakhtenbrot and Ya. M. Barzdin, Finite Automata, Behaviour and Synthesis. Nauka Publishers, Moscow 1970. (Russian; English translation: North-Holland, Amsterdam 1973)

    Google Scholar 

  86. R. Valk, Infinite behaviour of Petri nets. Theoret. Comput. Sci. 25 (1983), 311–341.

    Article  MathSciNet  MATH  Google Scholar 

  87. Vardi, M. Y., Wolper, P., Reasoning about infinite computations. Inform. Comput. 115 (1994) 1, 1–37.

    Article  MathSciNet  MATH  Google Scholar 

  88. K. Wagner, Eine Axiomatisierung der Theorie der regulären Folgenmengen. Elektron. Informationsverarb. Kybernetik EIK 12 (1976) 7, 337–354.

    MATH  Google Scholar 

  89. K. Wagner, Eine topologische Charakterisierung einiger Klassen regulärer Folgenmengen. Elektron. Informationsverarb. Kybernetik EIK 13(1977) 9, 473–487.

    MATH  Google Scholar 

  90. K. Wagner, On ω-regular sets. Inform. and Control 43 (1979), 123–177.

    Article  MathSciNet  MATH  Google Scholar 

  91. K. Wagner and G. Wechsung, Computational Complexity, Deutscher Verlag der Wissenschaften, Berlin 1986, and Reidel, Dordrecht 1986

    Google Scholar 

  92. R. van Wesep, Wadge degrees and descriptive set theory. In: “Cabal Semi- nar 76–77” (A. S. Kechris and Y. N. Moschovacis, eds.), Lect. Notes Math. 689, Springer-Verlag, Berlin 1978, 151–170.

    Chapter  Google Scholar 

  93. Th. Wilke, An algebraic theory for regular languages of finite and infinite words, Intern. J. Algebra and Computation 3, 447–489, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  94. A preliminary version appeared as: Th. Wilke, An Eilenberg theorem for oc-languages, in: Proc. of the 18th. Intern. Colloquium on Automata, Languages and Programming (J. Albert Leach, B. Monien and M. Rodriguez Artalejo, eds.), 588–599, Lect. Notes Comput. Sci. 510, Springer-Verlag, Berlin 1991.)

    Google Scholar 

  95. Th. Wilke and H. Yoo, Computing the Wadge degree, the Lifschitz degree, and the Rabin index of a regular language of infinite words in polynomial time. In: TAPSOFT ’95: Theory and Practice of Software Development (P. D. Mosses, M. Nielsen and M. I. Schwartzbach, eds.), Lect. Notes Comput. Sci. 915, Springer-Verlag, Berlin 1995, 288–302.

    Chapter  Google Scholar 

  96. H. Yamasaki, M. Takahashi and K. Kobayashi, Characterization of ω-regular languages by monadic second-order formulas, Theoret. Comput. Sci. 46 (1986) 1, 91–99.

    Article  MathSciNet  MATH  Google Scholar 

  97. H Yamasaki, Language-theoretical representations of ω-languages, Theoret. Comput. Sci. 66 (1989), 247–254.

    Article  MathSciNet  MATH  Google Scholar 

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Staiger, L. (1997). ω-Languages. In: Rozenberg, G., Salomaa, A. (eds) Handbook of Formal Languages. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59126-6_6

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