The mu-calculus and Model Checking

Chapter

Abstract

This chapter presents that part of the theory of the \(\mu\)-calculus that is relevant to the model-checking problem as broadly understood. The \(\mu\)-calculus is one of the most important logics in model checking. It is a logic with an exceptional balance between expressiveness and algorithmic properties.

The chapter describes at length the game characterization of the semantics of the \(\mu\)-calculus. It discusses the theory of the \(\mu\)-calculus starting with the tree-model property, and bisimulation invariance. Then it develops the notion of modal automaton: an automaton-based model behind the \(\mu\)-calculus. It gives a quite detailed explanation of the satisfiability algorithm, followed by results on alternation hierarchy, proof systems, and interpolation. Finally, the chapter discusses the relation of the \(\mu\)-calculus to monadic second-order logic as well as to some program and temporal logics. It also presents two extensions of the \(\mu\)-calculus that allow us to address issues such as inverse modalities.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    de Alfaro, L., Majumdar, R.: Quantitative solution of omega-regular games. J. Comput. Syst. Sci. 68(2), 374–397 (2004) MathSciNetMATHGoogle Scholar
  2. 2.
    Alur, R., Henzinger, T., Kupferman, O.: Alternating-time temporal logic. In: Int. Symp. on Foundations of Computer Science, pp. 100–109. IEEE, Piscataway (1997) Google Scholar
  3. 3.
    Andersen, H.: Model checking boolean graphs. Theor. Comput. Sci. 126(1), 3–30 (1994) MATHGoogle Scholar
  4. 4.
    Andersen, H.R.: Partial model checking. In: Ann. Symp. on Logic in Computer Science, pp. 398–407. IEEE, Piscataway (1995) Google Scholar
  5. 5.
    Andréka, H., van Benthem, J., Neméti, I.: Modal logics and bounded fragments of predicate logic. J. Philos. Log. 27, 217–274 (1998) MATHGoogle Scholar
  6. 6.
    Arnold, A.: The mu-calculus alternation-depth hierarchy is strict on binary trees. RAIRO Theor. Inform. Appl. 33, 329–339 (1999) MATHGoogle Scholar
  7. 7.
    Arnold, A., Crubille, P.: A linear time algorithm to solve fixpoint equations on transition systems. Inf. Process. Lett. 29, 57–66 (1988) MATHGoogle Scholar
  8. 8.
    Arnold, A., Niwiński, D.: Rudiments of \(\mu\)-Calculus. Elsevier, Amsterdam (2001) MATHGoogle Scholar
  9. 9.
    Arnold, A., Walukiewicz, I.: Nondeterministic controllers of nondeterministic processes. In: Flum, J., Grädel, E., Wilke, T. (eds.) Logic and Automata, Texts in Logic and Games, vol. 2, pp. 29–52. Amsterdam University Press, Amsterdam (2007) Google Scholar
  10. 10.
    Bekic, H.: Definable operation in general algebras, and the theory of automata and flowcharts. In: Jones, C.B. (ed.) Programming Languages and Their Definition—Hans Bekic (1936–1982). LNCS, vol. 177, pp. 30–55. Springer, Heidelberg (1984) Google Scholar
  11. 11.
    Benedikt, M., Segoufin, L.: Regular tree languages definable in FO and in FOmod. Trans. Comput. Log. 11(1), 4:1–4:32 (2009) MATHGoogle Scholar
  12. 12.
    Berwanger, D.: Game logic is strong enough for parity games. Stud. Log. 75(2), 205–219 (2003) MathSciNetMATHGoogle Scholar
  13. 13.
    Berwanger, D., Grädel, E., Kaiser, L., Rabinovich, R.: Entanglement and the complexity of directed graphs. Theor. Comput. Sci. 463, 2–25 (2012) MathSciNetMATHGoogle Scholar
  14. 14.
    Berwanger, D., Grädel, E., Lenzi, G.: The variable hierarchy of the mu-calculus is strict. Theory Comput. Syst. 40(4), 437–466 (2007) MathSciNetMATHGoogle Scholar
  15. 15.
    Berwanger, D., Serre, O.: Parity games on undirected graphs. Inf. Process. Lett. 112(23), 928–932 (2012) MathSciNetMATHGoogle Scholar
  16. 16.
    Bhat, G., Cleaveland, R.: Efficient model checking via the equational \(\mu\)-calculus. In: Clarke, E.M. (ed.) Ann. Symp. on Logic in Computer Science, pp. 304–312. IEEE, Piscataway (1996) Google Scholar
  17. 17.
    Björklund, H., Vorobyov, S.G.: A combinatorial strongly subexponential strategy improvement algorithm for mean payoff games. Discrete Appl. Math. 155(2), 210–229 (2007) MathSciNetMATHGoogle Scholar
  18. 18.
    Blackburn, R., de Rijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001) MATHGoogle Scholar
  19. 19.
    Bloem, R., Chatterjee, K., Jobstmann, B.: Graph games and reactive synthesis. In: Clarke, E.M., Henzinger, T.A., Veith, H., Bloem, R. (eds.) Handbook of Model Checking. Springer, Heidelberg (2018) Google Scholar
  20. 20.
    Blumensath, A., Kreutzer, S.: An extension to Muchnik’s theorem. J. Log. Comput. 13, 59–74 (2005) MathSciNetMATHGoogle Scholar
  21. 21.
    Bojanczyk, M., Segoufin, L., Straubing, H.: Piecewise testable tree languages. Log. Methods Comput. Sci. 8(3) (2012) Google Scholar
  22. 22.
    Bojanczyk, M., Straubing, H., Walukiewicz, I.: Wreath products of forest algebras, with applications to tree logics. Log. Methods Comput. Sci. 3, 19 (2012) MathSciNetMATHGoogle Scholar
  23. 23.
    Bojanczyk, M., Walukiewicz, I.: Characterizing EF and EX tree logics. Theor. Comput. Sci. 358(2–3), 255–272 (2006) MathSciNetMATHGoogle Scholar
  24. 24.
    Bojanczyk, M., Walukiewicz, I.: Forest algebras. In: Flum, J., Grädel, E., Wilke, T. (eds.) Logic and Automata, Texts in Logic and Games, vol. 2, pp. 107–132. Amsterdam University Press, Amsterdam (2007) Google Scholar
  25. 25.
    Bonatti, P.A., Lutz, C., Murano, A., Vardi, M.Y.: The complexity of enriched mu-calculi. Log. Methods Comput. Sci. 3, 11 (2008) MATHGoogle Scholar
  26. 26.
    Bouyer, P., Cassez, F., Laroussinie, F.: Timed modal logics for real-time systems: specification, verification and control. J. Log. Lang. Inf. 20, 169–203 (2011) MathSciNetMATHGoogle Scholar
  27. 27.
    Bradfield, J.: The modal mu-calculus alternation hierarchy is strict. Theor. Comput. Sci. 195, 133–153 (1997) MathSciNetMATHGoogle Scholar
  28. 28.
    Bradfield, J., Stirling, C.: Modal mu-calculi. In: Blackburn, P., van Benthem, J., Wolter, F. (eds.) The Handbook of Modal Logic, pp. 721–756. Elsevier, Amsterdam (2006) Google Scholar
  29. 29.
    Broadbent, C., Carayol, A., Ong, L., Serre, O.: Recursion schemes and logical reflection. In: Ann. Symp. on Logic in Computer Science, pp. 120–129. IEEE, Piscataway (2010) Google Scholar
  30. 30.
    Carayol, A., Wöhrle, S.: The Caucal hierarchy of infinite graphs in terms of logic and higher-order pushdown automata. In: Pandya, P.K., Radhakrishnan, J. (eds.) Intl. Conf. on Foundations of Software Technology and Theoretical Computer Science. LNCS, vol. 2914, pp. 112–124. Springer, Heidelberg (2003) MATHGoogle Scholar
  31. 31.
    Chatterjee, K., Henzinger, M.: An \({O}(n^{2})\) time algorithm for alternating Büchi games. In: Indyk, P. (ed.) Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, Philadelphia (2012) Google Scholar
  32. 32.
    Church, A.: Applications of recursive arithmetic to the problem of circuit synthesis. In: Summaries of the Summer Institute of Symbolic Logic, vol. I, pp. 3–50. Cornell University, Ithaca (1957) Google Scholar
  33. 33.
    Cleaveland, R., Steffen, B.: A linear model checking algorithm for the alternation-free modal \(\mu\)-caluclus. Form. Methods Syst. Des. 2, 121–147 (1993) MATHGoogle Scholar
  34. 34.
    Colcombet, T., Zdanowski, K.: A tight lower bound for determinization of transition labeled Büchi automata. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) International Colloquium on Automata, Languages and Programming. LNCS, vol. 5556, pp. 151–162. Springer, Heidelberg (2009) MATHGoogle Scholar
  35. 35.
    Condon, A.: The complexity of stochastic games. Inf. Comput. 96(2), 203–224 (1992) MathSciNetMATHGoogle Scholar
  36. 36.
    D’Agostino, G., Hollenberg, M.: Logical questions concerning the mu-calculus: interpolation, Lyndon and Łoś-Tarski. J. Symb. Log. 65(1), 310–332 (2000) MATHGoogle Scholar
  37. 37.
    D’Agostino, G., Lenzi, G.: On modal mu-calculus over reflexive symmetric graphs. J. Log. Comput. 23(3), 445–455 (2013) MATHGoogle Scholar
  38. 38.
    Dam, M.: CTL and ECTL as fragments of the modal \(\mu \)-calculus. Theor. Comput. Sci. 126(1), 77–96 (1994) MATHGoogle Scholar
  39. 39.
    Dawar, A., Grädel, E., Kreutzer, S.: Inflationary fixed points in modal logic. Trans. Comput. Log. 5(2), 282–315 (2004) MathSciNetMATHGoogle Scholar
  40. 40.
    Ebbinghaus, H.D., Flum, J.: Finite Model Theory. Springer, Heidelberg (1999) MATHGoogle Scholar
  41. 41.
    Ebbinghaus, H.D., Flum, J., Thomas, W.: Mathematical Logic. Springer, New York (1984) MATHGoogle Scholar
  42. 42.
    Emerson, E., Jutla, C., Sistla, A.: On model-checking for the mu-calculus and its fragments. Theor. Comput. Sci. 258(1–2), 491–522 (2001) MATHGoogle Scholar
  43. 43.
    Emerson, E.A.: Temporal and modal logic. In: Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 995–1072. Elsevier, Amsterdam (1990) Google Scholar
  44. 44.
    Emerson, E.A., Jutla, C.S.: The complexity of tree automata and logics of programs. In: Int. Symp. on Foundations of Computer Science, pp. 328–337. IEEE, Piscataway (1988) Google Scholar
  45. 45.
    Emerson, E.A., Jutla, C.S.: Tree automata, mu-calculus and determinacy. In: Int. Symp. on Foundations of Computer Science, pp. 368–377. IEEE, Piscataway (1991) Google Scholar
  46. 46.
    Emerson, E.A., Jutla, C.S.: The complexity of tree automata and logics of programs. SIAM J. Comput. 29(1), 132–158 (1999) MathSciNetMATHGoogle Scholar
  47. 47.
    Emerson, E.A., Lei, C.: Efficient model checking in fragments of propositional mu-calculus. In: Ann. Symp. on Logic in Computer Science, pp. 267–278. IEEE, Piscataway (1986) Google Scholar
  48. 48.
    Fischer, D., Grädel, E., Kaiser, L.: Model checking games for the quantitative mu-calculus. Theory Comput. Syst. 47, 696–719 (2010) MathSciNetMATHGoogle Scholar
  49. 49.
    Fisher, M., Ladner, R.: Propositional modal logic of programs. In: Hopcroft, J.E., Friedman, E.P., Harrison, M.A. (eds.) Annual Symp. on the Theory of Computing, pp. 286–294. ACM, New York (1977) Google Scholar
  50. 50.
    Fisher, M., Ladner, R.: Propositional dynamic logic of regular programs. J. Comput. Syst. Sci. 18, 194–211 (1979) MathSciNetMATHGoogle Scholar
  51. 51.
    Friedmann, O.: An exponential lower bound for the latest deterministic strategy iteration algorithms. Log. Methods Comput. Sci. 3, 23 (2011) MathSciNetMATHGoogle Scholar
  52. 52.
    Friedmann, O., Hansen, T.D., Zwick, U.: A subexponential lower bound for the random facet algorithm for parity games. In: Randall, D. (ed.) Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 202–216. SIAM, Philadelphia (2011) Google Scholar
  53. 53.
    Friedmann, O., Lange, M.: The modal mu-calculus caught off guard. In: Brünnler, K., Metcalfe, G. (eds.) Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX). LNCS, vol. 6793, pp. 149–163. Springer, Heidelberg (2011) Google Scholar
  54. 54.
    Gastin, P., Oddoux, D.: Fast LTL to Büchi automata translation. In: Berry, G., Comon, H., Finkel, A. (eds.) Intl. Conf. on Computer-Aided Verification (CAV). LNCS, vol. 2102, pp. 53–65. Springer, Heidelberg (2001) Google Scholar
  55. 55.
    Gimbert, H., Zielonka, W.: Perfect information stochastic priority games. In: Arge, L., Cachin, C., Jurdzinski, T., Tarlecki, A. (eds.) International Colloquium on Automata, Languages and Programming. LNCS, vol. 4596, pp. 850–861. Springer, Heidelberg (2007) Google Scholar
  56. 56.
    Grädel, E.: Guarded fixed point logics and the monadic theory of countable trees. Theor. Comput. Sci. 288(1), 129–152 (2002) MathSciNetMATHGoogle Scholar
  57. 57.
    Grädel, E., Hirsch, C., Otto, M.: Back and forth between guarded and modal logics. Trans. Comput. Log. 3(3), 418–463 (2002) MathSciNetMATHGoogle Scholar
  58. 58.
    Grädel, E., Kolaitis, P., Libkin, L., Marx, M., Spencer, J., Vardi, M., Venema, Y., Weinstein, S.: Finite Model Theory and Its Applications. Springer, Heidelberg (2007) MATHGoogle Scholar
  59. 59.
    Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games, vol. 2500. Springer, Heidelberg (2002) MATHGoogle Scholar
  60. 60.
    Grädel, E., Walukiewicz, I.: Guarded fixed point logic. In: Ann. Symp. on Logic in Computer Science, pp. 45–55. IEEE, Piscataway (1999) Google Scholar
  61. 61.
    Harel, D.: Dynamic logic. In: Gabbay, D., Guenther, F. (eds.) Handbook of Philosophical Logic, Vol. II, pp. 497–604. Reidel, Dordrecht (1984) MATHGoogle Scholar
  62. 62.
    Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. MIT Press, Cambridge (2000) MATHGoogle Scholar
  63. 63.
    Henzinger, T.A., Nicollin, X., Sifakis, J., Yovine, S.: Symbolic model checking for real-time systems. Inf. Comput. 111(2), 193–244 (1994) MathSciNetMATHGoogle Scholar
  64. 64.
    Hitchcock, P., Park, D.: Induction rules and termination proofs. In: Nivat, M. (ed.) International Colloquium on Automata, Languages and Programming, pp. 225–251 (1973) Google Scholar
  65. 65.
    Hoffman, A., Karp, R.: On nonterminating stochastic games. Manag. Sci. 12, 359–370 (1966) MathSciNetMATHGoogle Scholar
  66. 66.
    Immerman, N.: Descriptive Complexity. Springer, Heidelberg (1999) MATHGoogle Scholar
  67. 67.
    Janin, D., Lenzi, G.: On the relationship between monadic and weak monadic second order logic on arbitrary trees, with applications to the mu-calculus. Fundam. Inform. 61(3–4), 247–265 (2004) MathSciNetMATHGoogle Scholar
  68. 68.
    Janin, D., Walukiewicz, I.: Automata for the mu-calculus and related results. In: Wiedermann, J., Hájek, P. (eds.) International Symposium on Mathematical Foundations of Computer Science. LNCS, vol. 969, pp. 552–562. Springer, Heidelberg (1995) Google Scholar
  69. 69.
    Janin, D., Walukiewicz, I.: On the expressive completeness of the propositional mu-calculus with respect to monadic second order logic. In: Montanari, U., Sassone, V. (eds.) Intl. Conf. on Concurrency Theory (CONCUR). LNCS, vol. 1119, pp. 263–277. Springer, Heidelberg (1996) Google Scholar
  70. 70.
    Jurdziński, M.: Deciding the winner in parity games is in UP∩co-UP. Inf. Process. Lett. 68(3), 119–124 (1998) MathSciNetMATHGoogle Scholar
  71. 71.
    Jurdziński, M.: Small progress measures for solving parity games. In: Reichel, H., Tison, S. (eds.) Annual Symposium on Theoretical Aspects of Computer Science. LNCS, vol. 1770, pp. 290–301. Springer, Heidelberg (2000) Google Scholar
  72. 72.
    Jurdzinski, M., Paterson, M., Zwick, U.: A deterministic subexponential algorithm for solving parity games. SIAM J. Comput. 38(4), 1519–1532 (2008) MathSciNetMATHGoogle Scholar
  73. 73.
    Knapik, T., Niwinski, D., Urzyczyn, P.: Higher-order pushdown trees are easy. In: Nielsen, M., Engberg, U. (eds.) Intl. Conf. on Foundations of Software Science and Computational Structures (FoSSaCS). LNCS, vol. 2303, pp. 205–222. Springer, Heidelberg (2002) Google Scholar
  74. 74.
    Kobayashi, N.: Types and higher-order recursion schemes for verification of higher-order programs. In: Shao, Z., Pierce, B.C. (eds.) Ann. ACM Symp. on Principles of Programming Languages, pp. 416–428. ACM, New York (2009) Google Scholar
  75. 75.
    Kozen, D.: Results on the propositional mu-calculus. Theor. Comput. Sci. 27, 333–354 (1983) MATHGoogle Scholar
  76. 76.
    Kupferman, O.: Automata theory and model checking. In: Clarke, E.M., Henzinger, T.A., Veith, H., Bloem, R. (eds.) Handbook of Model Checking. Springer, Heidelberg (2018) Google Scholar
  77. 77.
    Libkin, L.: Elements of Finite Model Theory. Springer, Heidelberg (2004) MATHGoogle Scholar
  78. 78.
    Long, D.E., Browne, A., Clarke, E.M., Jha, S., Marrero, W.R.: An improved algorithm for the evaluation of fixpoint expressions. In: Dill, D.L. (ed.) Intl. Conf. on Computer-Aided Verification (CAV). LNCS, vol. 818, pp. 338–350. Springer, Heidelberg (1994) Google Scholar
  79. 79.
    Maksimova, L.L.: Absence of interpolation and of Beth’s property in temporal logics with “the next” operation. Sib. Math. J. 32(6), 109–113 (1991) MATHGoogle Scholar
  80. 80.
    Martin, D.: Borel determinacy. Ann. Math. 102, 363–371 (1975) MathSciNetMATHGoogle Scholar
  81. 81.
    McIver, A., Morgan, C.: Results on the quantitative \(\mu\)-calculus qM\(\mu\). Trans. Comput. Log. 8(1) (2007) Google Scholar
  82. 82.
    McMillan, K.L.: Interpolation and model checking. In: Clarke, E.M., Henzinger, T.A., Veith, H., Bloem, R. (eds.) Handbook of Model Checking. Springer, Heidelberg (2018) Google Scholar
  83. 83.
    McNaughton, R.: Infinite games played on finite graphs. Ann. Pure Appl. Log. 65, 149–184 (1993) MathSciNetMATHGoogle Scholar
  84. 84.
    Mio, M.: Game semantics for probabilistic mu-calculi. Ph.D. thesis, University of Edinburgh (2012) Google Scholar
  85. 85.
    Moschovakis, Y.: Elementary Induction on Abstract Structures. North-Holland, Amsterdam (1974) MATHGoogle Scholar
  86. 86.
    Moss, L.S.: Coalgebraic logic. Ann. Pure Appl. Log. 96, 277–317 (1999). Erratum published Ann. Pure Appl. Log. 99, 241–259 (1999) MathSciNetMATHGoogle Scholar
  87. 87.
    Mostowski, A.W.: Regular expressions for infinite trees and a standard form of automata. In: Skowron, A. (ed.) Fifth Symposium on Computation Theory. LNCS, vol. 208, pp. 157–168. Springer, Heidelberg (1984) Google Scholar
  88. 88.
    Mostowski, A.W.: Games with forbidden positions. Tech. Rep. 78, University of Gdansk (1991) Google Scholar
  89. 89.
    Muller, D., Schupp, P.: Alternating automata on infinite trees. Theor. Comput. Sci. 54, 267–276 (1987) MathSciNetMATHGoogle Scholar
  90. 90.
    Niwiński, D.: Fixed points vs. infinite generation. In: Ann. Symp. on Logic in Computer Science, pp. 402–409. IEEE, Piscataway (1988) Google Scholar
  91. 91.
    Obdrzálek, J.: Clique-width and parity games. In: Duparc, J., Henzinger, T.A. (eds.) Intl. Workshop Computer Science Logic (CSL). LNCS, vol. 4646, pp. 54–68. Springer, Heidelberg (2007) Google Scholar
  92. 92.
    Ong, C.H.L.: On model-checking trees generated by higher-order recursion schemes. In: Ann. Symp. on Logic in Computer Science, pp. 81–90. IEEE, Piscataway (2006) Google Scholar
  93. 93.
    Parikh, R.: The logic of games and its applications. Ann. Discrete Math. 24, 111–140 (1985) MathSciNetMATHGoogle Scholar
  94. 94.
    Park, D.: Finiteness is \(\mu\)-ineffable. Theor. Comput. Sci. 3, 173–181 (1976) MATHGoogle Scholar
  95. 95.
    Piterman, N.: From nondeterministic Büchi and Streett automata to deterministic parity automata. Log. Methods Comput. Sci. 3(3), 1–21 (2007) MathSciNetMATHGoogle Scholar
  96. 96.
    Place, T., Segoufin, L.: A decidable characterization of locally testable tree languages. Log. Methods Comput. Sci. 7(4) (2011) Google Scholar
  97. 97.
    Pratt, V.: A decidable \(\mu\)-calculus: preliminary report. In: Int. Symp. on Foundations of Computer Science, pp. 421–427. IEEE, Piscataway (1981) Google Scholar
  98. 98.
    Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Trans. Am. Math. Soc. 141, 1–23 (1969) MathSciNetMATHGoogle Scholar
  99. 99.
    Rabin, M.O.: Weakly definable relations and special automata. In: Mathematical Logic and Foundations of Set Theory, pp. 1–23 (1970) Google Scholar
  100. 100.
    Safra, S.: On the complexity of \(\omega\)-automata. In: Int. Symp. on Foundations of Computer Science. IEEE, Piscataway (1988) Google Scholar
  101. 101.
    Salwicki, A.: Formalized algorithmic languages. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 18, 227–232 (1970) MathSciNetMATHGoogle Scholar
  102. 102.
    Schewe, S.: Solving parity games in big steps. In: Arvind, V., Prasad, S. (eds.) Intl. Conf. on Foundations of Software Technology and Theoretical Computer Science. LNCS, vol. 485, pp. 449–460. Springer, Heidelberg (2007) Google Scholar
  103. 103.
    Schewe, S.: An optimal strategy improvement algorithm for solving parity and payoff games. In: Kaminski, M., Martini, S. (eds.) Intl. Workshop Computer Science Logic (CSL). LNCS, vol. 5213, pp. 369–384. Springer, Heidelberg (2008) Google Scholar
  104. 104.
    Schewe, S.: Büchi complementation made tight. In: Albers, S., Marion, J. (eds.) Annual Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics, vol. 3, pp. 661–672. Schloss Dagstuhl—Leibniz-Zentrum fuer Informatik, Dagstuhl (2009) Google Scholar
  105. 105.
    Scott, D., de Bakker, J.: A theory of programs. (1969). Unpublished notes, IBM, Vienna (1969) Google Scholar
  106. 106.
    Seidl, H.: Deciding equivalence of finite tree automata. SIAM J. Comput. 19, 424–437 (1990) MathSciNetMATHGoogle Scholar
  107. 107.
    Seidl, H.: Fast and simple nested fixpoints. Inf. Process. Lett. 59(6), 303–308 (1996) MathSciNetMATHGoogle Scholar
  108. 108.
    Seidl, H., Neumann, A.: On guarding nested fixpoints. In: Flum, J., Rodríguez-Artalejo, M. (eds.) Intl. Workshop Computer Science Logic (CSL). LNCS, vol. 1683, pp. 484–498. Springer, Heidelberg (1999) Google Scholar
  109. 109.
    Semenov, A.: Decidability of monadic theories. In: Chytil, M., Koubek, V. (eds.) International Symposium on Mathematical Foundations of Computer Science. LNCS, vol. 176, pp. 162–175. Springer, Heidelberg (1984) Google Scholar
  110. 110.
    Streett, R.S.: Propositional dynamic logic of looping and converse is elementarily decidable. Inf. Control 54, 121–141 (1982) MathSciNetMATHGoogle Scholar
  111. 111.
    Streett, R.S., Emerson, E.A.: The propositional mu-calculus is elementary. In: Paredaens, J. (ed.) International Colloquium on Automata, Languages and Programming. LNCS, vol. 172, pp. 465–472. Springer, Heidelberg (1984) Google Scholar
  112. 112.
    Streett, R.S., Emerson, E.A.: An automata theoretic procedure for the propositional mu-calculus. Inf. Comput. 81, 249–264 (1989) MathSciNetMATHGoogle Scholar
  113. 113.
    Thomas, W.: Languages, automata, and logic. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. III, pp. 389–455. Springer, Heidelberg (1997) Google Scholar
  114. 114.
    Thomas, W.: Infinite games and verification. In: Brinksma, E., Larsen, K.G. (eds.) Intl. Conf. on Computer-Aided Verification (CAV). LNCS, vol. 2404, pp. 58–64. Springer, Heidelberg (2002) Google Scholar
  115. 115.
    Thomas, W.: Constructing infinite graphs with a decidable MSO-theory. In: Rovan, B., Vojtás, P. (eds.) International Symposium on Mathematical Foundations of Computer Science. LNCS, vol. 2747, pp. 113–124. Springer, Heidelberg (2003) Google Scholar
  116. 116.
    Thomas, W.: Church’s problem and a tour through automata theory. In: Avron, A., Dershowitz, N., Rabinovich, A. (eds.) Pillars of Computer Science, Lecture Notes in Computer Science, vol. 4800, pp. 635–655. Springer, Heidelberg (2008) Google Scholar
  117. 117.
    van Benthem, J.: Modal Logic and Classical Logic. Bibliopolis, Napoli (1983) MATHGoogle Scholar
  118. 118.
    Vardi, M.: Reasoning about the past with two-way automata. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) International Colloquium on Automata, Languages and Programming. LNCS, vol. 1443, pp. 628–641. Springer, Heidelberg (1998) Google Scholar
  119. 119.
    Vardi, M.Y.: The Büchi complementation saga. In: Thomas, W., Weil, P. (eds.) Annual Symposium on Theoretical Aspects of Computer Science. LNCS, vol. 4393, pp. 12–22. Springer, Heidelberg (2007) Google Scholar
  120. 120.
    Vardi, M.Y., Wilke, T.: Automata: from logics to algorithms. In: Flum, J., Grädel, E., Wilke, T. (eds.) Logic and Automata, Texts in Logic and Games, vol. 2, pp. 629–736. Amsterdam University Press, Amsterdam (2007) Google Scholar
  121. 121.
    Venema, Y.: Automata and fixed point logic: a coalgebraic perspective. Inf. Comput. 204(4), 637–678 (2006) MathSciNetMATHGoogle Scholar
  122. 122.
    Vöge, J., Jurdziński, M.: A discrete strategy improvement algorithm for solving parity games (extended abstract). In: Emerson, E.A., Sistla, A.P. (eds.) Intl. Conf. on Computer-Aided Verification (CAV). LNCS, vol. 1855, pp. 202–215. Springer, Heidelberg (2000) Google Scholar
  123. 123.
    Walukiewicz, I.: Completeness of Kozen’s axiomatisation of the propositional \(\mu\)-calculus. Inf. Comput. 157, 142–182 (2000) MathSciNetMATHGoogle Scholar
  124. 124.
    Walukiewicz, I.: Monadic second order logic on tree-like structures. Theor. Comput. Sci. 257(1–2), 311–346 (2002) MathSciNetMATHGoogle Scholar
  125. 125.
    Zielonka, W.: Infinite games on finitely coloured graphs with applications to automata on infinite trees. Theor. Comput. Sci. 200, 135–183 (1998) MathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of EdinburghEdinburghUK
  2. 2.CNRS, University of BordeauxBordeauxFrance

Personalised recommendations