Logical specifications of infinite computations

  • Wolfgang Thomas
  • Helmut Lescow
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 803)


Starting from an identification of infinite computations with ω-words, we present a framework in which different classification schemes for specifications are naturally compared. Thereby we connect logical formalisms with hierarchies of descriptive set theory (e.g., the Borel hierarchy), of recursion theory, and with the hierarchy of acceptance conditions of ω-automata. In particular, it is shown in which sense these hierarchies can be viewed as classifications of logical formulas by the complexity measure of quantifier alternation. In this context, the automaton theoretic approach to logical specifications over ω-words turns out to be a technique to reduce quantifier complexity of formulas. Finally, we indicate some perspectives of this approach, discuss variants of the logical framework (first-order logic, temporal logic), and outline the effects which arise when branching computations are considered (i.e., when infinite trees instead of ω-words are taken as model of computation).


Infinite words ω-languages descriptive set theory Cantor space Borel hierarchy recursion theory Büchi automata acceptance conditions regular ω-languages monadic second-order logic infinite games temporal logic infinite trees 


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  1. [Arn83]
    A. Arnold. Topological characterizations of infinite behaviours of transition systems. In J. Diaz, editor, Automata, Languages and Programming, Volume 154 of LNCS, pages 28–38, Berlin, New York, Heidelberg, 1983. Springer-Verlag.Google Scholar
  2. [BL69]
    J. R. Büchi and L. H. Landweber. Solving sequential conditions by finitestate strategies. Trans. Amer. Math. Soc., 138:295–311, 1969.Google Scholar
  3. [Bü60]
    J. R. Büchi. Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlag. Math., 6:66–92, 1960.Google Scholar
  4. [Bü62]
    J. R. Büchi. On a decision method in restricted second order arithmetic. In Logic, Methodology and Philosophy of Science. Proceedings 1960 Intern. Congr., pages 1–11. Stanford Univ. Press, 1962.Google Scholar
  5. [Bü77]
    J. R. Büchi. Using determinacy of games to eliminate quantifiers. In M. Karpiński, editor, Fundamentals in Computation Theory, Volume 56 of LNCS, pages 367–378, Berlin, Heidelberg, New York, 1977. Springer-Verlag.Google Scholar
  6. [Bü83]
    J. R. Büchi. State strategies for games in F σδG δσ J. Symb. Logic, 48:1171–1198, 1983.Google Scholar
  7. [Car93]
    O. Carton. Mots Infinis, ω-Semigroupes et Topologie. PhD thesis, Université Paris 7, 1993.Google Scholar
  8. [Don70]
    J. Doner. Tree acceptors ans some of their applications. J. Comput. System Sci., 4:406–451, 1970.Google Scholar
  9. [EH93]
    J. Engelfriet and H. J. Hoogeboom. X-automata on ω-words. Theoretical Computer Science, 110:1–51, 1993.Google Scholar
  10. [Elg61]
    C. C. Elgot. Decision problems of finite automata design and related arithmetics. Trans. Amer. Math. Soc., 98:21–52, 1961.Google Scholar
  11. [Em90]
    E. A. Emerson. Temporal and modal logic. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, chapter 16, pages 995–1072. North-Holland, Amsterdam, 1990.Google Scholar
  12. [GS84]
    F. Gécseg and M. Steinby. Tree Automata. Akadémiai Kiadó, Budapest, 1984.Google Scholar
  13. [Haf87]
    T. Hafer. On the boolean closure of Büchi tree automaton definable sets of ω-trees. Aachener Inform. Ber. 87-16, RWTH Aachen, 1987.Google Scholar
  14. [Hin78]
    P. G. Hinman. Recursion-Theoretic Hierarchies. Springer-Verlag, Berlin, Heidelberg, New York, 1978.Google Scholar
  15. [Lan69]
    L. H. Landweber. Decision problems for ω-automata. Mathematical Systems Theory, 3:376–384, 1969.Google Scholar
  16. [LPZ85]
    O. Lichtenstein, A. Pnueli, and L. Zuck. The glory of the past. In R. Parikh, editor, Logics of Programs, Volume 193 of LNCS, pages 196–218, Berlin, Heidelberg, New York, 1985. Springer-Verlag.Google Scholar
  17. [Mar75]
    D. A. Martin. Borel determinacy. Ann. Math., 102:363–371, 1975.Google Scholar
  18. [McN66]
    R. McNaughton. Testing and generating infinite sequences by a finite automaton. Information and Control, 9:521–530, 1966.Google Scholar
  19. [McN92]
    R. McNaughton. Infinite games played on infinite graphs. Dep. of Computer Science, Technical Report 92-14, Rensselaer Polytechnic Institute, May 1992.Google Scholar
  20. [MLS90]
    S. Mac Lane and D. Siefkes, editors. The Collected Works of J. Richard Bucht. Springer-Verlag, Berlin, Heidelberg, NewYork, 1990.Google Scholar
  21. [Mos80]
    Y. N. Moschovakis. Descriptive Set Theory. North Holland, Amsterdam, 1980.Google Scholar
  22. [MP71]
    R. McNaughton and S. Papert. Counter-Free Automata. MIT Press, Cambridge, MA, 1971.Google Scholar
  23. [MP88]
    Z. Manna and A. Pnueli. The anchored version of the temporal framework. In J. W. de Bakker, W.-P. de Roever, and G. Rozenberg, editors, Linear Time, Branching Time and Partial Order in Logics and Model for Concurrency, Volume 354 of LNCS, pages 201–284, Berlin, Heidelberg, New York, 1988. Springer-Verlag.Google Scholar
  24. [MP92]
    Z. Manna and A. Pnueli. The Temporal Logic of Reactive and Concurrent Systems. Springer-Verlag, Berlin, Heidelberg, New York, 1992.Google Scholar
  25. [MS85]
    D. E. Muller and P. E. Schupp. The theory of ends, pushdown automata, and second-order logic. Theoretical Computer Science, 37:51–75, 1985.Google Scholar
  26. [Niw88]
    D. Niwinski. Fixed points vs. infinite generation. In Proc. 3rd Ann. IEEE Symp. on Logic in Computer Science, pages 402–409, 1988.Google Scholar
  27. [Per84]
    D. Perrin. Recent results on automata and infinte words. In M. P. Chytil and V. Koubek, editors, Mathematical Foundations of Computer Science, Volume 176 of LNCS, pages 134–148, Berlin, Heidelberg, New York, 1984. Springer-Verlag.Google Scholar
  28. [Pnu86]
    A. Pnueli. Application of temporal logic to the specification and verification of reactive systems: A survey of current trends. In J. W. de Bakker, W.-P. de Roever, and G. Rozenberg, editors, Current Trends in Concurrency, Volume 224 of LNCS, pages 510–584, Berlin, Heidelberg, New York, 1986. Springer-Verlag.Google Scholar
  29. [Rab69]
    M. O. Rabin. Decidability of second-order theories and automata on infinite trees. Transactions of the American Mathematical Society, 141:1–35, 1969.Google Scholar
  30. [Rab70]
    M. O. Rabin. Weakly definable relations and special automata. In Y. Bar-Hillel, editor, Mathematical Logic and Foundations in Set Theory, pages 1–23, Amsterdam, 1970. North-Holland.Google Scholar
  31. [Rog67]
    H. Rogers. Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York, 1967.Google Scholar
  32. [Sei92]
    S. Seibert. Quantifier hierarchies over word relations. In E. Börger & al., editor, Computer Science Logic, Volume 626 of LNCS, pages 329–338, Berlin, Heidelberg, New York, 1992. Springer-Verlag. Extended version to appear in Theor. Comp. Sci.Google Scholar
  33. [Sku93]
    J. Skurczyński. The Borel hierarchy is infinite in the class of regular sets of trees. Theoretical Computer Science, 112:413–418, 1993.Google Scholar
  34. [Sta86]
    L. Staiger. Hierarchies of recursive ω-languages. J. Inf Process. Cybern. EIK, 22:219–241, 1986.Google Scholar
  35. [Sta87]
    L. Staiger. Research in the theory of ω-languages. J. Inf Process. Cybern. EIK, 23:415–439, 1987.Google Scholar
  36. [Sta93]
    L. Staiger. Recursive automata on infinite words. In P. Enjalbert, A. Finkel, and K. W. Wagner, editors, STACS 93, Volume 665 of LNCS, pages 629–639, Berlin, Heidelberg, New York, 1993. Springer-Verlag.Google Scholar
  37. [Tho81]
    W. Thomas. A combinatorial approach to the theory of ω-automata. Inform & Control, 48:261–283, 1981.Google Scholar
  38. [Tho82a]
    W. Thomas. Classifying regular events in symbolic logic. J. Comput. System Sci., 25:360–376, 1982.Google Scholar
  39. [Tho82b]
    W. Thomas. A hierarchy of sets of infinite trees. In A. B. Cremers and H. P. Kriegel, editors, Theoretical Computer Science, Volume 145 of LNCS, pages 335–342, Berlin, Heidelberg, New York, 1982. Springer-Verlag.Google Scholar
  40. [Tho88a]
    W. Thomas. Automata and quantifier hierarchies. In J. E. Pin, editor, Formal Properties of finite Automata and Applications, Volume 386 of LNCS, pages 104–119, Berlin, Heidelberg, New York, 1988. Springer-Verlag.Google Scholar
  41. [Tho88b]
    W. Thomas. Safetyand liveness-properties in propositional temporal logic: Characterizations and decidability. In Mathematical Problems in Computation Theory, Volume 21 of Banach Center Publications, pages 403–417, Warsaw, 1988. PWN Polish Scientific Publishers.Google Scholar
  42. [Tho90]
    W. Thomas. Automata on infinite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, chapter 4, pages 131–191. North-Holland, Amsterdam, 1990.Google Scholar
  43. [TW68]
    J. W. Thatcher and J. B. Wright. Generalized finite automata with an application to the decision problem of second-order logic. Math. Systems Theory, 2:57–82, 1968.Google Scholar
  44. [Wag79]
    K. Wagner. On ω-regular sets. Information and Control, 43:123–177, 1979.Google Scholar
  45. [Wil93]
    T. Wilke. Locally threshold testable languages on infinite words. In P. Enjalbert, A. Finkel, and K. W. Wagner, editors, STACS 93, Volume 665 of LNCS, pages 607–616, Berlin, Heidelberg, New York, 1993. Springer-Verlag.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Wolfgang Thomas
    • 1
  • Helmut Lescow
    • 1
  1. 1.Institut für Informatik und Praktische MathematikChristian-Albrechts-Universität KielKiel

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