On fixed-point clones

  • Damian Niwiński
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 226)


Expressions involving least and greatest fixed point operators are interpreted in the power-set algebra of (possibly infinite) trees and also in some abstract models. Initiality of the tree algebra is established via a "Mezei-and Wright-like" result on interpretation of fixed point terms. Then a reduction of these terms to Rabin automata on infinite trees is shown which yields some decidability results. A connection is exhibited between the hierarchy of alternating least and greatest fixed points and the hierarchy of Rabin pair indices of automata.


Acceptance Condition Partial Tree Tree Language Recursive Program Infinite Path 
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  1. Arnold, A. & Nivat, M. (1977), Non deterministic recursive programs, in "Fundamentals of Computation Theory", pp. 12–21, LNCS No. 56, Springer Verlag, Heidelberg.Google Scholar
  2. deBakker, J. & deRoever, W.P. (1973), A Calculus for Recursive Program Schemes, First Int.Coll.on Automata, Languages and Programming.Google Scholar
  3. Emerson, A.E. & Clarke, E.C. (1980), Characterizing Correctness Properties of Parallel Programs using Fixpoints, Seventh Int.Coll. on Automata, Languages and Programming, 169–181.Google Scholar
  4. Engelfriet, J. & Schmidt, E.M. (1978), IO and OI, JCSS 15 (1977), 328–353 and 16 (1978), 67–99.Google Scholar
  5. Kozen, D. (1983), Results on the Propositional μ-Calculus, TCS 27, 333–54.Google Scholar
  6. Niwiński, D. (1984), Fixed-Point Characterization of Context-Free ∞-Languages, Inform. Contr. 61, 247–276.Google Scholar
  7. Niwiński, D. (1985), Equational μ-calculus, in "Computation Theory", LNCS No. 208, Springer Verlag, Heidelberg.Google Scholar
  8. Niwiński, D. (1986), A note on indices of Rabin pair automata, University of Warsaw, manuscript.Google Scholar
  9. Park, D. (1980), On the semantics of fair parallelism, in "Abstract Software specifications", LNCS No. 86, Springer Verlag, Heidelberg.Google Scholar
  10. Park, D. (1981), Concurrency and automata on infinite sequences, LNCS No. 104, Springer Verlag, Heidelberg, 167–183.Google Scholar
  11. Rabin, M.O. (1969), Decidability os second-order theories and automata on infinite trees, in Trans. Amer. Math. Soc. 141, 1–35.Google Scholar
  12. Rabin, M.O. (1970), Weakly definable relations and special automata, in Math. Logic and Found. of Set Theory (Ed. Bar-Hillel), North-Holland.Google Scholar
  13. Rabin, M.O. (1972), Automata on Infinite Objects and Church's Problem, in Proc. Regional AMS Conf. Series in Math. 13, 1–22.Google Scholar
  14. Takahashi, M. (1985), The Greatest Fixed-Points and Rational Omega-Tree Languages, Université Paris VII, manuscript.Google Scholar
  15. Vardi, M.Y. & Wolper, P. (1984), Automata-theoretic Techniques for Modal Logic of Programs, Report RJ 4450, IBM Research, Yorktown Heights.Google Scholar
  16. Wand, M. (1973), First Int.Coll. on Automata, Languages and Programming, 331–344.Google Scholar
  17. McNaughton, R. (1966), Testing and generating infinite sequences by a finite automaton, Inform. Contr. 9, 521–530.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Damian Niwiński
    • 1
  1. 1.Institute of MathematicsUniversity of WarsawWarsawPoland

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