On Guarding Nested Fixpoints

  • Helmut Seidl
  • Andreas Neumann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1683)


For every hierarchicalsystem of equations S over some complete and distributive lattice we construct an equivalent system with the same set of variables which additionally is guarded. The price to be paid is that the resulting right-hand sides may grow exponentially. We therefore present methods how the exponentialbl ow-up can be avoided. Especially, the loop structure of the variable dependence graph is taken into account. Also we prove that size O(m· S) suffices whenever S originates from a fixpoint expression where the nesting-depth of fixpoints is at most m. Finally, we sketch an application to regular tree pattern-matching.


Distributive Lattice Free Variable Complete Lattice Hierarchical System Variable Assignment 
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  1. 1.
    H.R. Andersen, C. Stirling, and G. Winskel. A Compositional Proof System for the Modal Mu-Calculus. In IEEE Conf. on Logic in Computer Science (LICS), 144–153, 1994.Google Scholar
  2. 2.
    A. Arnold and M. Nivat. Metric Interpretations of Infinite Trees and Semantics of Non-Deterministic Recursive Programs. Theoretical Computer Science (TCS), 11:181–205, 1980.CrossRefGoogle Scholar
  3. 3.
    H. Bekic. Definable Operations in General Algebras, and the Theory of Automata and Flowcharts. Technical Report, IBM Labor, Wien, 1967.Google Scholar
  4. 4.
    F. Bourdoncle. Efficient Chaotic Iteration Strategies with Widenings. In Int. Conf. on Formal Methods in Programming and their Applications, 128–141. LNCS 735, 1993.CrossRefGoogle Scholar
  5. 5.
    B. Courcelle. Basic Notions of Universal Algebra for Language Theory and Graph Grammars. Theoretical Computer Science (TCS), 163(1&2):1–54, 1996.MathSciNetCrossRefGoogle Scholar
  6. 6.
    C. Fecht and H. Seidl. A Faster Solver for General Systems of Equations. Science of Computer Programming (SCP), 1999. To appear.Google Scholar
  7. 7.
    A.C. Fong and J.D. Ullman. Finding the Depth of a Flow Graph. J. of Computer and System Sciences (JCSS), 15(3):300–309, 1977.MathSciNetCrossRefGoogle Scholar
  8. 8.
    M.S. Hecht. Flow Analysis of Computer Programs. North Holland, 1977.Google Scholar
  9. 9.
    R. Kaivola. A Simple Decision Method for the Linear Time Mu-Calculus. In Int. Workshop on Structures in ConcurrencyThe ory(STRICT) (ed. J. Desel), 190–204. Springer-Verlag, Berlin, 1995.CrossRefGoogle Scholar
  10. 10.
    R. Kaivola. Fixpoints for Rabin Tree Automata Make Complementation Easy. In 23rd Int. Coll. on Automata, Languages and Programming (ICALP), 312–323. LNCS 1099, 1996.CrossRefGoogle Scholar
  11. 11.
    R. Kaivola. Using Automata to Characterise Fixpoint Temporal Logics. PhD thesis, Dept. of Computer Science, Univ. of Edinburgh, 1996.Google Scholar
  12. 12.
    J.B. Kam and J.D. Ullman. Global Data Flow Analysis and Iterative Algorithms. J. of the Association for Computing Machinary(JA CM), 23(1):158–171, 1976.MathSciNetCrossRefGoogle Scholar
  13. 13.
    D. Kozen. Results on the Propositional μ-Calculus. Theoretical Computer Science (TCS), 27:333–354, 1983.MathSciNetCrossRefGoogle Scholar
  14. 14.
    A. Mader. Verification of Modal Properties Using Boolean Equation Systems. PhD thesis, TU München, 1997.Google Scholar
  15. 15.
    A. Neumann and H. Seidl. Locating Matches of Tree Patterns in Forests. Technical Report 98-08, University of Trier, 1998. Short version in 18th Int. Conf. on Foundations of Software Technologyand Theoretical Computer Science (FST&TCS), LNCS 1530, 134–145, 1998.CrossRefGoogle Scholar
  16. 16.
    D. Niwinski. Fixed Point Characterization of Infinite Behavior of Finite State Systems. Theoretical Computer Science (TCS), 189:1–69, 1997.MathSciNetCrossRefGoogle Scholar
  17. 17.
    D. Niwinski and I. Walukiewicz. Games for the μ-Calculus. Theoretical Computer Science (TCS), 163(1&2):99–116, 1996.MathSciNetCrossRefGoogle Scholar
  18. 18.
    H. Seidland D. Niwinski. On Distributive Fixpoint Expressions. TechnicalRep ort TR 98-04 (253), University of Warsaw, 1998. To appear in RAIRO.Google Scholar
  19. 19.
    R.E. Tarjan. Finding Dominators in Directed Graphs. SIAM J. of Computing, 2(3):211–216, 1974.MathSciNetCrossRefGoogle Scholar
  20. 20.
    I. Walukiewicz. Notes on the Propositional μ-Calculus: Completeness and Related Results. Technical Report NS-95-1, BRICS Notes Series, 1995.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Helmut Seidl
    • 1
  • Andreas Neumann
    • 1
  1. 1.FB IV - InformatikUniversität TrierTrierGermany

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