Precise Fixpoint Computation Through Strategy Iteration

  • Thomas Gawlitza
  • Helmut Seidl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4421)


We present a practical algorithm for computing least solutions of systems of equations over the integers with addition, multiplication with positive constants, maximum and minimum. The algorithm is based on strategy iteration. Its run-time (w.r.t. the uniform cost measure) is independent of the sizes of occurring numbers. We apply our technique to solve systems of interval equations. In particular, we show how arbitrary intersections as well as full interval multiplication in interval equations can be dealt with precisely.


Feasible Solution Complete Lattice Interval Analysis Full Multiplication Strategy Iteration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Bjorklund, H., Sandberg, S., Vorobyov, S.: Complexity of Model Checking by Iterative Improvement: the Pseudo-Boolean Framework . In: Broy, M., Zamulin, A.V. (eds.) PSI 2003. LNCS, vol. 2890, pp. 381–394. Springer, Heidelberg (2004)Google Scholar
  2. 2.
    Cochet-Terrasson, J., Gaubert, S., Gunawardena, J.: A Constructive Fixed Point Theorem for Min-Max Functions. Dynamics and Stability of Systems 14(4), 407–433 (1999)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Costan, A., et al.: A Policy Iteration Algorithm for Computing Fixed Points in Static Analysis of Programs. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 462–475. Springer, Heidelberg (2005)Google Scholar
  4. 4.
    Cousot, P., Cousot, R.: Static Determination of Dynamic Properties of Programs. In: Second Int. Symp. on Programming, pp. 106–130. Dunod, Paris (1976)Google Scholar
  5. 5.
    Cousot, P., Cousot, R.: Comparison of the Galois Connection and Widening/Narrowing Approaches to Abstract Interpretation. BIGRE (JTASPEFL ’91, Bordeaux) 74, 107–110 (1991)Google Scholar
  6. 6.
    Gawlitza, T., et al.: Polynomial Exact Interval Analysis Revisited. Technical report, TU München (2006)Google Scholar
  7. 7.
    Hoffman, A.J., Karp, R.M.: On Nonterminating Stochastic Games. Management Sci. 12, 359–370 (1966)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Howard, R.: Dynamic Programming and Markov Processes. Wiley, New York (1960)zbMATHGoogle Scholar
  9. 9.
    Knuth, D.E.: A Generalization of Dijkstra’s algorithm. Information Processing Letters (IPL) 6(1), 1–5 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Megiddo, N.: On the Complexity of Linear Programming. In: Bewley, T. (ed.) Advances in Economic Theory: 5th World Congress, pp. 225–268. Cambridge University Press, Cambridge (1987)Google Scholar
  11. 11.
    Miné, A.: Relational Abstract Domains for the Detection of Floating-Point Run-Time Errors. In: Schmidt, D. (ed.) ESOP 2004. LNCS, vol. 2986, pp. 3–17. Springer, Heidelberg (2004)Google Scholar
  12. 12.
    Miné, A.: Symbolic Methods to Enhance the Precision of Numerical Abstract Domains. In: Emerson, E.A., Namjoshi, K.S. (eds.) VMCAI 2006. LNCS, vol. 3855, pp. 348–363. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  13. 13.
    Puri, A.: Theory of Hybrid and Discrete Systems. PhD thesis, University of California, Berkeley (1995)Google Scholar
  14. 14.
    Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, New York (1994)zbMATHGoogle Scholar
  15. 15.
    Seidl, H.: Least and Greatest Solutions of Equations over \(\cal N\). Nordic Journal of Computing (NJC) 3(1), 41–62 (1996)MathSciNetGoogle Scholar
  16. 16.
    Su, Z., Wagner, D.: A Class of Polynomially Solvable Range Constraints for Interval Analysis Without Widenings. Theor. Comput. Sci (TCS) 345(1), 122–138 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Vöge, J., Jurdzinski, M.: A Discrete Strategy Improvement Algorithm for Solving Parity Games. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 202–215. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  18. 18.
    Zwick, U., Paterson, M.: The Complexity of Mean Payoff Games on Graphs. Theoretical Computer Science (TCS) 158(1-2), 343–359 (1996)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Thomas Gawlitza
    • 1
  • Helmut Seidl
    • 1
  1. 1.TU München, Institut für Informatik, I2, 85748 MünchenGermany

Personalised recommendations