Abstract
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.
Keywords
- 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.
Chapter PDF
References
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)
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)
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)
Cousot, P., Cousot, R.: Static Determination of Dynamic Properties of Programs. In: Second Int. Symp. on Programming, pp. 106–130. Dunod, Paris (1976)
Cousot, P., Cousot, R.: Comparison of the Galois Connection and Widening/Narrowing Approaches to Abstract Interpretation. BIGRE (JTASPEFL ’91, Bordeaux) 74, 107–110 (1991)
Gawlitza, T., et al.: Polynomial Exact Interval Analysis Revisited. Technical report, TU München (2006)
Hoffman, A.J., Karp, R.M.: On Nonterminating Stochastic Games. Management Sci. 12, 359–370 (1966)
Howard, R.: Dynamic Programming and Markov Processes. Wiley, New York (1960)
Knuth, D.E.: A Generalization of Dijkstra’s algorithm. Information Processing Letters (IPL) 6(1), 1–5 (1977)
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)
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)
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)
Puri, A.: Theory of Hybrid and Discrete Systems. PhD thesis, University of California, Berkeley (1995)
Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, New York (1994)
Seidl, H.: Least and Greatest Solutions of Equations over \(\cal N\). Nordic Journal of Computing (NJC) 3(1), 41–62 (1996)
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)
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)
Zwick, U., Paterson, M.: The Complexity of Mean Payoff Games on Graphs. Theoretical Computer Science (TCS) 158(1-2), 343–359 (1996)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Gawlitza, T., Seidl, H. (2007). Precise Fixpoint Computation Through Strategy Iteration. In: De Nicola, R. (eds) Programming Languages and Systems. ESOP 2007. Lecture Notes in Computer Science, vol 4421. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71316-6_21
Download citation
DOI: https://doi.org/10.1007/978-3-540-71316-6_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-71314-2
Online ISBN: 978-3-540-71316-6
eBook Packages: Computer ScienceComputer Science (R0)