Constraint-Based Linear-Relations Analysis

  • Sriram Sankaranarayanan
  • Henny B. Sipma
  • Zohar Manna
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3148)


Linear-relations analysis of transition systems discovers linear invariant relationships among the variables of the system. These relationships help establish important safety and liveness properties. Efficient techniques for the analysis of systems using polyhedra have been explored, leading to the development of successful tools like HyTech. However, existing techniques rely on the use of approximations such as widening and extrapolation in order to ensure termination. In an earlier paper, we demonstrated the use of Farkas Lemma to provide a translation from the linear-relations analysis problem into a system of constraints on the unknown coefficients of a candidate invariant. However, since the constraints in question are non-linear, a naive application of the method does not scale. In this paper, we show that by some efficient simplifications and approximations to the quantifier elimination procedure, not only does the method scale to higher dimensions, but also enjoys performance advantages for some larger examples.


Abstract Interpretation Liveness Property Inductive Assertion Farkas Lemma Symbolic Transition System 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bagnara, R., Hill, P.M., Ricci, E., Zaffanella, E.: Precise widening operators for convex polyhedra. In: Cousot, R. (ed.) SAS 2003. LNCS, vol. 2694, pp. 337–354. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Bagnara, R., Ricci, E., Zaffanella, E., Hill, P.M.: Possibly not closed convex polyhedra and the Parma Polyhedra Library. In: Hermenegildo, M.V., Puebla, G. (eds.) SAS 2002. LNCS, vol. 2477, pp. 213–229. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. 3.
    Bardin, S., Finkel, A., Leroux, J., Petrucci, L.: Fast: Fast accelereation of symbolic transition systems. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 118–121. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  4. 4.
    Besson, F., Jensen, T., Talpin, J.-P.: Polyhedral analysis of synchronous languages. In: Cortesi, A., Filé, G. (eds.) SAS 1999. LNCS, vol. 1694, pp. 51–69. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Collins, G.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)Google Scholar
  6. 6.
    Colón, M., Sankaranarayanan, S., Sipma, H.: Linear invariant generation using non-linear constraint solving. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 420–433. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Cousot, P., Cousot, R.: Abstract Interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: ACM Principles of Programming Languages, pp. 238–252 (1977)Google Scholar
  8. 8.
    Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among the variables of a program. In: ACM Principles of Programming, January 1978, pp. 84–97 (1978)Google Scholar
  9. 9.
    Fukuda, K., Prodon, A.: Double description method revisited. In Combinatorics and Computer Science. In: Deza, M., Manoussakis, I., Euler, R. (eds.) CCS 1995. LNCS, vol. 1120, pp. 91–111. Springer, Heidelberg (1996)Google Scholar
  10. 10.
    Halbwachs, N., Merchat, D., Parent-Vigouroux, C.: Cartesian factoring of polyhedra for linear relation analysis. In: Cousot, R. (ed.) SAS 2003. LNCS, vol. 2694, pp. 355–365. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. 11.
    Halbwachs, N., Proy, Y., Roumanoff, P.: Verification of real-time systems using linear relation analysis. Formal Methods in System Design 11(2), 157–185 (1997)CrossRefGoogle Scholar
  12. 12.
    Henzinger, T.A., Ho, P.: HyTech: The Cornell hybrid technology tool. In: Antsaklis, P.J., Kohn, W., Nerode, A., Sastry, S.S. (eds.) HS 1994. LNCS, vol. 999, pp. 265–293. Springer, Heidelberg (1995)Google Scholar
  13. 13.
    Manna, Z., Pnueli, A.: Temporal Verification of Reactive Systems: Safety. Springer, New York (1995)Google Scholar
  14. 14.
    Sankaranarayanan, S., Sipma, H.B., Manna, Z.: Petri net analysis using invariant generation. In: Dershowitz, N. (ed.) Verification: Theory and Practice. LNCS, vol. 2772, pp. 682–701. Springer, Heidelberg (2004)Google Scholar
  15. 15.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, Chichester (1986)zbMATHGoogle Scholar
  16. 16.
    Tarski, A.: A decision method for elementary algebra and geometry, p. 5. Univ. of California Press, Berkeley (1951)zbMATHGoogle Scholar
  17. 17.
    Weispfenning, V.: The complexity of linear problems in fields. Journal of Symbolic Computation 5(1-2), 3–27 (1988)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Sriram Sankaranarayanan
    • 1
  • Henny B. Sipma
    • 1
  • Zohar Manna
    • 1
  1. 1.Computer Science DepartmentStanford UniversityStanfordUSA

Personalised recommendations